Randomized Algorithms课件.pptVIP

Chapter 11;Randomized algorithms ;The closest pair problem ;A randomized algorithm for closest pair finding;Step 3: Construct four sets of squares T1, T2, T3 and T4 derived from T by doubling the mesh-size to 2? . Step 4: For each Ti, find the induced decomposition S=S1(i) ?S2(i) ? … ? Sj(i), 1?i ?4, where Sj(i) is a non-empty intersection of S with a square of Ti. Step 5: For each xp, xq?Sj(i), compute d(xp, xq). Let xa and xb be the pair of points with the shortest distance among these pairs. Return xa and xb as the closest pair.;6δ;6δ;X3;Time complexity;Analysis of Step 5; Construct T1, T2, …, T16 mesh-size: 4?0 The probability that each square in T falls into at least one square of Ti , i = 1, 2, …, 16 is The probability that ;Let the square in T0 with the largest number of elements among the 16 squares have k elements. N(T0) ? c0n = N(Ti ) ? cin;A randomized algorithm to test whether a number is prime. ;Randomized prime number testing algorithm;Examples for randomized prime number testing;Example 2: N = 11 Randomly choose 2, 5, 7 (1) 211-1=1024≡1 mod 11 j=1, (N-1)/2j==5 GCD(25-1, 11) = 1 W(2) does not hold . (2) 511-1=9765625≡1 mod 11 GCD(55-1, 11) = 11 W(5) does not hold . (3) 711-1=282475249≡1 mod 11 GCD(75-1, 11) = 1 W(7) does not hold . Thus, 11 is a prime number with the probability of correctness being at least 1-2-3= 7/8 .;Theorem for number theory ;Pattern matching;Binary representation;Fingerprints of binary strings;Examples for using fingerprints;e.g. X = 10110 , Y = 10011 , P = 3 Bp(X) = (22)3 = 1 Bp(Y(1)) = (19)3 = 1 ? X= Y(1) WRONG! If Bp(X) ? Bp(Y(i)), then X ? Y(i) . If Bp(X) = Bp(Y(i)), we may do a bit by bit checking or compute k different fingerprints by using k different prime numbers in {1,2,…nt2} . ;A randomized algorithm for pattern matching;Step 1: Randomly choose k prime numbers p1, p2, …, pk from {1,2,…,nt2}, where t = m - n + 1. Step 2: i = 1. Step 3: j = 1. Step 4: If B(X)