Csr2011 june14 09_30_grigoriev

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Csr2011 june14 09_30_grigoriev

  1. 1. Complexity lower bounds for algebraic computation trees Dima Grigoriev (Lille) CNRS 14/06/2011, Saint-PetersbourgDima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 1 / 23
  2. 2. Several non-linear lower bounds are known in algebraic complexityunlike its boolean counterpart Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 2 / 23
  3. 3. Computational modelsLet F be a ground field.Algebraic d-decision tree Input x = (x1 , . . . , xn ) ∈ F n is attributed to the root of tree T . To each node v of T (except leaves) a (testing) polynomial gv ∈ F [X1 , . . . , Xn ] with deg(gv ) ≤ d is assigned. The algebraic decision tree branches at node v according to whether gv (x) = 0. In case F = R one uses alternatively gv (x) > 0 as a condition of branching. To each leaf L an output ”accept” or ”reject” is assigned. Denote by SL ⊂ F n the set of inputs which arrive at L. They are pairwise disjoint and the set S ⊂ F n accepted by the algebraic decision tree is the union of SL for all leaves L assigned with ”accept”. The complexity Cd is the depth of T .For d = 1 they are called linear decision trees. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 3 / 23
  4. 4. Computational modelsLet F be a ground field.Algebraic d-decision tree Input x = (x1 , . . . , xn ) ∈ F n is attributed to the root of tree T . To each node v of T (except leaves) a (testing) polynomial gv ∈ F [X1 , . . . , Xn ] with deg(gv ) ≤ d is assigned. The algebraic decision tree branches at node v according to whether gv (x) = 0. In case F = R one uses alternatively gv (x) > 0 as a condition of branching. To each leaf L an output ”accept” or ”reject” is assigned. Denote by SL ⊂ F n the set of inputs which arrive at L. They are pairwise disjoint and the set S ⊂ F n accepted by the algebraic decision tree is the union of SL for all leaves L assigned with ”accept”. The complexity Cd is the depth of T .For d = 1 they are called linear decision trees. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 3 / 23
  5. 5. Computational modelsLet F be a ground field.Algebraic d-decision tree Input x = (x1 , . . . , xn ) ∈ F n is attributed to the root of tree T . To each node v of T (except leaves) a (testing) polynomial gv ∈ F [X1 , . . . , Xn ] with deg(gv ) ≤ d is assigned. The algebraic decision tree branches at node v according to whether gv (x) = 0. In case F = R one uses alternatively gv (x) > 0 as a condition of branching. To each leaf L an output ”accept” or ”reject” is assigned. Denote by SL ⊂ F n the set of inputs which arrive at L. They are pairwise disjoint and the set S ⊂ F n accepted by the algebraic decision tree is the union of SL for all leaves L assigned with ”accept”. The complexity Cd is the depth of T .For d = 1 they are called linear decision trees. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 3 / 23
  6. 6. Computational modelsLet F be a ground field.Algebraic d-decision tree Input x = (x1 , . . . , xn ) ∈ F n is attributed to the root of tree T . To each node v of T (except leaves) a (testing) polynomial gv ∈ F [X1 , . . . , Xn ] with deg(gv ) ≤ d is assigned. The algebraic decision tree branches at node v according to whether gv (x) = 0. In case F = R one uses alternatively gv (x) > 0 as a condition of branching. To each leaf L an output ”accept” or ”reject” is assigned. Denote by SL ⊂ F n the set of inputs which arrive at L. They are pairwise disjoint and the set S ⊂ F n accepted by the algebraic decision tree is the union of SL for all leaves L assigned with ”accept”. The complexity Cd is the depth of T .For d = 1 they are called linear decision trees. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 3 / 23
  7. 7. Computational modelsLet F be a ground field.Algebraic d-decision tree Input x = (x1 , . . . , xn ) ∈ F n is attributed to the root of tree T . To each node v of T (except leaves) a (testing) polynomial gv ∈ F [X1 , . . . , Xn ] with deg(gv ) ≤ d is assigned. The algebraic decision tree branches at node v according to whether gv (x) = 0. In case F = R one uses alternatively gv (x) > 0 as a condition of branching. To each leaf L an output ”accept” or ”reject” is assigned. Denote by SL ⊂ F n the set of inputs which arrive at L. They are pairwise disjoint and the set S ⊂ F n accepted by the algebraic decision tree is the union of SL for all leaves L assigned with ”accept”. The complexity Cd is the depth of T .For d = 1 they are called linear decision trees. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 3 / 23
  8. 8. Computational modelsLet F be a ground field.Algebraic d-decision tree Input x = (x1 , . . . , xn ) ∈ F n is attributed to the root of tree T . To each node v of T (except leaves) a (testing) polynomial gv ∈ F [X1 , . . . , Xn ] with deg(gv ) ≤ d is assigned. The algebraic decision tree branches at node v according to whether gv (x) = 0. In case F = R one uses alternatively gv (x) > 0 as a condition of branching. To each leaf L an output ”accept” or ”reject” is assigned. Denote by SL ⊂ F n the set of inputs which arrive at L. They are pairwise disjoint and the set S ⊂ F n accepted by the algebraic decision tree is the union of SL for all leaves L assigned with ”accept”. The complexity Cd is the depth of T .For d = 1 they are called linear decision trees. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 3 / 23
  9. 9. Computational modelsLet F be a ground field.Algebraic d-decision tree Input x = (x1 , . . . , xn ) ∈ F n is attributed to the root of tree T . To each node v of T (except leaves) a (testing) polynomial gv ∈ F [X1 , . . . , Xn ] with deg(gv ) ≤ d is assigned. The algebraic decision tree branches at node v according to whether gv (x) = 0. In case F = R one uses alternatively gv (x) > 0 as a condition of branching. To each leaf L an output ”accept” or ”reject” is assigned. Denote by SL ⊂ F n the set of inputs which arrive at L. They are pairwise disjoint and the set S ⊂ F n accepted by the algebraic decision tree is the union of SL for all leaves L assigned with ”accept”. The complexity Cd is the depth of T .For d = 1 they are called linear decision trees. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 3 / 23
  10. 10. Computational modelsLet F be a ground field.Algebraic d-decision tree Input x = (x1 , . . . , xn ) ∈ F n is attributed to the root of tree T . To each node v of T (except leaves) a (testing) polynomial gv ∈ F [X1 , . . . , Xn ] with deg(gv ) ≤ d is assigned. The algebraic decision tree branches at node v according to whether gv (x) = 0. In case F = R one uses alternatively gv (x) > 0 as a condition of branching. To each leaf L an output ”accept” or ”reject” is assigned. Denote by SL ⊂ F n the set of inputs which arrive at L. They are pairwise disjoint and the set S ⊂ F n accepted by the algebraic decision tree is the union of SL for all leaves L assigned with ”accept”. The complexity Cd is the depth of T .For d = 1 they are called linear decision trees. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 3 / 23
  11. 11. Algebraic computation treeThe difference with algebraic decision trees is that testing polynomialgv is calculated as gv = a ◦ b where operation ◦ ∈ {+, ×} a, b ∈ F ∪ {X1 , . . . , Xn } ∪ {gu } where u runs the nodes on the path from the root to v .In particular, deg(gv ) ≤ 2kv where kv is the depth of v .Denote by C the complexity of algebraic computation trees. For anyconstant d the model of computation trees is stronger than one ofd-decision trees.We study the complexities Cd (S), C(S) of the membership problem toS. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 4 / 23
  12. 12. Algebraic computation treeThe difference with algebraic decision trees is that testing polynomialgv is calculated as gv = a ◦ b where operation ◦ ∈ {+, ×} a, b ∈ F ∪ {X1 , . . . , Xn } ∪ {gu } where u runs the nodes on the path from the root to v .In particular, deg(gv ) ≤ 2kv where kv is the depth of v .Denote by C the complexity of algebraic computation trees. For anyconstant d the model of computation trees is stronger than one ofd-decision trees.We study the complexities Cd (S), C(S) of the membership problem toS. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 4 / 23
  13. 13. Algebraic computation treeThe difference with algebraic decision trees is that testing polynomialgv is calculated as gv = a ◦ b where operation ◦ ∈ {+, ×} a, b ∈ F ∪ {X1 , . . . , Xn } ∪ {gu } where u runs the nodes on the path from the root to v .In particular, deg(gv ) ≤ 2kv where kv is the depth of v .Denote by C the complexity of algebraic computation trees. For anyconstant d the model of computation trees is stronger than one ofd-decision trees.We study the complexities Cd (S), C(S) of the membership problem toS. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 4 / 23
  14. 14. Algebraic computation treeThe difference with algebraic decision trees is that testing polynomialgv is calculated as gv = a ◦ b where operation ◦ ∈ {+, ×} a, b ∈ F ∪ {X1 , . . . , Xn } ∪ {gu } where u runs the nodes on the path from the root to v .In particular, deg(gv ) ≤ 2kv where kv is the depth of v .Denote by C the complexity of algebraic computation trees. For anyconstant d the model of computation trees is stronger than one ofd-decision trees.We study the complexities Cd (S), C(S) of the membership problem toS. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 4 / 23
  15. 15. Algebraic computation treeThe difference with algebraic decision trees is that testing polynomialgv is calculated as gv = a ◦ b where operation ◦ ∈ {+, ×} a, b ∈ F ∪ {X1 , . . . , Xn } ∪ {gu } where u runs the nodes on the path from the root to v .In particular, deg(gv ) ≤ 2kv where kv is the depth of v .Denote by C the complexity of algebraic computation trees. For anyconstant d the model of computation trees is stronger than one ofd-decision trees.We study the complexities Cd (S), C(S) of the membership problem toS. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 4 / 23
  16. 16. Algebraic computation treeThe difference with algebraic decision trees is that testing polynomialgv is calculated as gv = a ◦ b where operation ◦ ∈ {+, ×} a, b ∈ F ∪ {X1 , . . . , Xn } ∪ {gu } where u runs the nodes on the path from the root to v .In particular, deg(gv ) ≤ 2kv where kv is the depth of v .Denote by C the complexity of algebraic computation trees. For anyconstant d the model of computation trees is stronger than one ofd-decision trees.We study the complexities Cd (S), C(S) of the membership problem toS. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 4 / 23
