Finding k Shortest Paths in Cayley Graphs of Finite Groups

Abstract

We present a new method for finding k shortest paths between any two vertices in the Cayley graph Cay(G,S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Cay}(G, S)$$\\end{document} of a finite group G with its generating set S closed under inverses. By using a reduced convergent rewriting system R for G, we first find the lexicographically minimal shortest path between two vertices in Cay(G,S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Cay}(G, S)$$\\end{document}. Then, by symmetrizing the length-preserving rules of R, we provide a polynomial time algorithm (in the size of certain rewrite rules, the lexicographically minimal shortest path, and k) for finding k shortest paths between two vertices in Cay(G,S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Cay}(G, S)$$\\end{document}. Our implementation of finding k shortest paths between two vertices in Cay(G,S)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {Cay}(G, S)$$\\end{document} is also discussed.