Sorting a bridge hand

Sorting a permutation by block moves is a task that every bridge player has to solve every time she picks up a new hand of cards. It is also a problem for the computational biologist, for block moves are a fundamental type of mutation that can explain why genes common to two species do not occur in the same order in the chromosome, It is not known whether there exists an optimal sorting procedure running in polynomial time. Bafna and Pevzner gave a polynomial time algorithm that sorts any permutation of length n in at most 3n/4 moves. Our new algorithm improves this to [(2n - 2)/3] for n greater than or equal to 9. For the reverse permutation, we give an exact expression for the number of moves needed, namely [(n + 1)/2]. Computations of Bafha and Pevzner up to n = 10 seemed to suggest that this is the worst case; but as it turns out, a first counterexample occurs for n = 13, i.e. the bridge player's case. Professional card players never sort by rank, only by suit. For this case, we give a complete answer to the optimal sorting problem. (C) 2001 Elsevier Science B.V. All rights reserved.