R. A. Borodin and J. E. Hopcroft, "Routing, merging and sorting on parallel models of computation, " in Proc. 14th ACM Symp. Theory Comput., San Francisco, CA. Apr. 1982, pp. 338-344.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the torus is both edge and vertexsymmetric, the overall problem size is reduced to O(CN) 5.1 Locality versus Worst case Throughput The first tradeoff we consider for oblivious routing algorithm design is how optimizing for the worst case affects the average distance a packet travels. From [15][16] 17] 18] a general trend is known for tori and other networks greedy, oblivious algorithms that attempt to maximize locality suffer from poor worst case performance. This trend can be quantified as a tradeoff by solving a series of routing algorithm design problems: for a particular ....

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

....or near worst case behavior [3] 4] However, for the example presented in Section IV, the traditional techniques overestimate the worst case throughput of the ROMM routing algorithm [3] by approximately 47 . Worst case characterization has also been approached from a theoretical perspective [5][6] 7] Despite providing strong results, these analyses do not provide exact throughput values for specific topologies and routing algo This work has been supported by an NSF Graduate Fellowship with supplement from Stanford University and under the MARCO Interconnect Focus Research Center. The ....

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

....or near worst case behavior [3] 4] However, for the example presented in Section iV, the traditional techniques overestimate the worst case throughput of the ROMM routing algorithm [3] by approximately 47 . Worst case characterization has also been approached from a theoretical perspective [5][6] 7] Despite providing strong results, these analyses do not provide exact throughput values for specific topologies and routing algo This work has been supported by an NSF Graduate Fellowship with supplement from Stanford University and under the MARCO interconnect Focus Research Center. The ....

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

....or near worst case behavior [3] 4] However, for the example presented in Section 5, the traditional techniques overestimate the worstcase throughput of the ROMM routing algorithm [3] by approximately 47 . Worst case characterization has also been approached from a theoretical perspective [5][6] 7] and while providing strong results, these analyses do not provide exact throughput values for specific topologies and routing algorithms. With the algorithms presented in this report, we hope to enable more quantitative studies of oblivious routing algorithms in the future. 2 2 ....

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

....processors [5] which is much worse than the O( # N) bound for oblivious permutation routing on two dimensional, unbounded queuesize meshes including # N # N processors. iii) An # # N) lower bound for oblivious permutation routing on any constant degree, unbounded queue size, N processor network [2, 3, 6]. It should be noted, however, that these lower bound proofs needed some supplementary conditions that might not seem so serious but are important for the proofs. In this paper, it is shown that the above lower bounds do not hold any more if those supplementary conditions are slightly relaxed. ....

....known for any constant degree, k queue size network including N processors under the pure condition [8] For 2D meshes, if we set k = # N , then that is equivalent to unbounded queue size. Our bound for this specific value of k is O(N 0.75 N 0. 25 ) O( # N ) which matches the lower bound of [2, 3, 6]. Our second result concerns with 3D meshes: In [5] an important exception was proved against the well known superiority of the 3D meshes over the 2D ones; oblivious permutation routing requires# N 2 3 ) steps over the 3D meshes including N processors under the following (not unusual, see the ....

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A. Borodin and J.E. Hopcroft, "Routing, merging, and sorting on parallel models of computation," J. Computer and System Sciences 30 (1985) 130145.

....expected time Theta(log n) and uses n= log n processors. Very fast parallel algorithms have been developed for several problems. We characterise very fast algorithms as those that run in time polynomial in log log n using a polynomial number of processors. Some examples of these can be found in [5,6,12,15]. It has been shown by Beame and Hastad [4] that sorting n items using polynomially many processors requires Omega Gammaqui n= log log n) time. Thus, sorting in time polynomial in log log n using polynomially many processors is impossible. Recently, attempts have been made to separate the list ....

....out using n 1 ffl processors in O( 1 ffl log log n) time. Proof: For each partition S, do the following: Divide the array into n ffl=2 blocks. Chain together the blocks that contain an element of S, and count them. The chaining can be done in time O(log log n) using n ffl=2 processors [6]. We can now assign n ffl=2 processors to each such block and continue the process. This process terminates when the block size drops to n ffl=2 i.e. in no more than 2=ffl steps, each step requiring time O(log log n) Each element is now assigned n ffl=2 processors to find its nearest right ....

Borodin A. and Hopcroft J.E., Routing merging and sorting on parallel models of computation, J of Comput. Syst. Sci., 30, 1985, pp 130-145.

