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- 1. Quantum Algorithms for Evaluating MIN-MAX Trees Richard Cleve Dmitry Gavinsky D. L. Yonge-Mallo Institute for Quantum Computing, University of Waterloo January 30, 2008 TQC – Tokyo, Japan
- 2. Motivation ● Why do we care about algorithms for MIN- MAX trees, anyway? trees ● What is so special about the quantum algorithms for MIN-MAX trees that I'm about to present? – The ideas behind them don't work in a classical setting! – Conversely, the classical ideas don't work in a quantum setting! 2
- 3. Why do we care about MIN-MAX trees? MIN-MAX trees arise in the analysis of deterministic games of perfect information between two players who alternate taking turns 3
- 4. Why do we care about MIN-MAX trees? MIN-MAX trees arise in the analysis of deterministic games of perfect information between two players who alternate taking turns MIN MAX MAX MAX 4 1 2 5 7 8 6 9 3 4
- 5. What is a MIN-MAX tree? ● internal nodes are MIN and MAX gates at alternating levels; ● leaves x1,...,xN take on values from some ordered set; ● value is value of root as a function of x1,...,xN. MIN MAX MAX MAX 4 1 2 5 7 8 6 9 3 5
- 6. Evaluating a MIN-MAX tree 4 MIN 4 8 9 MAX MAX MAX 4 1 2 5 7 8 6 9 3 6
- 7. Alpha-beta pruning 4 MIN 4 ⩾5 ⩾6 MAX MAX MAX 4 1 2 5 7 8 6 9 3 7
- 8. MIN-MAX trees and AND-OR trees ● An AND-OR tree is just a MIN-MAX tree restricted to the values {0,1}! {0,1} ● So MIN-MAX is at least as hard as AND-OR. 0 MIN 0 1 1 MAX MAX MAX 0 0 0 1 1 1 1 1 0 8
- 9. MIN-MAX trees and AND-OR trees ● An AND-OR tree is just a MIN-MAX tree restricted to the values {0,1}! {0,1} ● So MIN-MAX is at least as hard as AND-OR. 0 AND 0 1 1 OR OR OR 0 0 0 1 1 1 1 1 0 9
- 10. You can also turn MIN-MAX trees into AND-OR trees root root ⩾ v? MIN AND threshold v MAX OR xk xk ⩾ v 10
- 11. You can also turn MIN-MAX trees into AND-OR trees 4 root 0 root ⩾ 5? MIN AND threshold 5 4 8 9 0 1 1 MAX MAX MAX OR OR OR 4 1 2 5 7 8 6 9 3 0 0 0 1 1 1 1 1 0 This immediately suggests binary search... 11
- 12. Combining AND-OR and binary search 4 root 0 root ⩾ 5? MIN AND threshold 5 4 8 9 0 1 1 MAX MAX MAX OR OR OR 4 1 2 5 7 8 6 9 3 0 0 0 1 1 1 1 1 0 ● Is root ⩾ 5? No. 12
- 13. Combining AND-OR and binary search 4 root 1 root ⩾ 3? MIN AND threshold 3 4 8 9 1 1 1 MAX MAX MAX OR OR OR 4 1 2 5 7 8 6 9 3 1 0 0 1 1 1 1 1 1 ● Is root ⩾ 5? No. ● Is root ⩾ 3? Yes. 13
- 14. Combining AND-OR and binary search 4 root 1 root ⩾ 4? MIN AND threshold 4 4 8 9 1 1 1 MAX MAX MAX OR OR OR 4 1 2 5 7 8 6 9 3 1 0 0 1 1 1 1 1 0 } ● Is root ⩾ 5? No. ● Is root ⩾ 3? Yes. root = 4 ● Is root ⩾ 4? Yes. 14
- 15. Combining AND-OR and binary search We can consider two models of ordered non-binary data... ● in the input value query model, we have direct access to x1,...,xN through a black box; ● in the comparison query model, we are restricted to making comparisons of the form [xj < xk]. 15
- 16. Problems with combining AND-OR and binary search We need to find the midpoint of subintervals of the form [α, β]. In the comparison query model, the midpoint of an interval cannot be directly computed. In the input query model, if the numerical range is too large, the binary search may not converge in a logarithmic number of steps. 16
- 17. Saks-Wigderson algorithm Saks and Wigderson [SW86] showed that... ● the optimal classical randomized algorithm for AND-OR tree evaluation makes Θ(N0.7537...) queries; ● there is an algorithm for MIN-MAX tree evaluation which makes this number of queries, using AND-OR tree evaluation as a subroutine. 17
- 18. Saks-Wigderson algorithm MAX v MIN MIN AND MAX OR xk xk ⩾ v TN = 3/2 TN/2 + O(N0.7537...) This implies a Θ(N0.7537...) algorithm. 18
- 19. Quantum algorithm for AND-OR trees ● There is a lower bound of Ω(N1/2) [BS04] ● There is a “more-or-less” matching 1/2+ε algorithm that makes O(N ) queries [FGG07, CCJY07, A07+CRŠZ07] 19
