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Automation and Remote Control, Vol. 63, No. 9, 2002, pp. 15061514. Translated from Avtomatika i Telemekhanika, No. 9, 2002, pp. 153163.Original Russian Text Copyright c 2002 by Podlazov.

TECHNICAL DIAGNOSIS

Arbitrary Group Permutations on Hypercube

and Nonblocability of Cube-connected Cycles

V. S. Podlazov

Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, RussiaReceived January 3, 2002

AbstractFor packet switching of arbitrary group permutations on the hypercube and arbitrarypermutations on the cube-connected cycles with small number of node channels, methods ofconflictless realization were proposed, and their speed was considered.

1. INTRODUCTION

Works on the communication networks of highly-parallel multiprocessor computer systems focuson permutation of the data elements between N processors (network nodes). Consideration isusually given to the permutation where each network node before and after operation containsa single data element. Group permutation where each network node before and after operationcontains a group of r 2 data elements is given much less consideration. Group permutation canbe decomposed into a sequence of r conventional permutations, but decomposition is a nontrivialoperation of computational complexity O(rN).

Communication networks are usually divided into dynamic or dedicated and static or directnetworks [1]. The former are multistage (n-cube, inverse n-cube, omega, or ClosBenes) networksusually based on channel switching. The latter are characterized by rigid neighborhood of nodesand make use of packet switching (multiring, hypercube, multidimensional grid, or cube-connectedcycles).

Among the dynamic networks, only the ClosBenes ones are nonblockable on arbitrary permu-tations or, more correctly, conditionally nonblockable because any permutation is realized by anindividual schedule. Depending on the degree of parallelism, the algorithm of schedule compilationrequires from O(log22 N) to O(N log2 N) operations.

In practice, channel switching or its derivative mixed packet-channel switching (the wormholeand cutthrough techniques) are used without preliminary compilation of schedules, that is, withpossible blockings. This substantially contracts the effective width (parallelism) of the switch,that is, the mean number of data elements transmitted concurrently through it. For example, theeffective width of the n-cube on arbitrary permutations is only

N [1].

Among the static networks, full p-ary multiring and generalized p-ary hypercube are nonblock-able on arbitrary permutations. Nonblockability is attained by using packet switching and realizingany permutation according to unique static schedules structured as counter-forests [26]. At that,on a network with N = pr nodes and mG = r(p 1) input-output channels arbitrary permutationat each node is realized in n cycles obeying the following expression:

nG(N) =

(prb 1)(p 1)rb

+

(pre 1)(p 1)re

, where rb = dr/2e and re = br/2c. (1)

0005-1179/02/6309-1506$27.00 c 2002 MAIK Nauka/Interperiodica

ARBITRARY GROUP PERMUTATIONS 1507

For even r, (1) assumes a more convenient form

nG(N) = 2

2(N 1)

(p 1)r

= 2

2(N 1)mG

. (2)

According to (1), the number of cycles for arbitrary permutation is minimal and can vary only ifthe number of channels in nodes varies. In this case, the effective width of the hypercube is

N/NG 0.25N log2 N, (3)

which for N 256 is much greater than for the multistage n-cube.The full multiring and generalized hypercube have rather complicated nodes. There exist static

switches with nodes of much lower complexity for their comparable number. For the same numberof nodes N = pr, for example, the p-ary r-cube (multidimensional grid) has mrC = logpN input-output channels at each node, and the cube-connected cycles [7] have for N = r2r nodes onlymcC = 2 3 input-output channels at each node. One should discriminate between the cube-connected cycles and cyclic cubes [8] which are close parametrically and quite distinct structurally.Together with the hypercube, they are the Cayley graphs [9] and have smaller diameter as comparedwith the hypercube having a close number of nodes.

No deterministic methods of conflictless realization of arbitrary permutations with given cycledelays are known for these switches. This gives rise to the question whether transmission by thecounter-forest schedules is applicable to them and what are the delays reached in this case. Thepresent author obtained a positive answer for the toral multidimensional grids (p-ary r-cubes forN = pr) [10]. This paper proposes a method of realization of arbitrary permutations on cube-connected cycles and examines its characteristics.