  17. 17. Complexity lower bounds via topologicalinvariantsThere are several lower bounds for algebraic computation trees overthe field either F = C or F = R of the formC(S) ≥ log2 (ci ) − n, i = 1, 2, 3 where c1 is the maximum of the number of connected components in S and in F n S (Ben-Or [1983]); ¨ c2 is the Euler characteristic of S (Bjorner-Lovasz-Yao [1992]); c3 is the sum of Betti numbers (ranks of homological groups) of S ˜ (Yao [1994], Montana-Morais-Pardo [1996]).c1 , c2 ≤ c3 Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 5 / 23
  18. 18. Complexity lower bounds via topologicalinvariantsThere are several lower bounds for algebraic computation trees overthe field either F = C or F = R of the formC(S) ≥ log2 (ci ) − n, i = 1, 2, 3 where c1 is the maximum of the number of connected components in S and in F n S (Ben-Or [1983]); ¨ c2 is the Euler characteristic of S (Bjorner-Lovasz-Yao [1992]); c3 is the sum of Betti numbers (ranks of homological groups) of S ˜ (Yao [1994], Montana-Morais-Pardo [1996]).c1 , c2 ≤ c3 Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 5 / 23
  19. 19. Complexity lower bounds via topologicalinvariantsThere are several lower bounds for algebraic computation trees overthe field either F = C or F = R of the formC(S) ≥ log2 (ci ) − n, i = 1, 2, 3 where c1 is the maximum of the number of connected components in S and in F n S (Ben-Or [1983]); ¨ c2 is the Euler characteristic of S (Bjorner-Lovasz-Yao [1992]); c3 is the sum of Betti numbers (ranks of homological groups) of S ˜ (Yao [1994], Montana-Morais-Pardo [1996]).c1 , c2 ≤ c3 Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 5 / 23
  20. 20. Complexity lower bounds via topologicalinvariantsThere are several lower bounds for algebraic computation trees overthe field either F = C or F = R of the formC(S) ≥ log2 (ci ) − n, i = 1, 2, 3 where c1 is the maximum of the number of connected components in S and in F n S (Ben-Or [1983]); ¨ c2 is the Euler characteristic of S (Bjorner-Lovasz-Yao [1992]); c3 is the sum of Betti numbers (ranks of homological groups) of S ˜ (Yao [1994], Montana-Morais-Pardo [1996]).c1 , c2 ≤ c3 Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 5 / 23
  21. 21. Complexity lower bounds via topologicalinvariantsThere are several lower bounds for algebraic computation trees overthe field either F = C or F = R of the formC(S) ≥ log2 (ci ) − n, i = 1, 2, 3 where c1 is the maximum of the number of connected components in S and in F n S (Ben-Or [1983]); ¨ c2 is the Euler characteristic of S (Bjorner-Lovasz-Yao [1992]); c3 is the sum of Betti numbers (ranks of homological groups) of S ˜ (Yao [1994], Montana-Morais-Pardo [1996]).c1 , c2 ≤ c3 Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 5 / 23
  22. 22. Combinatorial problems EQUALITY is the set {(x1 , . . . , xn , y1 , . . . , yn ) ⊂ F 2n : {x1 , . . . , xn } = {y1 , . . . , yn }}; DISTINCTNESS is the set {(x1 , . . . , xn ) : xi = xj , i = j}; KNAPSACK is the set I⊂{1,...,n} {(x1 , . . . , xn ) : i∈I xi = 1}.Corollary C(EQUALITY ) n · logn (QuickSort (F = R), elementary symmetric functions (F = C) Strassen [1973], Ben-Or [1983]); C(DISTINCTNESS) n · logn (QuickSort (F = R), discriminant (F = C) Strassen [1973], Ben-Or [1983]); C(KNAPSACK ) ≥ Ω(n2 ) (Ben-Or [1983]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 6 / 23
  23. 23. Combinatorial problems EQUALITY is the set {(x1 , . . . , xn , y1 , . . . , yn ) ⊂ F 2n : {x1 , . . . , xn } = {y1 , . . . , yn }}; DISTINCTNESS is the set {(x1 , . . . , xn ) : xi = xj , i = j}; KNAPSACK is the set I⊂{1,...,n} {(x1 , . . . , xn ) : i∈I xi = 1}.Corollary C(EQUALITY ) n · logn (QuickSort (F = R), elementary symmetric functions (F = C) Strassen [1973], Ben-Or [1983]); C(DISTINCTNESS) n · logn (QuickSort (F = R), discriminant (F = C) Strassen [1973], Ben-Or [1983]); C(KNAPSACK ) ≥ Ω(n2 ) (Ben-Or [1983]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 6 / 23
  24. 24. Combinatorial problems EQUALITY is the set {(x1 , . . . , xn , y1 , . . . , yn ) ⊂ F 2n : {x1 , . . . , xn } = {y1 , . . . , yn }}; DISTINCTNESS is the set {(x1 , . . . , xn ) : xi = xj , i = j}; KNAPSACK is the set I⊂{1,...,n} {(x1 , . . . , xn ) : i∈I xi = 1}.Corollary C(EQUALITY ) n · logn (QuickSort (F = R), elementary symmetric functions (F = C) Strassen [1973], Ben-Or [1983]); C(DISTINCTNESS) n · logn (QuickSort (F = R), discriminant (F = C) Strassen [1973], Ben-Or [1983]); C(KNAPSACK ) ≥ Ω(n2 ) (Ben-Or [1983]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 6 / 23
  25. 25. Combinatorial problems EQUALITY is the set {(x1 , . . . , xn , y1 , . . . , yn ) ⊂ F 2n : {x1 , . . . , xn } = {y1 , . . . , yn }}; DISTINCTNESS is the set {(x1 , . . . , xn ) : xi = xj , i = j}; KNAPSACK is the set I⊂{1,...,n} {(x1 , . . . , xn ) : i∈I xi = 1}.Corollary C(EQUALITY ) n · logn (QuickSort (F = R), elementary symmetric functions (F = C) Strassen [1973], Ben-Or [1983]); C(DISTINCTNESS) n · logn (QuickSort (F = R), discriminant (F = C) Strassen [1973], Ben-Or [1983]); C(KNAPSACK ) ≥ Ω(n2 ) (Ben-Or [1983]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 6 / 23
  26. 26. Combinatorial problems EQUALITY is the set {(x1 , . . . , xn , y1 , . . . , yn ) ⊂ F 2n : {x1 , . . . , xn } = {y1 , . . . , yn }}; DISTINCTNESS is the set {(x1 , . . . , xn ) : xi = xj , i = j}; KNAPSACK is the set I⊂{1,...,n} {(x1 , . . . , xn ) : i∈I xi = 1}.Corollary C(EQUALITY ) n · logn (QuickSort (F = R), elementary symmetric functions (F = C) Strassen [1973], Ben-Or [1983]); C(DISTINCTNESS) n · logn (QuickSort (F = R), discriminant (F = C) Strassen [1973], Ben-Or [1983]); C(KNAPSACK ) ≥ Ω(n2 ) (Ben-Or [1983]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 6 / 23
  27. 27. Combinatorial problems EQUALITY is the set {(x1 , . . . , xn , y1 , . . . , yn ) ⊂ F 2n : {x1 , . . . , xn } = {y1 , . . . , yn }}; DISTINCTNESS is the set {(x1 , . . . , xn ) : xi = xj , i = j}; KNAPSACK is the set I⊂{1,...,n} {(x1 , . . . , xn ) : i∈I xi = 1}.Corollary C(EQUALITY ) n · logn (QuickSort (F = R), elementary symmetric functions (F = C) Strassen [1973], Ben-Or [1983]); C(DISTINCTNESS) n · logn (QuickSort (F = R), discriminant (F = C) Strassen [1973], Ben-Or [1983]); C(KNAPSACK ) ≥ Ω(n2 ) (Ben-Or [1983]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 6 / 23
  28. 28. Algebraic computation trees over positivecharacteristic p > 0Let F be an algebraically closed field of characteristic p > 0. For avariety S ⊂ F n introduce B(S) which replaces the sum of Bettinumbers (over C or R).Zeta function Denote Nk = Nk (S) = (S ∩ Fnk ). p Zeta function Z (S, t) = exp( 1≤k<∞ Nk t k /k) = P(t)/Q(t) is a rational function (due to Dwork, Deligne). Define B(S) = deg(P) + deg(Q).TheoremC(S) ≥ a · log(B(S)) − b · n for some constants a, b (Ben-Or [1994])CorollaryC(EQUALITY ), C(DISTINCTNESS) n · logn (Strassen [1973],Ben-Or [1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 7 / 23
  29. 29. Algebraic computation trees over positivecharacteristic p > 0Let F be an algebraically closed field of characteristic p > 0. For avariety S ⊂ F n introduce B(S) which replaces the sum of Bettinumbers (over C or R).Zeta function Denote Nk = Nk (S) = (S ∩ Fnk ). p Zeta function Z (S, t) = exp( 1≤k<∞ Nk t k /k) = P(t)/Q(t) is a rational function (due to Dwork, Deligne). Define B(S) = deg(P) + deg(Q).TheoremC(S) ≥ a · log(B(S)) − b · n for some constants a, b (Ben-Or [1994])CorollaryC(EQUALITY ), C(DISTINCTNESS) n · logn (Strassen [1973],Ben-Or [1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 7 / 23
  30. 30. Algebraic computation trees over positivecharacteristic p > 0Let F be an algebraically closed field of characteristic p > 0. For avariety S ⊂ F n introduce B(S) which replaces the sum of Bettinumbers (over C or R).Zeta function Denote Nk = Nk (S) = (S ∩ Fnk ). p Zeta function Z (S, t) = exp( 1≤k<∞ Nk t k /k) = P(t)/Q(t) is a rational function (due to Dwork, Deligne). Define B(S) = deg(P) + deg(Q).TheoremC(S) ≥ a · log(B(S)) − b · n for some constants a, b (Ben-Or [1994])CorollaryC(EQUALITY ), C(DISTINCTNESS) n · logn (Strassen [1973],Ben-Or [1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 7 / 23
  31. 31. Algebraic computation trees over positivecharacteristic p > 0Let F be an algebraically closed field of characteristic p > 0. For avariety S ⊂ F n introduce B(S) which replaces the sum of Bettinumbers (over C or R).Zeta function Denote Nk = Nk (S) = (S ∩ Fnk ). p Zeta function Z (S, t) = exp( 1≤k<∞ Nk t k /k) = P(t)/Q(t) is a rational function (due to Dwork, Deligne). Define B(S) = deg(P) + deg(Q).TheoremC(S) ≥ a · log(B(S)) − b · n for some constants a, b (Ben-Or [1994])CorollaryC(EQUALITY ), C(DISTINCTNESS) n · logn (Strassen [1973],Ben-Or [1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 7 / 23
  32. 32. Algebraic computation trees over positivecharacteristic p > 0Let F be an algebraically closed field of characteristic p > 0. For avariety S ⊂ F n introduce B(S) which replaces the sum of Bettinumbers (over C or R).Zeta function Denote Nk = Nk (S) = (S ∩ Fnk ). p Zeta function Z (S, t) = exp( 1≤k<∞ Nk t k /k) = P(t)/Q(t) is a rational function (due to Dwork, Deligne). Define B(S) = deg(P) + deg(Q).TheoremC(S) ≥ a · log(B(S)) − b · n for some constants a, b (Ben-Or [1994])CorollaryC(EQUALITY ), C(DISTINCTNESS) n · logn (Strassen [1973],Ben-Or [1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 7 / 23
  33. 33. Algebraic computation trees over positivecharacteristic p > 0Let F be an algebraically closed field of characteristic p > 0. For avariety S ⊂ F n introduce B(S) which replaces the sum of Bettinumbers (over C or R).Zeta function Denote Nk = Nk (S) = (S ∩ Fnk ). p Zeta function Z (S, t) = exp( 1≤k<∞ Nk t k /k) = P(t)/Q(t) is a rational function (due to Dwork, Deligne). Define B(S) = deg(P) + deg(Q).TheoremC(S) ≥ a · log(B(S)) − b · n for some constants a, b (Ben-Or [1994])CorollaryC(EQUALITY ), C(DISTINCTNESS) n · logn (Strassen [1973],Ben-Or [1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 7 / 23