....model on which the Ultracomputer is based. Although this idealized machine is not physically realizable, we show that a close approximation can be built. 2.1. The Model An idealized parallel processor, dubbed a paracomputer by Schwartz [23] and classified as a WRAM by Borodin and Hopcroft [1], consists of a number of autonomous processing elements (PEs) sharing a central memory (see also Fortune Wylie [5] and Snir [24] Every PE is permitted to read or write a shared memory cell each cycle. In particular, simultaneous reads and writes directed at the same memory cell are all ....

A. Borodin and J.E. Hopcroft, "Routing, merging and sorting on parallel models of computation", in Proc. 14th Annual ACM Symp. on Theory of Comp., May, 1982.

....uses O(n) operations and O(log n log n) time; it assumes that the input is provided in an array of length n. The algorithm makes use of the [AKS83 ] sorting network. However, if the algorithm is implemented on a CREW PRAM we do not need to use this sorting network; the sorting algorithm of [BH 82] will suffice. The advantage of the latter sorting algorithm is that the constants in the running time are much smaller. The previous best result for selection on the PRAM was the following: O(n) operations in time O(log n log log n) Vi 83a] Much previous work on parallel algorithms for the ....

A. Borodin and J. Hopcroft, "Routing, merging and sorting on parallel models of computation ", Proc. Fourteenth Annual Symp. on Theory of Computing, 338-344.

....1 packets in dn O(log 2 n) time steps on the d dimensional n d node torus and in 2dn O(log 2 n) time steps on the d dimensional n d node mesh, with high probability. This research was partially supported by the NEC Research Institute. 1 Introduction A number of researchers [1, 4, 3, 2, 5, 6, 7, 8, 10, 11, 14, 12] have suggested algorithms for routing packets in a network with the property that on each step, each node in the network sends all of the packets it received on the previous step along one of its outgoing edges (with at most one packet leaving per edge) Such schemes are generally referred to as ....

....behavior (i.e. without any independence assumptions) has proven to be difficult. Hajek [8] showed that for a natural algorithm on the N node hypercube, k packets with worst case destinations are delivered in 2k log N time steps. An algorithm for the hypercube suggested by Borodin and Hopcroft [4] was shown by Prager [15] to terminate in O(log N) steps on a special class of permutations. Feige and Raghavan [5] were the first to present an exact analysis of the average case and worst case behavior of hot potato algorithms for two and three dimensional tori. Feige and Raghavan also showed ....

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

....is maximum advance and achieves t p d p 2(k Gamma 1) Unfortunately, this result does not hold for mesh networks, since even a single packet does not use a shortest path. 1.5 Related Work Baran [3] is widely credited with having first proposed hot potato routing. Borodin and Hopcroft [7] proposed an algorithm for hot potato routing on the hypercube. Although they did not give a complete analysis of its behavior, they observed that experimentally the algorithm appears promising . Prager [25] showed that the Borodin13 Hopcroft algorithm stops in n steps on the 2 n nodes ....

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

....was shown that rt(T ) 3n for any tree T of n vertices. As a consequence, rt(G) 3n for any graph G of n vertices. A lot of work has been devoted to the study of packet routing problems. As it is natural, several routing models have been considered. However, most of the papers in the literature [2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17] consider the model in which, at any time step, all edges can carry a packet (bidirectional edges can carry one packet for each direction) Upfal [15] considered the model in which, at each step, each processor can either send or receive and only along one communication link. Meyer auf der Heide, ....

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

....bit reversal permutation 01001001 10010010. Each # corresponds to traversal of an edge in the hypercube. 0 1 0 0 1 0 0 1 # 1 1 0 0 1 0 0 1 # 1 0 0 0 1 0 0 1 # 1 0 0 1 1 0 0 1 # 1 0 0 1 0 0 0 1 # 1 0 0 1 0 0 1 1 # 1 0 0 1 0 0 1 0 Borodin and Hopcroft s bound The following result [8] says that there are no uniformly good deterministic algorithms for oblivious permutation routing: Theorem: For any deterministic, oblivious permutation routing algorithm, there is a permutation for which the routing takes Omega Gamma p N=d 3 ) steps. Example: For the leading bit ....

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

....n log c n processors, for some constant c . By Theorem 3.1, there exists an algorithm for merging two ordered lists of size n each which runs in o (loglogn ) time and performs at most n log c 2 n simultaneous comparisons in a comparison model. This contradicts the corresponding lower bound of [BHo 85] and [HH82 ] for merging in a parallel comparison model. This concludes the proof of Theorem 3.1. The proof of Theorem 3.2 is based on a lemma from [MW 85] Suppose we are given a strong CRCW PRAM algorithm for the ANN problem. Using Claim 3.1 below we show how to construct from this algorithm a ....

