DeepAI AI Chat
Log In Sign Up

Faster Evaluation of Subtraction Games

by   David Eppstein, et al.

Subtraction games are played with one or more heaps of tokens, with players taking turns removing from a single heap a number of tokens belonging to a specified subtraction set; the last player to move wins. We describe how to compute the set of winning heap sizes in single-heap subtraction games (for an input consisting of the subtraction set and maximum heap size n), in time Õ(n), where the Õ elides logarithmic factors. For multi-heap games, the optimal game play is determined by the nim-value of each heap; we describe how to compute the nim-values of all heaps of size up to n in time Õ(mn), where m is the maximum nim-value occurring among these heap sizes. These time bounds improve naive dynamic programming algorithms with time O(n|S|), because m<|S| for all such games. We apply these results to the game of subtract-a-square, whose set of winning positions is a maximal square-difference-free set of a type studied in number theory in connection with the Furstenberg-Sárközy theorem. We provide experimental evidence that, for this game, the set of winning positions has a density comparable to that of the densest known square-difference-free sets, and has a modular structure related to the known constructions for these dense sets. Additionally, this game's nim-values are (experimentally) significantly smaller than the size of its subtraction set, implying that our algorithm achieves a polynomial speedup over dynamic programming.


Universal Complexity Bounds Based on Value Iteration and Application to Entropy Games

We develop value iteration-based algorithms to solve in a unified manner...

On the Complexity of Solving Subtraction Games

We study algorithms for solving Subtraction games, which sometimes are r...

Some i-Mark games

Let S be a set of positive integers, and let D be a set of integers larg...

New Results for Domineering from Combinatorial Game Theory Endgame Databases

We have constructed endgame databases for all single-component positions...

Partition games are pure breaking games

Taking-and-breaking games are combinatorial games played on heaps of tok...

Taking-and-merging games as rewrite games

This work contributes to the study of rewrite games where positions are ...

Ann wins the nonrepetitive game over four letters and the erase-repetition game over six letters

We consider two games between two players Ann and Ben who build a word t...