  34. 34. Complexity upper bound for membership to alinear complexAn arrangement S ⊂ F n is a union of hyperplanes ∪i Hi . A face of S isan intersection of some of the hyperplanes. DISTINCTNESS andKNAPSACK are arrangements.TheoremLet S be either an arrangement or a (possibly unbounded) polyhedron(when F = R). Denote by N the number of faces (of all the dimensions)of S. Then C(S) ≤ O(n3 · logN) (Meyer auf der Heide [1985]).One can generalize the theorem to arbitrary linear complexes, i. e.the unions of polyhedra.CorollaryC(KNAPSACK ) ≤ O(n5 ) (Meyer auf der Heide [1985]).The corollary does not directly imply P=NP since the construction of analgebraic computation tree is non-uniform (depends on n). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 8 / 23
  35. 35. Complexity upper bound for membership to alinear complexAn arrangement S ⊂ F n is a union of hyperplanes ∪i Hi . A face of S isan intersection of some of the hyperplanes. DISTINCTNESS andKNAPSACK are arrangements.TheoremLet S be either an arrangement or a (possibly unbounded) polyhedron(when F = R). Denote by N the number of faces (of all the dimensions)of S. Then C(S) ≤ O(n3 · logN) (Meyer auf der Heide [1985]).One can generalize the theorem to arbitrary linear complexes, i. e.the unions of polyhedra.CorollaryC(KNAPSACK ) ≤ O(n5 ) (Meyer auf der Heide [1985]).The corollary does not directly imply P=NP since the construction of analgebraic computation tree is non-uniform (depends on n). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 8 / 23
  36. 36. Complexity upper bound for membership to alinear complexAn arrangement S ⊂ F n is a union of hyperplanes ∪i Hi . A face of S isan intersection of some of the hyperplanes. DISTINCTNESS andKNAPSACK are arrangements.TheoremLet S be either an arrangement or a (possibly unbounded) polyhedron(when F = R). Denote by N the number of faces (of all the dimensions)of S. Then C(S) ≤ O(n3 · logN) (Meyer auf der Heide [1985]).One can generalize the theorem to arbitrary linear complexes, i. e.the unions of polyhedra.CorollaryC(KNAPSACK ) ≤ O(n5 ) (Meyer auf der Heide [1985]).The corollary does not directly imply P=NP since the construction of analgebraic computation tree is non-uniform (depends on n). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 8 / 23
  37. 37. Complexity upper bound for membership to alinear complexAn arrangement S ⊂ F n is a union of hyperplanes ∪i Hi . A face of S isan intersection of some of the hyperplanes. DISTINCTNESS andKNAPSACK are arrangements.TheoremLet S be either an arrangement or a (possibly unbounded) polyhedron(when F = R). Denote by N the number of faces (of all the dimensions)of S. Then C(S) ≤ O(n3 · logN) (Meyer auf der Heide [1985]).One can generalize the theorem to arbitrary linear complexes, i. e.the unions of polyhedra.CorollaryC(KNAPSACK ) ≤ O(n5 ) (Meyer auf der Heide [1985]).The corollary does not directly imply P=NP since the construction of analgebraic computation tree is non-uniform (depends on n). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 8 / 23
  38. 38. Complexity upper bound for membership to alinear complexAn arrangement S ⊂ F n is a union of hyperplanes ∪i Hi . A face of S isan intersection of some of the hyperplanes. DISTINCTNESS andKNAPSACK are arrangements.TheoremLet S be either an arrangement or a (possibly unbounded) polyhedron(when F = R). Denote by N the number of faces (of all the dimensions)of S. Then C(S) ≤ O(n3 · logN) (Meyer auf der Heide [1985]).One can generalize the theorem to arbitrary linear complexes, i. e.the unions of polyhedra.CorollaryC(KNAPSACK ) ≤ O(n5 ) (Meyer auf der Heide [1985]).The corollary does not directly imply P=NP since the construction of analgebraic computation tree is non-uniform (depends on n). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 8 / 23
  39. 39. Complexity upper bound for membership to alinear complexAn arrangement S ⊂ F n is a union of hyperplanes ∪i Hi . A face of S isan intersection of some of the hyperplanes. DISTINCTNESS andKNAPSACK are arrangements.TheoremLet S be either an arrangement or a (possibly unbounded) polyhedron(when F = R). Denote by N the number of faces (of all the dimensions)of S. Then C(S) ≤ O(n3 · logN) (Meyer auf der Heide [1985]).One can generalize the theorem to arbitrary linear complexes, i. e.the unions of polyhedra.CorollaryC(KNAPSACK ) ≤ O(n5 ) (Meyer auf der Heide [1985]).The corollary does not directly imply P=NP since the construction of analgebraic computation tree is non-uniform (depends on n). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 8 / 23
  40. 40. Membership to a polyhedronLet P ⊂ Rn be a convex polyhedron (with N faces of all thedimensions). Then the sum of its Betti numbers equals 1.TheoremComplexity of linear decision trees C1 (P) ≥ log2 N (Rivest-Yao [1980])TheoremCd (P) ≥ Ω(logN), provided that N ≥ (dn)Ω(n) (G.-Karpinski-Vorobjov[1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 9 / 23
  41. 41. Membership to a polyhedronLet P ⊂ Rn be a convex polyhedron (with N faces of all thedimensions). Then the sum of its Betti numbers equals 1.TheoremComplexity of linear decision trees C1 (P) ≥ log2 N (Rivest-Yao [1980])TheoremCd (P) ≥ Ω(logN), provided that N ≥ (dn)Ω(n) (G.-Karpinski-Vorobjov[1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 9 / 23
  42. 42. Membership to a polyhedronLet P ⊂ Rn be a convex polyhedron (with N faces of all thedimensions). Then the sum of its Betti numbers equals 1.TheoremComplexity of linear decision trees C1 (P) ≥ log2 N (Rivest-Yao [1980])TheoremCd (P) ≥ Ω(logN), provided that N ≥ (dn)Ω(n) (G.-Karpinski-Vorobjov[1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 9 / 23
  43. 43. Pfaffian decision treesAs a testing function gv at a node v of a Pfaffian d-decision treeappears a Pfaffian function.Let v0 , . . . , vk be the nodes on the path from the root v0 to vk .dgvk = 1≤j≤n hkj (gv0 , . . . , gvk , X1 , . . . , Xn ) · dXjfor some polynomials hkj ∈ R[Z0 , . . . , Zk , X1 , . . . , Xn ], deg(hkj ) ≤ d. pfaffThe complexity denote by Cd .Examples of Pfaffian functions: polynomials, exp, log (on the√positive half-line), sin (on the interval (−π, π), X (on the positivehalf-line), X −1 on R {0}.Theorem Let S ⊂ Rn be semi-pfaffian (defined by inequalities of the form g ≥ 0 for Pfaffian functions g. Denote by c3 the sum of its Betti pfaff √ numbers. Then Cd ≥ Ω( logc3 ) (G.-Vorobjov [1994]); For a polyhedron P with N faces of all the dimensions pfaff 4 Cd (P) ≥ Ω( logN), provided that N ≥ (dn)Ω(n ·logd) (G.-Vorobjov [1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 10 / 23
  44. 44. Pfaffian decision treesAs a testing function gv at a node v of a Pfaffian d-decision treeappears a Pfaffian function.Let v0 , . . . , vk be the nodes on the path from the root v0 to vk .dgvk = 1≤j≤n hkj (gv0 , . . . , gvk , X1 , . . . , Xn ) · dXjfor some polynomials hkj ∈ R[Z0 , . . . , Zk , X1 , . . . , Xn ], deg(hkj ) ≤ d. pfaffThe complexity denote by Cd .Examples of Pfaffian functions: polynomials, exp, log (on the√positive half-line), sin (on the interval (−π, π), X (on the positivehalf-line), X −1 on R {0}.Theorem Let S ⊂ Rn be semi-pfaffian (defined by inequalities of the form g ≥ 0 for Pfaffian functions g. Denote by c3 the sum of its Betti pfaff √ numbers. Then Cd ≥ Ω( logc3 ) (G.-Vorobjov [1994]); For a polyhedron P with N faces of all the dimensions pfaff 4 Cd (P) ≥ Ω( logN), provided that N ≥ (dn)Ω(n ·logd) (G.-Vorobjov [1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 10 / 23
  45. 45. Pfaffian decision treesAs a testing function gv at a node v of a Pfaffian d-decision treeappears a Pfaffian function.Let v0 , . . . , vk be the nodes on the path from the root v0 to vk .dgvk = 1≤j≤n hkj (gv0 , . . . , gvk , X1 , . . . , Xn ) · dXjfor some polynomials hkj ∈ R[Z0 , . . . , Zk , X1 , . . . , Xn ], deg(hkj ) ≤ d. pfaffThe complexity denote by Cd .Examples of Pfaffian functions: polynomials, exp, log (on the√positive half-line), sin (on the interval (−π, π), X (on the positivehalf-line), X −1 on R {0}.Theorem Let S ⊂ Rn be semi-pfaffian (defined by inequalities of the form g ≥ 0 for Pfaffian functions g. Denote by c3 the sum of its Betti pfaff √ numbers. Then Cd ≥ Ω( logc3 ) (G.-Vorobjov [1994]); For a polyhedron P with N faces of all the dimensions pfaff 4 Cd (P) ≥ Ω( logN), provided that N ≥ (dn)Ω(n ·logd) (G.-Vorobjov [1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 10 / 23
  46. 46. Pfaffian decision treesAs a testing function gv at a node v of a Pfaffian d-decision treeappears a Pfaffian function.Let v0 , . . . , vk be the nodes on the path from the root v0 to vk .dgvk = 1≤j≤n hkj (gv0 , . . . , gvk , X1 , . . . , Xn ) · dXjfor some polynomials hkj ∈ R[Z0 , . . . , Zk , X1 , . . . , Xn ], deg(hkj ) ≤ d. pfaffThe complexity denote by Cd .Examples of Pfaffian functions: polynomials, exp, log (on the√positive half-line), sin (on the interval (−π, π), X (on the positivehalf-line), X −1 on R {0}.Theorem Let S ⊂ Rn be semi-pfaffian (defined by inequalities of the form g ≥ 0 for Pfaffian functions g. Denote by c3 the sum of its Betti pfaff √ numbers. Then Cd ≥ Ω( logc3 ) (G.-Vorobjov [1994]); For a polyhedron P with N faces of all the dimensions pfaff 4 Cd (P) ≥ Ω( logN), provided that N ≥ (dn)Ω(n ·logd) (G.-Vorobjov [1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 10 / 23
  47. 47. Pfaffian decision treesAs a testing function gv at a node v of a Pfaffian d-decision treeappears a Pfaffian function.Let v0 , . . . , vk be the nodes on the path from the root v0 to vk .dgvk = 1≤j≤n hkj (gv0 , . . . , gvk , X1 , . . . , Xn ) · dXjfor some polynomials hkj ∈ R[Z0 , . . . , Zk , X1 , . . . , Xn ], deg(hkj ) ≤ d. pfaffThe complexity denote by Cd .Examples of Pfaffian functions: polynomials, exp, log (on the√positive half-line), sin (on the interval (−π, π), X (on the positivehalf-line), X −1 on R {0}.Theorem Let S ⊂ Rn be semi-pfaffian (defined by inequalities of the form g ≥ 0 for Pfaffian functions g. Denote by c3 the sum of its Betti pfaff √ numbers. Then Cd ≥ Ω( logc3 ) (G.-Vorobjov [1994]); For a polyhedron P with N faces of all the dimensions pfaff 4 Cd (P) ≥ Ω( logN), provided that N ≥ (dn)Ω(n ·logd) (G.-Vorobjov [1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 10 / 23
