Direct Routing: Algorithms and Complexity

Abstract

Direct routing is the special case of bufferless routing where N packets, once injected into the network, must be delivered to their destinations without collisions. We give a general treatment of three facets of direct routing:(i) Algorithms. We present a polynomial time greedy direct algorithm which is worstcase optimal. We improve the bound of the greedy algorithm for special cases, by applying variants of the this algorithm to commonly used network topologies. In particular, we obtain near-optimal routing time for the tree, mesh, butterfly and hypercube.(ii) Complexity. By a reduction from Vertex Coloring, we show that optimal Direct Routing is inapproximable, unless P=NP.(iii) Lower Bounds for Buffering. We show that certain direct routing problems cannot be solved efficiently; in order to solve these problems, any routing algorithm needs buffers. We give non-trivial lower bounds on such buffering requirements for general routing algorithms. * A preliminary version of this

Packet Routing: Complexity and Algorithms

“…In particular, they show that for those trees the farthest-destinationfirst (FDF)-algorithm works optimally. Busch et al [5] study the direct routing problem, i.e., the problem of finding the shortest direct schedule. They give complexity results and algorithms for finding direct schedules.…”

“…In particular, they improve the algorithm by Raghavan et al mentioned above and present a (4/3)approximation. For coloring directed paths on bidirected trees (i.e., trees in which each edge represents two links, one in each direction) there are algorithms known which need at most 5 3 L colors where L denotes the maximum load on a directed link [8]. This is tight since there are instances which actually need 5 3 L colors [17].…”

“…For coloring directed paths on bidirected trees (i.e., trees in which each edge represents two links, one in each direction) there are algorithms known which need at most 5 3 L colors where L denotes the maximum load on a directed link [8]. This is tight since there are instances which actually need 5 3 L colors [17]. Gargano et al [14] investigate the problem of coloring all directed paths in a bidirected tree.…”

“…Having computed such a coloring, we present a new technique which constructs a direct schedule whose length is bounded by C + D − 1. In comparison, the best known algorithm for computing direct schedules for packet routing on general trees guarantees a schedule of length 2C + D − 2 [5]. (Note that the authors of [5] assume that the edges can be used in opposite directions at the same time.…”

Competitive Packet Routing with Priority Lists

“…A fundamental combinatorial optimization problem that has received considerable attention in the past (cf. [4,14,17,18,21,24]) is packet routing in graphs. We are given a set of packets, which may, for example, correspond to unit-sized messages/bits in a communication network.…”