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(1 2 5 7 9)

(3 4 6)

T(n) = 2 T(n/2) + (n-1)

worst case comparisons for merge

Unfortunately, d is not necessarily the smallest distance between all pairs of points in S1 and

S2 because a closer pair of points can lie on the opposite sides separating the line. When we

combine the two sets, we must examine such points. (Illustrate this on the diagram)

- 1. Chapter 4Chapter 4 Divide-and-ConquerDivide-and-Conquer Copyright © 2007 Pearson Addison-Wesley. All rights reserved.
- 2. 4-2Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Divide-and-ConquerDivide-and-Conquer The most-well known algorithm design strategy:The most-well known algorithm design strategy: 1.1. Divide instance of problem into two or more smallerDivide instance of problem into two or more smaller instancesinstances 2.2. Solve smaller instances recursivelySolve smaller instances recursively 3.3. Obtain solution to original (larger) instance by combiningObtain solution to original (larger) instance by combining these solutionsthese solutions
- 3. 4-3Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Divide-and-Conquer Technique (cont.)Divide-and-Conquer Technique (cont.) subproblem 2 of size n/2 subproblem 1 of size n/2 a solution to subproblem 1 a solution to the original problem a solution to subproblem 2 a problem of size n (instance) It general leads to a recursive algorithm!
- 4. 4-4Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Divide-and-Conquer ExamplesDivide-and-Conquer Examples Sorting: mergesort and quicksortSorting: mergesort and quicksort Binary tree traversalsBinary tree traversals Binary search (?)Binary search (?) Multiplication of large integersMultiplication of large integers Matrix multiplication: Strassen’s algorithmMatrix multiplication: Strassen’s algorithm Closest-pair and convex-hull algorithmsClosest-pair and convex-hull algorithms
- 5. 4-5Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 General Divide-and-Conquer RecurrenceGeneral Divide-and-Conquer Recurrence TT((nn) =) = aTaT((n/bn/b) +) + ff ((nn)) wherewhere ff((nn)) ∈∈ ΘΘ((nndd ),), dd ≥≥ 00 Master TheoremMaster Theorem: If: If a < ba < bdd ,, TT((nn)) ∈∈ ΘΘ((nndd )) IfIf a = ba = bdd ,, TT((nn)) ∈∈ ΘΘ((nndd loglog nn)) IfIf a > ba > bdd ,, TT((nn)) ∈∈ ΘΘ((nnloglog bb aa )) Note: The same results hold with O instead ofNote: The same results hold with O instead of ΘΘ.. Examples:Examples: TT((nn) = 4) = 4TT((nn/2) +/2) + nn ⇒⇒ TT((nn)) ∈∈ ?? TT((nn) = 4) = 4TT((nn/2) +/2) + nn22 ⇒⇒ TT((nn)) ∈∈ ?? TT((nn) = 4) = 4TT((nn/2) +/2) + nn33 ⇒⇒ TT((nn)) ∈∈ ?? ΘΘ((n^2n^2)) ΘΘ((n^2log nn^2log n)) ΘΘ((n^3n^3))
- 6. 4-6Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 MergesortMergesort Split array A[0..Split array A[0..nn-1] into about equal halves and make-1] into about equal halves and make copies of each half in arrays B and Ccopies of each half in arrays B and C Sort arrays B and C recursivelySort arrays B and C recursively Merge sorted arrays B and C into array A as follows:Merge sorted arrays B and C into array A as follows: • Repeat the following until no elements remain in one ofRepeat the following until no elements remain in one of the arrays:the arrays: – compare the first elements in the remainingcompare the first elements in the remaining unprocessed portions of the arraysunprocessed portions of the arrays – copy the smaller of the two into A, whilecopy the smaller of the two into A, while incrementing the index indicating the unprocessedincrementing the index indicating the unprocessed portion of that arrayportion of that array • Once all elements in one of the arrays are processed,Once all elements in one of the arrays are processed, copy the remaining unprocessed elements from the othercopy the remaining unprocessed elements from the other array into A.array into A.
