Improved Approximation Algorithm for the Number of Queries Necessary to Identify a Permutation

TL;DR

This paper introduces an improved algorithm that reduces the number of queries needed to identify a secret permutation in a variant of Mastermind, achieving near-optimal efficiency for the case where the number of colors equals the number of positions.

Contribution

It presents a new strategy that identifies the secret permutation in O(n log n) queries, nearly halving the previous query complexity for the case k=n, and extends analysis to cases where k>n.

Findings

Achieves O(n log n) query complexity for permutation identification.

Improves previous results by nearly a factor of 2 for the case k=n.

Extends analysis to cases where the number of colors exceeds the number of positions.

Abstract

In the past three decades, deductive games have become interesting from the algorithmic point of view. Deductive games are two players zero sum games of imperfect information. The first player, called "codemaker", chooses a secret code and the second player, called "codebreaker", tries to break the secret code by making as few guesses as possible, exploiting information that is given by the codemaker after each guess. A well known deductive game is the famous Mastermind game. In this paper, we consider the so called Black-Peg variant of Mastermind, where the only information concerning a guess is the number of positions in which the guess coincides with the secret code. More precisely, we deal with a special version of the Black-Peg game with n holes and k >= n colors where no repetition of colors is allowed. We present a strategy that identifies the secret code in O(n log n) queries. Our algorithm improves the previous result of Ker-I Ko and Shia-Chung Teng (1985) by almost a factor of 2 for the case k = n. To our knowledge there is no previous work dealing with the case k > n. Keywords: Mastermind; combinatorial problems; permutations; algorithms