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-JohnsonKnuth[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(K1)=0
or bw(K1)=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 K4 is immersed in the graph.
• K4 obstructs cutwidth 3
• Any arrangement of its vertices on a line
(Booth, Govindan, Langston, and
Ramachandramurthi [1992])
Examples
K4 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=K1,n
– B(G) ≥ χ(G)-1
• Attained when G=Kn
– B(G) ≥ n-1/2(2-[(2n-1)2-8e]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–McKee algorithm
• Algorithm to permute a symmetric matrix with small
bandwidth
– The reverse Cuthill–McKee algorithm (Alan
George)
• same algorithm but with the resulting index numbers
reversed.
• generally results in less fill-in than the CM ordering
Cuthill-McKee
Cuthill-McKee Ordering
Reverse Cuthill-McKee
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
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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. McKee. 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 11th 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
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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 MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/GraphBandwidth.html
West, Douglas Brent. Introduction to Graph Theory. Upper Saddle River, NJ: Prentice Hall, 2001.
Print.