lec7 - Outline 3 examples of the power of randomness CS151...

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1 CS151 Complexity Theory Lecture 7 April 19, 2011 April 19, 2011 2 Outline • 3 examples of the power of randomness – communication complexity – polynomial identity testing – complexity of finding unique solutions • randomized complexity classes April 19, 2011 3 1. Communication complexity • Goal : compute f(x, y) while communicating as few bits as possible between Alice and Bob • count number of bits exchanged (computation free) • at each step: one party sends bits that are a function of held input and received bits so far two parties: Alice and Bob function f:{0,1} n x {0,1} n {0,1} Alice holds x {0,1} n ; Bob holds y {0,1} n April 19, 2011 4 Communication complexity • simple function (equality): EQ(x, y) = 1 iff x = y • simple protocol: – Alice sends x to Bob (n bits) – Bob sends EQ(x, y) to Alice (1 bit) – total: n + 1 bits – (works for any predicate f) April 19, 2011 5 Communication complexity • Can we do better? – deterministic protocol? – probabilistic protocol ? • at each step: one party sends bits that are a function of held input and received bits so far and the result of some coin tosses • required to output f(x, y) with high probability over all coin tosses April 19, 2011 6 Communication complexity Theorem : no deterministic protocol can compute EQ(x, y) while exchanging fewer than n+1 bits. • Proof: – “input matrix”: X = {0,1} n Y = {0,1} n f(x,y)
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2 April 19, 2011 7 Communication complexity – assume 1 bit sent at a time, alternating (same proof works in general setting) – A sends 1 bit depending only on x: X = {0,1} n Y = {0,1} n inputs x causing A to send 1 inputs x causing A to send 0 April 19, 2011 8 Communication complexity – B sends 1 bit depending only on y and received bit: X = {0,1} n Y = {0,1} n inputs y causing B to send 1 inputs y causing B to send 0 April 19, 2011 9 Communication complexity – at end of protocol involving k bits of communication , matrix is partitioned into at most 2 k combinatorial rectangles – bits sent in protocol are the same for every input (x, y) in given rectangle – conclude: f(x,y) must be constant on each rectangle April 19, 2011 10 Communication complexity – any partition into combinatorial rectangles with constant f(x,y) must have 2 n + 1 rectangles – protocol that exchanges ≤ n bits can only create 2 n rectangles, so must exchange at least n+1 bits. X = {0,1} n Y = {0,1} n 1 1 1 1 0 0 Matrix for EQ: April 19, 2011 11 Communication complexity • protocol for EQ employing randomness? – Alice picks random prime p in {1. ..4n 2 }, sends: • p • (x mod p) – Bob sends: • (y mod p) – players output 1 if and only if: (x mod p) = (y mod p) April 19, 2011 12 Communication complexity – O(log n) bits exchanged – if x = y, always correct – if x ≠ y, incorrect if and only if: p divides |x – y| – # primes in range is ≥ 2n – # primes dividing |x – y| is ≤ n – probability incorrect ≤ 1/2 Randomness gives an exponential advantage!!
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lec7 - Outline 3 examples of the power of randomness CS151...

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