BandwidthCutwidth Matt Huddleston CS 594 Graph Theory Thursday
Bandwidth/Cutwidth Matt Huddleston CS 594: Graph Theory Thursday, March 27, 14
Definitions • When the vertices of a graph G are numbered with distinct integers, the dilation is the maximum difference between integers assigned to adjacent vertices.
More Definitions • The bandwidth B(G) of a graph G is the minimum dilation of a numbering of G. – Equivalently, it is the minimum matrix bandwidth among all possible adjacency matrices of graphs isomorphic to G. • A bandwidth numbering of G is a proper numbering f such that B(G) = Bf(G) – A proper numbering that achieves B(G) (West, 2001, p. 390)
More Definitions • The cutwidth of G = (V, E) is defined to be the maximum number of edges from E that must be cut between consecutive vertices in any linear layout of V.
Cutwidth Example (Graph: Martí, Pantrigo, Duarte, and Pardo. [2013])
Example
The Bandwidth Minimization Problem (BMP) • Finding the minimum bandwidth is NPhard (Papadimitriou [1976]) – Even for simple graphs • NP-Hard for trees • Also, NP-Hard for trees that are caterpillars with hair length 3. (Garey-Graham-Johnson. Knuth[1978]) • O(n log n) when hair length is 1 or 2
Example
BMP History • Originated in the late 50’s and early 60’s in many ways – Structural engineers tried to analyze steel frameworks by computer manipulation of thier ‘Structural’ matrices. – at Jet Propulsion Laboratory in Pasadena. harper and hales were trying to construct the codes to minimize the max and avg absolute errors. (Shikare and Waphare [2004])
More BMP History • In the 60’s Korfhage began work on the BMP • Shortly after Harary made the BMP public at a conference in Prague • Problem received notable attention until Papadimitriou found that the BMP for graphs and matrices is NP-Complete (Shikare and Waphare [2004])
The Cutwidth Minimization Problem (CMP) • Find a linear layout of a graph so the maximum linear cut is minimized. • In 1977 Gavril showed that this is a NPHard problem
BMP & CMP • Associated with many optimization problems in circuit layout • BMP – Standard cell design – Network addressing • CMP – – VLSI design network communications automatic graph drawings information retrieval
Examples/Facts • Bandwidth of an arbitrary tree cannot be determined • Ando, Kaneko and Gervacio gave the following bound. – If T is a tree with k pendent vertices, then B(T) ≤ |k/2| (ceiling on k/2)
Examples/Facts • The bandwidth of the singleton graph is not defined, but the conventions bw(K 1)=0 or bw(K 1)=1 are sometimes adopted. (Miller [1988]) • If H is a subgraph of G, then B(H) ≤ B(G) – An optimal labeling of G is also a labeling of H.
Examples • B(Pn ) = ? • A nontrivial graph has bandwidth 1 – Iff it can be ordered so that it’s nonconsecutive vertices are adjacent – Making all components paths
Examples • B(Cn ) = ? – Cn has n edges – there are only n-1 consecutive pairs among the numbers 0, . . , n-1. – Some adjacent pair of vertices of Cn must be labeled with non consecutive integers – B(Cn) = 2
Example 4 2 (Gross, Yellen [])
Bandwidth Special Cases (Weisstein)
Examples • Cutwidth approximation in linear time – Bounded width – Finite (unknown) list of obstruction tests – Most fundamental test is to determine whether K 4 is immersed in the graph. • K 4 obstructs cutwidth 3 • Any arrangement of its vertices on a line (Booth, Govindan, Langston, and Ramachandramurthi [1992])
Examples K 4 immersion (Booth, Govindan, Langston, and Ramachandramurthi [1992])
Bounds on the Bandwidth • Graphs that have similar values of a certain parameter may have bandwidth arbitrarily far apart. – This shows that bandwidth is an independent parameter in its own right. – Bounds can still be important in certain graphs.
