The Congestion of n-cube Layout on a Rectangular Grid

Save this PDF as:

Size: px
Start display at page:

Transcription

1 The Congestion of n-cube Layout on a Rectangular Grid S.L. Bezrukov J.D. Chavez L.H. Harper M. Röttger U.-P. Schroeder Abstract We consider the problem of embedding the n-dimensional cube into a rectangular grid with 2 n vertices in such a way as to minimize the congestion, the maximum number of edges along any point of the grid. After presenting a short solution for the cutwidth problem of the n-cube (in which the n-cube is embedded into a path), we show how to extend the results to give an exact solution for the congestion problem. 1 Introduction Let G = (V, E) represent a graph with vertex set V and edge set E. We may think of G as representing the wiring diagram of an electronic circuit, with the vertices representing components and the edges representing wires connecting them. A (vertex) numbering of G is a function η : V 1, 2,..., V }, which is one-to-one (and therefore onto). A numbering may be thought of as an embedding of G into a linear chassis. The cutwidth of G, with respect to η, cw(g; η), is the maximum number of wires which pass any point on the linear chassis: cw(g; η) = max (v, w) E η(v) l < η(w)}. l Then, the cutwidth of G is cw(g) = min cw(g; η). η The cutwidth problem is to find the value of cw(g). While the cutwidth problem is, in general, NP-complete [2], the solution is known for the n-cube and for some other families of graphs, such as the product of complete graphs (see e.g. [5]). In this paper, we extend that problem Department of Mathematics and Computer Science, University of Paderborn, Germany Department of Mathematics, California State University, San Bernardino, CA Department of Mathematics, University of California, Riverside, CA

2 by taking the host graph to be a rectangular grid. We shall adopt the term congestion when considering grids as host graphs, and maintain the use of the term cutwidth when the host graph is a path. We denote by Q n the n-cube defined as the cartesian product of n edges. After presenting background material and a simple proof for the solution of the cutwidth problem for Q n in section 2, the main result of this paper is presented in section, where we give the solution of the congestion problem for the n-cube. 2 Background Given a graph, G, and a numbering, η, then for each l, 0 l V, define S l (η) = η 1 (1,..., l}). Thus, S l (η) is the set of the first l vertices of G to be numbered by η. For S V, define θ G (S) = (v, w) E v S, w S}, θ G (l) = min η θ G (S l (η)). For a fixed value of l, the problem of minimizing the value of θ G (S l (η)) over all numberings, η, can be thought of as a discrete isoperimetric problem. In fact, resulting isoperimetric theorems have proven quite useful. In [4] (see also []) it was shown that the solution of the isoperimetric problem for θ provides a lower bound for the wirelength problem for a graph. When G is the graph of an n-cube Q n, the lower bound is sharp, with the numbering, lex, corresponding to the lexicographic ordering of the vertices, providing θ Qn (l) = θ Qn (S l (lex)) for each l. 2.1 The cutwidth of the n-cube Given a graph, G, let η : V 1, 2,..., m}, where m = V, be a numbering of the vertices of G. Then, cw(g; η) can be thought of as the maximum number of wires which pass any point on the linear chassis. Let the point be between l and l + 1. Since every wire which has one end numbered less than or equal to l, and the other greater than or equal to l + 1 must pass this point, cw(g; η) = max 0 l m θ G(S l (η)). And so the cutwidth of G is defined by cw(g) = max 0 l m θ G(l). (1) Theorem 1 cw(q n ) = 2 n+1 2 if n is even 2 n+1 1 if n is odd. 2

3 Proof. Since we know that for Q n, the numbering given by the lexicographic ordering of the vertices minimizes θ Qn (S l ) for each l (see [4]), (1) implies cw(q n ) = max 0 l 2 n θ Q n (S l (lex)). The keys to computing cw(q n ) and related parameters are the following observations about θ. For a graph G and S V, θ G (S) = θ G (S c ), where S c = V \ S. Thus, θ G (l) = θ G ( V l). (2) That is, θ Qn (l) is symmetric about 2 n 1. Its maximum does not generally occur at 2 n 1, however, we need only look at l 2 n 1 in searching for the maximum. Furthermore, on Q n we have the following recursion. θ Qn (S l (lex)) = 2l + θqn 2(S l (lex)) if 0 l 2 n 2 2 n 1 + θ Qn 2 (S l 2 n 2(lex)) if 2 n 2 l 2 n 1. () For the recursion given in (), observe that its maximum value is achieved at some l from 2 n 2 to 2 n 1. To see that this is true, for each l, 0 l 2 n 2, let l = l + 2 n 2. Then we see that θ Qn (S l (lex)) = 2 n 1 + θ Qn 2 (S l 2 n 2(lex)) = 2 n 1 + θ Qn 2 (S l (lex)) 2l + θ Qn 2 (S l (lex)) = θ Qn (S l (lex)). From (2) and () we obtain the recurrence relation, 0 if n = 0 cw(q n ) = 1 if n = 1 2 n 1 + cw(q n 2 ) if n 2. Solving this recurrence we have the Theorem. Thus, cw(q n ) 4 2n 1 as n. 2.2 Example The wiring diagram that results from embedding Q 4 into a linear chassis (using the lexicographic numbering) is shown in Figure 1. Note that the cutwidth of Q 4 is 10, with that value being achieved at several points (but not at l = 2 ). In general, the maximum value of θ Qn (l) does not occur at l = 2 n 1. Where, then, does it occur? Let l M (n) = minl θ Qn (l) = max 0 l 2 n θ Q n (l)}.

