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Information Processing Letters 23 (1986) 5-9 North-Holland 20 July 1986 L O N G E D G E S IN T H E L A Y O U T S O F S H U F F L E - E X C H A N G E A N D C U B E - C O N N E C T E D C Y C L E S G R A P H S Ferng-Ching L I N and W e i - K u a n S H I H

Department of Computer Science and Information Engineering, College of Engineering, National Taiwan Unioersity, Taipei, Taiwan, Republic of China

Communicated by K. Mehlhorn Received March 1985

Revised 4 September 1985

A direct method is devised to prove, without information-theoretic arguments, the ~(N2/log2N) wire area lower bound for the shuffle-exchange and cube-connected cycles graphs. We further show the high occurrence of long edges in two ways: (1) In any layout, there are f~(N/log N) edges whose lengths are at least N/32 log2N. (2) The edges whose lengths are at least N/64 log2N occupy fi(N2/log2N) wire area.

Keywords: Graph layout, wire area, long edges, lower bound, shuffle-exchange, cube-connected cycles, path-edge, communica- tion power of edge, VLSI

1. Introduction

In VLSI computation, if a solution to some p r o b l e m is in the form of a c o m m u n i c a t i o n graph of processors, an efficient layout is desired to i m p l e m e n t the graph. Minirni7.irtg wire area is a critical concern d u e to the fact that layouts con- suming a larger a m o u n t of chip area are m o r e expensive to fabricate and less reliable. Speed of course is a n o t h e r critical factor in chip perfor- mance. Long wires raise p r o p a g a t i o n delays which can slow d o w n the clock a n d hence reduce the t h r o u g h p u t of the system. These are two of the major VLSI layout problems w h i c h have stimu- lated considerable interest in theoretical study [1]. F o r the formal model of V L S I graph layouts, we shall a d o p t the simple a n d widely accepted grid model [6,7]. Layouts are a s s u m e d to be on rectangular grids formed by horizontal and verti- cal grid lines which are spaced apart by unit intervals. The nodes of a graph are located only at the intersections of grid l i n e s - - t h e grid points. Edges are routed as wires t h r o u g h the grid lines to

c o n n e c t nodes. A n y two edges are n o t allowed to overlap for any distance, a n d an edge cannot overlap any n o d e which is n o t an e n d - n o d e of that edge. The layout area is defined to be the n u m b e r of all grid points in the grid. The wire area, however, is the c o u n t of those grid points covered by the edges.

T h e shuffle-exchange [5] a n d cube-connected cycles [4] graphs are two pre-eminent structures for parallel c o m p u t a t i o n because of their eco- n o m i c interconnection patterns. In them, the h a r d w a r e cost a n d efficiency are traded off in search of the best c o m p r o m i s e for a contemplated range of applications. T h o m p s o n [6] first showed that any layout of a graph which computes an N - p o i n t Fourier transform in T steps requires f ~ ( N 2 / T 2) chip area. This implies that any layout o f the N - n o d e shuffle-exchange or cube-connected cycles graph requires f~(N2/log2N) chip area. For the proofs, see also [7]. Leighton [3] then proved an analogous b u t stronger lower b o u n d statement for the wire area. H e also p r o v e d that any graph which c o m p u t e s an N - p o i n t Fourier transform 0020-0190/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland) 5

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Volume 23, Number 1 I N F O R M A T I O N PROCESSING LE'Iq'ERS 20 July 1986

must have a wire which crosses f i ( N / T 2) other wires. This, in particular, means that any layout of the N-node shuffle-exchange or cube-connected cycles graph contains a wire of f~(N/log2N) length.

All the lower b o u n d results mentioned above are indirectly derived through information-theo- retic (area-time tradeoff) arguments. In this paper, we shall devise a method to prove these lower bounds directly. The basic idea is roughly as fol- lows. If we separate some or all of the nodes of the graph into two parts, between them there must be at least a certain amount of communicating paths. According to the interconnection pattern of the graph, removing one edge can only destroy some communicating paths, hence there must be at least a certain number of edges connecting these two parts. By using this kind of information, we are not only able to derive wire area lower bounds but also lower bounds on the n u m b e r of long edges. In fact, we strengthen Leighton's result by showing that there are f~(N/log N) edges which are at least N / 3 2 log2N long. Furthermore, we show that the wire area occupied by the edges whose lengths are at least N / 6 4 log2N is as large (up to a constant) as the whole wire area.

