Collecting Coupons with Random Initial Stake
Benjamin Doerr, Carola Doerr

TL;DR
This paper precisely analyzes the coupon collector problem with a random initial set, revealing it is roughly half a round faster than starting with half the coupons, with implications for randomized local search algorithms.
Contribution
The paper provides a detailed analysis of the coupon collector problem with a random initial set, deriving exact expected times and implications for heuristic optimization.
Findings
Expected rounds: $nH_{n/2} - 1/2 ext{ (plus small error)}$
Random initial stake speeds up collection by about half a round
Implication for local search: $nH_{n/2} - 1/2$ iterations to find optima
Abstract
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 , where denotes the th harmonic number when is even, and when is odd. Consequently, the coupon collector with random initial stake is by half a round faster than the one starting with exactly coupons (apart from additive terms). This result implies that classic simple heuristic called \emph{randomized local search} needs an expected number of iterations to find the optimum of any monotonic function defined on bit-strings of…
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Taxonomy
TopicsOptimization and Search Problems · Algorithms and Data Compression · Machine Learning and Algorithms
