Congestion and Dilation, Similarities and

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We survey general results on congestion and dilation, and their special cases ... mum load on the edges (communication links) of H and dilation is the maximum.
Congestion and Dilation, Similarities and Differences: a Survey A. R ASPAUD University of Bordeaux I, France ´ O. S YKORA Loughborough University, The United Kingdom I. V RT ’ O Slovak Academy of Sciences, Slovakia

Abstract We survey general results on congestion and dilation, and their special cases (cyclic) cutwidth and (cyclic) bandwidth, with the emphasis on the similarity and the duality of both parameters. Keywords Bandwidth, Congestion, Cutwidth, Embedding, Dilation

1 Introduction Recently diverse properties and invariants of interconnection networks (not only those of parallel machines) have been studied and a lot of interesting results have been shown (e.g. see [16, 36]). One of the important features of an interconnection network is its ability to efficiently simulate programs written for other architectures. Such a simulation problem can be mathematically formulated as a graph embedding. Informally, the graph embedding problem is to label the vertices of a “guest” graph (e.g. a communication graph of processes and relations between the processes) G by distinct vertices of a “host” graph (an interconnection network) H. The quality of the embedding corresponding to the efficiency of the simulation depends mainly on the time delay of the communication among processors. The delay is caused by sum of congestion and dilation [29]. Congestion is the maximum load on the edges (communication links) of H and dilation is the maximum distance in H between adjacent vertices (processes) in G.  This

work has been supported in part by grant No. 02/7007/20 of VEGA.

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There are many other motivations for studying congestions and dilations and, in particular, their special cases. The cutwidth (bandwidth) is the congestion (dilation) if the host graph is a path. The cutwidth problem naturally arises in VLSI layouts, when processing elements are placed on a line and wires are routed above the line [9, 34]. Cutwidth of a graph G also provides a lower bound for the product of the maximum degree of the graph and the ratio of queue and stack numbers used as basic mechanisms for laying out the graph in a linear order [22]. The bandwidth problem is equivalent to the question of how to permute rows and columns of the adjacency matrix of a graph such that all 1s are as close to the main diagonal as possible. The problem has a practical impact and has been known for about 50 years. The bandwidth problem for the hypercube was first studied in [21] in connection with error correcting codes. A number of upper and lower bounds are known which relate dilation to various graph invariants, however the exact values or even estimates are only known for a small number of classes of graphs. Most of them are surveyed in [11, 15, 28, 35, 36]. The results include mutual embeddings between meshes, trees, paths, hypecubes, and other typical graphs representing interconnection networks. Similar situations are found with congestion and cutwidth results. Survey works are in [12, 28]. Known results include complete binary trees [30], hypercubes [5, 37], products of complete graphs [33] and 2-dimensional meshes [41, 42]. Approximations of cutwidths of shuffle exchange graphs and related networks are in [4, 39] and an estimation of the cutwidth of mesh of trees is in [44]. The aim of this survey paper is not to describe all known result on the congestion and the dilation. We omit the results for specific graphs and concentrate on the general relation, bounds and inequalities for the studied parameters and their special cases stressing the similarity and duality between them. It turns out that many claims concerning the congestion have a counterpart claim for the dilation. The paper is organized as follows. The next section contains basic definitions. The section 3 describes the survey results divided into basic inequalities, lower bound arguments, upper bounds for product graphs, mutual relations and some complexity aspects. We conclude with a few open questions.