- 20. Quantum algorithm for AND-OR trees ● There is a lower bound of Ω(N1/2) [BS04] ● There is a “more-or-less” matching 1/2+ε algorithm that makes O(N ) queries [FGG07, CCJY07, A07+CRŠZ07] The “obvious question”... Do these results generalize to MIN-MAX tree evaluation? 20
- 21. “Quantum Saks-Wigderson” MAX v MIN MIN AND MAX OR xk xk ⩾ v 0.5 0.7537... TN = 3/2 TN/2 + O(N ) This implies an O(N0.5850...) algorithm. 21
- 22. Can we do better? ● We could try to analyze the AND-OR tree algorithm and try to apply it directly to MIN-MAX trees... ● A better idea: idea perform a binary search... search root root ⩾ v? MIN AND pivot v MAX OR xk xk ⩾ v 22
- 23. Can we do better? But haven't we already ● We could try to analyze the Aestablished that this ND-OR tree algorithm and try to apply it directlyapproach isAX trees... to MIN-M full of problems? ● A better idea: idea perform a binary search... search root root ⩾ v? MIN AND pivot v MAX OR xk xk ⩾ v 23
- 24. Solution: use random pivots ● A better idea: idea perform a binary search using random pivots. pivots ● Classically, finding a random pivot is as hard as searching, which can take Ω(N) queries to do even once! ● We have a quantum algorithm to find a pivot with cost O(√N): Grover's search! search 24
- 25. Quantum algorithm for evaluating MIN-MAX trees ● A better idea: idea perform a binary search using random pivots. pivots root root ⩾ v? MIN AND random pivot v MAX OR xk xk ⩾ v 25
- 26. Quantum algorithm for evaluating MIN-MAX trees ● The algorithm runs for O(log N) stages. ● Each stage costs O(√N loglog N). To amplify the subroutines to lower the error probability to O(1/log(N))... 26
- 27. Quantum algorithm for evaluating MIN-MAX trees ● The algorithm runs for O(log N) stages. ● Each stage costs O(√N loglog N). It turns out that this is unnecessary! (Using a trick involving a stack...) 27
- 28. Quantum algorithm for evaluating MIN-MAX trees ● The algorithm runs for O(log N) stages. ● Each stage costs O(√N). This gives a quantum algorithm for evaluating MIN-MAX trees... Total cost: O(√N log N) This is O(N1/2+ε) for an arbitrarily small constant ε. 28
- 29. Obtaining the optimal move ● If the values of the leaves x1,...,xN are distinct, this is easy. ● Otherwise, we can use the quantum minimum/maximum finding algorithm [DH96]. MIN MAX MAX MAX 4 1 2 5 7 8 6 9 3 29
- 30. Summary ● Classically, the Saks-Wigderson reduction Classically from MIN-MAX to AND-OR uses ϴ(N0.7537...) queries. ● Calling the quantum AND-OR subroutine results in an O(N0.5850...) algorithm, which is not optimal! ● The classical algorithms are based on examining the subtrees of the tree. 30
- 31. Summary ● Our quantum algorithm performs a binary search using random pivots and requires 1/2+ε O(N ) queries, which is (close to) optimal. optimal ● Conversely, binary search is too costly for a classical algorithm. – The ideas behind the quantum algorithm don't work in the classical setting! 31
- 32. Summary (chart) Classical: ϴ(N0.7537...) Quantum: O(N1/2+ε) ● Binary search is too ● Uses binary search costly ● Based on evaluating ● Based on evaluating subtrees of the MIN-MAX the entire tree as an tree AND-OR tree, with different thresholds ● Doesn't get full ● Gets full speedup from speedup from quantum quantum AND-OR and AND-OR subroutine Grover's search subroutines 32
- 33. The Moral of the Story What works in the classical setting may fail to work in the quantum setting. What fails to work in the classical setting may work very well in the quantum setting. To develop quantum algorithms, one must be willing to abandon classical intuitions! 33
- 34. Thanks! References: ● [CGY07] Quantum Algorithms for Evaluating MIN-MAX Trees. arXiv:quant-ph/0710.5794 ● [FGG07] A Quantum Algorithm for the Hamiltonian NAND Tree. arXiv:quant-ph/0702144 ● [A+CRŠZ07] Every NAND formula on N variables can be evaluated in time O(N1/2+ε). arXiv:quant-ph/0703015 34

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