## Randomized routing and sorting on fixed-connection networks (1994)

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Venue: | JOURNAL OF ALGORITHMS |

Citations: | 88 - 13 self |

### BibTeX

@ARTICLE{Leighton94randomizedrouting,

author = {F. T. Leighton and Bruce M. Maggs and Abhiram G. Ranade and Satish B. Rao},

title = {Randomized routing and sorting on fixed-connection networks},

journal = {JOURNAL OF ALGORITHMS},

year = {1994}

}

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### OpenURL

### Abstract

This paper presents a general paradigm for the design of packet routing algorithms for fixed-connection networks. Its basis is a randomized on-line algorithm for scheduling any set of N packets whose paths have congestion c on any bounded-degree leveled network with depth L in O(c + L + log N) steps, using constant-size queues. In this paradigm, the design of a routing algorithm is broken into three parts: (1) showing that the underlying network can emulate a leveled network, (2) designing a path selection strategy for the leveled network, and (3) applying the scheduling algorithm. This strategy yields randomized algorithms for routing and sorting in time proportional to the diameter for meshes, butterflies, shuffle-exchange graphs, multidimensional arrays, and hypercubes. It also leads to the construction of an area-universal network: an N-node network with area Θ(N) that can simulate any other network of area O(N) with slowdown O(log N).

### Citations

1308 | Introduction to Parallel Algorithms and Architectures: Arrays - Leighton - 1992 |

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A measure of asymptotic efficiency of tests of a hypothesis based on the sum of observations
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(Show Context)
Citation Context ...ial has probability p of success. Then we have Pr[S = s] = \Gamma r s \Delta p s (1 \Gamma p) r\Gammas . We estimate the area under the tails of this binomial distribution using a Chernoff-type bound =-=[8]-=-. Following Angluin and Valiant [3] we have Pr[Ssfl 1 rp]se \Gamma(1\Gammafl 1 ) 2 rp=2 Pr[Ssfl 2 rp]se \Gamma(1\Gammafl 2 ) 2 rp=3 for 0sfl 1s1 and 1sfl 2s2. In our application r = n, p = p M=n, fl 1... |

668 |
Universal classes of hash functions
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- 1977
(Show Context)
Citation Context ...re that no other packet for the same destination will arrive later in another queue. Thus, at most one packet for each destination traverses an edge. We assign ranks using the universal hash function =-=[7]-=- rank(x) = // m\Gamma1 X i=0 a i x i ! mod P ! mod R which maps a destination x 2 [0; P \Gamma 1] to a rank in [0; R \Gamma 1] with k-wise independence. Here P is a prime number greater than the total... |

501 | Sorting networks and their applications
- Batcher
- 1968
(Show Context)
Citation Context ...e to route any permutation of N packets in O(log N) steps by pipelining \Theta(log N) such permutations. Unfortunately, no efficient parallel algorithm for finding the paths is known. In 1968 Batcher =-=[4]-=- devised an elegant and practical parallel algorithm for sorting N packets on an N-node shuffle-exchange network in log 2 N steps 1 using queues of size 1. The algorithm can be used to route any permu... |

279 |
Probabilistic construction of deterministic algorithms: Approximating packing integer programs
- Raghavan
- 1988
(Show Context)
Citation Context ...h depth L, a setting of ranks that ensures delivery in time O(c+ L + log N) can be found can be found off-line deterministically in time 2 O(c+L+logN) . The proof uses the Raghavan--Spencer technique =-=[27, 33]-=- to sequentially find a setting of the ranks so that no bad event corresponding to a delay sequence occurs. One application is in preparing simulations by volume and area-universal networks off-line s... |

256 |
Fast Probabilistic Algorithms for Hamiltonian Circuits and Matchings
- Angluin, Valiant
- 1979
(Show Context)
Citation Context ...hen we have Pr[S = s] = \Gamma r s \Delta p s (1 \Gamma p) r\Gammas . We estimate the area under the tails of this binomial distribution using a Chernoff-type bound [8]. Following Angluin and Valiant =-=[3]-=- we have Pr[Ssfl 1 rp]se \Gamma(1\Gammafl 1 ) 2 rp=2 Pr[Ssfl 2 rp]se \Gamma(1\Gammafl 2 ) 2 rp=3 for 0sfl 1s1 and 1sfl 2s2. In our application r = n, p = p M=n, fl 1 = 1=2, and fl 2 = 3=2. For any fix... |