- 7. 4-7Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Pseudocode of MergesortPseudocode of Mergesort
- 8. 4-8Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Pseudocode of MergePseudocode of Merge Time complexity: ΘΘ((p+qp+q)) = ΘΘ((nn)) comparisons
- 9. 4-9Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Mergesort ExampleMergesort Example 8 3 2 9 7 1 5 4 8 3 2 9 7 1 5 4 8 3 2 9 7 1 5 4 8 3 2 9 7 1 5 4 3 8 2 9 1 7 4 5 2 3 8 9 1 4 5 7 1 2 3 4 5 7 8 9 The non-recursive version of Mergesort starts from merging single elements into sorted pairs.
- 10. 4-10Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Analysis of MergesortAnalysis of Mergesort All cases have same efficiency:All cases have same efficiency: ΘΘ((nn loglog nn)) Number of comparisons in the worst case is close toNumber of comparisons in the worst case is close to theoretical minimum for comparison-based sorting:theoretical minimum for comparison-based sorting: loglog22 nn!! ≈≈ nn loglog22 nn - 1.44- 1.44nn Space requirement:Space requirement: ΘΘ((nn) () (notnot in-place)in-place) Can be implemented without recursion (bottom-up)Can be implemented without recursion (bottom-up) T(n) = 2T(n/2) +T(n) = 2T(n/2) + ΘΘ(n), T(1) = 0(n), T(1) = 0
- 11. 4-11Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 QuicksortQuicksort Select aSelect a pivotpivot (partitioning element) – here, the first element(partitioning element) – here, the first element Rearrange the list so that all the elements in the firstRearrange the list so that all the elements in the first ss positions are smaller than or equal to the pivot and all thepositions are smaller than or equal to the pivot and all the elements in the remainingelements in the remaining n-sn-s positions are larger than orpositions are larger than or equal to the pivot (see next slide for an algorithm)equal to the pivot (see next slide for an algorithm) Exchange the pivot with the last element in the first (i.e.,Exchange the pivot with the last element in the first (i.e., ≤≤)) subarray — the pivot is now in its final positionsubarray — the pivot is now in its final position Sort the two subarrays recursivelySort the two subarrays recursively p A[i]≤p A[i]≥p
- 12. 4-12Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Partitioning AlgorithmPartitioning Algorithm Time complexity: ΘΘ((r-lr-l)) comparisons or i > r or j = l<
- 13. 4-13Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Quicksort ExampleQuicksort Example 5 3 1 9 8 2 4 75 3 1 9 8 2 4 7 2 3 1 4 5 8 9 7 1 2 3 4 5 7 8 9 1 2 3 4 5 7 8 9 1 2 3 4 5 7 8 9 1 2 3 4 5 7 8 9
- 14. 4-14Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Analysis of QuicksortAnalysis of Quicksort Best case: split in the middleBest case: split in the middle —— ΘΘ((nn loglog nn)) Worst case: sorted array! —Worst case: sorted array! — ΘΘ((nn22 )) Average case: random arraysAverage case: random arrays —— ΘΘ((nn loglog nn)) Improvements:Improvements: • better pivot selection: median of three partitioningbetter pivot selection: median of three partitioning • switch to insertion sort on small subfilesswitch to insertion sort on small subfiles • elimination of recursionelimination of recursion These combine to 20-25% improvementThese combine to 20-25% improvement Considered the method of choice for internal sorting of largeConsidered the method of choice for internal sorting of large files (files (nn ≥≥ 10000)10000) T(n) = T(n-1) +T(n) = T(n-1) + ΘΘ((nn))
