LEIGHTON, T. 1990. Average case analysis of greedy routing algorithms on arrays. In Proceedings of the 2nd Annual ACM Symposium on Parallel Algorithms and Architectures (Island of Crete, Greece, July 2-- 6). ACM, New York, pp. 2--10.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....related to the asymptotic behavior of this chain. We note in passing that similar Markov chains have been an object of study in queueing theory for over four decades (see, e.g. 13] in computer science they have also received attention, for example in the stochastic analysis of packet routing [10]. Despite this extensive body of research few general analytical tools exist, and even the simplest questions, such as whether such a chain is ergodic, seem hard to answer. The chief exception is the method of constructive use of Lyapunov functions, developed in recent years mainly by Malyshev, ....

F.T. Leighton. Average case analysis of greedy routing algorithms on arrays. In Proceedings of the 2nd Annual ACM Symposium on Parallel Algorithms and Architectures, 1990, pp. 2--10.

....step when M = M, and M = ffl 1, P 1. We compare lower bounds obtained for random destination routing with experiments on simulation of the EREW PRAM on a mesh. In our experiments we use a model of a mesh with small (size one) first in first out (FIFO) input and output queues. Leighton [7], showed that the greedy algorithm for random destination routing has a maximal number of packets queued at any edge which is at most 4 with probability 1 Gamma O( log 4 p ) where P is the number of processors in a 2D mesh. The probability that any particular packet is delayed Delta steps ....

Leighton, T.,"Average case analysis of greedy routing algorithms on arrays" Proc. ACM Symp. on Parallel Algorithms and Architectures, 1990, pp. 2-10.

....to make sure that the network does not become overloaded. Overloaded networks perform poorly) Typical ow control methods include: Nodes must negotiate network bandwidth before they are allowed to inject packets. Nodes must wait for a long deterministic adversarial [22, 6] or random [19, 16, 21, 25] duration between injections. Nodes must await acknowledgments of previous packets before injecting new packets. Current real world ow control mechanisms use the rst and third approach. An overview of the current state of the art can be found in the books of Gouda [13] and Keshav [18] For ....

F. T. Leighton. Average case analysis of greedy routing algorithms on arrays. In A.-S. ACM-SIGARCH, editor, Proceedings of the 2nd Annual ACM Symposium on Parallel Algorithms and Architectures, pages 2-10, Island of Crete, Greece, July 1990. ACM Press.

....performance of a dynamic algorithm is a function of the inter arrival distribution. The goal is to develop algorithms that perform close to optimal for any inter arrival distribution. Several recent articles have addressed the dynamic routing problem, in the context of packet routing on arrays [7, 10, 5, 2], on the hypercube and the butterfly [13] and general networks [12] Except for [2] the analyses in these works assume a Poisson arrival distribution and require unbounded queues in the routing switches (though some works give a high probability bound on the size of the queue used [7, 5] ....

.... [7, 10, 5, 2] on the hypercube and the butterfly [13] and general networks [12] Except for [2] the analyses in these works assume a Poisson arrival distribution and require unbounded queues in the routing switches (though some works give a high probability bound on the size of the queue used [7, 5]) Unbounded queues allow the application of some tools from queuing theory (see [3, 4] and help reduce the correlation between events in the system, thus simplifying the analysis at the cost of a less realistic model. Here we focus on analyzing dynamic packet routing in networks with bounded ....

[Article contains additional citation context not shown here]

T. Leighton. Average case analysis of greedy routing algorithms on arrays. Procs. of the Second Annual A CM Symp. on Parallel Algorithms and Architectures. Pages 2-10, 1990.

....very good bounds for multidimensional k k routing. For cut through routing we considerably improve the known bounds for the number of steps, namely approximately by a factor of 2. Makedon and Simvonis [10] gave the first good bounds for cut through routing, which is also discussed by Leighton [9]. Rajasekaran and Raghavachari [12] show that their analysis for the general k k routing problem also holds for the cut through routing problem. Both use randomized algorithms. We make some very general considerations for one dimensional routing assuming no specific conflict resolution strategy ....