249 |
Universal schemes for parallel communication
- Valiant, Brebner
- 1981
(Show Context)
Citation Context ...he O(c +L+ log N) scheduling algorithm to route N packets on a p N \Theta p N mesh in O( p N) steps using constant-size queues. Although O( p N)-step routing algorithms for the mesh were known before =-=[13, 14, 39]-=-, they all have more complicated path selection strategies. In an n \Theta n mesh, each node has a distinct label (x; y), where x is its column and y is its row, and 0sx; ysn \Gamma 1. Thus, an n \The... |

224 |
Fat-trees: Universal networks for hardware-efficient supercomputing
- Leiserson
- 1985
(Show Context)
Citation Context ...-dimensional array with maximum side length M in O(kM) steps. In Section 7, we show how to adapt the scheduling algorithm to route a set of messages with load factorsin O(+ log M) steps on a fat-tree =-=[21]-=- with root capacity M . The fat-tree routing algorithm leads to the construction of an areauniversalsnetwork: an N-node network with area \Theta(N ) that can simulate any other network of area O(N) wi... |

206 |
A scheme for fast parallel communication
- Valiant
- 1982
(Show Context)
Citation Context ...stant factors in Upfal's algorithm. There has also been great success in the development of efficient randomized packet routing algorithms. The study of randomized algorithms was pioneered by Valiant =-=[38]-=- who showed how to route any permutation of N packets in O(log N) steps on an N-node hypercube with queues of size O(log N) at each node. Valiant's idea was to route each packet to a randomly-chosen i... |

195 |
How to Emulate Shared Memory
- Ranade
- 1987
(Show Context)
Citation Context ...ty at least 1 \Gamma 1=N k for any fixed constant k, where N is the number of packets. Valiant's result was improved in a succession of papers by Aleliunas [2], Upfal [36], Pippenger [26], and Ranade =-=[29]-=-. Aleliunas and Upfal developed the notion of a delay path and showed how to route on the shuffle-exchange and butterfly networks (respectively) in O(log N) steps with queues of size O(log N ). Pippen... |

167 |
Tight bounds on the complexity of parallel sorting
- Leighton
- 1985
(Show Context)
Citation Context ... destination. No better deterministic algorithm was found until 1983, when Ajtai, Koml'os, and Szemer'edi [1] solved a classic open problem by constructing an O(log N)-depth sorting network. Leighton =-=[16]-=- then used this O(N log N )- node network to construct a degree 3 N-node network capable of solving any N-packet routing problem in O(log N) steps using queues of size 1. Although this result is optim... |

133 |
Mathematical Theory of Connecting Networks and Telephone Traffic
- Beneš
- 1965
(Show Context)
Citation Context ...right place at the right time is an important, interesting, and challenging problem. Not surprisingly, it has also been the subject of a great deal of research. 1.1 Past work on routing In 1965 Benes =-=[6]-=- showed that the inputs and outputs of an N-node Benes network (two back-to-back butterfly networks) can be connected in any permutation by a set of disjoint paths. Shortly thereafter Waksman [40] dev... |

120 |
A logarithmic time sort for linear size networks
- Reif, Valiant
- 1987
(Show Context)
Citation Context ...s on fixed-connection networks. In his fundamental paper, Batcher [4] showed that an algorithm for sorting on a network can usually be converted into an algorithm for packet routing. Reif and Valiant =-=[32]-=-, on the other hand, described a method for converting a routing algorithm into a randomized sorting algorithm. As a consequence, they derived randomized sorting algorithms for hypercubes and butterfl... |

112 |
Sorting in c log n parallel steps
- Ajtai, Komlos, et al.
- 1983
(Show Context)
Citation Context ...he same destination can be combined to form a single packet should they meet en route to their destination. No better deterministic algorithm was found until 1983, when Ajtai, Koml'os, and Szemer'edi =-=[1]-=- solved a classic open problem by constructing an O(log N)-depth sorting network. Leighton [16] then used this O(N log N )- node network to construct a degree 3 N-node network capable of solving any N... |

104 | A complexity theory for VLSI
- Thompson
- 1980
(Show Context)
Citation Context ... congestion is c = O( + log M ). Each packet travels a distance of L = 2m = 2 log M . The packets are then scheduled using the algorithm from Section 2. Let us now consider the VLSI area requirements =-=[34]-=- of fat-trees. A fat-tree with root capacity M and \Theta(M 2 ) nodes has a layout with area O(M 2 log 2 M) that is obtained by embedding the fat-tree in the tree of meshes [15]. The nodes of the tree... |

