A New Algorithm for Finding Minimal Cycle-Breaking Sets of Turns in a Graph

Abstract

We consider the problem of constructing a minimal cycle-breaking set of turns for a given undirected graph. This problem is important for deadlock-free wormhole routing in computer and communication networks, such as Networks of Workstations. The proposed Cycle Breaking algorithm, or CB algorithm, guarantees that the constructed set of prohibited turns is minimal and that the fraction of the prohibited turns does not exceed 1/3 for any graph. The computational complexity of the proposed algorithm is O(N 2 ∆), where N is the number of vertices, and ∆ is the maximum node degree. The memory complexity of the algorithm is O(N ∆).We provide lower bounds on the minimum size of cycle-breaking sets for connected graphs. Further, we construct minimal cycle-breaking sets and establish bounds on the minimum fraction of prohibited turns for two important classes of graphs, namely, t-partite graphs and graphs with small degrees. The upper bounds are tight and demonstrate the optimality of the CB algorithm for certain classes of graphs. Results of computer simulations illustrate the superiority of the proposed CB algorithm as compared to the well-known and the widely used Up/Down technique.

Minimal Sets of Turns for Breaking Cycles in Graphs Modeling Networks

“…Consider a set of R cycles in G such that no more than r cycles are covered by the same turn. Then [18], the number of prohibited turns ZðGÞ and fraction of prohibited turns zðGÞ satisfy the following inequalities:…”

“…In this section, we present an algorithm, called the SCB algorithm, which is much simpler than and at least as efficient as those in [17], [18], [33].…”

“…After Glass and Ni showed it for regular topologies such as meshes and tori, this conclusion was confirmed by other authors [15], [16], and [17] for irregular topologies as well. Experimental data [17], [18] show that there is a considerable gain of approximately 7-8 percent in the maximum sustainable throughput in the network, for each percentage point reduction in the fraction of prohibited turns.…”

“…Instead of employing spanning trees, they elaborate a procedure to label all nodes and to prohibit or permit turns according to this labeling. The algorithms have been shown to outperform dramatically the Up à =Down à algorithm in terms of the fraction of prohibited turns, average distance between nodes, and saturation load (see [17], [18]). …”

“…However, some of them are not applicable to the general case. A new and different approach to the problem of deadlock prevention by turn prohibition was developed in [5], [18], [29] (the TP and CB algorithms). These algorithms can be called tree-free algorithms.…”