2 Basic Definitions and Notations First we introduce the concept of embedding. Let G1 = (VG1 ; EG1 ) and G2 = be graphs such that jVG1 j  jVG2 j. An embedding of G1 in G2 is a couple of mappings (φ; ψ) satisfying

(VG2 ; EG2 )

φ : VG1 ψ : EG1

! VG

2

is an injection

! f set of all

paths in G2 g;

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such that if uv 2 EG1 then ψ(uv) is a path between φ(u) and φ(v). Define the congestion of an edge e 2 EG2 under the embedding (φ; ψ) as cg(G1 ; G2 ; φ; ψ; e) = jf f

2 EG

1

: e 2 ψ( f )gj;

and the congestion of G1 in G2 as cg(G1 ; G2 ) = min max fcg(G1; G2 ; φ; ψ; e)g (φ;ψ) e2EG2 Define the average congestion of an embedding (φ; ψ) as acg(G1 ; G2 ; φ; ψ) =

1 jEG2 j uv2∑EG cg(G1; G2 ; φ; ψ; e) 2

and the average congestion of G1 in G2 as acg(G1 ; G2 ) = min facg(G1 ; G2 ; φ; ψ)g: (φ;ψ)

A special case of congestion of 2KjVG j ; which is the complete graph where each edge is doubled, in G is called edge forwarding index (for more details see [24]). Define the dilation of an edge e 2 EG1 under the embedding (φ; ψ) as dil (G1 ; G2 ; φ; ψ; e) = jψ(e)j and the dilation of G1 in G2 as dil (G1 ; G2 ) = min max fdil (G1 ; G2 ; φ; ψ; e)g: (φ;ψ) e2EG1 Define the average dilation of an embedding (φ; ψ) as adil (G1 ; G2 ; φ; ψ) =

1 jEG1 j e2∑EG dil (G1 ; G2 ; φ; ψ; e) 1

and the average dilation of G1 in G2 as adil (G1 ; G2 ) = min fadil (G1 ; G2 ; φ; ψ)g: (φ;ψ)

Let Pn and Cn denote the n vertex path and cycle, respectively. Let G = (V; E ) be an n vertex graph. Define the cutwidth of G as cw(G) = cg(G; Pn ); and the bandwidth of G as bd (G) = dil (G; Pn ):

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We will use special notations also for the closely related concepts of cyclic cutwidth and cyclic bandwidth: ccw(G) = cg(G; Cn ); cbd (G) = dil (G; Cn ): In a similar way as above we can define the average (cyclic) cutwidth and the average (cyclic) bandwidth. For a set A  VG let ∂e (A) denote the set of all edges having one end in A and the second end in VG A : = Ag ∂e (A) = fuv 2 EG : u 2 A; v 2

We call ∂e (A) the edge boundary of A. Similarly, for a set A  VG let ∂v (A) denote the set of all vertices from A which have some neighbouring vertices in VG A : ∂v (A) = fu 2 VG

A : uw 2 EG ; w 2 Ag

We call ∂v (A) the vertex boundary of A: The Cartesian product G1  G2 of graphs G1 and G2 is a graph on the vertex set VG1  VG2 ; in which (i; j); (r; s) are adjacent iff either i = r and js 2 EG2 or j = s and ir 2 EG1 : The Kronecker product G1 G2 of graphs G1 and G2 is defined as follows: V (G1 G2 ) = VG1  VG2 and E (G1  G2) = f(u; x)(v; y) : uv 2 EG1 and xy 2 EG2 g. Let ∆(G) be the maximal degree of G and ∆¯ be the average degree of G. Let diam(G) denote the diameter of G.

3 Relations, Bounds and Complexity Issues In this section we survey relations, general bounds for the dilation and the congestion, their special cases and some algorithmic aspects. We emphasize the similarities and the duality between the studied parameters.

3.1 Basic Inequalities We start with a simple but useful inequality: Lemma 1 If G1 is a subgraph of G2 and H2 is a subgraph of H1 obtained by deleting a subset of edges of H1 ; then cg(G1 ; H1 )  cg(G2 ; H2 ); dil (G1 ; H1 )  dil (G2 ; H2 ):

