Jop F. Sibeyn

Packet Routing in Fixed-Connection Networks

We survey routing problems on fixed-connection networks. We consider many aspects of the routing problem and provide known theoretical results for various communication models. We focus on (partial) permutation, k-relation routing, routing to random destinations, dynamic routing, isotonic routing, fault tolerant routing, and related sorting results. We also provide a list of unsolved problems and numerous references.

Engineering an External Memory Minimum Spanning Tree Algorithm

We develop an external memory algorithm for computing minimum spanning trees. The algorithm is considerably simpler than previously known external memory algorithms for this problem and needs a factor of at least four less I/Os for realistic inputs.

Matching the bisection bound for routing and sorting on the mesh

In this paper we present randomized algorithms for kk routing, k-k sorting, and cut through routing on the mesh connected processor array. In these three problems, each processor is assumed to contain k packets at the beginning and k packets are destined for any processor node with k ≥ 1. We give two different algorithms for k-k routing that run in kn 2 +o(kn) and k2 n+o(kn) routing steps respectively. We also show that k-k sorting can be accomplished within k2 n + n + o(kn) steps and that cut through routing can be done in kn 2 + n k + 3 2n+ o(kn) steps. The stated resource bounds hold with high probability and for any k ≥ 8. The best known previous algorithms take almost twice as many routing steps in any case. For k ≤ 8 we derive new bounds which come close to the optimum. kn 2 is a known lower bound for all three problems, the bisection bound. Hence, our algorithms are very nearly optimal. All the above mentioned algorithms have optimal queue length, namely k + o(k). These algorithms also extend to higher dimensional meshes. The achieved improvements are made possible by novel algorithmic and analytical techniques.

Algorithms and experiments for the webgraph

In this paper we present an experimental study of the properties of web graphs. We study a large crawl from 2001 of 200M pages and about 1.4 billion edges made available by the WebBase project at Stanford [19], and synthetic graphs obtained by the large scale simulation of stochastic graph models for the Webgraph. This work has required the development and the use of external and semi-external algorithms for computing properties of massive graphs, and for the large scale simulation of stochastic graph models. We report our experimental findings on the topological properties of such graphs, describe the algorithmic tools developed within this project and report the experiments on their time performance.

Gossiping on meshes and tori

Algorithms for performing gossiping on one- and higher-dimensional meshes are presented. As a routing model, the practically important wormhole routing is assumed. We especially focus on the trade-off between the start-up time and the transmission time. For one-dimensional arrays and rings, we give a novel lower bound and an asymptotically optimal gossiping algorithm for all choices of the parameters involved. For two-dimensional meshes and tori, a simple algorithm composed of one-dimensional phases is presented. For an important range of packet and mesh sizes, it gives clear improvements upon previously developed algorithms. The algorithm is analyzed theoretically and the achieved improvements are also convincingly demonstrated by simulations, as well as an implementation on the Paragon. On the Paragon, our algorithm even outperforms the gossiping routine provided in the NX message-passing library. For higher-dimensional meshes, we give algorithms which are based on an interesting generalization of the notion of a diagonal. These algorithms are analyzed theoretically, as well as by simulation.

Better trade-offs for parallel list ranking

An earlier parallel list ranking algorithm performs well for problem sizes N that are extremely large in comparison to the number of PUS P. However, no existing algorithm gives good performance for reasonable loads. We present a novel family of algorithms, that achieve a better trade-off between the number of start-ups and the routing volume. We have implemented them on an Intel Paragon, and they turn out to considerably outperform all earlier algorithms: with P = 2 the sequential algorithm is already beaten for IV = 25,000; for P = 100 and N = 107, the speed-up is 21, and for N = 108 it even reaches 30. A modification of one of our algorithms solves a theoretical question: we show that on one-dimensional processor arrays, list ranking can be solved with a number of steps equal to the diameter of the network.

Randomized Routing on Meshes with Buses

We give algorithms and lower bounds for the problem of routing k-permutations on d-dimensional MIMD meshes with row and column buses We prove a lower bound for routing permutations (the case k=1) on d-dimensional meshes. For d=2, 3 and 4 the lower bound is respectively 0.69·n, 0.72·n and 0.76·n steps; the bound increases monotonically with d and is at least (1−1/d)·n steps for all d≥5. Previously, a bisection argument had been used to show that for all d≥1, 0.66·n steps axe required for this problem (i.e., the lower bound did not increase with increasing d). These lower bounds hold for off-line routing as well. We give a general algorithm that routes k-permutations on d-dimensional meshes in min{(2−1/d) · k · n, max{4/3 · d · n, k · n/3}} + o(d · k · n) steps, for all k, d≥1. This improves considerably on previous results for many values of k and d. In particular, the routing time for permutations is bounded by 2 · n, for all 1≤d<n1/3, and the routing time is optimal for all k≥4 · d. More specialized algorithms have better performance for routing on 2-dimensional meshes. A simple algorithm routes 2-permutations in 1.39 · n steps, and a more sophisticated one routes permutations in 0.78 · n steps. This is the first algorithm that routes permutations on the 2-dimensional mesh in less than n steps. The algorithms are randomized, on-line and achieve the given routing times with high probability.

Randomized Multipacket Routing and Sorting on Meshes

We consider the problem of routing and sorting ond-dimensionaln×...× mesh connected computers. Each of the processing units initially holdsk packets. We present randomized algorithms that solve these problems with (1+o(1))·max{2·d·n,k·n/2} communication steps. On a torus these problems are solved twice as fast. Thus we match the bisection bound up to lower-order terms, for allk≥4·d. Earlier algorithms required some additional Θ(n) steps or more, and were more complicated. With 2·d·n extra steps our algorithm can also route in the cut-through routing model.

Deterministic Permutation Routing on Meshes

New deterministic algorithms for routing permutations on two-dimensional meshes are developed. On ann×narray, one algorithm runs in the optimal 2·n?2 steps, with maximum queue length 32. Another algorithm runs in near-optimal time, 2·n+O(1) steps, with a maximum queue length of only 12.

Solving fundamental problems on sparse-meshes

A sparse-mesh, which has PUs on the diagonal of a two-dimensional grid only, is a cost effective distributed memory machine. Variants of this machine have been considered before, but none are as simple and pure as a sparse-mesh. Various fundamental problems (routing, sorting, list ranking) are analyzed, proving that sparse-meshes have great potential. It is shown that on a two-dimensional n/spl times/n sparse-mesh, which has n PUs, for h=/spl omega/(n/sup /spl epsiv///spl middot/log n), h-relations can be routed in (h+o(h))//spl epsiv/ steps. The results are extended for higher dimensional sparse-meshes. On a d-dimensional n x/spl middot//spl middot//spl middot/x n sparse-mesh, with h=/spl omega/(n/sup /spl epsiv///spl middot/log n), h-relations are routed in (6/spl middot/(d-1)//spl epsiv/-4)/spl middot/(h+o(h)) steps.