  48. 48. Pfaffian decision treesAs a testing function gv at a node v of a Pfaffian d-decision treeappears a Pfaffian function.Let v0 , . . . , vk be the nodes on the path from the root v0 to vk .dgvk = 1≤j≤n hkj (gv0 , . . . , gvk , X1 , . . . , Xn ) · dXjfor some polynomials hkj ∈ R[Z0 , . . . , Zk , X1 , . . . , Xn ], deg(hkj ) ≤ d. pfaffThe complexity denote by Cd .Examples of Pfaffian functions: polynomials, exp, log (on the√positive half-line), sin (on the interval (−π, π), X (on the positivehalf-line), X −1 on R {0}.Theorem Let S ⊂ Rn be semi-pfaffian (defined by inequalities of the form g ≥ 0 for Pfaffian functions g. Denote by c3 the sum of its Betti pfaff √ numbers. Then Cd ≥ Ω( logc3 ) (G.-Vorobjov [1994]); For a polyhedron P with N faces of all the dimensions pfaff 4 Cd (P) ≥ Ω( logN), provided that N ≥ (dn)Ω(n ·logd) (G.-Vorobjov [1994]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 10 / 23
  49. 49. Probabilistic decision and computation treesLet {Ti }1≤i≤s be a collection of either algebraic d-decision or,respectively computation trees with attributed probabilitiesp1 , . . . , ps , 1≤i≤s pi = 1. It is called probabilistic algebraicd-decision or, respectively probabilistic computation tree.A probabilistic tree accepts set S ⊂ F n if for any input x ∈ F n theoutput is correct with probability > 2/3. probThe complexity Cd is defined as expectation 1≤i≤s pi · Cd (Ti ).Similarly, one defines C prob .One can also consider a continuous distribution of trees.Speed-up by probabilistic computation trees C prob (EQUALITY ) n (Burgisser-Karpinski-Lickteig [1992]); ¨ Consider polynomials f (Z ) = 1≤i≤n (Z − xi ), h(Z ) = 1≤i≤n (Z − yi ). Sets {x1 , . . . , xn } = {y1 , . . . , yn } ⇔ f ≡ h; Choose randomly z0 ∈ F and test whether f (z0 ) = h(z0 ). If yes, return ”accept”, else ”reject”. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 11 / 23
  50. 50. Probabilistic decision and computation treesLet {Ti }1≤i≤s be a collection of either algebraic d-decision or,respectively computation trees with attributed probabilitiesp1 , . . . , ps , 1≤i≤s pi = 1. It is called probabilistic algebraicd-decision or, respectively probabilistic computation tree.A probabilistic tree accepts set S ⊂ F n if for any input x ∈ F n theoutput is correct with probability > 2/3. probThe complexity Cd is defined as expectation 1≤i≤s pi · Cd (Ti ).Similarly, one defines C prob .One can also consider a continuous distribution of trees.Speed-up by probabilistic computation trees C prob (EQUALITY ) n (Burgisser-Karpinski-Lickteig [1992]); ¨ Consider polynomials f (Z ) = 1≤i≤n (Z − xi ), h(Z ) = 1≤i≤n (Z − yi ). Sets {x1 , . . . , xn } = {y1 , . . . , yn } ⇔ f ≡ h; Choose randomly z0 ∈ F and test whether f (z0 ) = h(z0 ). If yes, return ”accept”, else ”reject”. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 11 / 23
  51. 51. Probabilistic decision and computation treesLet {Ti }1≤i≤s be a collection of either algebraic d-decision or,respectively computation trees with attributed probabilitiesp1 , . . . , ps , 1≤i≤s pi = 1. It is called probabilistic algebraicd-decision or, respectively probabilistic computation tree.A probabilistic tree accepts set S ⊂ F n if for any input x ∈ F n theoutput is correct with probability > 2/3. probThe complexity Cd is defined as expectation 1≤i≤s pi · Cd (Ti ).Similarly, one defines C prob .One can also consider a continuous distribution of trees.Speed-up by probabilistic computation trees C prob (EQUALITY ) n (Burgisser-Karpinski-Lickteig [1992]); ¨ Consider polynomials f (Z ) = 1≤i≤n (Z − xi ), h(Z ) = 1≤i≤n (Z − yi ). Sets {x1 , . . . , xn } = {y1 , . . . , yn } ⇔ f ≡ h; Choose randomly z0 ∈ F and test whether f (z0 ) = h(z0 ). If yes, return ”accept”, else ”reject”. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 11 / 23
  52. 52. Probabilistic decision and computation treesLet {Ti }1≤i≤s be a collection of either algebraic d-decision or,respectively computation trees with attributed probabilitiesp1 , . . . , ps , 1≤i≤s pi = 1. It is called probabilistic algebraicd-decision or, respectively probabilistic computation tree.A probabilistic tree accepts set S ⊂ F n if for any input x ∈ F n theoutput is correct with probability > 2/3. probThe complexity Cd is defined as expectation 1≤i≤s pi · Cd (Ti ).Similarly, one defines C prob .One can also consider a continuous distribution of trees.Speed-up by probabilistic computation trees C prob (EQUALITY ) n (Burgisser-Karpinski-Lickteig [1992]); ¨ Consider polynomials f (Z ) = 1≤i≤n (Z − xi ), h(Z ) = 1≤i≤n (Z − yi ). Sets {x1 , . . . , xn } = {y1 , . . . , yn } ⇔ f ≡ h; Choose randomly z0 ∈ F and test whether f (z0 ) = h(z0 ). If yes, return ”accept”, else ”reject”. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 11 / 23
  53. 53. Probabilistic decision and computation treesLet {Ti }1≤i≤s be a collection of either algebraic d-decision or,respectively computation trees with attributed probabilitiesp1 , . . . , ps , 1≤i≤s pi = 1. It is called probabilistic algebraicd-decision or, respectively probabilistic computation tree.A probabilistic tree accepts set S ⊂ F n if for any input x ∈ F n theoutput is correct with probability > 2/3. probThe complexity Cd is defined as expectation 1≤i≤s pi · Cd (Ti ).Similarly, one defines C prob .One can also consider a continuous distribution of trees.Speed-up by probabilistic computation trees C prob (EQUALITY ) n (Burgisser-Karpinski-Lickteig [1992]); ¨ Consider polynomials f (Z ) = 1≤i≤n (Z − xi ), h(Z ) = 1≤i≤n (Z − yi ). Sets {x1 , . . . , xn } = {y1 , . . . , yn } ⇔ f ≡ h; Choose randomly z0 ∈ F and test whether f (z0 ) = h(z0 ). If yes, return ”accept”, else ”reject”. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 11 / 23
  54. 54. Probabilistic decision and computation treesLet {Ti }1≤i≤s be a collection of either algebraic d-decision or,respectively computation trees with attributed probabilitiesp1 , . . . , ps , 1≤i≤s pi = 1. It is called probabilistic algebraicd-decision or, respectively probabilistic computation tree.A probabilistic tree accepts set S ⊂ F n if for any input x ∈ F n theoutput is correct with probability > 2/3. probThe complexity Cd is defined as expectation 1≤i≤s pi · Cd (Ti ).Similarly, one defines C prob .One can also consider a continuous distribution of trees.Speed-up by probabilistic computation trees C prob (EQUALITY ) n (Burgisser-Karpinski-Lickteig [1992]); ¨ Consider polynomials f (Z ) = 1≤i≤n (Z − xi ), h(Z ) = 1≤i≤n (Z − yi ). Sets {x1 , . . . , xn } = {y1 , . . . , yn } ⇔ f ≡ h; Choose randomly z0 ∈ F and test whether f (z0 ) = h(z0 ). If yes, return ”accept”, else ”reject”. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 11 / 23
  55. 55. Probabilistic decision and computation treesLet {Ti }1≤i≤s be a collection of either algebraic d-decision or,respectively computation trees with attributed probabilitiesp1 , . . . , ps , 1≤i≤s pi = 1. It is called probabilistic algebraicd-decision or, respectively probabilistic computation tree.A probabilistic tree accepts set S ⊂ F n if for any input x ∈ F n theoutput is correct with probability > 2/3. probThe complexity Cd is defined as expectation 1≤i≤s pi · Cd (Ti ).Similarly, one defines C prob .One can also consider a continuous distribution of trees.Speed-up by probabilistic computation trees C prob (EQUALITY ) n (Burgisser-Karpinski-Lickteig [1992]); ¨ Consider polynomials f (Z ) = 1≤i≤n (Z − xi ), h(Z ) = 1≤i≤n (Z − yi ). Sets {x1 , . . . , xn } = {y1 , . . . , yn } ⇔ f ≡ h; Choose randomly z0 ∈ F and test whether f (z0 ) = h(z0 ). If yes, return ”accept”, else ”reject”. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 11 / 23
  56. 56. Lower bounds for probabilistic algebraic treesThe topological methods and differential-geometric methods fail forprobabilistic trees (another evidence is the latter example).Let S ⊂ F n be a polyhedron (when F = R) or an arrangementdetermined by m hyperplanes having N ≥ mΩ(n) faces.Theorem prob1) Cd (S) ≥ Ω(logN) for constant d (G.-Karpinski-Meyer auf derHeide-Smolensky [1995]);2) C prob (S) ≥ Ω(logN) (G. [1997]).Corollary prob prob1) Cd (DISTINCTNESS) n · logn, Cd (KNAPSACK ) ≥ Ω(n2 )(G.-Karpinski-Meyer auf der Heide-Smolensky [1995]);2) C prob (DISTINCTNESS) n · logn, C prob (KNAPSACK ) ≥ Ω(n2 ) (G.).EQUALITY is a union of n-dimensional planes in F 2n (not anarrangement), thus the Theorem is not applicable. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 12 / 23
  57. 57. Lower bounds for probabilistic algebraic treesThe topological methods and differential-geometric methods fail forprobabilistic trees (another evidence is the latter example).Let S ⊂ F n be a polyhedron (when F = R) or an arrangementdetermined by m hyperplanes having N ≥ mΩ(n) faces.Theorem prob1) Cd (S) ≥ Ω(logN) for constant d (G.-Karpinski-Meyer auf derHeide-Smolensky [1995]);2) C prob (S) ≥ Ω(logN) (G. [1997]).Corollary prob prob1) Cd (DISTINCTNESS) n · logn, Cd (KNAPSACK ) ≥ Ω(n2 )(G.-Karpinski-Meyer auf der Heide-Smolensky [1995]);2) C prob (DISTINCTNESS) n · logn, C prob (KNAPSACK ) ≥ Ω(n2 ) (G.).EQUALITY is a union of n-dimensional planes in F 2n (not anarrangement), thus the Theorem is not applicable. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 12 / 23
  58. 58. Lower bounds for probabilistic algebraic treesThe topological methods and differential-geometric methods fail forprobabilistic trees (another evidence is the latter example).Let S ⊂ F n be a polyhedron (when F = R) or an arrangementdetermined by m hyperplanes having N ≥ mΩ(n) faces.Theorem prob1) Cd (S) ≥ Ω(logN) for constant d (G.-Karpinski-Meyer auf derHeide-Smolensky [1995]);2) C prob (S) ≥ Ω(logN) (G. [1997]).Corollary prob prob1) Cd (DISTINCTNESS) n · logn, Cd (KNAPSACK ) ≥ Ω(n2 )(G.-Karpinski-Meyer auf der Heide-Smolensky [1995]);2) C prob (DISTINCTNESS) n · logn, C prob (KNAPSACK ) ≥ Ω(n2 ) (G.).EQUALITY is a union of n-dimensional planes in F 2n (not anarrangement), thus the Theorem is not applicable. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 12 / 23
  59. 59. Lower bounds for probabilistic algebraic treesThe topological methods and differential-geometric methods fail forprobabilistic trees (another evidence is the latter example).Let S ⊂ F n be a polyhedron (when F = R) or an arrangementdetermined by m hyperplanes having N ≥ mΩ(n) faces.Theorem prob1) Cd (S) ≥ Ω(logN) for constant d (G.-Karpinski-Meyer auf derHeide-Smolensky [1995]);2) C prob (S) ≥ Ω(logN) (G. [1997]).Corollary prob prob1) Cd (DISTINCTNESS) n · logn, Cd (KNAPSACK ) ≥ Ω(n2 )(G.-Karpinski-Meyer auf der Heide-Smolensky [1995]);2) C prob (DISTINCTNESS) n · logn, C prob (KNAPSACK ) ≥ Ω(n2 ) (G.).EQUALITY is a union of n-dimensional planes in F 2n (not anarrangement), thus the Theorem is not applicable. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 12 / 23