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Given three graphs G1 ; G2 and G3 : By embedding of G1 into G2 followed by an embedding of G2 into G3 we get an embedding of G1 into G3 . Then: Lemma 2 cg(G1 ; G3 )  cg(G1 ; G2 )cg(G2 ; G3 ); dil (G1 ; G3 )  dil (G1 ; G2 )dil (G2 ; G3 ): The above inequalities belong to folklore but they can be used efficiently in two directions: either as an upper bound argument for the embeddings of G1 into G3 or as a lower bound argument for embedding of G1 into G2 or G2 into G3 , provided that the other embeddings and the corresponding parameters are known. An intermediate consequence is: Corollary 1 ccw(G)  cw(G)  2ccw(G) cbd (G)  bd (G)  2cbd (G): For average congestion and dilation we have similar claims [17]. Lemma 3 acg(G1 ; G3 )  cg(G1 ; G2 )acg(G2 ; G3 ); adil (G1 ; G3 )  adil (G1 ; G2 )dil (G2 ; G3 ):

3.2 Lower Bound Arguments One of the first lower bound arguments described in [40] is based on the maximum degrees of the graphs and average degree of the graph G1 : Theorem 1 Let G1 and G2 be graphs. Then dil (G1 ; G2 ) 

adil (G1 ; G2 ) 

log(∆(G1 )) ; log(∆(G2 ))

log(∆¯ (G1 )) log(∆(G2 ))

3:

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Another lower bound argument from [40] is based on the “volume” consideration. Define the following quantities for any graph G and nonnegative integer k: Let VolG (v; k) be the set of all vertices in a distance at most k from v in G. Let VolG (k) = max VolG (v; k): v2G Theorem 2 Consider the optimal embedding of G1 in G2 with respect to the dilation. Let the function α : N ! N be such that, for all k 2 N ; α(k) is the smallest integer for which VolG2 (α(k))  VolG1 (k): Then dil (G1 ; G2 )  max

 α(k) 

k

k

:

The same argument can be slightly simplified and strengthened for the bandwidth of a graph G [13]. Theorem 3

 jV j H bd (G)  max

1 diam(H )

 ;

where the maximum is taken over any subgraph H of the graph G: This argument is called the local density argument. For example it was used in [2] to determine the bandwidth of caterpillars. We do not know on the similar bound as given in Theorem 2 for the cutwidth. But in [8] it was proved: Theorem 4

8 min fj∂ (A)jg 9 = < AjAjV=Gi e cg(G H )  max i : min AVH fj∂e (A)jg ; ;

jAj=i

The lower bound is tight for embedding of the hypercube in some special meshes. If we restrict ourselves to the cutwidth and the bandwidth then we obtain nice dual results [20, 21]. Theorem 5 Let G = (V; E ) be a graph. Then cw(G) = max minfj∂e (A)jg; i

AV jAj=i

bd (G)  max minfj∂v (A)jg: i

AV jAj=i

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Note the equality in the result for the cutwidth. These results demonstrate a very close relation of the cutwidth and bandwidth problem to the so called edge resp. vertex isoperimetric problem in graphs [3, 6], which is in fact the problem of finding minima for all i in the right hand sides in the above relations. The next lemma gives a simple but achievable lower bound argument for the cutwidth and the bandwidth. Lemma 4 cw(G)  ∆(G)=2; bd (G)  ∆(G)=2: We conclude this subsection by a volume based criterion for the average dilation from [17]. Lemma 5 Consider an optimal embedding of the k-vertex star G1 into a graph G2 . Define r = minfradius(H ) : H is a k=2 H

vertex induced subgraph of G2 g:

Then adil (G1 ; G2 )  r=4:

3.3 Upper Bounds for Product Graphs The lower bounds from the previous part can give good estimates for our parameters, especially for some product graphs. The upper bounds are usually obtained by the following inequalities. Lemma 6 For arbitrary graphs G1 = (V1 ; E1 ) and G2 = (V2 ; E2 ) cw(G1  G2 )  minfcw(G1 )jV2 j + cw(G2 ); cw(G2 )jV1 j + cw(G1 )g; bd (G1  G2 )  minfbd (G1)jV2 j; bd (G2 )jV1 jg: The proposition given below was proved independently in [23, 25]. Lemma 7 Let G1 ; G2 ; H1 ; and H2 be graphs. Then it holds: cg(G1  H1 ; G2  H2 )  maxfcg(G1 ; G2 ); cg(H1 ; H2 )g; dil (G1  H1 ; G2  H2)  maxfdil (G1 ; G2 ); dil (H1 ; H2 )g;