96 |
Complexity issues in VLSI
- Leighton
- 1983
(Show Context)
Citation Context ...VLSI area requirements [34] of fat-trees. A fat-tree with root capacity M and \Theta(M 2 ) nodes has a layout with area O(M 2 log 2 M) that is obtained by embedding the fat-tree in the tree of meshes =-=[15]-=-. The nodes of the tree of meshes in this layout are separated by a distance of log M in both the horizontal and vertical directions. Thus, the \Theta(log M ) space for the chain associated with each ... |

88 |
Sorting on a mesh-connected parallel computer
- Thompson, Kung
- 1977
(Show Context)
Citation Context ...ch network considered, the algorithm runs in time proportional to the diameter of the network, and uses constant-size queues. Such algorithms were previously known only for bounded-dimensional arrays =-=[16, 35]-=-. 1.4 Outline of the results The basis of most of the results in this paper is a proof that a variant of Ranade's algorithm can be used to schedule any set of N packets whose paths have congestion c o... |

75 |
Universal packet routing algorithms
- Leighton, Maggs, et al.
- 1988
(Show Context)
Citation Context ...s traditionally been the hard part of routing. Curiously, however, we have found that by ignoring the underlying network and the method of path selection, Stage 2 actually becomes easier to solve! In =-=[20]-=- for example, Leighton, Maggs, and Rao show that for any set of packets whose paths have congestion c and dilation d, in any network, there is a schedule of length O(c + d) in which at most one packet... |

73 | A permutation network
- Waksman
(Show Context)
Citation Context ...enes [6] showed that the inputs and outputs of an N-node Benes network (two back-to-back butterfly networks) can be connected in any permutation by a set of disjoint paths. Shortly thereafter Waksman =-=[40]-=- devised a simple sequential algorithm for finding the paths in O(N) time. Given the paths, it is straightforward to route a set of packets from the inputs to the outputs an N-node Benes network in an... |

70 |
Parallel Permutation and Sorting Algorithms and a New Generalized Connection Network
- Nassimi, Sahni
- 1982
(Show Context)
Citation Context ...N) time on an N log Nnode butterfly using constant-size queues. time. The candidates are then sorted in O(log M) time using a simple deterministic algorithm based on counting due to Nassimi and Sahni =-=[25]-=-. After the candidates are sorted, every (log N)th one in the sorted order is chosen to be a splitter. This oversampling technique, due to Reif, ensures that each of the intervals contains approximate... |

64 |
Efficient Schemes for Parallel Communication
- Upfal
- 1984
(Show Context)
Citation Context ...robability to mean with probability at least 1 \Gamma 1=N k for any fixed constant k, where N is the number of packets. Valiant's result was improved in a succession of papers by Aleliunas [2], Upfal =-=[36]-=-, Pippenger [26], and Ranade [29]. Aleliunas and Upfal developed the notion of a delay path and showed how to route on the shuffle-exchange and butterfly networks (respectively) in O(log N) steps with... |

60 |
An O(logN) Deterministic Packet Routing Scheme. STOC
- Upfal
- 1998
(Show Context)
Citation Context ...cal use. Hence, the effort to find fast deterministic algorithms has continued. Recently Upfal discovered an O(log N)-step algorithm for routing on an expander-based network called the multibutterfly =-=[37]-=-. The algorithm solves the routing problem directly without reducing it to sorting, and the constant factors are much smaller than those of the AKS-based algorithms. In [18], Leighton and Maggs show t... |

56 |
Parallel Hashing --- an Efficient Implementation of Shared Memory
- Karlin, Upfal
- 1986
(Show Context)
Citation Context ...lating shared-memory machines on bounded-degree networks. A shared-memory machine with a large address space can be emulated by randomly hashing the memory locations to the nodes of a butterfly as in =-=[11]-=- and [29]. The hashing ensures that the congestion of the packets implementing each memory access step is small. The algorithm from Section 2 is used to schedule the movements of these packets. Given ... |