  60. 60. Lower bounds for probabilistic algebraic treesThe topological methods and differential-geometric methods fail forprobabilistic trees (another evidence is the latter example).Let S ⊂ F n be a polyhedron (when F = R) or an arrangementdetermined by m hyperplanes having N ≥ mΩ(n) faces.Theorem prob1) Cd (S) ≥ Ω(logN) for constant d (G.-Karpinski-Meyer auf derHeide-Smolensky [1995]);2) C prob (S) ≥ Ω(logN) (G. [1997]).Corollary prob prob1) Cd (DISTINCTNESS) n · logn, Cd (KNAPSACK ) ≥ Ω(n2 )(G.-Karpinski-Meyer auf der Heide-Smolensky [1995]);2) C prob (DISTINCTNESS) n · logn, C prob (KNAPSACK ) ≥ Ω(n2 ) (G.).EQUALITY is a union of n-dimensional planes in F 2n (not anarrangement), thus the Theorem is not applicable. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 12 / 23
  61. 61. Lower bounds for probabilistic algebraic treesThe topological methods and differential-geometric methods fail forprobabilistic trees (another evidence is the latter example).Let S ⊂ F n be a polyhedron (when F = R) or an arrangementdetermined by m hyperplanes having N ≥ mΩ(n) faces.Theorem prob1) Cd (S) ≥ Ω(logN) for constant d (G.-Karpinski-Meyer auf derHeide-Smolensky [1995]);2) C prob (S) ≥ Ω(logN) (G. [1997]).Corollary prob prob1) Cd (DISTINCTNESS) n · logn, Cd (KNAPSACK ) ≥ Ω(n2 )(G.-Karpinski-Meyer auf der Heide-Smolensky [1995]);2) C prob (DISTINCTNESS) n · logn, C prob (KNAPSACK ) ≥ Ω(n2 ) (G.).EQUALITY is a union of n-dimensional planes in F 2n (not anarrangement), thus the Theorem is not applicable. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 12 / 23
  62. 62. Lower bounds for probabilistic algebraic treesThe topological methods and differential-geometric methods fail forprobabilistic trees (another evidence is the latter example).Let S ⊂ F n be a polyhedron (when F = R) or an arrangementdetermined by m hyperplanes having N ≥ mΩ(n) faces.Theorem prob1) Cd (S) ≥ Ω(logN) for constant d (G.-Karpinski-Meyer auf derHeide-Smolensky [1995]);2) C prob (S) ≥ Ω(logN) (G. [1997]).Corollary prob prob1) Cd (DISTINCTNESS) n · logn, Cd (KNAPSACK ) ≥ Ω(n2 )(G.-Karpinski-Meyer auf der Heide-Smolensky [1995]);2) C prob (DISTINCTNESS) n · logn, C prob (KNAPSACK ) ≥ Ω(n2 ) (G.).EQUALITY is a union of n-dimensional planes in F 2n (not anarrangement), thus the Theorem is not applicable. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 12 / 23
  63. 63. Probabilistic computation trees over algebraicallyclosed fields of zero characteristicLet S = H1 ∪ · · · ∪ Hm ⊂ F n be an arrangement.TheoremAssume that for a certain c > 0 any subarrangement Hi1 ∪ · · · ∪ Hi cmcontaining cm hyperplanes, has ≥ N faces (of all the dimensions).Then C prob (S) ≥ Ω(logN − 2 · n) (G. [1997]).Corollary C prob (DISTINCTNESS) n · logn; C prob (KNAPSACK ) ≥ Ω(n2 ). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 13 / 23
  64. 64. Probabilistic computation trees over algebraicallyclosed fields of zero characteristicLet S = H1 ∪ · · · ∪ Hm ⊂ F n be an arrangement.TheoremAssume that for a certain c > 0 any subarrangement Hi1 ∪ · · · ∪ Hi cmcontaining cm hyperplanes, has ≥ N faces (of all the dimensions).Then C prob (S) ≥ Ω(logN − 2 · n) (G. [1997]).Corollary C prob (DISTINCTNESS) n · logn; C prob (KNAPSACK ) ≥ Ω(n2 ). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 13 / 23
  65. 65. Probabilistic computation trees over algebraicallyclosed fields of zero characteristicLet S = H1 ∪ · · · ∪ Hm ⊂ F n be an arrangement.TheoremAssume that for a certain c > 0 any subarrangement Hi1 ∪ · · · ∪ Hi cmcontaining cm hyperplanes, has ≥ N faces (of all the dimensions).Then C prob (S) ≥ Ω(logN − 2 · n) (G. [1997]).Corollary C prob (DISTINCTNESS) n · logn; C prob (KNAPSACK ) ≥ Ω(n2 ). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 13 / 23
  66. 66. Analytic and topological decision treesIf a decision tree admits arbitrary polynomials as testing functions wecall it topological tree and its complexity denote by Ctop ≤ C, Cd(Shub-Smale).I. e. computations are gratis, only branchings are counted.If a decision tree admits arbitrary analytic functions as testing functionswe call it analytic tree and its complexity denote by Can ≤ Ctop .MAX ORTANT = Rn = {(x1 , . . . , xn ) : x1 ≥ 0, . . . , xn ≥ 0} ⇔ + MAX= = {(x1 , . . . , xn ) : x1 = max{x1 , . . . , xn }} is a weaker problem than MAX: to compute max{x1 , . . . , xn } by means of a modification of a decision (or computation) tree in which the outputs of leaves are functions (polynomials) rather than labels ”accept” or ”reject”. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 14 / 23
  67. 67. Analytic and topological decision treesIf a decision tree admits arbitrary polynomials as testing functions wecall it topological tree and its complexity denote by Ctop ≤ C, Cd(Shub-Smale).I. e. computations are gratis, only branchings are counted.If a decision tree admits arbitrary analytic functions as testing functionswe call it analytic tree and its complexity denote by Can ≤ Ctop .MAX ORTANT = Rn = {(x1 , . . . , xn ) : x1 ≥ 0, . . . , xn ≥ 0} ⇔ + MAX= = {(x1 , . . . , xn ) : x1 = max{x1 , . . . , xn }} is a weaker problem than MAX: to compute max{x1 , . . . , xn } by means of a modification of a decision (or computation) tree in which the outputs of leaves are functions (polynomials) rather than labels ”accept” or ”reject”. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 14 / 23
  68. 68. Analytic and topological decision treesIf a decision tree admits arbitrary polynomials as testing functions wecall it topological tree and its complexity denote by Ctop ≤ C, Cd(Shub-Smale).I. e. computations are gratis, only branchings are counted.If a decision tree admits arbitrary analytic functions as testing functionswe call it analytic tree and its complexity denote by Can ≤ Ctop .MAX ORTANT = Rn = {(x1 , . . . , xn ) : x1 ≥ 0, . . . , xn ≥ 0} ⇔ + MAX= = {(x1 , . . . , xn ) : x1 = max{x1 , . . . , xn }} is a weaker problem than MAX: to compute max{x1 , . . . , xn } by means of a modification of a decision (or computation) tree in which the outputs of leaves are functions (polynomials) rather than labels ”accept” or ”reject”. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 14 / 23
  69. 69. Analytic and topological decision treesIf a decision tree admits arbitrary polynomials as testing functions wecall it topological tree and its complexity denote by Ctop ≤ C, Cd(Shub-Smale).I. e. computations are gratis, only branchings are counted.If a decision tree admits arbitrary analytic functions as testing functionswe call it analytic tree and its complexity denote by Can ≤ Ctop .MAX ORTANT = Rn = {(x1 , . . . , xn ) : x1 ≥ 0, . . . , xn ≥ 0} ⇔ + MAX= = {(x1 , . . . , xn ) : x1 = max{x1 , . . . , xn }} is a weaker problem than MAX: to compute max{x1 , . . . , xn } by means of a modification of a decision (or computation) tree in which the outputs of leaves are functions (polynomials) rather than labels ”accept” or ”reject”. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 14 / 23
  70. 70. C1 (MAX ) ≤ n − 1Theorem Can (MAX =) = n − 1 (Rabin [1972]; ˜ corrected proof Montana-Pardo-Recio [1994]; short proof G.-Karpinski-Smolensky [1995]); prob Cd (MAX =) ≥ n/(2 · d), C prob (MAX =) ≥ n/4 (G.-K.-S. [1995]); prob prob Ctop (MAX =) ≤ Cn (MAX =) ≤ O(log2 n) (G.-K.-S. [1995]); prob prob Ctop (MAX ) ≤ Cn (MAX ) ≤ O(log2 n) (Ben-Or [1996]; ≤ O(log5 n) G.-K.-S. [1995]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 15 / 23
  71. 71. C1 (MAX ) ≤ n − 1Theorem Can (MAX =) = n − 1 (Rabin [1972]; ˜ corrected proof Montana-Pardo-Recio [1994]; short proof G.-Karpinski-Smolensky [1995]); prob Cd (MAX =) ≥ n/(2 · d), C prob (MAX =) ≥ n/4 (G.-K.-S. [1995]); prob prob Ctop (MAX =) ≤ Cn (MAX =) ≤ O(log2 n) (G.-K.-S. [1995]); prob prob Ctop (MAX ) ≤ Cn (MAX ) ≤ O(log2 n) (Ben-Or [1996]; ≤ O(log5 n) G.-K.-S. [1995]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 15 / 23
  72. 72. C1 (MAX ) ≤ n − 1Theorem Can (MAX =) = n − 1 (Rabin [1972]; ˜ corrected proof Montana-Pardo-Recio [1994]; short proof G.-Karpinski-Smolensky [1995]); prob Cd (MAX =) ≥ n/(2 · d), C prob (MAX =) ≥ n/4 (G.-K.-S. [1995]); prob prob Ctop (MAX =) ≤ Cn (MAX =) ≤ O(log2 n) (G.-K.-S. [1995]); prob prob Ctop (MAX ) ≤ Cn (MAX ) ≤ O(log2 n) (Ben-Or [1996]; ≤ O(log5 n) G.-K.-S. [1995]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 15 / 23
  73. 73. C1 (MAX ) ≤ n − 1Theorem Can (MAX =) = n − 1 (Rabin [1972]; ˜ corrected proof Montana-Pardo-Recio [1994]; short proof G.-Karpinski-Smolensky [1995]); prob Cd (MAX =) ≥ n/(2 · d), C prob (MAX =) ≥ n/4 (G.-K.-S. [1995]); prob prob Ctop (MAX =) ≤ Cn (MAX =) ≤ O(log2 n) (G.-K.-S. [1995]); prob prob Ctop (MAX ) ≤ Cn (MAX ) ≤ O(log2 n) (Ben-Or [1996]; ≤ O(log5 n) G.-K.-S. [1995]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 15 / 23
  74. 74. C1 (MAX ) ≤ n − 1Theorem Can (MAX =) = n − 1 (Rabin [1972]; ˜ corrected proof Montana-Pardo-Recio [1994]; short proof G.-Karpinski-Smolensky [1995]); prob Cd (MAX =) ≥ n/(2 · d), C prob (MAX =) ≥ n/4 (G.-K.-S. [1995]); prob prob Ctop (MAX =) ≤ Cn (MAX =) ≤ O(log2 n) (G.-K.-S. [1995]); prob prob Ctop (MAX ) ≤ Cn (MAX ) ≤ O(log2 n) (Ben-Or [1996]; ≤ O(log5 n) G.-K.-S. [1995]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 15 / 23
  75. 75. Topological complexity of the range searchingFor f1 , . . . , fm ∈ R[X1 , . . . , Xn ] we say that a decision/computation treeT solves the RANGE SEARCHING problem if any two inputs x, y ∈ Rnwith different sign vectors(sgn(f1 ), . . . , sgn(fn ))(x) = (sgn(f1 ), . . . , sgn(fn ))(y)arrive to different leaves of T . Denote by N the number of sign vectors.TheoremCtop (RANGE SEARCHING) logN (G. [1998]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 16 / 23
  76. 76. Topological complexity of the range searchingFor f1 , . . . , fm ∈ R[X1 , . . . , Xn ] we say that a decision/computation treeT solves the RANGE SEARCHING problem if any two inputs x, y ∈ Rnwith different sign vectors(sgn(f1 ), . . . , sgn(fn ))(x) = (sgn(f1 ), . . . , sgn(fn ))(y)arrive to different leaves of T . Denote by N the number of sign vectors.TheoremCtop (RANGE SEARCHING) logN (G. [1998]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 16 / 23
