Shuffling by Semi-random Transpositions

In the cyclic-to-random shuffle, we are given n cards arranged in a circle. At step k, we exchange the kth card along the circle with a uniformly chosen random card. The problem of determining the mixing time of the cyclic-to-random shuffle was raised by Aldous and Diaconis in 1986. Mironov used this shuffle as a model for the cryptographic system known as RC4, and proved an upper bound of O(n log n) for the mixing time. We prove a matching lower bound, thus establishing that the mixing time is indeed of order /spl Theta/(n log n). We also prove an upper bound of O(n log n) for the mixing time of any "semirandom transposition shuffle", i.e., any shuffle in which a random card is exchanged with another card chosen according to an arbitrary (deterministic or random) rule. To prove our lower bound, we exhibit an explicit complex-valued test function which typically takes very different values for permutations arising from few iterations of the cyclic-to-random-shuffle and for uniform random permutations. Perhaps surprisingly, the proof hinges on the fact that the function e/sup z/ - 1 has nonzero fixed points in the complex plane. A key insight from our work is the importance of complex analysis tools for uncovering structure in nonreversible Markov chains.