Edge- and Node-Disjoint Paths in P Systems

Abstract

In this paper, we continue our development of algorithms used for topological network discovery. We present native P system versions of two fundamental problems in graph theory: finding the maximum number of edge-and node-disjoint paths between a source node and target node. We start from the standard depth-first-search maximum flow algorithms, but our approach is totally distributed, when initially no structural information is available and each P system cell has to even learn its immediate neighbors. For the node-disjoint version, our P system rules are designed to enforce node weight capacities (of one), in addition to edge capacities (of one), which are not readily available in the standard network flow algorithms.

BFS Solution for Disjoint Paths in P Systems

“…Rather than actually splitting the intermediate P cells, we simulate this by ad-hoc cell rules. This approach could be in other distributed networks, where nodes cannot be split [3]. Essentially, node splitting prevents more than one unit flow to pass through an intermediate node [8].…”

“…Figure 3 illustrates a scenario when one cell, y, is visited twice, first on its entry and then on its exit node [3]. Assume that path π = s.x.y.z.t, is established.…”

“…Section 4 discusses the specifics of the disjoint paths problem in P systems. Section 5 reviews the previous DFS-based solution [3] and sets out our faster BFS-based solution. Section 6 presents the P system rules for the disjoint paths algorithm using BFS.…”

“…there is no central node and only local channels (between structural neighbours) are allowed. In 2010, Dinneen, Kim and Nicolescu [3] proposed the first P solution, as a distributed version of the Ford-Fulkerson algorithm, based on depth-first search (DFS). This solution searches by visiting nodes sequentially, which is not always efficient.…”

“…This paper continues the research on determining a maximum cardinality set of edge-and node-disjoint paths between a source cell and a target cell in P systems. We review the previous solution [3], based on depth-first search (DFS), and we propose a faster solution, based on breadth-first search (BFS), which leverages the parallel and distributed characteristics of P systems. The runtime complexity shows that, our BFSbased solution performs better than the DFS-based solution, in terms of P steps.…”