2. HYPERCUBE, CUBE-CONNECTED CYCLES,AND GROUP PERMUTATION ON THE HYPERCUBE

The ordinaryor binaryhypercube has N = 2r nodes. Each node has an r-digit binary

number x = xr1 . . . xi . . . x0, where xi [0, 1] and x =r1i=0

xi2i. Any two nodes in hypercube with

numbers differing in one and only one ith position are connected by a duplex channel regardedas that of the ith dimension (i [0, r 1]). A formal length 2i is assigned to the channel of ithdimension. The nodes with the numbers having ith positions xi and yi are connected by a channelfrom xi to yi of formal length 2i if and only if (xi + 1) mod 2 = yi.

We characterize the hypercube by a set of formal channel lengths

{SmG} = {1S = 1, 2S, . . . mGS},

where 1S < 2S < . . . < mGS, mG = r, and i+1S = 2i (0 i r 1).A route from the node with the number x = xr1 . . . xi . . . x0 to the node with the number

y = yr1 . . . yi . . . y0 has the decomposition (dr1, . . . , d0) for di [0, 1] if (xi + di) mod 2 = yi issatisfied for each i. The data element moving over the hypercube uses one cycle to pass the channelof the ith dimension if di = 1, and does not move along the channel of the ith dimension if di = 0.

The formal length d =r1i=0

di2i is assigned to the route with the decomposition (dr1, . . . , d0).

Passage of any data element along any route in the hypercube is defined by the route schedulecharacterized by the formal route length. It defines the sequence of passing the channels whoselengths are involved in the decomposition of this route and the numbers of cycles in which these

AUTOMATION AND REMOTE CONTROL Vol. 63 No. 9 2002

1508 PODLAZOV

Fig. 1. Three-dimensional cube-connected cycles. The channels of the original hypercube are shown by bold lines.

channels are passed. These cycles can be nonadjacent, that is, alternate with cycles where elementsstay in nodes without moving. These cycles are treated as passage of zero-length channel.

The method of conflictless realization of arbitrary permutation is based on using a static counter-forest schedule where any two route schedules coinciding in a cycle with nonzero length of the passedchannel coincide either in all preceding or all succeeding cycles. This schedule enables conflictlessrealization of arbitrary permutation in the number of cycles obeying (1) for p = 2. It is constructedas a direct Cartesian product of the initial and final unilateral schedules of much smaller size.Table 1 shows examples of such schedules for N = 256. For a greater number of nodes, examplescan be found in [26, 10]. In the initial schedule, the channels from the first half of the set SmG areused, and in the final schedule, those from the second half are used. The route schedules coincidingin a cycle coincide in all preceding cycles in the initial schedule and in all succeeding cycles in thefinal schedule.

The cube-connected cycles of dimensionality r with N = r2r nodes are obtained from theordinary 2r-node hypercube by replacing each node by a group of r nodes enumerated within eachgroup from [0, r 1] and connected by a unilateral ring channel (ring) or two counter-rings. Eachnode of any group is connected with a node of the same name (number) of another group bya duplex channel of the original hypercube whose number of dimension coincides with the nodenumber. Figure 1 depicts an example of three-dimensional cube-connected cycles where the nodesof each group are connected by a pair of counter-rings. In this case, each node has only three

Table 1. Unilateral schedules for the hypercube with N = 256 nodes

Initial schedule Final schedule

L \ T 1 2 3 4 5 6 7 8 T \ L0 01 1 16 162 2 32 323 1 2 32 16 484 4 64 645 4 1 16 64 806 2 4 64 32 967 4 1 2 32 16 64 1128 8 128 1289 8 1 16 128 144

10 2 8 128 32 16011 2 8 1 16 128 32 17612 4 8 128 64 19213 8 1 4 64 16 128 20814 2 8 4 64 128 32 22415 4 1 2 8 128 32 16 64 240L \ T 1 2 3 4 5 6 7 8 T \ L

AUTOMATION AND REMOTE CONTROL Vol. 63 No. 9 2002

ARBITRARY GROUP PERMUTATIONS 1509

input-output channels independently of the dimensionality of hypercube. If the nodes of a groupare connected by one ring, there are only two such channels.

Solution of the problem of arbitrary permutation on cube-connected cycles by necessity requiressolution of the problem of group arbitrary permutation on the ordinary hypercube. This becomesevident if each group of the nodes of cube-connected cycles is folded into a node of the ordinaryhypercube retaining the data elements contained in the nodes of the group. Therefore, we considera method of realizing group permutation on hypercube where each data element of any group istransmitted according to a conflictless schedule (Table 1) intended for the ordinary hypercube.Cycles of data element transmission from any node having the same names (numbers) are unitedin a hypercycle having the number of its component cycles. In any hypercycle, the data elementsare transmitted from any node in an arbi