4 Figure 1: Q 4 embedded in a linear chassis Then l M (n) = 0 if n = 0 1 if n = 1, and l M (n) 2 n 1 by (2). In fact, l M (n) = 2 n 2 + l M (n 2) by (). Therefore, Thus, l M (n) 2n as n. l M (n) = 2 n 1 if n is even 2 n +1 if n is odd. From this one sees that the set of l s where θ Qn (l) takes its maximum value, cw(q n ) 4 2n 1, approximates a Cantor set. Furthermore, if we let then #(n) = l θ Qn (l) = cw(q n )}, 0 if n = 0 #(n) = 1 if n = 1 if n = 2. The recursion () implies that for n > 2 the maximum of θ Qn (l) is achieved at some l, 2 n 2 < l < 2 n 1. Since θ Qn (l) is symmetric about 2 n 1, one has #(n) = 2 #(n 2). Therefore, #(n) = = n if n > 0 and even 2 n 1 2 if n odd 2 2 n if n > 0 and even n if n odd. The congestion of the n-cube In this section we consider the problem of minimizing the congestion of the n-cube. The congestion problem is of particular interest in rectilinear network layout design (see [1] for a nice summary). As with the cutwidth, the congestion, con(g : H), is the minimum over all η : V G V H of the maximum number of wires that pass any point on the (host) graph. Note that in addition to 4

5 arranging the vertices on the host graph (n! possibilities as in the linear case), the wires must be laid out on the rectangular grid in such a way as to minimize the number of wires along any point. Our strategy is to obtain a lower bound for con(g : H), based on the known solutions of isoperimetric problems, and then show that the lower bound is achieved. Let G and H be graphs and let η : V G V H be a one-to-one function. We assume that H is connected, so that θ H (l) 0 if 0 < l < n. Lemma 1 θ G (l) con(g : H) max 1 l<n θ H (l). Proof. Suppose that T is any subset of V H. Then θ H (T ) con(g : H) θ G (η 1 (T )), since each of the edges from η 1 (T ) to η 1 (T c ) must be assigned to a path from T to T c, which contains at least one edge counted by θ H (T ). Each such edge e V H can have at most con(g : H) edges of E G assigned to it, so the inequality must hold. For the left hand side of this inequality, the worst case is that T minimizes θ H for its cardinality, so θ H (l) con(g : H) = θ H (T ) con(g : H) θ G (η 1 (T )) θ G (l). We apply this result to calculate con(q n : F ), where F = P 2 n 1 P 2 n d, with d 2, n n d = n and n 1 n d. Theorem 2 con(q n : F ) = cw(q nd ). Proof. Denote n = n n d 1 and let l = 2 n 1 2 n n + ( 1) n n +1 2 n. Note that 2 n 2 l < 2 n 1 and l = a 2 n for some integer a. Consider the 2 n 1 2 n d 1 a subgrid T of F. One has θ F (l) θ F (T ) = 2 n. Also, by an argument similar to that for identity (), θ Qn (l) = 2 n n n if n d is odd 2 n n n +1 if n d is even. 5

6 Therefore, θ Qn (l) θ F (l) 1 2 n n n if n d is odd 2 n 2 n n n +1 if n d is even = n d 1 if n d is odd n d 1 if n d is even 2 n d +1 1 if n = d is odd 2 n d +1 2 if n d is even = cw(q nd ). Then, by Lemma 1, con(q n : F ) cw(q nd ). On the other hand, let η : Q n F equal the product of the numberings lex i : Q ni P 2 n i, i = 1,..., d, that solve their corresponding cutwidth problems. Running all wires along one dimension of F, we have con(q n : F ) max i cw(q ni ) and so the theorem. 4 Conclusion An important characteristic of the n-cube is that it is factorable (in many ways) as a product of lower dimensional cubes. One may wonder if the above results apply to all graphs that may be factored as products of subgraphs. The following example shows that that is not the case. Consider the product of the complete graphs K 4 and K 2, shown in Figure 2. This graph may be embedded into a 2 4 grid as shown in Figure. But con(k 4 K 2 : P 2 P 4 ) = < 4 = cw(k 4 ) b a Figure 2: K 4 K 2 Our results might be extended for solving the cutwidth or congestion problem for other families of graphs (with the host graph a path or a grid). Another possible direction for extension would be to solve the congestion problem for the n-cube with a different host graph, such as a cycle. References [1] F.R.K. Chung. Labelings of Graphs, Graph Theory, Academic Press,