- 15. 4-15Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Binary SearchBinary Search Very efficient algorithm for searching inVery efficient algorithm for searching in sorted arraysorted array:: KK vsvs A[0] . . . A[A[0] . . . A[mm] . . . A[] . . . A[nn-1]-1] IfIf K =K = A[A[mm], stop (successful search); otherwise, continue], stop (successful search); otherwise, continue searching by the same method in A[0..searching by the same method in A[0..mm-1] if-1] if K <K < A[A[mm]] and in A[and in A[mm+1..+1..nn-1] if-1] if K >K > A[A[mm]] ll ←← 0;0; rr ←← nn-1-1 whilewhile ll ≤≤ rr dodo mm ←← ((ll++rr)/2)/2 ifif K =K = A[A[mm] return] return mm else ifelse if K <K < A[A[mm]] rr ←← mm-1-1 elseelse ll ←← mm+1+1 return -1return -1
- 16. 4-16Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Analysis of Binary SearchAnalysis of Binary Search Time efficiencyTime efficiency • worst-case recurrence:worst-case recurrence: CCww ((nn) = 1 +) = 1 + CCww(( nn/2/2 ),), CCww (1) = 1(1) = 1 solution:solution: CCww((nn) =) = loglog22((nn+1)+1) This is VERY fast:This is VERY fast: e.g., Ce.g., Cww(10(1066 ) = 20) = 20 Optimal for searching a sorted arrayOptimal for searching a sorted array Limitations: must be a sorted array (not linked list)Limitations: must be a sorted array (not linked list) Bad (degenerate) example of divide-and-conquerBad (degenerate) example of divide-and-conquer because only one of the sub-instances is solvedbecause only one of the sub-instances is solved Has a continuous counterpart calledHas a continuous counterpart called bisection methodbisection method for solvingfor solving equations in one unknownequations in one unknown ff((xx)) == 0 (see Sec. 12.4)0 (see Sec. 12.4)
- 17. 4-17Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Binary Tree AlgorithmsBinary Tree Algorithms Binary tree is a divide-and-conquer ready structure!Binary tree is a divide-and-conquer ready structure! Ex. 1: Classic traversals (preorder, inorder, postorder)Ex. 1: Classic traversals (preorder, inorder, postorder) AlgorithmAlgorithm InorderInorder((TT)) ifif TT ≠≠ ∅∅ aa aa InorderInorder((TTleftleft)) b c b cb c b c print(root ofprint(root of TT)) d ed e d ed e InorderInorder((TTrightright)) Efficiency:Efficiency: ΘΘ((nn). Why?). Why? Each node is visited/printed once.
- 18. 4-18Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Binary Tree Algorithms (cont.)Binary Tree Algorithms (cont.) Ex. 2: Computing the height of a binary treeEx. 2: Computing the height of a binary tree T TL R hh((TT) = max{) = max{hh((TTLL),), hh((TTRR)} + 1 if)} + 1 if TT ≠≠ ∅∅ andand hh((∅∅) = -1) = -1 Efficiency:Efficiency: ΘΘ((nn). Why?). Why?
