Recall the definition of a universal family of hash functions: Let U = {x1,...,xn} denote the domain of inputs and {0,..., m 1} denote the outputs for a family Hof hash functions. H is called universal, if for every distinct pair x y EU, the number of functions hH satisfying h(x) = = h(y) is H m' a) Let {h1, h2, h3} be a family of hash functions that maps {x1, x2, x3, x4, x5} to {0, 1,2} as shown in the table below. Show whether this is universal or why not. h h2 h3 x1 0 1 2 x2 1 1 0 x3 1 2 1 x4 0 2 0 X5 1 0 2 b) Given h {x1,...,xn} {0,..., m -1} where h is part of a universal family of hash functions, conclude that the expected chain length (the number of inputs that hash to a specific value) is O(n/m). c) Show that if m = O(n), the expected search time is O(1).