Bounds on Bandwidth • For any graph G, – B(G) ≥ [Δ(G)/2] • Attained when G=K 1, n – B(G) ≥ χ(G)-1 • Attained when G=Kn – B(G) ≥ n-1/2(2 -[(2 n-1)2 -8 e]1/2) (Dewdney) • For this n and e denote the number of edges and vertices in G, respectively. • Attained when G=Kn
More Bounds on Bandwidth • Local density bound (Chung [1988]) • Every numbering of G contains a subgraph of G • On every subgraph H – Two numbers differing by at least n(H) – 1 are used – Some edge on a path between corresponding vertices has dilation at least n(H) – 1 divided by distance between
More Bounds on Bandwidth B(G) > maxk min|S|=k |δS| • For all k, some set S must be the first k vertices in the optimal numbering of G • Bandwidth must be at least |δS| – Vertex among δS that has the least label has an edge dilation at least |δS| to its neighborhood above S. (Harper [1966])
Other Applications • Matrix handling – The Cuthill–Mc. Kee algorithm • Algorithm to permute a symmetric matrix with small bandwidth – The reverse Cuthill–Mc. Kee algorithm (Alan George) • same algorithm but with the resulting index numbers reversed. • generally results in less fill-in than the CM ordering
Cuthill-Mc. Kee Ordering Reverse Cuthill-Mc. Kee Ordering
Examples • Bandwidth Coloring Problem – This problem generalizes the classical vertex coloring problem. • Let G=(V, E) be a graph with a strictly positive integer weight dij associated to each edge (i, j) ∈ E. • A k-coloring c of G labels each vertex i ∈ V with an integer c(i) ∈ {1, 2, . . . , k} (called color) in such a way that |c(i) - c(j)| ≥ dij for all (i, j) ∈ E. • The bandwidth coloring problem (BCP) consists of finding a k coloring with the smallest value of k.
Other Applications • University of Ottawa published “Generalized Gene Adjacencies, Graph Bandwidth, and Clusters in Yeast Evolution” – Present a “parameterized definition of gene clusters that allows us to control emphasis placed on conserved order within a cluster” – This turns out to be closely related to the bandwidth of a graph
Homework 1. What is the bandwidth of the following graph:
Homework • 2. Compute the bandwidth of Kn.
Homework • 3. What is the bandwidth of the following graph:
References • • • Ando, Kiyoshi, Atsusi Kaneko, and Severino Gervacio. "The Bandwidth of a Tree with K Leaves Is at Most. " Discrete Mathematics 150. 1 -3 (1996): 403 -06. Print. Beineke, Lowell W. , and Robin J. Wilson. Selected Topics in Graph Theory 3. London: Academic, 1988. Print. Booth, H. D. ; Govindan, R. ; Langston, M. A. ; Ramachandramurthi, S. , "Cutwidth approximation in linear time, " VLSI, 1992. , Proceedings of the Second Great Lakes Symposium on , vol. , no. , pp. 70, 73, 28 -29 Feb 1992 Chung F. R. K. , Labellings of graphs. In Selected Topics in Graph Theory, Vol. 3. (ed. L. W. Beinke and R. J. Wilson) Acad. Press (1988), 151 -168. Cuthill, E. , and J. Mc. Kee. Reducing the Bandwidth of Sparse Symmetric Matrices. Ft. Belvoir: Defense Technical Information Center, 1969. Print. Garey, M. R. , R. L. Graham, D. S. Johnson, and D. E. Knuth. "Complexity Results for Bandwidth Minimization. " SIAM Journal on Applied Mathematics 34. 3 (1978): 477. Print. Gavril, F. Some NP-complete problems on graphs. In Proceedings of the 11 th conference on information Sciences and Systems, 91 -95, 1977. Gross, Jonathan L. , and Jay Yellen. Graph Theory and Its Applications. Boca Raton: Chapman & Hall/CRC, 2006. Print. Harper L. J. , Optimal numberings and isoperimetric problems on graphs. J. Comb. Th. 1 (1966), 385 -393.
More References • • Martí, Rafael, Juan J. Pantrigo, Abraham Duarte, and Eduardo G. Pardo. "Branch and Bound for the Cutwidth Minimization Problem. " Computers & Operations Research 40. 1 (2013): 137 -49. Print. Miller, Z. "A Linear Algorithm for Topological Bandwidth with Degree-Three Trees. " SIAM J. Comput. 17, 1018 -1035, 1988. Qian Zhu; Adam, Z. ; Choi, V. ; Sankoff, D. , "Generalized Gene Adjacencies, Graph Bandwidth, and Clusters in Yeast Evolution, " Computational Biology and Bioinformatics, IEEE/ACM Transactions on , vol. 6, no. 2, pp. 213, 220, April-June 200 Papadimitriou, Ch. H. "The NP-Completeness of the Bandwidth Minimization Problem. " Computing 16. 3 (1976): 263 -70. Print. R. R. Korfhage, Numberings of the vertices of graphs, Computer Science Department, Technical Report #5, Purdue University, Lafayette, IN (1966) Shikare, M. M. , and B. N. Waphare. Combinatorial Optimization. New Delhi: Narosa Pub. House, 2004. Print. Weisstein, Eric W. "Graph Bandwidth. " From Math. World--A Wolfram Web Resource. http: //mathworld. wolfram. com/Graph. Bandwidth. html West, Douglas Brent. Introduction to Graph Theory. Upper Saddle River, NJ: Prentice Hall, 2001. Print.
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