Constant Queue Route on a Mesh

Packet routing is an important problem in parallel computation since a single step of inter-processor communication can be thought of as a packet routing task. In this paper we present an optimal algorithm for packet routing on a mesh-connected computer. Two important criteria for judging a routing algorithm will be 1) its run time, i.e., the number of parallel steps it takes for the last packet to reach its destination, and 2) its queue size, i.e., the maximum number of packets that any node will have to store at any time during routing. We present a 2n - 2 step routing algorithm for an n x n mesh that requires a queue size of only 58. The previous best known result is a routing algorithm with the same time bound but with a queue size of 672. The time bound of 2n - 2 is optimal. A queue size of 672 is rather large for practical use. We believe that the queue size of our algorithm is practical. The improvement in the queue size is possible due to (from among other things) a new 3s + o(s) sorting algorithm for an s x s mesh

Randomized Routing and Sorting on the Reconfigurable Mesh

In this paper we demonstrate the power of reconfiguration by presenting efficient randomized algorithms for both packet routing and sorting on a reconfigurable mesh connected computer (referred to simply as the mesh from hereon). The run times of these algorithms are better than the best achievable time bounds on a conventional mesh. In particular, we show that permutation routing problem can be solved on a linear array of size n in 3/4n steps, whereas n-1 is the best possible run time without reconfiguration. We also show that permutation routing on an n x n reconfigurable mesh can be done in time n + o(n)using a randomized algorithm or in time 1.25n + o(n) deterministically. In contrast, 2n-2 is the diameter of a conventional mesh and hence routing and sorting will need at least 2n-2 steps on a conventional mesh. In addition we show that the problem of sorting can be solved in time n+ o(n). All these time bounds hold with high probability. The bisection lower bound for both sorting and routing on the mesh is n/2, and hence our algorithms have nearly optimal time bounds

Randomized Algorithms For Packet Routing on the Mesh

Packet routing is an important problem of parallel computing since a fast algorithm for packet routing will imply 1) fast inter-processor communication, and 2) fast algorithms for emulating ideal models like PRAMs on fixed connection machines.There are three different models of packet routing, namely 1) Store and forward, 2) Multipacket, and 3) Cut through. In this paper we provide a survey of the best known randomized algorithms for store and forward routing, k-k routing, and cut through routing on the Mesh Connected Computers

A Comparison of Meshes With Static Buses and Unidirectional Wrap-Arounds

We investigate the relative computational powers of a mesh with static buses and a mesh with unidirectional wrap-mounds. A mesh with unidirectional wraparounds is a torus with the restriction that any wraparoundlink of the architecture can only transmit data in one of the two directions at any clock tick. We show that the problem of packet routing can be solved as efficiently on a linear array with unidirectional wrap-around link as on a linear array with a broadcast bus. We also present a routing algorithm for a twcdimensional torus with unidirectional wraparound links whose run time is close to that of the best known algorithm for routing on a mesh with broadcast buses in each dimension. In addition, we show that on a mesh with broadcast buses, sorting can be done in time that is essentially the same as the time needed for packet routing

Routing with locality in partitioned-bus meshes

We show that adding partitioned-buses (as opposed to long buses that span an entire row or column) to ordinary meshes can reduce the routing time by approximately one-third for permutation routing with locality. A matching time lower bound is also proved. The result can be generalized to multi-packet routing.published_or_final_versio

\u3cem\u3ek-k\u3c/em\u3e Routing, \u3cem\u3ek-k\u3c/em\u3e Sorting, and Cut Through Routing on the Mesh

In this paper we present randomized algorithms for k-k routing, k-k sorting, and cut through routing. The stated resource bounds hold with high probability. The algorithm for k-k routing runs in [k/2]n+o(kn) steps. We also show that k-k sorting can be accomplished within [k/2] n+n+o(kn) steps, and cut through routing can be done in [3/4]kn+[3/2]n+o(kn) steps. The best known time bounds (prior to this paper) for all these three problems were kn+o(kn). [kn/2] is a known lower bound for all the three problems (which is the bisection bound), and 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

Sprinklers: A Randomized Variable-Size Striping Approach to Reordering-Free Load-Balanced Switching

Internet traffic continues to grow exponentially, calling for switches that can scale well in both size and speed. While load-balanced switches can achieve such scalability, they suffer from a fundamental packet reordering problem. Existing proposals either suffer from poor worst-case packet delays or require sophisticated matching mechanisms. In this paper, we propose a new family of stable load-balanced switches called "Sprinklers" that has comparable implementation cost and performance as the baseline load-balanced switch, but yet can guarantee packet ordering. The main idea is to force all packets within the same virtual output queue (VOQ) to traverse the same "fat path" through the switch, so that packet reordering cannot occur. At the core of Sprinklers are two key innovations: a randomized way to determine the "fat path" for each VOQ, and a way to determine its "fatness" roughly in proportion to the rate of the VOQ. These innovations enable Sprinklers to achieve near-perfect load-balancing under arbitrary admissible traffic. Proving this property rigorously using novel worst-case large deviation techniques is another key contribution of this work

General Routing Algorithms for Star Graphs

In designing algorithms for a specific parallel architecture, a programmer has to cope with topological and cardinality variations. Both these problems always increase the programmer\u27s effort. However, an ideal shared memory abstract parallel model called the parallel random access machine (PRAM) [KRUS86, KRUS88] that avoids these problems and also simple-to-program has been proposed. Unfortunately, the PRAM does not seem to be realizable in the present or even foreseeable technologies. On the other hand, a packet routing technique can be employed to simulate the PRAM on a feasible parallel architecture without significant loss of efficiency. The problem of routing is also important due to its intrinsic significance in distributed processing and its important role in the simulations among parallel models. The routing problem is defined as follows: Given a specific network and a set of packets of information in which a packet is an (origin, destination) pair. To start with, the packets are placed on their origins, one per node. These packets must be routed in parallel to their own destinations such that at most one packet passes through any link of the network at any time and all packets arrive at their destinations as quickly as possible. We are interested in a special case of the general routing problem called permutation routing in which the destinations form some permutation of the origins. A routing algorithm is said to be oblivious if the path taken by each packet is only dependent on its source and destination. An oblivious routing strategy is preferable since it will lead to a simple control structure for the individual processing elements. Also oblivious routing algorithms can be used in a distributed environment. In this paper we are concerned with only oblivious routing strategies