  77. 77. Lower bound for probabilistic analytic treesWe have seen that for MAX tossing a coin can (exponentially)speed-up computation. Here is an example of a set for which it is notthe case.Let an integer q = 2m . Consider setMODq = {(x1 , . . . , xn ) ∈ Rn : 1≤i≤n xi = 0, q | {i : xi < 0}}.Theorem prob √Can (MODq ) ≥ Ω( n) (G.-Karpinski-Smolensly [1995]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 17 / 23
  78. 78. Lower bound for probabilistic analytic treesWe have seen that for MAX tossing a coin can (exponentially)speed-up computation. Here is an example of a set for which it is notthe case.Let an integer q = 2m . Consider setMODq = {(x1 , . . . , xn ) ∈ Rn : 1≤i≤n xi = 0, q | {i : xi < 0}}.Theorem prob √Can (MODq ) ≥ Ω( n) (G.-Karpinski-Smolensly [1995]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 17 / 23
  79. 79. Lower bound for probabilistic analytic treesWe have seen that for MAX tossing a coin can (exponentially)speed-up computation. Here is an example of a set for which it is notthe case.Let an integer q = 2m . Consider setMODq = {(x1 , . . . , xn ) ∈ Rn : 1≤i≤n xi = 0, q | {i : xi < 0}}.Theorem prob √Can (MODq ) ≥ Ω( n) (G.-Karpinski-Smolensly [1995]). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 17 / 23
  80. 80. Size of decision treesSo far, we studied the depth of decision/computation trees. Sincedepth ≥ log(size), the bounds on the size are stronger.Consider setEXACTn = {(x1 , . . . , x2·n ) ∈ R2·n : 1≤i≤2·n xi = 0, {i : xi < 0} = n}.TheoremFor analytic decision trees sizean (EXACTn ) ≥ 2n /n (G.-K.-S. [1995]).For linear decision trees size1 (MAX =) ≤ 2 · n;Alternatively to usual (binary) trees one can consider ternary decisiontrees in which a node v with a testing function gv branches into 3nodes according to whether gv < 0, or gv = 0, or gv > 0.TheoremTernary sized (MAX =) ≥ 2cd ·n for d-decision trees, cd > 0(G.-Karpinski-Yao [1994])Open question: size (MAX) for binary decision/computation trees? Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 18 / 23
  81. 81. Size of decision treesSo far, we studied the depth of decision/computation trees. Sincedepth ≥ log(size), the bounds on the size are stronger.Consider setEXACTn = {(x1 , . . . , x2·n ) ∈ R2·n : 1≤i≤2·n xi = 0, {i : xi < 0} = n}.TheoremFor analytic decision trees sizean (EXACTn ) ≥ 2n /n (G.-K.-S. [1995]).For linear decision trees size1 (MAX =) ≤ 2 · n;Alternatively to usual (binary) trees one can consider ternary decisiontrees in which a node v with a testing function gv branches into 3nodes according to whether gv < 0, or gv = 0, or gv > 0.TheoremTernary sized (MAX =) ≥ 2cd ·n for d-decision trees, cd > 0(G.-Karpinski-Yao [1994])Open question: size (MAX) for binary decision/computation trees? Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 18 / 23
  82. 82. Size of decision treesSo far, we studied the depth of decision/computation trees. Sincedepth ≥ log(size), the bounds on the size are stronger.Consider setEXACTn = {(x1 , . . . , x2·n ) ∈ R2·n : 1≤i≤2·n xi = 0, {i : xi < 0} = n}.TheoremFor analytic decision trees sizean (EXACTn ) ≥ 2n /n (G.-K.-S. [1995]).For linear decision trees size1 (MAX =) ≤ 2 · n;Alternatively to usual (binary) trees one can consider ternary decisiontrees in which a node v with a testing function gv branches into 3nodes according to whether gv < 0, or gv = 0, or gv > 0.TheoremTernary sized (MAX =) ≥ 2cd ·n for d-decision trees, cd > 0(G.-Karpinski-Yao [1994])Open question: size (MAX) for binary decision/computation trees? Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 18 / 23
  83. 83. Size of decision treesSo far, we studied the depth of decision/computation trees. Sincedepth ≥ log(size), the bounds on the size are stronger.Consider setEXACTn = {(x1 , . . . , x2·n ) ∈ R2·n : 1≤i≤2·n xi = 0, {i : xi < 0} = n}.TheoremFor analytic decision trees sizean (EXACTn ) ≥ 2n /n (G.-K.-S. [1995]).For linear decision trees size1 (MAX =) ≤ 2 · n;Alternatively to usual (binary) trees one can consider ternary decisiontrees in which a node v with a testing function gv branches into 3nodes according to whether gv < 0, or gv = 0, or gv > 0.TheoremTernary sized (MAX =) ≥ 2cd ·n for d-decision trees, cd > 0(G.-Karpinski-Yao [1994])Open question: size (MAX) for binary decision/computation trees? Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 18 / 23
  84. 84. Size of decision treesSo far, we studied the depth of decision/computation trees. Sincedepth ≥ log(size), the bounds on the size are stronger.Consider setEXACTn = {(x1 , . . . , x2·n ) ∈ R2·n : 1≤i≤2·n xi = 0, {i : xi < 0} = n}.TheoremFor analytic decision trees sizean (EXACTn ) ≥ 2n /n (G.-K.-S. [1995]).For linear decision trees size1 (MAX =) ≤ 2 · n;Alternatively to usual (binary) trees one can consider ternary decisiontrees in which a node v with a testing function gv branches into 3nodes according to whether gv < 0, or gv = 0, or gv > 0.TheoremTernary sized (MAX =) ≥ 2cd ·n for d-decision trees, cd > 0(G.-Karpinski-Yao [1994])Open question: size (MAX) for binary decision/computation trees? Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 18 / 23
  85. 85. Size of decision treesSo far, we studied the depth of decision/computation trees. Sincedepth ≥ log(size), the bounds on the size are stronger.Consider setEXACTn = {(x1 , . . . , x2·n ) ∈ R2·n : 1≤i≤2·n xi = 0, {i : xi < 0} = n}.TheoremFor analytic decision trees sizean (EXACTn ) ≥ 2n /n (G.-K.-S. [1995]).For linear decision trees size1 (MAX =) ≤ 2 · n;Alternatively to usual (binary) trees one can consider ternary decisiontrees in which a node v with a testing function gv branches into 3nodes according to whether gv < 0, or gv = 0, or gv > 0.TheoremTernary sized (MAX =) ≥ 2cd ·n for d-decision trees, cd > 0(G.-Karpinski-Yao [1994])Open question: size (MAX) for binary decision/computation trees? Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 18 / 23
  86. 86. Size of decision treesSo far, we studied the depth of decision/computation trees. Sincedepth ≥ log(size), the bounds on the size are stronger.Consider setEXACTn = {(x1 , . . . , x2·n ) ∈ R2·n : 1≤i≤2·n xi = 0, {i : xi < 0} = n}.TheoremFor analytic decision trees sizean (EXACTn ) ≥ 2n /n (G.-K.-S. [1995]).For linear decision trees size1 (MAX =) ≤ 2 · n;Alternatively to usual (binary) trees one can consider ternary decisiontrees in which a node v with a testing function gv branches into 3nodes according to whether gv < 0, or gv = 0, or gv > 0.TheoremTernary sized (MAX =) ≥ 2cd ·n for d-decision trees, cd > 0(G.-Karpinski-Yao [1994])Open question: size (MAX) for binary decision/computation trees? Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 18 / 23
  87. 87. Communication complexity of decision treesCommunication treeLet F = R (also F = C can be studied). Let input variables be dividedinto 2 groups: X1 , . . . , Xn1 and Y1 , . . . , Yn2 . An input (x, y ) ∈ Rn1 +n2 .To each node v with an even (respectively, odd) depth a calculatingpolynomial av ∈ R[X1 , . . . , Xn1 ] (respectively, bv ∈ R[Y1 , . . . , Yn2 ]) isattached.Also a family of testing polynomials {gvi }1≤i≤Nv is assigned.There are 2 players; the first one has access to x and the second to y .Let v0 , . . . , vk = v be the nodes on the path from the root v0 to vk = vand k be odd (for definiteness).Then at v the second player computes bv (y), transmits it to the firstplayer and branches according to the vector{sgn(gvi (av0 (x), bv1 (y ), av2 (x), . . . , avk −1 (x), bvk (y)}1≤i≤NvCommunication complexity CC is the depth of the communicationtree.One defines also probabilistic communication complexity CC prob . Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 19 / 23
  88. 88. Communication complexity of decision treesCommunication treeLet F = R (also F = C can be studied). Let input variables be dividedinto 2 groups: X1 , . . . , Xn1 and Y1 , . . . , Yn2 . An input (x, y ) ∈ Rn1 +n2 .To each node v with an even (respectively, odd) depth a calculatingpolynomial av ∈ R[X1 , . . . , Xn1 ] (respectively, bv ∈ R[Y1 , . . . , Yn2 ]) isattached.Also a family of testing polynomials {gvi }1≤i≤Nv is assigned.There are 2 players; the first one has access to x and the second to y .Let v0 , . . . , vk = v be the nodes on the path from the root v0 to vk = vand k be odd (for definiteness).Then at v the second player computes bv (y), transmits it to the firstplayer and branches according to the vector{sgn(gvi (av0 (x), bv1 (y ), av2 (x), . . . , avk −1 (x), bvk (y)}1≤i≤NvCommunication complexity CC is the depth of the communicationtree.One defines also probabilistic communication complexity CC prob . Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 19 / 23
  89. 89. Communication complexity of decision treesCommunication treeLet F = R (also F = C can be studied). Let input variables be dividedinto 2 groups: X1 , . . . , Xn1 and Y1 , . . . , Yn2 . An input (x, y ) ∈ Rn1 +n2 .To each node v with an even (respectively, odd) depth a calculatingpolynomial av ∈ R[X1 , . . . , Xn1 ] (respectively, bv ∈ R[Y1 , . . . , Yn2 ]) isattached.Also a family of testing polynomials {gvi }1≤i≤Nv is assigned.There are 2 players; the first one has access to x and the second to y .Let v0 , . . . , vk = v be the nodes on the path from the root v0 to vk = vand k be odd (for definiteness).Then at v the second player computes bv (y), transmits it to the firstplayer and branches according to the vector{sgn(gvi (av0 (x), bv1 (y ), av2 (x), . . . , avk −1 (x), bvk (y)}1≤i≤NvCommunication complexity CC is the depth of the communicationtree.One defines also probabilistic communication complexity CC prob . Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 19 / 23
  90. 90. Communication complexity of decision treesCommunication treeLet F = R (also F = C can be studied). Let input variables be dividedinto 2 groups: X1 , . . . , Xn1 and Y1 , . . . , Yn2 . An input (x, y ) ∈ Rn1 +n2 .To each node v with an even (respectively, odd) depth a calculatingpolynomial av ∈ R[X1 , . . . , Xn1 ] (respectively, bv ∈ R[Y1 , . . . , Yn2 ]) isattached.Also a family of testing polynomials {gvi }1≤i≤Nv is assigned.There are 2 players; the first one has access to x and the second to y .Let v0 , . . . , vk = v be the nodes on the path from the root v0 to vk = vand k be odd (for definiteness).Then at v the second player computes bv (y), transmits it to the firstplayer and branches according to the vector{sgn(gvi (av0 (x), bv1 (y ), av2 (x), . . . , avk −1 (x), bvk (y)}1≤i≤NvCommunication complexity CC is the depth of the communicationtree.One defines also probabilistic communication complexity CC prob . Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 19 / 23