53 |
Meshes with reconfigurable buses
- Miller
- 1988
(Show Context)
Citation Context ...estination. As a consequence, no scheme for simulating shared-bus networks was known. A network that can simulate in O(1) steps each step of any shared-bus network area of equal area was presented in =-=[24]-=-. However, the connections in that network are not fixed, but instead nodes communicate via reconfigurable buses. 8 Sorting on butterflies In this section we present a randomized algorithm for sorting... |

52 |
Deadlock Free Message Routing In Multiprocessor Interconnection Networks
- Dally, Seitz
- 1987
(Show Context)
Citation Context ...ake assumptions about packet lengths. Consequently, it is the most studied model in the literature. Other models of interest are the circuit switching model [12] and the cut-through or wormhole model =-=[9]-=-. These models arise in practice and are also of theoretical interest, although less is known about them. Although our results have some limited applications in these models, we will primarily concern... |

51 | Randomized routing on fat-trees
- Greenberg, Leiserson
- 1989
(Show Context)
Citation Context ...e class with N nodes has area O(N) and can, with high probability, simulate in O(log N) steps each step of any network of area O(N). The networks are based on the fat-trees of Greenberg and Leiserson =-=[10]-=- and the simulation uses the packet routing algorithm from Section 2. Leiserson was the first to display a class of networks that could efficiently simulate any other network of the same area or volum... |

50 |
The Fluent abstract machine
- Ranade, Bhatt, et al.
- 1988
(Show Context)
Citation Context ...ve, it is possible for each node to send outgoing message packets as soon as the corresponding packets arrive on all incoming edges. In fact message combining can also be made to work with pipelining =-=[30, 31]-=-. It is possible to show that Theorem 2.12 still applies. The analysis involves constructing a delay sequence and a counting argument similar that for Theorem 2.9. 4 3 2 1 0 0 1 2 3 4 column row Figur... |

32 |
A 2n \Gamma 2 Step Algorithm for Routing in an n \Theta n Array with Constant Size Queues
- Leighton, Makedon, et al.
- 1989
(Show Context)
Citation Context ...g N) steps using constant-size queues. Most recently, Leighton, Makedon, and Tollis discovered a deterministic algorithm for routing any permutation in 2 p N \Gamma 2 steps using constant-size queues =-=[19]-=-, thus achieving the optimal time bound in the worst case. 1.2 Our approach to routing One deficiency with the state-of-the-art in packet routing is that aside from Valiant's paradigm of first routing... |

30 |
Randomized parallel communication
- Aleliunas
- 1982
(Show Context)
Citation Context ...with high probability to mean with probability at least 1 \Gamma 1=N k for any fixed constant k, where N is the number of packets. Valiant's result was improved in a succession of papers by Aleliunas =-=[2]-=-, Upfal [36], Pippenger [26], and Ranade [29]. Aleliunas and Upfal developed the notion of a delay path and showed how to route on the shuffle-exchange and butterfly networks (respectively) in O(log N... |

27 | Fast algorithms for routing around faults in multibutterflies and randomly-wired splitter networks
- Leighton, Maggs
- 1992
(Show Context)
Citation Context ... called the multibutterfly [37]. The algorithm solves the routing problem directly without reducing it to sorting, and the constant factors are much smaller than those of the AKS-based algorithms. In =-=[18]-=-, Leighton and Maggs show that the multibutterfly is fault tolerant and improve the constant factors in Upfal's algorithm. There has also been great success in the development of efficient randomized ... |

27 |
Mathematical Theory of Connecting Networks and Telephone Traffic
- Beneˇs
- 1965
(Show Context)
Citation Context ...ight place at the right time is an important, interesting, and challenging problem. Not surprisingly, it has also been the subject of a great deal of research. 1.1 Past work on routing In 1965 Beneˇs =-=[6]-=- showed that the inputs and outputs of an N-node Beneˇs network (two back-to-back butterfly networks) can be connected in any permutation by a set of disjoint paths. Shortly thereafter Waksman [40] de... |

26 |
A Measure of Asymptotic E ciency for Tests of a Hypothesis Based on the Sum of Observations
- Cherno
- 1952
(Show Context)
Citation Context ...noulli trials where each trial has probability p of success. Then we havePr[S = s] = , r s ps (1 , p) r,s .Weestimate the area under the tails of this binomial distribution using a Cherno -type bound =-=[8]-=-. Following Angluin and Valiant [3] we have Pr[S 1rp] e ,(1, 1) 2 rp=2 Pr[S 2rp] e ,(1, 2) 2 rp=3 for 0 1 1 and 1 2 2. In our application r = n, p = p M=n, 1 =1=2, and 2 =3=2. For any xed constant k3,... |