A. Borodin and J.E. Hopcroft, "Routing, merging, and sorting on parallel models of comparison", J. of Comp. and System Sci., 30 (1985), pp. 130-145.

....it was shown that rt(T ) 3n for any tree T of n vertices. As a consequence, rt(G) 3n for any graph G of n vertices. A lot of work has been devoted to the study of packet routing problems. As is natural, several routing models have been considered. However, most of the papers in the literature [2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 17, 18] consider the model in which, at any time step, all edges can carry a packet (bidirectional edges can carry one packet for each direction) Upfal [16] considered the model in which, at each step, each processor can either send or receive and only along one communication link. Meyer auf der Heide, ....

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

....first used the partitioning strategy to handle the problem of merging two sorted sequences; he obtained an O(log log n) time algorithm based on the parallel comparison tree model of computation. Kruskal achieved the same O(log log n) time bounds on a PRAM in [Kr] Finally, Borodin and Hopcroft in [BH] proved an Omega Gamma 12 log n) lower bound for the problem, and thus the O(log log n) time optimality. The procedure p merge presented below takes as inputs two sorted lists A and B and outputs the sorted sequence C, the merged list of A and B. The p merge procedure requires the repeated use ....

Borodin A., Hopcroft J.E., "Routing, Merging and Sorting on Parallel Models of Computation ", J. Comput. System Sci., 30:130-145, 1985.

....The power of randomness in computation is a major issue in all aspects of computer science, and is yet to be fully understood. There are many cases in which there are tremendous gaps between the complexities, or even possibilities of deterministic and randomized computations (e.g. routing [BH,V], Byzantine agreements [FLP,B,FL,FM] Communication Complexity [Y2,PS,MS] A method of smoothing these gaps is to measure the quantity of randomization of an algorithm, thus substituting the qualitative question Is the algorithm deterministic or randomized by the quantitative question ....

Borodin, A., and J. E. Hopcroft, "Routing, Merging, and Sorting on Parallel Models of Computing", Journal of Computer and System Science 30, pp. 130145, 1985.

....according to load on network , it selects routing paths based on the local load characteristics, thus, adaptive routers can diffuse local congestion by exploiting alternative paths to a destination. They are also quite fault tolerant . One of the simplest routing algorithms is oblivious [BoHo 85], where the routing of a message is completely determined by the target address and source address in SCI packets. Oblivious routers require relatively simple logic to route messages and to guarantee the three essential properties of every router, i.e. freedom from deadlock, livelock and ....

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

....that rt(T ) 3n for any tree T of n vertices. As a consequence, rt(G) 3n for any graph G of n vertices. A lot of work has been devoted to the study of packet routing problems. As it is natural, several routing models have been considered. However, most of the papers 146 in the literature [2, 5, 6, 7, 8, 9, 11, 12, 13, 15, 17] consider the model in which, at any time step, all edges can carry a packet (bidirectional edges can carry one packet for each direction) Upfal [16] considered the model in which, at each step, each processor can either send or receive and only along one communication link. Meyer auf der Heide, ....

A. Borodin and J.E. Hopcroft, "Routing, Merging , and Sorting on Parallel Models of Computation", Journal of Computer and System Sciences, Vol. 30, 1985, pp. 130-145. 160

....slope) of each function in constant time. Next, we merge the sets of breakpoints for f 1 and f 2 according to their x coordinate, but keep track of whether each came from f 1 or f 2 ; call these respectively type 1 and type 2 breakpoints. The merging can be done efficiently by the algorithm of [4]. Next, using standard pointer doubling techniques, each type 1 (resp. type 2) point can determine the previous and following type 2 (resp. type 1) point. Once this information is available, each point can determine in O(1) time whether it lies below or on g(x) Finally, knowing the type of its ....

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

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R. A. Borodin and J. E. Hopcroft, "Routing, merging and sorting on parallel models of computation, " in Proc. 14th ACM Symp. Theory Comput., San Francisco, CA. Apr. 1982, pp. 338-344.

No context found.

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

No context found.

Borodin and Hopcroft, "Routing, Merging, and Sorting on Parallel Models of Computation," Proc. 14th Annual ACM Symposium on Theory of Computing, 1982, pp.338-344.

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A. Borodin and J. E. Hopcroft, "Routing, merging, and sorting on parallel models of computation," J. Comput. System Sci. 30(1. 1..

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Alan Borodin and John E. Hopcroft, "Routing, Merging, and Sorting on Parallel Models of Computation ", 14th Annual ACM Symposium on Theory of Computing, pp. 338-344, 1982.

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

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