2. Communication power of edges

In order to extract the communication ability of a given graph, we shall consider 'path-edges' which are paths properly selected from the origi- nal graph. The concept of 'path-edge' will become explicit when we deal with those two particular graphs in the next section. We need a few defini- tions for describing the communication power of edges. 'Edges' used in this section really are 'path-edges' when applied to concrete examples in the next section.

Definition 2.1. The diameter D of a given graph is

the smallest integer such that, for any two nodes i and j, i can be connected to j in at most D steps.

Definition 2.2. Let e be any edge, 1 ~< d ~< D, 1 ~< h d. A communication circle can be defined as the

set of node pairs

C(e, d, h ) = { (i, j ) l i and j are nodes such that there is a d-step path from i to j, with e as the hth step in the path}.

Definition 2.3. If e is an edge, we define the

pairing set of e as

P(e) = {(i, j)Ji and j are nodes such that there is a path from i to j through e and the number of steps in the path is at most D}.

To serve our purpose, we should constrain the graphs to be of bounded degree. In this paper, we are only interested in the case that the in-degree and out-degree of each node are bounded by 2, although the results obtained here can be ex- tended naturally. The following easy lemma can be used to bound from above the size of com- munication circles and pairing sets.

Lemma 2.4. I f the diameter is D and the degree bound is 2, then, for any e, 1 ~< d ~< D, 1 ~< h ~< d,

(a) I C(e, d, h) I ~< 2 a - 1, (b) I P(e) I ~ D(2 D - 1).

Proof. (a) In C(e, d, h), let m count the starting nodes with multiplicity and n count the ending nodes with multiplicity. It is clear that m ~< 2 h-1 and n ~< 2 d- h. So, IC(e, d, h) l ~ m n ~ 2h-12 d - h = 2 d-1. (b) Since a ( e ) = [,.J C(e, d, h), l < d ~ < D , l < h < d we have D D

IP(e) I ~< E d2d-a ~< D E 2d-a

d = l d = l

~< D(2D-- 1). []

If we purposely choose a set of nodes S to be observed, we can define the communication power of an edge relative to that set as the number of communicating pairs through that edge with both starting and ending nodes in S. Also, the diameter,

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Volume 23, Number 1 INFORMATION PROCESSING LETTERS 20 July 1986 if necessary, should be reduced to fit the situation

of the set of observed nodes.

i ° 2P %1

Definition 2.5. Suppose S is a set of observed nodes and e is any edge in the graph. We define the c o m m u n i c a t i o n power of e relative to S as Ps(e) = { (i, j ) l i , j ~ S and (i, j) ~ P(e) }, a n d define

R s = max IPs(e)I.

e

3. Shuffle-exchange and cube-connected cycles

The shuffle-exchange graph consists of N = 2" nodes as shown for n = 3 in Fig. 1. Each n o d e is associated with a unique n-bit binary n u m b e r . N o d e i is linked to j via a shuffle edge (uni-direc- tional d a s h e d line) if j is the left cyclic shift of i. T w o nodes i and j are linked via an exchange edge (bi-directional solid line) if i and j differ only in the rightmost bit. According to the interconnec- tion pattern, any two nodes can be c o n n e c t e d through a p a t h of at most 2n = 2 log N steps. ] This is n o t good enough, and we should c o m b i n e edges to lower the diameter somewhat. If we con- sider the shuffle-then-exchange and shuffle-only p a t h s as the 'path-edges', we arrive at a new graph whose diameter is log N only. We shall observe all nodes in the graph, so Ps(e) = P(e) for any e, and hence R s ~< N log N by L e m m a 2.4(b).

T h e cube-connected cycles graph consists of N = n log n nodes, organized as log n ranks of n nodes each. One example for n = 2 3 is shown in Fig. 2. Let us denote the ith node on the rth rank by p~. N o d e P~i is linked to node Pr+l,i via a cycle edge (uni-directional dashed line), where r is taken as an integer m o d log n. N o d e p,~ is also linked to po o n the same rank via a cross edge (bi-direc- tional solid line) if i a n d j only differ in the rth bit from the right. We can select any rank, say rank 0, as the set of observed nodes S. A n y two nodes in S can be connected through a path of at m o s t 2 log n steps. Again, this is n o t good enough, a n d we should consider the cycle-then-cross a n d cycle-only

Fig. 1. Shuffle-exchange graph.