22 |
Routing and sorting on mesh connected arrays
- Kunde
- 1988
(Show Context)
Citation Context ...or routing in meshes. The randomized algorithm of Valiant and Brebner can be used to route any permutation of N packets on a p N \Theta p N mesh in O( p N) steps using queues of size O(log N ). Kunde =-=[14] showed ho-=-w to route deterministically in (2 + ") p N steps using queues of size O(1="). Also, Krizanc, Rajasekaran, and Tsantilis [13] showed how to randomly route any permutation in 2 p N + O(log N)... |

21 |
Parallel Communication with Limited Buffers
- Pippenger
- 1984
(Show Context)
Citation Context ...an with probability at least 1 \Gamma 1=N k for any fixed constant k, where N is the number of packets. Valiant's result was improved in a succession of papers by Aleliunas [2], Upfal [36], Pippenger =-=[26]-=-, and Ranade [29]. Aleliunas and Upfal developed the notion of a delay path and showed how to route on the shuffle-exchange and butterfly networks (respectively) in O(log N) steps with queues of size ... |

20 |
Increasing the size of a network by a constant factor can increase performance by more than a constant factor
- Koch
- 1988
(Show Context)
Citation Context ... the various PRAM models, and it does not make assumptions about packet lengths. Consequently, it is the most studied model in the literature. Other models of interest are the circuit switching model =-=[12]-=- and the cut-through or wormhole model [9]. These models arise in practice and are also of theoretical interest, although less is known about them. Although our results have some limited applications ... |

14 |
Fluent Parallel Computation
- Ranade
- 1989
(Show Context)
Citation Context ...ve, it is possible for each node to send outgoing message packets as soon as the corresponding packets arrive on all incoming edges. In fact message combining can also be made to work with pipelining =-=[30, 31]-=-. It is possible to show that Theorem 2.12 still applies. The analysis involves constructing a delay sequence and a counting argument similar that for Theorem 2.9. 4 3 2 1 0 0 1 2 3 4 column row Figur... |

12 |
Communication-efficient parallel graph algorithms for distributed random-access machines
- Leiserson, Maggs
- 1988
(Show Context)
Citation Context ...iversal networks called the distributed random-access machine (DRAM). A host of conservative algorithms for tree and graph problems for the exclusive-read exclusive-write (EREW) DRAM are presented in =-=[22]-=-. Recently we discovered conservative concurrent-read concurrent-write (CRCW) algorithms that require fewer steps for some of these problems [23]. Until now, however, no efficient fat-tree routing alg... |

9 |
Fat-trees: Universal networks for hardware-e cient supercomputing
- Leiserson
- 1985
(Show Context)
Citation Context ...dimensional array with maximum side length M in O(kM) steps. In Section 7, we show how to adapt the scheduling algorithm to route a set of messages with load factor in O( + log M) steps on a fat-tree =-=[21]-=- with root capacity M. The fat-tree routing algorithm leads to the construction of an areauniversal network: an N-node network with area (N) that can simulate any other network of area O(N) with slowd... |

3 |
Locality in Parallel Computation
- Maggs
- 1989
(Show Context)
Citation Context ...ata carried by the two packets may be added together. Combining is used in the emulation of concurrentread concurrent-write parallel random-access machines [29] and distributed random-access machines =-=[23]-=-. If the congestion is to remain a lower bound when combining is allowed, then its definition must be modified slightly. The new congestion of an edge is the number of different destinations for which... |

3 |
y parallel processor overview
- Butter
- 1986
(Show Context)
Citation Context ...ng cycles (roughly the time to route a random permutation) or petit cycles (the time to perform an atomic step of the routing algorithm). Similarly, the performance of a machine like the BBN Butter y =-=[5]-=- is substantially in uenced by the speed and rate of successful delivery of its router. Packet routing also provides an important bridge between theoretical computer science and applied computer scien... |

2 |
Is the shuffle-exchange better than the butterfly? Unpublished manuscript
- Raghunathan, Saran
(Show Context)
Citation Context ...ice recursive structure possessed by the butterfly. However we can use the following theorem to sort on the shuffle-exchange graph. A similar theorem was proven independently by Raghunathan and Saran =-=[28]-=-. Theorem 9.1 There is an embedding of M= log M distinct M log M-node butterfly graphs in an N = M 2 -node shuffle-exchange graph with load l = 2, congestion c = O(1), and dilation d = 3. Proof: We as... |

