Common intervals in permutations

An interval of a permutation is a consecutive substring consisting of consecutive symbols. For example, 4536 is an interval in the permutation 71453682. These arise in genetic applications. For the applications, it makes sense to generalize so as to allow gaps of bounded size δ -1, both in the locations and the symbols. For example, 4527 has gaps bounded by 1 (since 3 and 6 are missing) and is therefore a δ -interval of 389415627 for δ =2. After analyzing the distribution of the number of intervals of a uniform random permutation, we study the number of 2-intervals. This is exponentially large, but tightly clustered around its mean. Perhaps surprisingly, the quenched and annealed means are the same. Our analysis is via a multivariate generating function enumerating pairs of potential 2-intervals by size and intersection size.\par

[1]  J. Wolfowitz,et al.  Additive Partition Functions and a Class of Statistical Hypotheses , 1942 .

[2]  Heinz Mühlenbein,et al.  Evolution algorithms in combinatorial optimization , 1988, Parallel Comput..

[3]  L. Gordon,et al.  Poisson Approximation and the Chen-Stein Method , 1990 .

[4]  Jens Stoye,et al.  Algorithms for Finding Gene Clusters , 2001, WABI.

[5]  Jens Stoye,et al.  Finding All Common Intervals of k Permutations , 2001, CPM.

[6]  J. Wolfowitz.  Note on Runs of Consecutive Elements , 1944 .

[7]  Gilles Didier,et al.  Common Intervals of Two Sequences , 2003, WABI.

[8]  Takeaki Uno,et al.  Fast Algorithms to Enumerate All Common Intervals of Two Permutations , 1997, Algorithmica.

[9]  R. Pemantle,et al.  Asymptotics of Multivariate Sequences, part I. Smooth points of the singular variety , 2000 .

[10]  Jens Stoye,et al.  Common intervals and sorting by reversals: a marriage of necessity , 2002, ECCB.

[11]  Mathieu Raffinot,et al.  The Algorithmic of Gene Teams , 2002, WABI.

[12]  Yuliy Baryshnikov,et al.  Convolutions of inverse linear functions via multivariate residues , 2004 .

[13]  I. Kaplansky.  The Asymptotic Distribution of Runs of Consecutive Elements , 1945 .

[14]  Mark C. Wilson,et al.  Asymptotics of Multivariate Sequences: I. Smooth Points of the Singular Variety , 2002, J. Comb. Theory, Ser. A.

[15]  Isao Ono,et al.  An Efficient Genetic Algorithm for Job Shop Scheduling Problems , 1995, International Conference on Genetic Algorithms.

[16]  Mark C. Wilson,et al.  Asymptotics of Multivariate Sequences II: Multiple Points of the Singular Variety , 2004, Combinatorics, Probability and Computing.

[17]  Jens Stoye,et al.  On the Similarity of Sets of Permutations and Its Applications to Genome Comparison , 2006, J. Comput. Biol..

[18]  L. Gordon,et al.  Two moments su ce for Poisson approx-imations: the Chen-Stein method , 1989 .

[19]  Kellogg S. Booth,et al.  Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-Tree Algorithms , 1976, J. Comput. Syst. Sci..