Citations: in and J. E. Hopcroft. Routing, merging, and sorting on parallel models of computation

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A. Borodin and J. E. Hopcroft. Routing, merging, and sorting on parallel models of computation. Journal of Computer and System Science, 30:130--145, 1985.

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Worst-case Traffic for Oblivious Routing Functions - Brian Towles And (2002) (1 citation) (Correct)

....or near worst case behavior [5, 12] However, for the example presented in Section 5, the traditional techniques overestimate the worst case throughput of the ROMM routing algorithm [12] by approximately 47 . Worstcase characterization has also been approached from a theoretical perspective [1, 6, 9, 11]. Despite providing strong results, these analysis do not provide exact throughput values for specific topologies and routing algorithms. With the algorithms presented in this paper, we hope to enable more quantitative studies of oblivious routing algorithms in the future. 2. PRELIMINARIES 2.1 ....

A. Borodin and J. Hopcroft. Routing, merging, and sorting on parallel models of computation. Journal of Computer and System Sciences, 30:130--145, 1985.

Online Oblivious Routing - Bansal, Blum, Chawla, Meyerson (2003) (1 citation) (Correct)

....al[1] who give a log n competitive centralized algorithm. Awerbuch et al.[3] achieve the same competitive ratio in a distributed setting. In the oblivious model, it is well known that deterministic approaches (which must select a single path for each demand) perform very poorly in the worst case [6, 11]. This motivates the use of randomization. A randomized oblivious routing algorithm specifies a probability distribution on the paths between i and j for every (i; j) pair. We will view this as selecting a multi commodity flow on the graph, and routing each packet probabilistically according to ....

A. Borodin and J. E. Hopcroft. Routing, merging, and sorting on parallel models of computation. Journal of Computer and System Sciences, 30(1):130--145, 1985.

Almost Optimal Permutation Routing on Hypercubes - Vöcking (Correct)

....one might assume that nodes are connected by undirected edges. All of our results hold in this slightly more restrictive model, too. Some details in the proofs, however, need to be adapted. Hopcroft show the following lower bound on the congestion of oblivious routing in arbitrary graphs [3]. For every deterministic, oblivious routing strategy on a network with n nodes and maximum node degree c, there is a permutation that produces a congestion of at least p n= c 3=2 ) Kaklamanis et al. 7] sligthly improve this lower bound to p n= 2c) Furthermore, Kaklamanis et al. give an ....

A. Borodin and J. E. Hopcroft. Routing, merging, and sorting on parallel models of computation. Journal of Computer and System Science, 42:130-145, 1985.

Experimental Evaluation of Hot-Potato Routing.. - Bartzis.. (2000) (3 citations) (Correct)

....packet chooses a path from its source to its destination and proceeds along this path; if it arrives at a node and finds its prefered outgoing edge occupied, it is stored in a queue of packets wishing to use that edge. The first hot potato algorithm was proposed by Baran [2] Borodin and Hopcroft [7], Prager [17] and Hajek [12] presented algorithms for hypercubes. Hot potato routing algorithms for 2 dimensional meshes and tori were proposed by Bar Noy et al. 3] Ben Aroya et al. 4] Newman and Schuster [16] Kaufman et al. 14] Feige and Raghavan [11] Kaklamanis et al. 13] Borodin ....

A. Borodin and J. Hopcroft. Routing, Merging, and Sorting on Parallel Models of Computation. Journal of Computer and System Sciences, 30:130--145, 1985.

CoPa: a Parallel Programming Language for Collections - Suciu, Tannen (1998) (Correct)

.... Gamma from Gamma where blocks, even nested, have T = O(1) Taking the first step into account too, we can map any CoPa function into a BVRAM by increasing their complexities to O(T ) O(W 1 ) for that, notice that T W for any CoPa expression) This 2 A result by Borodin and Hopcroft [1] shows that an arbitrary permutation cannot be computed on a butterfly network using oblivious algorithms. 21 step is unlikely to be improved: we prove the following lower bound, which is also of independent interest. CoPa seq cannot express sorting in T = O(1) and W = O(n) 4.1 Implementing ....