- 19. 4-19Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Multiplication of Large IntegersMultiplication of Large Integers Consider the problem of multiplying two (large)Consider the problem of multiplying two (large) nn-digit integers-digit integers represented by arrays of their digits such as:represented by arrays of their digits such as: A = 12345678901357986429 B = 87654321284820912836A = 12345678901357986429 B = 87654321284820912836 The grade-school algorithm:The grade-school algorithm: aa11 aa22 …… aann bb11 bb22 …… bbnn ((dd1010))dd1111dd1212 …… dd11nn ((dd2020))dd2121dd2222 …… dd22nn … … … … … … …… … … … … … … ((ddnn00))ddnn11ddnn22 …… ddnnnn Efficiency:Efficiency: ΘΘ((nn22 ) single-digit multiplications) single-digit multiplications
- 20. 4-20Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 First Divide-and-Conquer AlgorithmFirst Divide-and-Conquer Algorithm A small example: AA small example: A ∗∗ B where A = 2135 and B = 4014B where A = 2135 and B = 4014 A = (21·10A = (21·1022 + 35), B = (40 ·10+ 35), B = (40 ·1022 + 14)+ 14) So, ASo, A ∗∗ B = (21 ·10B = (21 ·1022 + 35)+ 35) ∗∗ (40 ·10(40 ·1022 + 14)+ 14) = 21= 21 ∗∗ 40 ·1040 ·1044 + (21+ (21 ∗∗ 14 + 3514 + 35 ∗∗ 40) ·1040) ·1022 + 35+ 35 ∗∗ 1414 In general, if A = AIn general, if A = A11AA22 and B = Band B = B11BB22 (where A and B are(where A and B are nn-digit,-digit, AA11, A, A22, B, B11,,BB22 areare n/n/2-digit numbers),2-digit numbers), AA ∗∗ B = AB = A11 ∗∗ BB11·10·10nn + (A+ (A11 ∗∗ BB22 + A+ A22 ∗∗ BB11) ·10) ·10n/n/22 + A+ A22 ∗∗ BB22 Recurrence for the number of one-digit multiplications M(Recurrence for the number of one-digit multiplications M(nn):): M(M(nn) = 4M() = 4M(nn/2), M(1) = 1/2), M(1) = 1 Solution: M(Solution: M(nn) =) = nn22
- 21. 4-21Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Second Divide-and-Conquer AlgorithmSecond Divide-and-Conquer Algorithm AA ∗∗ B = AB = A11 ∗∗ BB11·10·10nn + (A+ (A11 ∗∗ BB22 + A+ A22 ∗∗ BB11) ·10) ·10n/n/22 + A+ A22 ∗∗ BB22 The idea is to decrease the number of multiplications from 4 to 3:The idea is to decrease the number of multiplications from 4 to 3: (A(A11 + A+ A22 )) ∗∗ (B(B11 + B+ B22 ) = A) = A11 ∗∗ BB11 + (A+ (A11 ∗∗ BB22 + A+ A22 ∗∗ BB11) + A) + A22 ∗∗ BB2,2, I.e., (AI.e., (A11 ∗∗ BB22 + A+ A22 ∗∗ BB11) = (A) = (A11 + A+ A22 )) ∗∗ (B(B11 + B+ B22 ) - A) - A11 ∗∗ BB11 - A- A22 ∗∗ BB2,2, which requires only 3 multiplications at the expense of (4-1) extrawhich requires only 3 multiplications at the expense of (4-1) extra add/sub.add/sub. Recurrence for the number of multiplications M(Recurrence for the number of multiplications M(nn):): M(M(nn) = 3) = 3MM((nn/2), M(1) = 1/2), M(1) = 1 Solution: M(Solution: M(nn)) = 3= 3loglog 22nn == nnloglog 2233 ≈≈ nn1.5851.585 What if we count both multiplications and additions?
- 22. 4-22Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Example of Large-Integer MultiplicationExample of Large-Integer Multiplication 21352135 ∗∗ 40144014 = (21*10^2 + 35) * (40*10^2 + 14) = (21*40)*10^4 + c1*10^2 + 35*14 where c1 = (21+35)*(40+14) - 21*40 - 35*14, and 21*40 = (2*10 + 1) * (4*10 + 0) = (2*4)*10^2 + c2*10 + 1*0 where c2 = (2+1)*(4+0) - 2*4 - 1*0, etc. This process requires 9 digit multiplications as opposed to 16.