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A similar situation holds for the Kronecker product, which was firstly done in [23] and recently reproved in [27]. Lemma 8 Let G1 ; G2 ; H1 ; and H2 be graphs. Then cg(G1 H1 ; G2 H2 )  maxfcg(G1 ; G2 ); cg(H1 ; H2 )g; dil (G1 H1 ; G2 H2)  maxfdil (G1 ; G2 ); dil (H1 ; H2 )g;

3.4 Mutual Relations A general inequality between congestion and dilation was proved by [1]. Theorem 6 It holds cg(G1 ; G2 )  ∆(G2 )dil (G1 ;G2 ) ; Cutwidth and bandwidth are tied with the following relation: Lemma 9 cw(G) 

∆(G)bd (G) : 2

In [26] a sufficient condition for equality of bandwidth and cyclic bandwidth was shown. Let X and Y be two cycles which intersect in a path P of length at least 1. The sum of the cycles X ; Y is the cycle Z obtained from the graph X [ Y by deleting edges and inner vertices of P: For more cycles the sum is defined inductively. A set of cycles in a graph G is called basic if any cycle in G either belongs to the basic set or can be expressed as a sum of cycles of the basic set. Trivially, the set of all cycles is a basic set. Theorem 7 Suppose the set of all cycles in G of length at most jVG j=bd (G) is a basic set. Then cbd (G) = bd (G): There are several standard graphs which satisfy the above condition, including trees, meshes and hypercubes. However there is no sufficient condition for the equality of the cutwidth and the cyclic cutwidth. A completely different situation is with average congestion and average dilation. Both parameters are strongly related by the following well known equality: Lemma 10 Let G1 and G2 be graphs such that jVG1 j = jVG2 j: Then acg(G1 ; G2 )jEG2 j = adil (G1 ; G2 )jEG1 j;

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3.5 Complexity Issues To find an embedding with optimal congestions, dilations and other parameters is a difficult problem, even in simple cases. The bandwidth problem of an arbitrary graph was shown to be NP-hard [38]. The problem remains NP-hard even if G is restricted to the class of trees with ∆(G)  3. For any fixed k, the bd (G) = k problem always has a polynomial solution. For the decision version of the cyclic bandwidth the NP-completeness was proved in [31]. The cutwidth problem is NP-hard for general graphs, [18], but solvable in polynomial time for trees [45]. The average bandwidth which, according to Lemma 10, is equivalent to the average cutwidth is better known as the optimal linear arrangement, or wirelength problem. The problem is NP-hard [19] and solvable in polynomial time for trees [15, 43].

4 Conclusions In this paper we surveyed results on congestion and dilation, parameters which measures the quality of graph embeddings. We preferred general results to specific estimates for typical graphs. Our aim was to emphasize the similarity and duality of both parameters as well as to point out the places where they are quite different. Finally we mention a few open problems. Find a sufficient condition which guarantees equality of the cutwidth and the cyclic cutwidth. There are a few graphs for which equality holds, including trees [14, 32] and m  n meshes with m  n + 2, [42]. A similar question can be studied for the average bandwidth and the average cyclic bandwidth. So far the equality has only been proved for trees [7] and hypercubes [10].

References [1] F. S. Annexstein, M. Baumslag, and A. L. Rosenberg. Group action graphs and parallel architectures. SIAM Journal of Computing, 19:544–569, 1990. [2] S. F. Assmann, B. W. Peck, M. M. Syslo, and J. Zak. The bandwidth of caterpillars with hairs of length 1 and 2. SIAM J. Algebraic and Discrete Methods, 2:387-393, 1981. [3] B. Bollob´as. Combinatorics, Chapter 16., Isoperimetric Problems. Cambridge Uni. Press, Cambridge 1986.

