13th International Conference on Membrane Computing - MTA Sztaki

<strong>13th</strong> <strong>International</strong> <strong>Conference</strong> on Membrane Computing, CMC13,
Budapest, Hungary, August 28 - 31, 2012. Proceedings, pages 171 - 197.
Fast Distributed DFS Solutions for Edge-disjoint
Paths in Digraphs
Hossam ElGindy 1 , Radu Nicolescu 2 and Huiling Wu 3,∗
1 School of Computer Science and Engineering, University of New South Wales,
Sydney, Australia
elgindyh@cse.unsw.edu.au
2,3 Department of Computer Science, University of Auckland,
Private Bag 92019, Auckland, New Zealand
r.nicolescu@auckland.ac.nz, hwu065@aucklanduni.ac.nz
Abstract. We present two new synchronous distributed message-based
depth-first search (DFS) based algorithms, Algorithms C and D, to
compute a maximum cardinality set of edge-disjoint paths, between a
source node and a target node in a digraph. We compare these new algorithms
with our previous implementation of the classical algorithm,
Algorithm A, and our previous improvement, Algorithm B [12]. Algorithm
B improved the search speed by discarding “dead” (useless) cells
found on failed search branches. Using a different idea, Algorithm C
discards “dead” cells found on both successful and failed branches. To
asses the merits of this idea, we also propose a restricted version of Algorithm
C, called Algorithm C ∗ , which intentionally discards “dead” cells
found on successful branches only. Interestingly, Algorithm C ∗ detects
all dead cells detected by Algorithm B and a few more, but is not able to
prune all of them in real time (it will prune all, if allowed to run longer).
Thus, there are cases when one of these two algorithms is more appropriate
than the other. We further propose Algorithm D, which combines
best features of Algorithms B and C. Conceptually, all our algorithms
run in O(nd) distributed synchronous steps, where n is the number of
nodes and d is the outdegree of the source node. Empirical results show
that, on a set of random digraphs, our algorithms are faster than the
classical Algorithm A: B by 41.0%, C by 41.8%, C ∗ by 38.0% and D
by 42.1%. All these improved algorithms have been inspired and guided
by a P system modelling exercise, but are suitable for any distributed
implementation. To achieve the maximum theoretical performance, our
P systems specification uses high-level generic rules applied in matrix
grammar mode.
Keywords: edge-disjoint paths, depth-first search, network flow, distributed
systems, P systems, generic rules, matrix grammars
1 Introduction
The edge-disjoint paths problem finds a maximum cardinality set of edge-disjoint
paths between a source node and a target node in a digraph [9]. Alternative
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