Rank 2 Rank I Rank 0 ,' ! ' ' I I I t ' i ' ~-"L>.,~'-~-~ I I ' , _ . . - ~ ~ ~ "{ i " r ~'!" I t I ] I ~ I ~, I ~, I ' , I I I I I I d I I I ~ , i . L . - - i ~ , ~ l,..k-'-~..k ~ 2 . r - r % L ~ L~ ) L V ~ V L Y LY L Y ~ Fig. 2. Cube-connected cycles graph.

paths as the 'path-edges' and reduce the diameter to log n. It is n o t hard to see that, for any edge e, Ps(e) = C(e, log n, h) for some h. Therefore, from L e m m a 2.4(a), we know that R s < 2 l°g"-I = ½n.

4. Wire area and long edges

We will use a " # " sign partition technique due to Leighton [3] to prove wire area lower b o u n d s a n d lower b o u n d s o n long edges. W e observe the d i a g r a m in Fig. 3, where a given g r a p h is laid out arbitrarily. We slice it vertically by two boundaries b 1 a n d b 2 with a single j o g each, so that

{ISI

observed n o d e s are to the left o f b a, a n d the same n u m b e r to the right of b 2. Similarly, we divide the layout horizontally by another two boundaries, with ~

I SI

observed nodes above and below the top a n d b o t t o m boundaries. We assume, without loss of generality, that the width w of the center rectangle is as great as the height.

b, b W Isl18 IsllB IS l18 h Isl/8

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Volume 23, Number 1 I N F O R M A T I O N PROCESSING LETTERS 20 July 1986

Lemma 4.1. w >~ IS 12/32Rs •

Proof. The number of observed nodes in the center rectangle is at least ½1S [, and there are at least

¼ I SI

in the rest area. So, there are at least

1_ 21SI × ¼ 1 S 1 = 1 1 S I 2 node pairs across the pe-

rimeter of the center rectangle. By the definition of R s we know that the perimeter should cut at least

IS I2/8Rs

edges. Hence,

4w>t I S I 2 / 8 R s or w>~ I S I 2 / 3 2 R s • [] Lemma 4.2. The wire area of any layout is ~2( I SI 4/R%).

Proof. Any vertical boundary between bl and b2 must cut at least (I

IS I × -~ IS

[ ) / R s = IS 12/64Rs edges. Consequently, the occupied wire area be- tween b~ and b 2 is at least

w( IS [ 2 / 6 4 R s ) = IS 14/(32 x 64R2). []

at least one edge whose length is at least w / D . But such an edge can communicate at most R s node pairs. We thus conclude that there are at least (-~

IS I × ~ IS

I ) / R s = IS [ 2/64Rs edges whose lengths are at least w / D . []

Theorem 4.5. Any layout of N-node shuffle-ex-

change or cube-connected cycles graph contains

f ~ ( N / l o g N ) edges whose lengths are at least N / 3 2 log2N.

Proof. The proof immediately follows from Lem- mas 4.1 and 4.4. []

For both graphs, if we take the sum of the lengths of the long edges, we find that the wire area occupied by the long edges is fl(n2/log3N), which is slightly smaller than the lower bound of the Whole wire area by a factor of log N. Nonethe- less, we can match up these two bounds by consid- ering half-long edges also.

Theorem 4.3. Any layout of the N-node shuffle-ex-

change or cube-connected cycles graph occupies

~2 (N 2/log 2 N) wire area.

Lemma 4.6. In any layout, the wire area occupied

by the edges whose lengths are at least w / 2 D is

f~( IS 14/REs). Proof. For the shuffle-exchange graph, as dis-

cussed in the previous section, we have R s N log N, where IS I = N. By Lemma 4.2, we get the desired wire area lower bound. For the cube-connected cycles graph, IS1 = n and R s ½n, therefore the wire area, by Lemma 4.2, is f~(n2), i.e., f~(N2/log2N). []

As regards Fig. 3 again, an observed node left to ba can be connected to any observed node right

t o b E by a path whose length is at least w. The

average edge length in the path is at least w / D if the number of steps is not greater than D. So we are sure that there must be one edge in the path whose length is at least w / D . Such an edge is considered to be long.