- 23. 4-23Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Conventional Matrix MultiplicationConventional Matrix Multiplication Brute-force algorithmBrute-force algorithm cc0000 cc0101 aa0000 aa0101 bb0000 bb0101 = *= * cc1010 cc1111 aa1010 aa1111 bb1010 bb1111 aa0000 * b* b0000 + a+ a0101 * b* b1010 aa0000 * b* b0101 + a+ a0101 * b* b1111 == aa1010 * b* b0000 + a+ a1111 * b* b1010 aa1010 * b* b0101 + a+ a1111 * b* b1111 8 multiplications 4 additions Efficiency class in general: Θ (n3 )
- 24. 4-24Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Strassen’s Matrix MultiplicationStrassen’s Matrix Multiplication Strassen’s algorithm for two 2x2 matrices (1969):Strassen’s algorithm for two 2x2 matrices (1969): cc0000 cc0101 aa0000 aa0101 bb0000 bb0101 = *= * cc1010 cc1111 aa1010 aa1111 bb1010 bb1111 mm11 + m+ m44 - m- m55 + m+ m77 mm33 + m+ m55 == mm22 + m+ m44 mm11 + m+ m33 - m- m22 + m+ m66 mm11 = (a= (a0000 + a+ a1111)) ** (b(b0000 + b+ b1111)) mm22 = (a= (a1010 + a+ a1111)) ** bb0000 mm33 = a= a0000 ** (b(b0101 - b- b1111)) mm44 = a= a1111 ** (b(b1010 - b- b0000)) mm55 = (a= (a0000 + a+ a0101)) ** bb1111 mm66 = (a= (a1010 - a- a0000)) ** (b(b0000 + b+ b0101)) mm = (a= (a - a- a )) ** (b(b + b+ b )) 7 multiplications 18 additions
- 25. 4-25Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Strassen’s Matrix MultiplicationStrassen’s Matrix Multiplication Strassen observed [1969] that the product of two matrices canStrassen observed [1969] that the product of two matrices can be computed in general as follows:be computed in general as follows: CC0000 CC0101 AA0000 AA0101 BB0000 BB0101 = *= * CC1010 CC1111 AA1010 AA1111 BB1010 BB1111 MM11 + M+ M44 - M- M55 + M+ M77 MM33 + M+ M55 == MM22 + M+ M44 MM11 + M+ M33 - M- M22 + M+ M66
- 26. 4-26Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Formulas for Strassen’s AlgorithmFormulas for Strassen’s Algorithm MM11 = (A= (A0000 + A+ A1111)) ∗∗ (B(B0000 ++ BB1111)) MM22 = (A= (A1010 + A+ A1111)) ∗∗ BB0000 MM33 = A= A0000 ∗∗ (B(B0101 -- BB1111)) MM44 = A= A1111 ∗∗ (B(B1010 -- BB0000)) MM55 = (A= (A0000 + A+ A0101)) ∗∗ BB1111 MM66 = (A= (A1010 - A- A0000)) ∗∗ (B(B0000 ++ BB0101)) MM77 = (A= (A0101 - A- A1111)) ∗∗ (B(B1010 ++ BB1111))
- 27. 4-27Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Analysis of Strassen’s AlgorithmAnalysis of Strassen’s Algorithm IfIf nn is not a power of 2, matrices can be padded with zeros.is not a power of 2, matrices can be padded with zeros. Number of multiplications:Number of multiplications: M(M(nn) = 7M() = 7M(nn/2), M(1) = 1/2), M(1) = 1 Solution: M(Solution: M(nn)) = 7= 7loglog 22nn == nnloglog 2277 ≈≈ nn2.8072.807 vs.vs. nn33 of brute-force alg.of brute-force alg. Algorithms with better asymptotic efficiency are known but theyAlgorithms with better asymptotic efficiency are known but they are even more complex and not used in practice.are even more complex and not used in practice. What if we count both multiplications and additions?
- 28. 4-28Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Closest-Pair Problem by Divide-and-ConquerClosest-Pair Problem by Divide-and-Conquer Step 0 Sort the points by x (list one) and then by y (list two).Step 0 Sort the points by x (list one) and then by y (list two). Step 1 Divide the points given into two subsets SStep 1 Divide the points given into two subsets S11 and Sand S22 by aby a vertical linevertical line xx == cc so that half the points lie to the left or onso that half the points lie to the left or on the line and half the points lie to the right or on the line.the line and half the points lie to the right or on the line.