....Unfortunately, the packets from one row that want to go to the same column, accumulate between the two phases in one single queue, which may have size n. The algorithm performs not much better if the packets are allowed to move vertically as soon as they reached the destination column. Leighton [9] has shown that on the average this greedy strategy performs very well (queue size 5) since the extreme situation described above is very unlikely. Since the strategy is nice and easy to analyse, it serves in almost all packet routing algorithms as a basis. Randomlzed vertical distribution of ....

Leighton, T., 'Average Case Analysis of Greedy Routing Algorithms on Arrays,' Proc. ACM Symposium on Parallel Algorithms and Architectures, SPAA 90, ACM Press, pp. 2-10, 1990.

....to make sure that the network does not become overloaded. Overloaded networks perform poorly) Typical ow control methods include: # Nodes must negotiate network bandwidth before they are allowed to inject packets. # Nodes must wait for a long deterministic adversarial [22, 6] or random [19, 16, 21, 25] duration between injections. # Nodes must await acknowledgments of previous packets before injecting new packets. Current real world ow control mechanisms use the rst and third approach. An overview of the current state of the art can be found in the books of Gouda [13] and Keshav [18] For ....

F. T. Leighton. Average case analysis of greedy routing algorithms on arrays. In A.-S. ACM-SIGARCH, editor, ########### ## ### ### ###### ### ######### ## ######## ########## ### #############, pages 2-10, Island of Crete, Greece, July 1990. ACM Press.

....network, P . 11 With respect to delays, what differentiates packet routing networks from each other is the location and duration of bottlenecks. A bottleneck is a part of the network through which only one packet can pass at a time. For example, the theoretical computer science community [44], 36] 81] generally considers each wire to be a bottleneck. Specifically, in their definition it takes some unit constant amount of time for a packet to traverse each wire and only one packet may traverse any given wire at a time. If a packet arrives at a wire which is currently being used, ....

....is commonly called permutation routing because the packets are being permuted among the hosts. The situation we usually refer to where packets arrive continually from outside the network is known as dynamic packet routing) Examples of research on static packet routing networks are [44], 45] 82] 80] 4] 77] 28] 3] 16] All of these are specific to a particular network and a particular routing scheme. They mostly concentrate on the problem of permutation routing, and use the Chernoff bound approach. Some research on static packet routing networks applies to general ....

[Article contains additional citation context not shown here]

Tom Leighton. Average case analysis of greedy routing algorithms on arrays. In 2nd Annual ACM Symposium on Parallel Algorithms and Architectures, pages 2--10, July 1990.

....for example, builds 4 on techniques used in the proof of the static result of Leighton, Maggs, and Rao [32] the protocol, however, itself has a simple description that is independent of the analysis. 2. 4 Continuous packet routing Without any explicit use of queuing theory results, Leighton [31] analyzes one bend routing on n n arrays; the paths in one bend routing are acyclic and in fact one bend routing on arrays turns out to be easier to analyze than routing on cyclic networks such as rings. Leighton considers the case where each injected packet has a random destination 4 and ....

....in queuing theory) of a xed, time invariant input distribution (e.g. say a Poisson or constant rate arrival process) for each possible request. It also subsumes the case of oblivious packet routing for packets that are generated at each node with randomly chosen destinations (e.g. as studied in [30, 31, 49]) To di erentiate such stochastic adversaries from the non stochastic adversaries de ned above, we will refer to the non stochastic adversaries as deterministic adversaries. Clearly, deterministic adversaries are a special case of stochastic adversaries. To the best of our knowledge, these ....

F.T. Leighton. Average case analysis of greedy routing algorithms on arrays. Proceeding of the Second Annual ACM Symposium on Parallel Algorithms and Architectures, 1990.

