Problem 3. For n = N, let b = b1b2 ..... b2 be a binary string with b1 + b2 + +bn ≥ 1 (that is, b1 = 1 for at least one index i), and let v = (C1, C2, ..., Cn) be a value vector with entries in N. An improvement operation on the value vector v with respect to the binary string b consists of the following two steps. 1. Choose an index i from the set {1, 2,...,n} with probability Сі c1+c2++Cn' == 1. 2. For a chosen index i, replace ci by ci - 1 if bi = 0 and replace ci by ci +1 if bi Design an efficient algorithm that computes the expected value of each entry of the vector value v after m improvement operations, and explain the worst time complexity of your algorithm in terms of m and n.

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