  91. 91. Communication complexity of decision treesCommunication treeLet F = R (also F = C can be studied). Let input variables be dividedinto 2 groups: X1 , . . . , Xn1 and Y1 , . . . , Yn2 . An input (x, y ) ∈ Rn1 +n2 .To each node v with an even (respectively, odd) depth a calculatingpolynomial av ∈ R[X1 , . . . , Xn1 ] (respectively, bv ∈ R[Y1 , . . . , Yn2 ]) isattached.Also a family of testing polynomials {gvi }1≤i≤Nv is assigned.There are 2 players; the first one has access to x and the second to y .Let v0 , . . . , vk = v be the nodes on the path from the root v0 to vk = vand k be odd (for definiteness).Then at v the second player computes bv (y), transmits it to the firstplayer and branches according to the vector{sgn(gvi (av0 (x), bv1 (y ), av2 (x), . . . , avk −1 (x), bvk (y)}1≤i≤NvCommunication complexity CC is the depth of the communicationtree.One defines also probabilistic communication complexity CC prob . Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 19 / 23
  92. 92. Communication complexity of decision treesCommunication treeLet F = R (also F = C can be studied). Let input variables be dividedinto 2 groups: X1 , . . . , Xn1 and Y1 , . . . , Yn2 . An input (x, y ) ∈ Rn1 +n2 .To each node v with an even (respectively, odd) depth a calculatingpolynomial av ∈ R[X1 , . . . , Xn1 ] (respectively, bv ∈ R[Y1 , . . . , Yn2 ]) isattached.Also a family of testing polynomials {gvi }1≤i≤Nv is assigned.There are 2 players; the first one has access to x and the second to y .Let v0 , . . . , vk = v be the nodes on the path from the root v0 to vk = vand k be odd (for definiteness).Then at v the second player computes bv (y), transmits it to the firstplayer and branches according to the vector{sgn(gvi (av0 (x), bv1 (y ), av2 (x), . . . , avk −1 (x), bvk (y)}1≤i≤NvCommunication complexity CC is the depth of the communicationtree.One defines also probabilistic communication complexity CC prob . Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 19 / 23
  93. 93. Communication complexity of decision treesCommunication treeLet F = R (also F = C can be studied). Let input variables be dividedinto 2 groups: X1 , . . . , Xn1 and Y1 , . . . , Yn2 . An input (x, y ) ∈ Rn1 +n2 .To each node v with an even (respectively, odd) depth a calculatingpolynomial av ∈ R[X1 , . . . , Xn1 ] (respectively, bv ∈ R[Y1 , . . . , Yn2 ]) isattached.Also a family of testing polynomials {gvi }1≤i≤Nv is assigned.There are 2 players; the first one has access to x and the second to y .Let v0 , . . . , vk = v be the nodes on the path from the root v0 to vk = vand k be odd (for definiteness).Then at v the second player computes bv (y), transmits it to the firstplayer and branches according to the vector{sgn(gvi (av0 (x), bv1 (y ), av2 (x), . . . , avk −1 (x), bvk (y)}1≤i≤NvCommunication complexity CC is the depth of the communicationtree.One defines also probabilistic communication complexity CC prob . Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 19 / 23
  94. 94. Lower bound on communication complexityRn1 +n2 = {(x, y ) ∈ Rn1 +n2 : ∀i, j xi > 0, yj > 0} ⊂ Rn1 +n2 = Rn1 +n2 . > + >Proposition CC(Rn1 +n2 ), CC(Rn1 +n2 ) = n1 + n2 ; > + CC prob (Rn1 +n2 ) ≤ 4, > CC prob (Rn1 +n2 ) ≤ logO(1) (n1 + n2 ). +Lower bound on probabilistic communication complexityPolyhedron S = {(x, y) ∈ R2·n : ∀i xi + yi > 0}.Arrangement Q = 1≤i≤n {Xi + Yi = 0} ⊂ R2·n .TheoremCC prob (S), CC prob (Q), CC prob (EQUALITY ), CC prob (KNAPSACK ) ≥ n(G. [2006]).Problem; to obtain a lower bound for communication trees withbranching condition {sgn(gvi (av0 (x), av2 (x), . . . , avk −1 (x), y ))}1≤i≤Nv Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 20 / 23
  95. 95. Lower bound on communication complexityRn1 +n2 = {(x, y ) ∈ Rn1 +n2 : ∀i, j xi > 0, yj > 0} ⊂ Rn1 +n2 = Rn1 +n2 . > + >Proposition CC(Rn1 +n2 ), CC(Rn1 +n2 ) = n1 + n2 ; > + CC prob (Rn1 +n2 ) ≤ 4, > CC prob (Rn1 +n2 ) ≤ logO(1) (n1 + n2 ). +Lower bound on probabilistic communication complexityPolyhedron S = {(x, y) ∈ R2·n : ∀i xi + yi > 0}.Arrangement Q = 1≤i≤n {Xi + Yi = 0} ⊂ R2·n .TheoremCC prob (S), CC prob (Q), CC prob (EQUALITY ), CC prob (KNAPSACK ) ≥ n(G. [2006]).Problem; to obtain a lower bound for communication trees withbranching condition {sgn(gvi (av0 (x), av2 (x), . . . , avk −1 (x), y ))}1≤i≤Nv Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 20 / 23
  96. 96. Lower bound on communication complexityRn1 +n2 = {(x, y ) ∈ Rn1 +n2 : ∀i, j xi > 0, yj > 0} ⊂ Rn1 +n2 = Rn1 +n2 . > + >Proposition CC(Rn1 +n2 ), CC(Rn1 +n2 ) = n1 + n2 ; > + CC prob (Rn1 +n2 ) ≤ 4, > CC prob (Rn1 +n2 ) ≤ logO(1) (n1 + n2 ). +Lower bound on probabilistic communication complexityPolyhedron S = {(x, y) ∈ R2·n : ∀i xi + yi > 0}.Arrangement Q = 1≤i≤n {Xi + Yi = 0} ⊂ R2·n .TheoremCC prob (S), CC prob (Q), CC prob (EQUALITY ), CC prob (KNAPSACK ) ≥ n(G. [2006]).Problem; to obtain a lower bound for communication trees withbranching condition {sgn(gvi (av0 (x), av2 (x), . . . , avk −1 (x), y ))}1≤i≤Nv Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 20 / 23
  97. 97. Lower bound on communication complexityRn1 +n2 = {(x, y ) ∈ Rn1 +n2 : ∀i, j xi > 0, yj > 0} ⊂ Rn1 +n2 = Rn1 +n2 . > + >Proposition CC(Rn1 +n2 ), CC(Rn1 +n2 ) = n1 + n2 ; > + CC prob (Rn1 +n2 ) ≤ 4, > CC prob (Rn1 +n2 ) ≤ logO(1) (n1 + n2 ). +Lower bound on probabilistic communication complexityPolyhedron S = {(x, y) ∈ R2·n : ∀i xi + yi > 0}.Arrangement Q = 1≤i≤n {Xi + Yi = 0} ⊂ R2·n .TheoremCC prob (S), CC prob (Q), CC prob (EQUALITY ), CC prob (KNAPSACK ) ≥ n(G. [2006]).Problem; to obtain a lower bound for communication trees withbranching condition {sgn(gvi (av0 (x), av2 (x), . . . , avk −1 (x), y ))}1≤i≤Nv Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 20 / 23
  98. 98. Lower bound on communication complexityRn1 +n2 = {(x, y ) ∈ Rn1 +n2 : ∀i, j xi > 0, yj > 0} ⊂ Rn1 +n2 = Rn1 +n2 . > + >Proposition CC(Rn1 +n2 ), CC(Rn1 +n2 ) = n1 + n2 ; > + CC prob (Rn1 +n2 ) ≤ 4, > CC prob (Rn1 +n2 ) ≤ logO(1) (n1 + n2 ). +Lower bound on probabilistic communication complexityPolyhedron S = {(x, y) ∈ R2·n : ∀i xi + yi > 0}.Arrangement Q = 1≤i≤n {Xi + Yi = 0} ⊂ R2·n .TheoremCC prob (S), CC prob (Q), CC prob (EQUALITY ), CC prob (KNAPSACK ) ≥ n(G. [2006]).Problem; to obtain a lower bound for communication trees withbranching condition {sgn(gvi (av0 (x), av2 (x), . . . , avk −1 (x), y ))}1≤i≤Nv Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 20 / 23
  99. 99. Lower bound on communication complexityRn1 +n2 = {(x, y ) ∈ Rn1 +n2 : ∀i, j xi > 0, yj > 0} ⊂ Rn1 +n2 = Rn1 +n2 . > + >Proposition CC(Rn1 +n2 ), CC(Rn1 +n2 ) = n1 + n2 ; > + CC prob (Rn1 +n2 ) ≤ 4, > CC prob (Rn1 +n2 ) ≤ logO(1) (n1 + n2 ). +Lower bound on probabilistic communication complexityPolyhedron S = {(x, y) ∈ R2·n : ∀i xi + yi > 0}.Arrangement Q = 1≤i≤n {Xi + Yi = 0} ⊂ R2·n .TheoremCC prob (S), CC prob (Q), CC prob (EQUALITY ), CC prob (KNAPSACK ) ≥ n(G. [2006]).Problem; to obtain a lower bound for communication trees withbranching condition {sgn(gvi (av0 (x), av2 (x), . . . , avk −1 (x), y ))}1≤i≤Nv Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 20 / 23
  100. 100. Lower bound on communication complexityRn1 +n2 = {(x, y ) ∈ Rn1 +n2 : ∀i, j xi > 0, yj > 0} ⊂ Rn1 +n2 = Rn1 +n2 . > + >Proposition CC(Rn1 +n2 ), CC(Rn1 +n2 ) = n1 + n2 ; > + CC prob (Rn1 +n2 ) ≤ 4, > CC prob (Rn1 +n2 ) ≤ logO(1) (n1 + n2 ). +Lower bound on probabilistic communication complexityPolyhedron S = {(x, y) ∈ R2·n : ∀i xi + yi > 0}.Arrangement Q = 1≤i≤n {Xi + Yi = 0} ⊂ R2·n .TheoremCC prob (S), CC prob (Q), CC prob (EQUALITY ), CC prob (KNAPSACK ) ≥ n(G. [2006]).Problem; to obtain a lower bound for communication trees withbranching condition {sgn(gvi (av0 (x), av2 (x), . . . , avk −1 (x), y ))}1≤i≤Nv Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 20 / 23
  101. 101. Parallel networksTo the root of a network input x ∈ Rn is attributed.To each node v of depth k lead edges from (at most) two nodes v1 , v2of depth k − 1.To each node v a computing polynomial fv ∈ R[X1 , . . . , xn ] and aboolean indicator bv are attached.Informally, bv = 1 (”active”) means that one of the processors islocated in v . We impose the condition that the number of ”active” v ofdepth k is bounded by p ≤ 2k .fv = A ◦ B where ◦ ∈ {+, ×} and A, B ∈ F ∪ {X1 , . . . , Xn } ∪ {fv1 , fv2 }To v two boolean functions Bv 1 , Bv 2 are also attached. if both v1 , v2 are ”passive” then v is ”passive” as well; if v1 , v2 are ”active” then bv = Bv 2 (sgn(fv1 (x)), sgn(fv2 (x))); if only vi , i = 1, 2 is ”active” then bv = Bv 1 (sgn(fvi (x))).Exactly one node w with the largest depth is ”active”, sgn(fw (x)) istreated as an output of the parallel network.The parallel complexity PC is the depth of the network. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 21 / 23
  102. 102. Parallel networksTo the root of a network input x ∈ Rn is attributed.To each node v of depth k lead edges from (at most) two nodes v1 , v2of depth k − 1.To each node v a computing polynomial fv ∈ R[X1 , . . . , xn ] and aboolean indicator bv are attached.Informally, bv = 1 (”active”) means that one of the processors islocated in v . We impose the condition that the number of ”active” v ofdepth k is bounded by p ≤ 2k .fv = A ◦ B where ◦ ∈ {+, ×} and A, B ∈ F ∪ {X1 , . . . , Xn } ∪ {fv1 , fv2 }To v two boolean functions Bv 1 , Bv 2 are also attached. if both v1 , v2 are ”passive” then v is ”passive” as well; if v1 , v2 are ”active” then bv = Bv 2 (sgn(fv1 (x)), sgn(fv2 (x))); if only vi , i = 1, 2 is ”active” then bv = Bv 1 (sgn(fvi (x))).Exactly one node w with the largest depth is ”active”, sgn(fw (x)) istreated as an output of the parallel network.The parallel complexity PC is the depth of the network. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 21 / 23
