Question

_____________________________

' ' 2 . {1 ..2:.L-.' '1 On Monday September 30 we will cover universal hashing: a randomized hash function' 1s a function h: U > 2,, selected at random, such that Pr[h[11] = 1'] = 1 /p for all 11 E U and all 1' E Zp, and also Pr[h[11]= h[11]]= 1/p for any two elements 11 -, 1:. Here 2,, = {0, 1,. . . ,p 1} is the set of integers mod p and U is our universe of elements we want to hash. In class we considered storing a set S C U of 11 = |S I elements using this hash function (storing elements 11 in an array at position h[11]), and showed that if 11 5 p than the expected number of elements colliding with any element 11 is at most 1. This problem considers extensions of this idea. (a) (5 points) Even with the random choice of a hash function h as we have done in class, with high probability there will be some collisions. Suppose 11 = p, and let 11,- be the 2number of elements 11 E S that hash to 1', that is 11, = |{u E S | h(11) 1'}|. Show that Pr[f 3112 > 4p]_