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Proceedings in Informatics

[4] D. Barth, F. Pellegrini, A. Raspaud, and J. Roman. On bandwidth, cutwidth and quotient graphs. RAIRO Informatique, Th´eorique et Applications, 29:487–508, 1995. [5] A. Bel Hala. Congestion optimale du plongement de l’ypercube H (n) dans la chaineP(2n). RAIRO Informatique, Th´eorique et Applications 27:1–17, 1993. [6] S. L. Bezrukov. Edge isoperimetric problems on graphs. In Graph Theory and Combinatorial Biology, Bolyai Soc. Math. Stud. 7, (L. Lov´asz, A. Gyarf´as, G.O.H. Katona, A. Recski, L. Sz´ekely eds.), Budapest, 1999. [7] S. Bezrukov, and U. P. Schroeder. The cyclic wirelength of trees. Discr. Appl. Math., 87:275–277, 1998. [8] S. L. Bezrukov, J. D. Chavez, L. H. Harper, M. R¨ottger, and U.-P. Schroeder. The congestion of the n-cube layout on a rectangular grid. Dicrete Mathematics, 213:75–94, 2000. [9] S. N. Bhat, and F. T. Leighton. A framework for solving VLSI graph layout problems. Computer and System Science, 28:300-343, 1984. [10] Ching-Jung Guu. The circular wirelength problem for hypercubes. Ph.D. Thesis, University of California, Riverside, 1997. [11] P. Z. Chinn, J. Chv´atalov´a, A. K. Dewdney, and N. E. Gibbs. The bandwidth problem for graphs and matrices - a survey. Journal of Graph Theory, 6:223–254, 1982. [12] F. R. K. Chung. Labelings of graphs. In Selected Topics in Graph Theory 3, (L.Beineke and R.Wilson eds.), Academic Press, New York, 151–168, 1988. [13] J. Chv´atalov´a. Optimal labeling of a product of two paths. Discrete Mathematics, 11:249–253, 1975. [14] J. D. Chavez, and R. Trapp. The cyclic cutwidth of trees. Discrete Appl. Math. 87:25–32, 1998. [15] F. R. K. Chung On optimal linear arrangements of trees. Comput. Math. Appl., 10:43–60, 1984. [16] J. de Rumeur. Communications dans les r´eseaux de processeurs. Masson, Paris, 1994. [17] K. Diks, H. N. Djidjev, O. S´ykora, and I. Vrt’o. Edge separators of planar and outerplanar graphs with applications. J. of Algorithms, 14:258-279, 1993.

Congestion and Dilation

11

[18] F. Gavril. Some NP complete problems on graphs. In Proc. of the 11th Conf. on Information Sciences and Systems, Johns Hopkins University, Baltimore, MD, 91–95, 1977. [19] M. R. Garey, D. S. Johnson, and L. Stockmayer. Some simplified NP complete graph problems. Theoretical Computer Science, 1:237-267, 1976. [20] L. H. Harper. Optimal assignment of number to vertices. J. Soc. Ind. Appl. Math., 12:131–135, 1964. [21] L. H. Harper. Optimal numberings and isoperimetric problems on graphs. J. Combinatorial Theory, 1:385–393, 1966. [22] L. S. Heath, F. T. Leighton, and A. L. Rosenberg. Comparing queues and stacks as mechanisms for laying out graphs. SIAM J. Discrete Math., 5:398– 412, 1992. [23] U. Hendrik, and M. Stiebitz. On the bandwidth of graph products. EIK, 114–125, 1992. [24] M. C. Heydemann, J. C. Meyer, and D. Sotteau. On forwarding indices of networks. Discrete Applied Math., 23:103-123, 1989. [25] C.-T. Ho, and S. L. Johnson. Embedding meshes in boolean cubes by graph decomposition. Journal of Parallel and Distributed Computing, 8:325–339, 1990. [26] J. Hromkoviˇc, V. M¨uller, O. S´ykora, and I. Vrt’o. On embeddings in cycles. Information and Computation, 118:302-305, 1995. [27] T. H. Lai, and W. White. Embeddings pyramids in hypercubes. Research Report No. 41, CISRC, Ohio State University, Columbus, Ohio. [28] F. T. Leighton. Introduction to Parallel Algorithms and Architectures. Morgan Kaufmann, San Mateo, 1992. [29] F. T. Leighton, B. Maggs, and S. Rao. Universal packet routing algorithms. In Proc. of the 29th Annual IEEE Symposium of the Foundations of Computer Science, IEEE Press, 256–269, 1988. [30] T. Lengauer. Upper and lower bounds for the min-cut linear arrangement of trees. SIAM J. Algebraic and Discrete Methods, 3:99–113, 1982. [31] J. Y. T. Leung, O. Vornberger, and O. D. Witthoff. On some variants of the bandwidth minimization problem. SIAM J. Computing, 13:650–667, 1984. [32] Y. Lin, O. S´ykora, and I. Vrt’o. On cyclic cutwidths. manuscript, 1998.

