randomized algorithm

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1. . . . . . . . Selected Topics of Randomness and Computation ਾࣟ᪻ ACM Honored Class 2010 July 10, 2012 ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 1 / 9
2. Outline . . . 1 Non-determinism vs. Randomness . . . 2 Polynomial Identity Testing . . . 3 Randomizing against adversary . . . 4 Pseudo Random-number Generator ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 2 / 9
3. Review of NTM,PTM,BPP TM. A deterministic machine, configuration graph. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 3 / 9
4. Review of NTM,PTM,BPP TM. A deterministic machine, configuration graph. NTM, introducing non-determinism, the NP/co-NP language, polynomial hierachy. Is there a sequence of choice? ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 3 / 9
5. Review of NTM,PTM,BPP TM. A deterministic machine, configuration graph. NTM, introducing non-determinism, the NP/co-NP language, polynomial hierachy. Is there a sequence of choice? PTM. How large is this fraction of choice. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 3 / 9
6. Review of NTM,PTM,BPP TM. A deterministic machine, configuration graph. NTM, introducing non-determinism, the NP/co-NP language, polynomial hierachy. Is there a sequence of choice? PTM. How large is this fraction of choice. BPP, error reduction, any algorithm that’s correct in majority cases, can be extended, with no more than some extra polynomial time, to something that’s correct in almost all cases. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 3 / 9
7. Polynomial Identity Testing . Problem 1. . . . . . . . . Given degree d, n-variate polynomial P(x),decide if P(x) is identically 0. (over some field F) Expand and compare? ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 4 / 9
8. Polynomial Identity Testing . Problem 1. . . . . . . . . Given degree d, n-variate polynomial P(x),decide if P(x) is identically 0. (over some field F) Expand and compare? How good can randomized test could be? ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 4 / 9
9. Polynomial Identity Testing . Problem 1. . . . . . . . . Given degree d, n-variate polynomial P(x),decide if P(x) is identically 0. (over some field F) Expand and compare? How good can randomized test could be? . Lemma 1. . . . . . . . . (Schwartz–Zippel) Pick x1, . . . xn uniformly at random, then Pr(P(x1, . . . , xn) = 0) ≤ d ∥F∥ . . Proof. . . . . . . . . By induction. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 4 / 9
10. Randomized Communication Complexity The problem EQ, deterministic algorithm takes at least O(n) bit of messages for absolute certainty. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 5 / 9
11. Randomized Communication Complexity The problem EQ, deterministic algorithm takes at least O(n) bit of messages for absolute certainty. Randomized EQ with public coin, only O(log n) messages needed. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 5 / 9
12. Randomized Communication Complexity The problem EQ, deterministic algorithm takes at least O(n) bit of messages for absolute certainty. Randomized EQ with public coin, only O(log n) messages needed. Any shared n-bit random string can be simulated by a private string protocol that takes O(log n) extra bits. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 5 / 9
13. Randomizing against adversary Evil adversary ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 6 / 9
14. Randomizing against adversary Evil adversary DOS vulnerabilities due to hash collision. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 6 / 9
15. Randomizing against adversary Evil adversary DOS vulnerabilities due to hash collision. compiler killer. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 6 / 9
16. Universal Hashing . Remark 2. . . . . . . . . With universal hashing, with m slots and n keys, for any fixed x, E(#collisions with x) = n/m (1) Universality is good. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 7 / 9
17. Universal Hashing . Remark 2. . . . . . . . . With universal hashing, with m slots and n keys, for any fixed x, E(#collisions with x) = n/m (1) Universality is good. You can’t expect more. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 7 / 9
18. Randomized Qsort worst-case O(n2) for a deterministic one. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 8 / 9
19. Randomized Qsort worst-case O(n2) for a deterministic one. T(n) < Cn log n with high probability (C depends on the probability you wanna attain) for a randomized one. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 8 / 9
20. Randomized Qsort worst-case O(n2) for a deterministic one. T(n) < Cn log n with high probability (C depends on the probability you wanna attain) for a randomized one. expected O(n log n) for a randomized one. . Proof. . . . . . . . . On the blackboard. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 8 / 9
21. Pseudo Random-number Generator . Problem 2. . . . . . . . . (Open) BPP= P? How to derandomize a Randomized algorithm? ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 9 / 9
22. Pseudo Random-number Generator . Problem 2. . . . . . . . . (Open) BPP= P? How to derandomize a Randomized algorithm? Derandomizing via a PRG. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 9 / 9
23. Pseudo Random-number Generator . Problem 2. . . . . . . . . (Open) BPP= P? How to derandomize a Randomized algorithm? Derandomizing via a PRG. Derandomizing BPP. ਾࣟ᪻ (ACM Honored Class 2010) Selected Topics of Randomness and Computation July 10, 2012 9 / 9