Set Matrices and The Path/Cycle Problem

Presentation of set matrices and demonstration of their efficiency as a tool using the path/cycle problem. Introduction Set matrices are matrices whose elements are sets. The matrices comprise abilities of data storing and processing. That makes them a promising combinatorial structure. To prove the concept, this work applies the matrices to the path/cycle problem, see [1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, and many others]. The problem may be generalized as a problem to find all paths and all cycles of all length in form of vertex pairs (start, finish). That is a NP-hard problem because any of its solutions will include a solution of the Hamiltonian path/cycle problems [5]. This presentation uses set matrices to realize the following plan to solve the generalized problem: present the walk length dynamics with a generative grammar, but include in the grammar’s production rules some path/cycle filters in order to deplete the resulting walk language to the indication of path/cycle’s presence/absence, only. The design’s idea may be traced back trough the dynamic programming, the Ramsey theory, the formal language theory, and to the icosian calculus [16, 17]. Realization of the design requires to maintain a set of visited/unvisited vertices and to use that set as a filter in production of the next generation of walks. Set matrices satisfy the requirements. Sorting/factoring of the visited/unvisited vertices into vertex pairs (start, finish) creates a set matrix analog of the adjacency matrix. And the especially designed powers of the set matrix create an analytic path/cycle filter. The path/cycle language’s specification gets a realization in form of the easy-to-check properties of the elements of the adjacency set matrix’s powers. The factoring of the set of visited/unvisited vertices into vertex pairs (start, finish) may be seen as a walk coloring where colors are the factor-sets. Then, the family of algorithms realizing the design can be parametrized with the following four extreme strategies: to color the walks with sets of the visited/unvisited start/finish vertices. Work [18] describes a walk coloring with the unvisited vertices. This work deploys walk coloring with the visited vertices. Worst case for the algorithms is a complete graph. For a complete graph with n vertices, the algorithms perform n iterations and, on each of these iterations, O(n)-time processing for each of the n vertex pairs. That totals in time O(n) needed for the algorithms to find all paths and all cycles of all length in the form of vertex pairs (start, finish). Date: September 17, 2007. 2000 Mathematics Subject Classification. Primary 05C38, Secondary 68R10. 1