12

Proceedings in Informatics

[33] J. H. Lindsey II. Assignment of numbers to vertices. American Mathematical Monthly, 7:508–516, 1964. [34] A. D. Lopez, and H. F. S. Law. A dense gate matrix layout method for MOS VLSI. IEEE Trans. Electron. Devices, 27:1671–1675, 1980. [35] Y.-L. Lai, and K. Williams. A survey of solved problems and applications on bandwidth edgesum, and profile of graphs. J. Graph Theory, 31:75–94, 1999. [36] B. Monien, and H. Sudborough. Embedding one interconnection network in another. Comput. Suppl., 7:257–282, 1990. [37] K. Nakano, W. Chen, T. Masuzawa, K. Hagihara, and N. Tokura. Cutwidth and bisection width of hypercube graph. IEICE Transactions, J73-A:856– 862, 1990, (in Japanese). [38] CH. H. Papadimitriou. The NP-completeness of the bandwidth minimization problem. Computing, 16:263–270, 1976. [39] A. Raspaud, O. S´ykora, and I. Vrt’o. Cutwidth of the de Bruijn graphs. RAIRO, 6:509–514, 1995. [40] A. Rosenberg, L. Snyder. Bounds on the cost of data encodings. Mathematical System Theory, 12:9–39, 1978. [41] J. D. Rolim, O. S´ykora, I. Vrt’o. Optimal cutwidth of meshes. In Proc. 21th Intl. Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science 1017, Springer Verlag, Berlin, 252–264, 1995. [42] H. Schr¨oder, O. S´ykora, I. Vrˇto. Cyclic cutwidths of the mesh. In SOFSEM’99, Lecture Notes in Computer Science 1725, Springer Verlag, Berlin, 443–452, 1999. [43] Y. Shiloach. A minimum linear arrangement algorithm for undirected trees. SIAM J. Comput., 8:15–32, 1979. [44] I. Vrt’o. Optimal cutwidth of the mesh of d-ary trees. In Proc. Europar’97, Lecture Notes in Computer Science 1300, Springer Verlag, Berlin, 242–245, 1997. [45] M. Yannakakis. A polynomial algorithm for the Min Cut Linear Arrangement of trees. J. ACM, 32:950–988, 1985. Andr´e Raspaud is with LaBRI, University of Bordeaux I, Cours de la liberation, 33405 Talence, France. E-mail: [email protected]

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Ondrej S´ykora is with the Department of Computer Science, Loughborough University, Loughborough, Leicestershire, LE11 3TU, The United Kingdom. E-mail: [email protected] Imrich Vrt’o is with the Department of Informatics, Institute of Mathematics, Slovak Academy of Sciences, D´ubravsk´a 9, 840 00 Bratislava, Slovak Republic. E-mail: [email protected]