- 29. 4-29Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Closest Pair by Divide-and-Conquer (cont.)Closest Pair by Divide-and-Conquer (cont.) Step 2 Find recursively the closest pairs for the left and rightStep 2 Find recursively the closest pairs for the left and right subsets.subsets. Step 3 SetStep 3 Set dd = min{= min{dd11,, dd22}} We can limit our attention to the points in the symmetricWe can limit our attention to the points in the symmetric vertical strip of width 2vertical strip of width 2dd as possible closest pair. Let Cas possible closest pair. Let C11 and Cand C22 be the subsets of points in the left subset Sbe the subsets of points in the left subset S11 and ofand of the right subset Sthe right subset S22, respectively, that lie in this vertical, respectively, that lie in this vertical strip. The points in Cstrip. The points in C11 and Cand C22 are stored in increasingare stored in increasing order of theirorder of their yy coordinates, taken from the second list.coordinates, taken from the second list. Step 4 For every pointStep 4 For every point PP((xx,,yy) in C) in C11, we inspect points in, we inspect points in CC22 that may be closer tothat may be closer to PP thanthan dd. There can be no more. There can be no more thanthan 6 such points6 such points (because(because dd ≤≤ dd22)!)!
- 30. 4-30Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Closest Pair by Divide-and-Conquer: Worst CaseClosest Pair by Divide-and-Conquer: Worst Case The worst case scenario is depicted below:The worst case scenario is depicted below:
- 31. 4-31Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Efficiency of the Closest-Pair AlgorithmEfficiency of the Closest-Pair Algorithm Running time of the algorithm (without sorting) is:Running time of the algorithm (without sorting) is: T(T(nn) = 2T() = 2T(nn/2) + M(/2) + M(nn), where M(), where M(nn)) ∈∈ ΘΘ((nn)) By the Master Theorem (withBy the Master Theorem (with aa = 2,= 2, bb = 2,= 2, dd = 1)= 1) T(T(nn)) ∈∈ ΘΘ((nn loglog nn)) So the total time isSo the total time is ΘΘ((nn loglog nn).).
- 32. 4-32Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Quickhull AlgorithmQuickhull Algorithm Convex hullConvex hull: smallest convex set that includes given points. An: smallest convex set that includes given points. An O(n^3) bruteforce time is given in Levitin, Ch 3.O(n^3) bruteforce time is given in Levitin, Ch 3. Assume points are sorted byAssume points are sorted by xx-coordinate values-coordinate values IdentifyIdentify extreme pointsextreme points PP11 andand PP22 (leftmost and rightmost)(leftmost and rightmost) ComputeCompute upper hullupper hull recursively:recursively: • find pointfind point PPmaxmax that is farthest away from linethat is farthest away from line PP11PP22 • compute the upper hull of the points to the left of linecompute the upper hull of the points to the left of line PP11PPmaxmax • compute the upper hull of the points to the left of linecompute the upper hull of the points to the left of line PPmaxmaxPP22 ComputeCompute lower hulllower hull in a similar mannerin a similar manner P1 P2 Pmax
- 33. 4-33Copyright © 2007 Pearson Addison-Wesley. All rights reserved. A. Levitin “Introduction to the Design & Analysis of Algorithms,” 2nd ed., Ch. 4 Efficiency of Quickhull AlgorithmEfficiency of Quickhull Algorithm Finding point farthest away from lineFinding point farthest away from line PP11PP22 can be done incan be done in linear timelinear time Time efficiency:Time efficiency: T(n) = T(x) + T(y) + T(z) + T(v) + O(n),T(n) = T(x) + T(y) + T(z) + T(v) + O(n), where x + y + z +v <= n.where x + y + z +v <= n. • worst case:worst case: ΘΘ((nn22 )) T(n) = T(n-1) + O(n)T(n) = T(n-1) + O(n) • average case:average case: ΘΘ((nn) (under reasonable assumptions about) (under reasonable assumptions about distribution of points given)distribution of points given) If points are not initially sorted byIf points are not initially sorted by xx-coordinate value, this-coordinate value, this can be accomplished incan be accomplished in O(O(nn loglog nn) time) time SeveralSeveral O(O(nn loglog nn)) algorithms for convex hull are knownalgorithms for convex hull are known

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