....difference poses a world of subtle difficulties in adapting queueing theory to continuous packet routing. 2. Probabilistic analyses in which a Poisson or Bernoulli process generates packets bound for random destinations; the passage time through an edge is a constant as in packet routing (see [9], for example) The resulting analyses are technically difficult, and yield results that are specific to particular networks and queueing disciplines; moreover, as in queueing theoretic approaches, they rely heavily on the underlying probabilistic assumptions which determine the injection ....

....Corollary 9 Let G denote an arbitrary directed graph and S an arbitrary stochastic adversary that injects shortest paths with rate 1 , for some 0, and variance bounded by . Then LIS is stable against S. 5 Meshes, Trees, and DAG s An argument that builds on Lemmas 2 and 4 in Leighton [9] yields the following theorem. 1 The proof of Lemma 2 may appear to be independent of the variance of the stochastic adversary. Actually, this is not really the case. The variance does show up in the application of the Martingale tail inequality and it is still the case that our bound will not ....

T. LEIGHTON. Average case analysis of greedy routing algorithms on arrays. Prooceeding of the Second Annual ACM Symposium on Parallel Algorithms and Architectures, 1990.

....performance of a dynamic algorithm is a function of the inter arrival distribution. The goal is to develop algorithms that perform close to optimal for any inter arrival distribution. Several recent articles have addressed the dynamic routing problem, in the context of packet routing on arrays [7, 10, 5, 2], on the hypercube and the butter y [13] and general networks [12] Except for [2] the analyses in these works assume a Poisson arrival distribution and require unbounded queues in the routing switches (though some works give a high probability bound on the size of the queue used [7, 5] ....

.... [7, 10, 5, 2] on the hypercube and the butter y [13] and general networks [12] Except for [2] the analyses in these works assume a Poisson arrival distribution and require unbounded queues in the routing switches (though some works give a high probability bound on the size of the queue used [7, 5]) Unbounded queues allow the application of some tools from queuing theory (see [3, 4] and help reduce the correlation between events in the system, thus simplifying the analysis at the cost of a less realistic model. Here we focus on analyzing dynamic packet routing in networks with bounded ....

[Article contains additional citation context not shown here]

T. Leighton. Average case analysis of greedy routing algorithms on arrays. Procs. of the Second Annual ACM Symp. on Parallel Algorithms and Architectures. Pages 2-10, 1990.

.... Especially for meshes and tori, a class of hot potato routing algorithms that has received much attention is that of algorithms that make packets follow paths that approximate their natural greedy path (i.e. the path utilized by the greedy routing algorithm in the store and forward model [15]) Such algorithms were proposed and analyzed in [13] In this paper we describe the implementation of four hot potato routing algorithms on the 2 dimensional torus network. The algorithms are either greedy or have the property that approximate the greedy path. Moreover, they are simple in the ....

F.T. Leighton. Average Case Analysis of Greedy Routing Algorithm on Arrays. In Proc. of the 2nd Annual ACM Symposium on Parallel Algorithms and Architectures, pp. 2--10, 1990.

....very good bounds for multidimensional k k routing. For cut through routing we considerably improve the known bounds for the number of steps, namely approximately by a factor of 2. Makedon and Simvonis [10] gave the first good bounds for cut through routing, which is also discussed by Leighton [8]. Rajasekaran and Raghavachari [12] show that their analysis for the general k k routing problem also holds for the cut through routing problem. Both use randomized algorithms. We make some very general considerations for one dimensional routing assuming no specific conflict resolution strategy ....

....Unfortunately, the packets 5 from one row that want to go to the same column, accumulate between the two phases in one single queue, which may have size n. The algorithm performs not much better if the packets are allowed to move vertically as soon as they reached the destination column. Leighton [8] has shown that on the average this greedy strategy performs very well (queue size 5) since the extreme situation described above is very unlikely. Since the strategy is nice and easy to analyse, it serves in almost all packet routing algorithms as a basis. Randomized vertical distribution of ....

