We consider algorithms for many-to-many hot potato routing. In hot potato (deflection) routing a packet cannot be buffered, and is therefore always moving until it reaches its destination. We give optimal and nearly optimal deterministic algorithms for many-tomany packet routing in commonly occurring networks such as the hypercube, meshes and tori of various dimensions and sizes, trees and hypercubic networks such as the butterfly. All these algorithms are analyzed using a charging scheme that may be applicable to other algorithms as well. Moreover, all bounds hold in a dynamic setting in which packets can be injected at arbitrary times.

Amir Ben-Dor Shai Halevi y Assaf Schuster z January 21, 1994 Abstract In this work we study the problem of packet routing in synchronous networks of processors, in which at most one packet can traverse any communication link in each time step. We consider a class of algorithms known as hot-potato or deflection routing algorithms. The important characteristic of these algorithms is that they use no buffer space for storing delayed packets. Each packet, unless already arrived to its destination, must leave the processor at the step following its arrival. The main advantage in hot-potato routing is that there is no need to store delayed packets in the processors, and therefore, the processors can be much simpler, and contain less hardware. This work is concerned with greedy routing, in which a packet is bound to use an out-going link in the direction of its destination, whenever such a link is available. In this way, greediness guarantees that, unless some global congestion forbids...

We study the problem of centrally scheduling multiple messages in a linear network, when each message has both a release time and a deadline. We show that the problem of transmitting optimally many messages is NP-hard, both when messages may be buffered in transit and when they may not be; for either case, we present efficient algorithms that produce approximately optimal schedules. In particular, our bufferless scheduling algorithm achieves throughput that is within a factor of two of optimal. We show that buffering can improve throughput in general by a logarithmic factor (but no more), but that in several significant special cases, such as when all messages can be released immediately, buffering can help by only a small constant factor. Finally, we show how to convert our centralized, offline bufferless schedules to equally productive fully...

We present the rst hot-potato routing algorithm for the n × n mesh whose running time on any "hard" (i.e., n)) "many-to-one" batch routing problem is, with high probability, within a polylogarithmic factor of optimal. For any instance I of a batch routing problem, there exists a well-known lower bound LBI based on maximum path length and maximum congestion. If LBI is n), our algorithm solves I with high probability in time O(LBI log 3 n). The algorithm is distributed and greedy, and it makes use of a new routing technique based on multi-bend paths, a departure from paths using a constant number of bends used in prior hot-potato algorithms.

We present a novel greedy hot-potato routing algorithm for the 2-dimensional n × n mesh or torus. This algorithm uses randomization to adjust packet priorities. For any permutation problem or random destination problem, it ensures that each packet reaches its destination in asymptotically optimal expected O(n) steps, and all packets reach their destinations in O(n ln n) steps with high probability, an improvement over the previously-known deterministic upper bound of O(n²) for greedy algorithms. For a general batch problem, with high probability all packets reach their destination nodes in at most O(m ln n) steps, where m = min(mr ; mc ), where mr and mc are respectively the maximum number of packets targeted to a single row or column.

We study packet routing problems, in which we are given a set of N packets which will be sent on preselected paths with congestion C and dilation D. For store-and-forward routing, in which nodes have buffers for packets in transit, there are routing algorithms with performance that matches the lower bound Ω(C + D). Motivated from optical networks, we study hot-potato routing in which the nodes are bufferless. Due to the lack of buffers, in hot-potato routing the packets may be delayed more than in store-and-forward routing. An interesting question is how much is the performance of routing algorithms affected from the absence of buffers. Here, we answer this question for the class of leveled networks, in which the nodes are partitioned into L + 1 distinct levels. We present a randomized hot-potato routing algorithm for leveled networks, which routes the packets in O((C +L) ln 9 (LN)) time with high probability. For routing problems with dilation Ω(L), and where N is a polynonial in L, this bound is within polylogarithmic factors of the lower bound Ω(C + L). Our algorithm demonstrates that for such routing problems the benefit from using buffers is no more than polylogarithmic; thus, hot-potato routing is an efficient way to route packets in leveled networks. In hot-potato routing, due to the lack of buffers, the packets may not be able to remain on their preselected paths during the course of routing (while in store-and-forward routing the packets remain on their preselected paths). However, in our algorithm the actual path that each packet follows contains its original preselected path; thus the lower bound Ω(C + L) is also a lower bound for the new paths. Our algorithm is distributed, that is, routing decisions are taken locally at each node while packets are routed in the network. To our knowledge, this is the first hot-potato algorithm designed and analyzed, in terms of congestion and dilation, for leveled networks.

An adaptive routing algorithm is one in which the path a packet takes from its source to its destination may depend on other packets it encounters. Such algorithms potentially avoid network bottlenecks by routing packets around "hot spots." Minimal adaptive routing algorithms have the additional advantage that the path each packet takes is a shortest one. For a large class of minimal adaptive routing algorithms, we present an \Omega# n 2 =k 2 ) bound on the worst case time to route a static permutation of packets on an n 2 n mesh or torus with nodes that can hold up to k 1 packets each. This is the first nontrivial lower bound on adaptive routing algorithms. The argument extends to more general routing problems, such as the h-h routing problem. It also extends to a large class of dimension order routing algorithms, yielding an \Omega# n 2 =k) time bound. To complement these lower bounds, we present two upper bounds. One is an O(n 2 =k) time dimension order routing...

We present the first dynamic hot-potato routing algorithm that does not require any form of explicit flow control: a node may inject a message into the network (n × n mesh) whenever a link is free. In the worst case, a node may have to wait an expected O(n) time before it has a free link. If destinations are chosen uniformly at random, this algorithm guarantees delivery in an expected O(n) time steps. Both measures are optimal up to a constant factor.

In this paper we present an extensive study of many-to-many routing on trees under the matching routing model. Our study includes on-line and off-line algorithms. We present an asymptotically optimal on-line algorithm which routes k packets to their destination within d(k \Gamma 1) + d \Delta dist routing steps, where d is the degree of tree T on which the routing takes place and dist is the maximum distance any packet has to travel. We also present an off-line algorithm that solves the same problem within 2(k \Gamma 1)+dist steps. The analysis of our algorithms is based on the establishment of a close relationship between the matching and the hot-potato routing models that allows us to apply tools which were previously used exclusively in the analysis of hot-potato routing.

Recently, Chinn, Leighton, and Tompa [10] presented lower bounds for store-and-forward permutation routing algorithms on the n × n mesh with bounded buffer size and where a packet must take a shortest (or minimal) path to its destination. We extend their analysis to algorithms that are nearly minimal. We also apply this technique to the domain of hot potato algorithms, where there is no storage of packets and the shortest path to a destination is not assumed (and is in general impossible). We show that "natural" variants and "improvements" of several algorithms in the literature perform poorly in the worst case. As a result, we identify algorithmic features that are undesirable for worst case hot potato permutation routing. Recent works in hot potato routing have tried to define simple and greedy classes of algorithms. We show that when an algorithm is too simple and too greedy, its performance in routing permutations is poor in the worst case. Specifically, the technique of [10] ...