We formulate the Euclidean algorithm using directed graphs on integer points in the plane and operations on a particular semigroup of two by two matrices. The properties of the graphs and the semigroup provide surprisingly effective tools for solving certain classical diophantine equations, among other applications.

SummaryConsider a system of n players in which each initially starts on a different team. At each time step, we select an individual winner and an individual loser randomly and the loser joins the winner's team. The resulting Markov chain and stochastic matrix clearly have one absorbing state, in which all players are on the same team, but the combinatorics along the way are surprisingly elegant. The expected number of time steps until each team is eliminated is a ratio of binomial coefficients. When a team is eliminated, the probabilities that the players are configured in various partitions of n into t teams are given by multinomial coefficients. The expected value of the time to absorbtion is (n - 1)2 steps. The results depend on elementary combinatorics, linear algebra, and the theory of Markov chains.

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Given an mtimesn array of k single random error correction (or erasure) codewords, each having length l such that mn=kl, we construct optimal interleaving schemes that provide the maximum burst error correction power such that an arbitrarily shaped error burst of size t can be corrected for the largest possible value of t. We show that for all such mtimesn arrays, the maximum possible interleaving distance, or equivalently, the largest value of t such that an arbitrary error burst of size up to t can be corrected, is bounded by lfloorradic2krfloor if kleslceil(min{m,n}) 2 /2rceil, and by min{m,n}+lfloor(k-lceil(min{m,n}) 2 /2rceil)/min{m,n}rfloor if kgeslceil(min{m,n}) 2 /2rceil. We generalize the cyclic shifting algorithm developed by the authors in a previous paper and construct, in several special cases, optimal interleaving arrays achieving these upper bounds. Additionally, for codewords of variable lengths, we solve a related array coloring problem for which the same upper bounds hold and can be achieved

Given an m times n array of k single random error correction (or erasure) codewords, each having length l such that mn = kl, we construct optimal interleaving schemes that provide the maximum burst error correction power such that an arbitrarily shaped error burst of size t can be corrected for the largest possible value of t. We show that for all such m times n arrays, the maximum possible interleaving distance, or equivalently, the largest value of t such that an arbitrary error burst of size up to t can be corrected, is bounded by lfloorradic2krfloor if k les lceil(min{m, n}) 2 /2rceil, and by min{m, n} + lfloor(k - lceil(min{m, n}) 2 /2rceil) / min{m, n}rfloor if k ges lceil(min{m, n}) 2 /2rceil. We generalize the cyclic shifting algorithm developed by the authors in a previous paper and construct, in several special cases, optimal interleaving arrays achieving these upper bounds

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