Leighton, T., `Average Case Analysis of Greedy Routing Algorithms on Arrays,' Proc. ACM Symposium on Parallel Algorithms and Architectures, SPAA 90, ACM Press, pp. 2-10.

....O(log 3 n) computation time. MIMD Meshes. Now we consider permutation routing on MIMD meshes. 2 Delta n Gamma 2 steps is a lower bound because of the diameter of the mesh. Leighton has shown that for random inputs the greedy algorithm performs very well, even the queue sizes remain small [56]. Achieving good results for all permutations is harder. The first near optimal algorithms were presented in [47, 89] Then Leighton, Makedon and Tollis [58] proved the impossible : a deterministic algorithm running in 2 Delta n Gamma 2 steps with constant size queues. Though constant, the ....

....in every step (we consider a MIMD mesh) this implies that the system will get more and more congested, eventually resulting in infinite delays, for p 4=n. Leighton has shown that for all p 4=n routing the packets greedily along row and column to their destinations normally works fine [56]. If several packets are competing for the use of a connection, then the one that has to move farthest in this direction is given priority. Theorem 8 If the packet arrival rate in an n Theta n MIMD mesh is less than the network capacity 4=n, then the maximum delay incurred by any packet in any ....

[Article contains additional citation context not shown here]

Leighton, T., `Average Case Analysis of Greedy Routing Algorithms on Arrays,' Proc. 2nd Symposium on Parallel Algorithms and Architectures, pp. 2--10, ACM, 1990.

....mesh. It is well known that dimension order paths can be used to route any permutation on the n Theta n mesh in 2n Gamma 2 steps, matching the diameter lower bound (see Leighton [Lei92, pages 159 162] Unfortunately, this algorithm requires Theta(n) size queues at each node. Leighton [Lei90] proves that if each packet has a random destination i.e. the routing problem is not necessarily a permutation then 7 with high probability all packets will be delivered in 2n O(log 2 n) steps and none of the queues ever contains more than four packets. However, this average case ....

T. Leighton. Average case analysis of greedy routing algorithms on arrays. In Proceedings of the 1990 ACM Symposium on Parallel Algorithms and Architectures, pages 2--10, July 1990.

....for example, builds 4 on techniques used in the proof of the static result of Leighton, Maggs, and Rao [32] the protocol, however, itself has a simple description that is independent of the analysis. 2. 4 Continuous packet routing Without any explicit use of queuing theory results, Leighton [31] analyzes one bend routing on n Theta n arrays; the paths in one bend routing are acyclic and in fact one bend routing on arrays turns out to be easier to analyze than routing on cyclic networks such as rings. Leighton considers the case where each injected packet has a random destination 4 and ....

....queuing theory) of a fixed, time invariant input distribution (e.g. say a Poisson or constant rate arrival process) for each possible request. It also subsumes the case of oblivious packet routing for packets that are generated at each node with randomly chosen destinations (e.g. as studied in [30, 31, 49]) To differentiate such stochastic adversaries from the non stochastic adversaries defined above, we will refer to the non stochastic adversaries as deterministic adversaries. Clearly, deterministic adversaries are a special case of stochastic adversaries. To the best of our knowledge, these ....

F.T. Leighton. Average case analysis of greedy routing algorithms on arrays. Proceeding of the Second Annual ACM Symposium on Parallel Algorithms and Architectures, 1990.

....instead of the sort and unshuffle operation. We believe that the analysis is somewhat simpler in the deterministic case. We consider two different greedy routing schemes, which we refer to as the standard greedy and the extended greedy routing scheme. In the standard greedy routing scheme [65], every packet moves greedily towards its destination along edges of increasing dimension. In the case of edge contentions, priority is given to the 126 packet with the farthest distance to travel. In the extended greedy routing scheme, several permutations are simultaneously routed by running d ....