A.Borodin and J.Hopcroft. Routing, merging, and sorting on parallel models of computation. JCSS, 30(1):130--145, 1985.

An Optimal Hypercube Algorithm for the All Nearest Smaller.. - Kravets, Plaxton (1994) (Correct)

....of a monotone polygon, preprocessing for answering range minimum queries in constant time, reconstructing a binary tree from its inorder and either preorder or postorder labelings, and matching parenthesis. Furthermore, two fundamental problems can be reduced to ANSV: i) merging two sorted lists [5, 8], and (ii) finding the maximum of n elements [11] The ANSV problem is easy to solve sequentially in O(n) time using a stack. Berkman, Schieber, and Vishkin [4] give the following PRAM algorithms for the ANSV problem: i) an O(lg n) time (n= lg n) processor CREW PRAM algorithm, and (ii) an O(lg ....

A. Borodin and J. E. Hopcroft. Routing, merging, and sorting on parallel models of computation. Journal of Computer and System Sciences, 30:130--145, 1985.

Packet Routing in Multiprocessor Networks - Chinn (1995) (Correct)

....of portions of Chapters 2 and 3 has appeared previously [CLT94] A preliminary version of portions of Chapter 4 has also appeared previously [Chi] This chapter concludes by surveying some of the known results for permutation routing. 1. 1 Routing with Unbounded Queues Borodin and Hopcroft [BH85] prove an Omega Gamma p N=d 3=2 ) time bound for routing the worst case permutation on any N node, degree d network using any oblivious routing algorithm. Kaklamanis et al. KKT90] improve the bound to Omega Gamma p N=d) These results are useful for networks such as the hypercube, whose ....

....to that outlink. After this greedy scheduling, any unscheduled packets are assigned to available outlinks in an arbitrary way. Let us call our algorithm GreedyHP. GreedyHP is destination exchangeable, and it is intended to approximate the hot potato algorithm (suggested by Borodin and Hopcroft [BH85] that for each node randomly picks a scheduling of the outlinks that maximizes the number of packets that advance. This latter algorithm is currently impractical because computing a random maximal matching is expensive, either in time or in hardware complexity. Unlike the Chaos algorithm, ....

A. Borodin and J. E. Hopcroft. Routing, merging, and sorting on parallel models of computation. Journal of Computer and System Sciences, 30:130--145, 1985.

On Bufferless Routing of Variable Length Messages in.. - Bhatt, Bilardi.. (1996) (6 citations) (Correct)

....by the source and destination PEs of the message. This convention, which contrasts with typical studies of packet routing (see, e.g. 13] and wormhole routing, reduces our message routing problems to pure scheduling problems. The many respects in which our computing model differs from that of [2] renders the conclusions in that paper about the inevitable inefficiency of oblivious routing irrelevant for our study. 1 A processor network (or, processor array) is a parallel architecture comprising a set of identical processing elements (PEs) that communicate across an interconnection ....

A. Borodin and J.E. Hopcroft (1985): Routing, merging, and sorting on parallel models of computation. J. Comp. Syst. Sci. 30, 130-145.

Minimal Adaptive Routing on the Mesh with Bounded Queue Size - Chinn, Leighton, Tompa (1994) (4 citations) (Correct)

....at most one message and receives at most one message. At the very least, a good routing algorithm should be able to route permutations efficiently. This introduction concludes by surveying some of the known results for permutation routing. 1.1. Routing with Unbounded Queues Borodin and Hopcroft [3] prove an Omega# p N=d 3=2 ) time bound for routing the worst case permutation on any N node, degree d network using any oblivious routing algorithm (i.e. the path a packet takes depends only on its source and destination) Kaklamanis et al. 13] improve the bound to Omega# p N=d) These ....

A. Borodin and J. E. Hopcroft. Routing, merging, and sorting on parallel models of computation. Journal of Computer and System Sciences, 30:130--145, 1985.