  103. 103. Parallel networksTo the root of a network input x ∈ Rn is attributed.To each node v of depth k lead edges from (at most) two nodes v1 , v2of depth k − 1.To each node v a computing polynomial fv ∈ R[X1 , . . . , xn ] and aboolean indicator bv are attached.Informally, bv = 1 (”active”) means that one of the processors islocated in v . We impose the condition that the number of ”active” v ofdepth k is bounded by p ≤ 2k .fv = A ◦ B where ◦ ∈ {+, ×} and A, B ∈ F ∪ {X1 , . . . , Xn } ∪ {fv1 , fv2 }To v two boolean functions Bv 1 , Bv 2 are also attached. if both v1 , v2 are ”passive” then v is ”passive” as well; if v1 , v2 are ”active” then bv = Bv 2 (sgn(fv1 (x)), sgn(fv2 (x))); if only vi , i = 1, 2 is ”active” then bv = Bv 1 (sgn(fvi (x))).Exactly one node w with the largest depth is ”active”, sgn(fw (x)) istreated as an output of the parallel network.The parallel complexity PC is the depth of the network. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 21 / 23
  104. 104. Parallel networksTo the root of a network input x ∈ Rn is attributed.To each node v of depth k lead edges from (at most) two nodes v1 , v2of depth k − 1.To each node v a computing polynomial fv ∈ R[X1 , . . . , xn ] and aboolean indicator bv are attached.Informally, bv = 1 (”active”) means that one of the processors islocated in v . We impose the condition that the number of ”active” v ofdepth k is bounded by p ≤ 2k .fv = A ◦ B where ◦ ∈ {+, ×} and A, B ∈ F ∪ {X1 , . . . , Xn } ∪ {fv1 , fv2 }To v two boolean functions Bv 1 , Bv 2 are also attached. if both v1 , v2 are ”passive” then v is ”passive” as well; if v1 , v2 are ”active” then bv = Bv 2 (sgn(fv1 (x)), sgn(fv2 (x))); if only vi , i = 1, 2 is ”active” then bv = Bv 1 (sgn(fvi (x))).Exactly one node w with the largest depth is ”active”, sgn(fw (x)) istreated as an output of the parallel network.The parallel complexity PC is the depth of the network. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 21 / 23
  105. 105. Parallel networksTo the root of a network input x ∈ Rn is attributed.To each node v of depth k lead edges from (at most) two nodes v1 , v2of depth k − 1.To each node v a computing polynomial fv ∈ R[X1 , . . . , xn ] and aboolean indicator bv are attached.Informally, bv = 1 (”active”) means that one of the processors islocated in v . We impose the condition that the number of ”active” v ofdepth k is bounded by p ≤ 2k .fv = A ◦ B where ◦ ∈ {+, ×} and A, B ∈ F ∪ {X1 , . . . , Xn } ∪ {fv1 , fv2 }To v two boolean functions Bv 1 , Bv 2 are also attached. if both v1 , v2 are ”passive” then v is ”passive” as well; if v1 , v2 are ”active” then bv = Bv 2 (sgn(fv1 (x)), sgn(fv2 (x))); if only vi , i = 1, 2 is ”active” then bv = Bv 1 (sgn(fvi (x))).Exactly one node w with the largest depth is ”active”, sgn(fw (x)) istreated as an output of the parallel network.The parallel complexity PC is the depth of the network. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 21 / 23
  106. 106. Parallel networksTo the root of a network input x ∈ Rn is attributed.To each node v of depth k lead edges from (at most) two nodes v1 , v2of depth k − 1.To each node v a computing polynomial fv ∈ R[X1 , . . . , xn ] and aboolean indicator bv are attached.Informally, bv = 1 (”active”) means that one of the processors islocated in v . We impose the condition that the number of ”active” v ofdepth k is bounded by p ≤ 2k .fv = A ◦ B where ◦ ∈ {+, ×} and A, B ∈ F ∪ {X1 , . . . , Xn } ∪ {fv1 , fv2 }To v two boolean functions Bv 1 , Bv 2 are also attached. if both v1 , v2 are ”passive” then v is ”passive” as well; if v1 , v2 are ”active” then bv = Bv 2 (sgn(fv1 (x)), sgn(fv2 (x))); if only vi , i = 1, 2 is ”active” then bv = Bv 1 (sgn(fvi (x))).Exactly one node w with the largest depth is ”active”, sgn(fw (x)) istreated as an output of the parallel network.The parallel complexity PC is the depth of the network. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 21 / 23
  107. 107. Parallel networksTo the root of a network input x ∈ Rn is attributed.To each node v of depth k lead edges from (at most) two nodes v1 , v2of depth k − 1.To each node v a computing polynomial fv ∈ R[X1 , . . . , xn ] and aboolean indicator bv are attached.Informally, bv = 1 (”active”) means that one of the processors islocated in v . We impose the condition that the number of ”active” v ofdepth k is bounded by p ≤ 2k .fv = A ◦ B where ◦ ∈ {+, ×} and A, B ∈ F ∪ {X1 , . . . , Xn } ∪ {fv1 , fv2 }To v two boolean functions Bv 1 , Bv 2 are also attached. if both v1 , v2 are ”passive” then v is ”passive” as well; if v1 , v2 are ”active” then bv = Bv 2 (sgn(fv1 (x)), sgn(fv2 (x))); if only vi , i = 1, 2 is ”active” then bv = Bv 1 (sgn(fvi (x))).Exactly one node w with the largest depth is ”active”, sgn(fw (x)) istreated as an output of the parallel network.The parallel complexity PC is the depth of the network. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 21 / 23
  108. 108. Parallel networksTo the root of a network input x ∈ Rn is attributed.To each node v of depth k lead edges from (at most) two nodes v1 , v2of depth k − 1.To each node v a computing polynomial fv ∈ R[X1 , . . . , xn ] and aboolean indicator bv are attached.Informally, bv = 1 (”active”) means that one of the processors islocated in v . We impose the condition that the number of ”active” v ofdepth k is bounded by p ≤ 2k .fv = A ◦ B where ◦ ∈ {+, ×} and A, B ∈ F ∪ {X1 , . . . , Xn } ∪ {fv1 , fv2 }To v two boolean functions Bv 1 , Bv 2 are also attached. if both v1 , v2 are ”passive” then v is ”passive” as well; if v1 , v2 are ”active” then bv = Bv 2 (sgn(fv1 (x)), sgn(fv2 (x))); if only vi , i = 1, 2 is ”active” then bv = Bv 1 (sgn(fvi (x))).Exactly one node w with the largest depth is ”active”, sgn(fw (x)) istreated as an output of the parallel network.The parallel complexity PC is the depth of the network. Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 21 / 23
  109. 109. kInformally: at depth k the network can contain ≤ 22 nodes withpotential possibility of their use in computation, while for everyparticular input x just ≤ 2k are involved (the processors are located inthem).Which nodes to use in a computation for x is determined by theboolean indicators.Lower bound on the parallel complexityFor set S ⊂ Rn parallel complexity PC(S) ≥ Ω( (logN)/n) where N is either the sum of Betti numbers of S (Mulmuley [1994], ˜ Montana-Morais-Pardo [1996].) or the number of faces of S of all the dimensions when S is a polyhedron, provided that N ≥ nΩ(n) (G. [1996]) Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 22 / 23
  110. 110. kInformally: at depth k the network can contain ≤ 22 nodes withpotential possibility of their use in computation, while for everyparticular input x just ≤ 2k are involved (the processors are located inthem).Which nodes to use in a computation for x is determined by theboolean indicators.Lower bound on the parallel complexityFor set S ⊂ Rn parallel complexity PC(S) ≥ Ω( (logN)/n) where N is either the sum of Betti numbers of S (Mulmuley [1994], ˜ Montana-Morais-Pardo [1996].) or the number of faces of S of all the dimensions when S is a polyhedron, provided that N ≥ nΩ(n) (G. [1996]) Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 22 / 23
  111. 111. kInformally: at depth k the network can contain ≤ 22 nodes withpotential possibility of their use in computation, while for everyparticular input x just ≤ 2k are involved (the processors are located inthem).Which nodes to use in a computation for x is determined by theboolean indicators.Lower bound on the parallel complexityFor set S ⊂ Rn parallel complexity PC(S) ≥ Ω( (logN)/n) where N is either the sum of Betti numbers of S (Mulmuley [1994], ˜ Montana-Morais-Pardo [1996].) or the number of faces of S of all the dimensions when S is a polyhedron, provided that N ≥ nΩ(n) (G. [1996]) Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 22 / 23
  112. 112. kInformally: at depth k the network can contain ≤ 22 nodes withpotential possibility of their use in computation, while for everyparticular input x just ≤ 2k are involved (the processors are located inthem).Which nodes to use in a computation for x is determined by theboolean indicators.Lower bound on the parallel complexityFor set S ⊂ Rn parallel complexity PC(S) ≥ Ω( (logN)/n) where N is either the sum of Betti numbers of S (Mulmuley [1994], ˜ Montana-Morais-Pardo [1996].) or the number of faces of S of all the dimensions when S is a polyhedron, provided that N ≥ nΩ(n) (G. [1996]) Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 22 / 23
  113. 113. kInformally: at depth k the network can contain ≤ 22 nodes withpotential possibility of their use in computation, while for everyparticular input x just ≤ 2k are involved (the processors are located inthem).Which nodes to use in a computation for x is determined by theboolean indicators.Lower bound on the parallel complexityFor set S ⊂ Rn parallel complexity PC(S) ≥ Ω( (logN)/n) where N is either the sum of Betti numbers of S (Mulmuley [1994], ˜ Montana-Morais-Pardo [1996].) or the number of faces of S of all the dimensions when S is a polyhedron, provided that N ≥ nΩ(n) (G. [1996]) Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 22 / 23
  114. 114. Let a linear complex S ⊂ Rn be given as a boolean combination oflinear inequalities Lj ≥ 0, 1 ≤ j ≤ m.Upper bound on the parallel complexity √p, CP(S) ≤ O( logm · loglogm · 2n ) (G. [1996]).Thus, the bound is interesting for small n. The following corollaryprovides a nearly (sharp) quadratic gap between the parallelcomplexity and the usual (sequential) one.CorollaryLet S ⊂ R2 be an m-gon. Then C(S) logm (Steele-Yao [1982], Meyer auf der Heide [1985]); √ √ Ω( logm) ≤ PC(S) ≤ O( logm · loglogm) (Mulmuley [1994], G. [1996]).Open problem: obtain an upper bound on the parallel complexity formore general semi-algebraic sets (rather than for linear complexes). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 23 / 23
  115. 115. Let a linear complex S ⊂ Rn be given as a boolean combination oflinear inequalities Lj ≥ 0, 1 ≤ j ≤ m.Upper bound on the parallel complexity √p, CP(S) ≤ O( logm · loglogm · 2n ) (G. [1996]).Thus, the bound is interesting for small n. The following corollaryprovides a nearly (sharp) quadratic gap between the parallelcomplexity and the usual (sequential) one.CorollaryLet S ⊂ R2 be an m-gon. Then C(S) logm (Steele-Yao [1982], Meyer auf der Heide [1985]); √ √ Ω( logm) ≤ PC(S) ≤ O( logm · loglogm) (Mulmuley [1994], G. [1996]).Open problem: obtain an upper bound on the parallel complexity formore general semi-algebraic sets (rather than for linear complexes). Dima Grigoriev (CNRS) Complexity lower bounds: algebraic trees 14.6.11 23 / 23

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