2 |
Parallel Hashing | an E cient Implementation of Shared Memory
- Karlin, Upfal
- 1986
(Show Context)
Citation Context ...ulating shared-memory machines on bounded-degree networks. A shared-memory machine with a large address space can be emulated by randomly hashing the memory locations to the nodes of a butter y as in =-=[11]-=- and [29]. The hashing ensures that the congestion of the packets implementing each memory access step is small. The algorithm from Section 2 is used to schedule the movements of these packets. Given ... |

2 |
A 2N −2 step algorithm for routing in an N ×N mesh
- Leighton, Makedon, et al.
- 1989
(Show Context)
Citation Context ... O(log N) steps using constant-size queues. Most recently, Leighton, Makedon, and Tollis discovered a deterministic algorithm for routing any permutation in 2 p N , 2 steps using constant-size queues =-=[19]-=-, thus achieving the optimal time bound in the worst case. 1.2 Our approach to routing One de ciency with the state-of-the-art in packet routing is that aside from Valiant's paradigm of rst routing to... |

2 |
Meshes with recon gurable buses
- Miller, Prasanna-Kumar, et al.
- 1988
(Show Context)
Citation Context ...estination. As a consequence, no scheme for simulating shared-bus networks was known. A network that can simulate in O(1) steps each step of any shared-bus network area of equal area was presented in =-=[24]-=-. However, the connections in that network are not xed, but instead nodes communicate via recon gurable buses. 36s8 Sorting on butter ies In this section we present a randomized algorithm for sorting ... |

2 |
Parallel communication with limited bu ers
- Pippenger
- 1984
(Show Context)
Citation Context ...ity to mean with probability at least 1 , 1=N k for any xed constant k, where N is the numberofpackets. Valiant's result was improved in a succession of papers by Aleliunas [2], Upfal [36], Pippenger =-=[26]-=-, and Ranade [29]. Aleliunas and Upfal developed the notion of a delay path and showed how to route on the shu e-exchange and butter y networks (respectively) in O(log N) steps with queues of size O(l... |

1 |
Optimal routing algorithms for mesh-connected processor arrays
- Tsantilis
- 1988
(Show Context)
Citation Context ... p N mesh in O( p N) steps using queues of size O(log N ). Kunde [14] showed how to route deterministically in (2 + ") p N steps using queues of size O(1="). Also, Krizanc, Rajasekaran, and =-=Tsantilis [13]-=- showed how to randomly route any permutation in 2 p N + O(log N) steps using constant-size queues. Most recently, Leighton, Makedon, and Tollis discovered a deterministic algorithm for routing any pe... |

1 |
Communication-e cient parallel graph algorithms for distributed random-access machines
- Leiserson, Maggs
- 1988
(Show Context)
Citation Context ...iversal networks called the distributed random-access machine (DRAM). A host of conservative algorithms for tree and graph problems for the exclusive-read exclusive-write (EREW) DRAM are presented in =-=[22]-=-. Recently we discovered conservative concurrent-read concurrent-write (CRCW) algorithms that require fewer steps for some of these problems [23]. Until now, however, no e cient fat-tree routing algor... |

1 |
Is the shu e-exchange better than the butter y? Unpublished manuscript
- Raghunathan, Saran
(Show Context)
Citation Context ...e nice recursive structure possessed by the butter y. However we can use the following theorem to sort on the shu e-exchange graph. A similar theorem was proven independently by Raghunathan and Saran =-=[28]-=-. Theorem 9.1 There isanembedding of M= log M distinct M log M-node butter y graphs in an N = M 2 -node shu e-exchange graph with load l =2, congestion c = O(1), and dilation d =3. Proof: We assume th... |

1 |
cient schemes for parallel communication
- unknown authors
- 1982
(Show Context)
Citation Context ...th high probability to mean with probability at least 1 , 1=N k for any xed constant k, where N is the numberofpackets. Valiant's result was improved in a succession of papers by Aleliunas [2], Upfal =-=[36]-=-, Pippenger [26], and Ranade [29]. Aleliunas and Upfal developed the notion of a delay path and showed how to route on the shu e-exchange and butter y networks (respectively) in O(log N) steps with qu... |