All Nearest Smaller Values on the Hypercube - Kravets, Plaxton (1996) (3 citations) (Correct)

....i w j for all j i (resp. w r w i and w i w j for all i j r) The ANSV problem has been identified as an important sub problem in the design of many efficient parallel algorithms [1] 3] 5] Furthermore, two fundamental problems can be reduced to ANSV: i) merging two sorted lists [4], 7] and (ii) finding the maximum of n elements [10] The ANSV problem is easy to solve sequentially in O(n) time using a stack. Berkman et al. 3] and later Chen [5] give optimal ANSV algorithms for CREW, CRCW, and EREW PRAMs. The previous hypercube algorithms for the ANSV problem have optimal ....

A. Borodin and J. E. Hopcroft. Routing, merging, and sorting on parallel models of computation. Journal of Computer and System Sciences, 30:130--145, 1985.

How much can hardware help routing? (Extended Abstract) - Borodin, al. (1994) (Correct)

....to be as strong as ARM, and showing others to be strictly weaker. Much is known about permutation routing in a small degree network. In particular, for oblivious routing, we know that p n steps are asymptotically optimal for deterministic routing on the butterfly. See, Borodin and Hopcroft [4] and Kaklamanis et al. 7] On the other hand, Valiant s two phase randomized algorithm runs in optimal Theta(log n) time for sparse networks [3, 12, 14] For adaptive routing, there are Theta(log n) deterministic routing algorithms based on AKS sorting [2] which can be implemented on a ....

....by our results is that the single port model presents difficult 4 challenges that were ignored in previous work, because of their focus on low degree networks. 2 Oblivious Routing 2. 1 Oblivious Deterministic Routing Tight Bounds The following lower bounds are derived in Borodin and Hopcroft [4] and in Kaklamanis et al. 7] 1) For any n and d, and for any deterministic single port oblivious routing algorithm, there is a permutation requiring Omega Gamma q n=d) steps. 2) For any n and d, and for any deterministic multi port oblivious routing algorithm, there is a permutation ....

A. Borodin and J. E. Hopcroft. Routing, merging, and sorting on parallel models of computation. Journal of Computer and System Sciences, 30:130--145, 1985.

On the Theory of Interconnection Networks for Parallel Computers - Upfal (1994) (1 citation) (Correct)

....of the butterfly is only log N . Since simple greedy algorithms perform poorly in the worst case, a challenging question is whether there is a routing scheme that is simple to implement, uses only local information, and performs well on any permutation. A lower bound of Borodin and Hopcroft [6] shows that if we restrict the discussion to deterministic algorithms, then an algorithm that performs well in the worst case can not be too simple. Borodin and Hopcroft defined the class of oblivious routing algorithms: a routing algorithm is oblivious if the path of each packet does not depend ....

....each packet does not depend on the origin or destination of any other packet in the system. They show that any deterministic oblivious algorithm for routing on a low degree network performs poorly in the worst case. The following theorem [21] is a slight improvement of the original lower bound in [6]: Theorem 1. Let G be an N node network with maximum degree d, and let A be an oblivious routing algorithm for G. There is an N packet permutation (N ) such that algorithm A routes (N ) on G in Omega ( p N=d) parallel steps. A recent result [8] has shown that the above bound is optimal, ....

A. Borodin and J. E. Hopcroft. Routing, merging, and sorting on parallel models of computation. Journal of Computer and System Sciences, 30:130--145, 1985.

An Optimal Hypercube Algorithm for the All Nearest Smaller.. - Kravets, Plaxton (1994) (Correct)

....of a monotone polygon, preprocessing for answering range minimum queries in constant time, reconstructing a binary tree from its inorder and either preorder or postorder labelings, and matching parenthesis. Furthermore, two fundamental problems can be reduced to ANSV: i) merging two sorted lists [5, 10], and (ii) finding the maximum of n elements [13] The ANSV problem is easy to solve sequentially in O(n) time using a stack. Berkman, Schieber, and Vishkin [4] give the following PRAM algorithms for the ANSV problem: i) an O(lg n) time (n= lg n) processor CREW PRAM algorithm, and (ii) an O(lg lg ....

A. Borodin and J. E. Hopcroft. Routing, merging, and sorting on parallel models of computation. Journal of Computer and System Sciences, 30:130--145, 1985.

Optimal Oblivious Path Selection on the Mesh - Costas Busch Malik (2005) (Correct)