Lemma 4.4. In any layout, there are

(ISI2/Rs)

edges whose lengths are at least w / D .

Proof. In any path connecting an observed n o d e left to b 1 to an observed node right to b2, there is

Proof. Use the same layout partition as before. In any path connecting an observed node left to b 1 and an observed node right to b 2, if the total length of the path is L (>~ w) and there are y ( < D) edges whose lengths are less than w / 2 D , then the length sum of the edges whose lengths are at least

w / 2 D >/L - y w / 2 D >i L - y L / 2 D = L ( D + ( D - y)/2D)>~½L.

Recall that in the proof of Lemma 4.2 we estimate the whole wire area ( =

2(IS 14/R%))

by this kind of paths. Since at least half of the length of these paths is occupied by those long and half-long edges, we have proven the lemma. []

Theorem 4.7. In any layout of N-node shuffle-ex-

change or cube-connected cycles graph, the wire area occupied by the edges whose lengths are at least

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Volume 23, Number 1 INFORMATION PROCESSING LETI'ERS 20 July 1986

5. Concluding remarks

A n o t h e r popular interconnection-economic structure is the butterfly (or unfold shuffle-ex- change) graph [5], which is illustrated in Fig. 4. Unlike the cube-connected cycles graph, its cycle edges, instead of wrapped, are ended at the extra top rank. Besides, the cycle-then-cross and cycle- only paths in the cube-connected cycles graph are now considered as edges instead of 'path-edges' in the butterfly graph. Without altering the matter of layout complexity, we can view the vertical-up edges as bi-directional, and consider each vertical- up path from Plogn+l,i t o P0i as a 'path-edge'. If we observe rank 0, the relative diameter is log n + 1 and R s ~< 2 l°gn+1-1 = n. As a consequence, all the lower bounds previously obtained for the cube- connected cycles graph also apply to the butterfly graph.

The layout depicted in Fig. 4 is already chip area optimal for the butterfly graph. Chip area optimal layouts for the shuffle-exchange and cube-connected cycles graphs can be found in [2] and [4] respectively. As mentioned in [3], the wire area is worth minimizing because chips with lower

Ran k 3

Ronk 2

Ronk I

Rank 0

Fig. 4. Butterfly graph.

wire density will be less likely to be ruined by localized random errors. Unfortunately, the wire area is usually as large (up to a constant0 as the chip area. We redo the proof by baring down the nature of the interconnection patterns to avoid using any information-theoretic argument.

Along this line of proving process, we establish some lower bounds for the long edges. Our results indicate that there are somehow two very long edges which occupy at least a constant portion of the whole chip area. Drivers, which cost extra area, must be added to the long edges to either synchronize or speed up the system. Since long edges are 'almost everywhere', it will surely take a lot of engineering efforts to put these graphs on chips.

References

[1] S.N. Bhatt and F.T. Leighton, A framework for solving VLSI graph layout problems, J. Comput. System Sci. 28 (2) (1984) 300-343.

[2] D. Kleitman, F.T. Leighton, M. Lepley and G.L. Miller, An asymptotically optimal layout for the shuffle-exchange graph, J. Comput. System Sci. 26 (3) (1983) 339-361. [3] F.T. Leighton, New lower bound techniques for VLSI,

Math. System Thex)ry 17 (1) (1984) 47-70.

[4] F.P. Preparata and J. VuiUemin, The cube-connected cycles: A versatile network for parallel computation, Comm. ACM 24 (5) (1981) 300-309.

[5] H.S. Stone, Parallel processing with the perfect shuffle, IEEE Trans. Comput. C-20 (2) (1971) 153-161.

[6] C.D. Thompson, A complexity theory for VLSI, Ph.D. Thesis, Computer Science Dept., Carnegie-Mellon Univ., 1980.

[7] J.D. Ullman, Computational Aspects of VLSI (Computer Science Press, Rockville, MD, 1984).

數據

Fig.  1.  Shuffle-exchange graph.
Fig.  4.  Butterfly graph.

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