Collecting Coupons with Random Initial Stake

Motivated by a problem in the theory of randomized search heuristics, we give a very precise analysis for the coupon collector problem where the collector starts with a random set of coupons (chosen uniformly from all sets). We show that the expected number of rounds until we have a coupon of each type is nHn/2 − 1/2± o(1), where Hn/2 denotes the (n/2)th harmonic number when n is even, and Hn/2 := (1/2)H⌊n/2⌋+(1/2)H⌈n/2⌉ when n is odd. Consequently, the coupon collector with random initial stake is by half a round faster than the one starting with exactly n/2 coupons (apart from additive o(1) terms). This result implies that classic simple heuristic called randomized local search needs an expected number of nHn/2 − 1/2 ± o(1) iterations to find the optimum of any monotonic function defined on bit-strings of length n.