....algorithms for k k sorting in [47, 59] However, these results cannot be extended to the case of distance optimal routing. 127 For the standard greedy routing scheme, it is easy to see that one unshuffle permutation can be routed distance optimally on d dimensional meshes and tori. Leighton [65] has shown that this is also the case for a single random permutation, with high probability. In fact, his result shows that it is unlikely that any packet is delayed by more than O(lg n) steps. For the extended greedy routing scheme, we can show the following result. Lemma 5.2.1 Up to 2d ....

F. T. Leighton. Average case analysis of greedy routing algorithms on arrays. In Proceedings of the 2nd Annual ACM Symposium on Parallel Algorithms and Architectures, pages 2--10, July 1990.

....Thus, in virtual cut through, the buffer size at a node is proportional to the worm length, and when the head of a message is blocked in a given node, the entire message can fold into the buffer of that node, rather than occupying many nodes and edges and delaying many other messages. Leighton [24] conducted a probabilistic analysis of virtual cut through on meshes. A nice feature of Leighton s result is that it deals with the greedy algorithm, which is of great practical interest: there are no priorities assigned to messages, and each message simply keeps moving whenever it can, along the ....

F.T. Leighton. Average case analysis of greedy routing algorithms on arrays. In Second Annual ACM Symposium on Parallel Algorithms and Architectures, pages 2--10. 1990.

No context found.

LEIGHTON, T. 1990. Average case analysis of greedy routing algorithms on arrays. In Proceedings of the 2nd Annual ACM Symposium on Parallel Algorithms and Architectures (Island of Crete, Greece, July 2-- 6). ACM, New York, pp. 2--10.

No context found.

T. Leighton. Average case analysis of greedy routing algorithms on arrays. In Proc. of the 2nd Annual ACM Symp. on Parallel Algorithms and Architectures, 1990.

No context found.

T. Leighton. Average case analysis of greedy routing algorithms on arrays. In Proc. of the 2nd Annual ACM Symp. on Parallel Algorithms and Architectures, 1990.

....evidence that the bounds in Theorem 1.1 hold for this case as well, but no proof has yet been found. Finally, keeping with our Markov arrival and transit time assumptions, it would be interesting to study asymptotic behavior in the generalization of rings to toroidal arrays of processors (see Leighton (1990, 1992) Much is known about regular (open) arrays, as can be seen from the recent work of Mitzenmacher (1994) who gives references to the earlier work on this problem. But the analysis of toroidal arrays seems to require different methods. Acknowledgment We are grateful to I. Telatar and A. ....

Leighton, F. T. (1990), "Average Case Analysis of Greedy Routing Algorithms on Arrays," Proc. 2nd Ann. ACM Symp. Parallel Algs. Arch., 2--10.

....algorithm of choice in practice. It is well known that dimension order paths can be used to route any permutation on the n2n mesh in 2n 0 2 steps, matching the diameter lower bound (see Leighton [16, pages 159 162] Unfortunately, this algorithm requires 2(n) size queues at each node. Leighton [17] proves that if each packet has a random destination i.e. the routing problem is not necessarily a permutation then with high probability all packets will be delivered in 2 2n O(log 2 n) steps and none of the queues ever contains more than four packets. However, this average case ....

T. Leighton. Average case analysis of greedy routing algorithms on arrays. In Proceedings of the 1990 ACM Symposium on Parallel Algorithms and Architectures, pages 2--10, July 1990.

No context found.

LEIGHTON, F. T. 1990. Average case analysis of greedy routing algorithms on arrays. In Proceedings of the 2nd Annual ACM Symposium on Parallel Algorithms and Architectures (Island of Crete, Greece, July 2-- 6). ACM, New York, pp. 2--10.

No context found.

F. T. Leighton. Average case analysis of greedy routing algorithms on arrays. In Symposium on Parallel Algorithms and Architecture, pages 2#10, 1990.

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