7 a 4a 4b b 1a 2a 2b 1b Figure : K 4 K 2 embedded in P 2 P 4 [2] M.R. Garey and D.S. Johnson. Computers and Intractability, A Guide to the Theory of NP-Completeness, Freeman, San Francisco, [] L.H. Harper and J.D. Chavez. Global Methods of Combinatorial Optimization, to appear in The Encyclopedia of Mathematics, Cambridge University Press. [4] L.H. Harper. Optimal assignments of numbers to vertices, J. Society Industrial and Applied Mathematics 12 (1964), [5] K. Nakano. Linear Layouts of Generalized Hypercubes, Lecture Notes in Computer Science 790, Springer-Verlag,

On the Bandwidth of 3-dimensional Hamming graphs

On the Bandwidth of 3-dimensional Hamming graphs J. Balogh S.L. Bezrukov L. H. Harper A. Seress Abstract This paper presents strategies for improving the known upper and lower bounds for the bandwidth

The edge slide graph of the n-dimensional cube

The edge slide graph of the n-dimensional cube Howida AL Fran Institute of Fundamental Sciences Massey University, Manawatu 8th Australia New Zealand Mathematics Convention December 2014 Howida AL Fran

On the k-path cover problem for cacti

On the k-path cover problem for cacti Zemin Jin and Xueliang Li Center for Combinatorics and LPMC Nankai University Tianjin 300071, P.R. China zeminjin@eyou.com, x.li@eyou.com Abstract In this paper we

Available online at ScienceDirect. Procedia Computer Science 74 (2015 ) 47 52

Available online at www.sciencedirect.com ScienceDirect Procedia Computer Science 74 (05 ) 47 5 International Conference on Graph Theory and Information Security Fractional Metric Dimension of Tree and

Finding the Shortest Move-Sequence in the Graph-Generalized 15-Puzzle is NP-Hard

Finding the Shortest Move-Sequence in the Graph-Generalized 15-Puzzle is NP-Hard Oded Goldreich Abstract. Following Wilson (J. Comb. Th. (B), 1975), Johnson (J. of Alg., 1983), and Kornhauser, Miller and

Every tree contains a large induced subgraph with all degrees odd

Every tree contains a large induced subgraph with all degrees odd A.J. Radcliffe Carnegie Mellon University, Pittsburgh, PA A.D. Scott Department of Pure Mathematics and Mathematical Statistics University

Best Monotone Degree Bounds for Various Graph Parameters

Best Monotone Degree Bounds for Various Graph Parameters D. Bauer Department of Mathematical Sciences Stevens Institute of Technology Hoboken, NJ 07030 S. L. Hakimi Department of Electrical and Computer

Single machine parallel batch scheduling with unbounded capacity

Workshop on Combinatorics and Graph Theory 21th, April, 2006 Nankai University Single machine parallel batch scheduling with unbounded capacity Yuan Jinjiang Department of mathematics, Zhengzhou University

SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH

31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,

Lecture 6: Approximation via LP Rounding

Lecture 6: Approximation via LP Rounding Let G = (V, E) be an (undirected) graph. A subset C V is called a vertex cover for G if for every edge (v i, v j ) E we have v i C or v j C (or both). In other

Two General Methods to Reduce Delay and Change of Enumeration Algorithms

ISSN 1346-5597 NII Technical Report Two General Methods to Reduce Delay and Change of Enumeration Algorithms Takeaki Uno NII-2003-004E Apr.2003 Two General Methods to Reduce Delay and Change of Enumeration

All trees contain a large induced subgraph having all degrees 1 (mod k)

All trees contain a large induced subgraph having all degrees 1 (mod k) David M. Berman, A.J. Radcliffe, A.D. Scott, Hong Wang, and Larry Wargo *Department of Mathematics University of New Orleans New

Fundamentals of Media Theory

Fundamentals of Media Theory ergei Ovchinnikov Mathematics Department an Francisco tate University an Francisco, CA 94132 sergei@sfsu.edu Abstract Media theory is a new branch of discrete applied mathematics

On the Relationship between Classes P and NP

Journal of Computer Science 8 (7): 1036-1040, 2012 ISSN 1549-3636 2012 Science Publications On the Relationship between Classes P and NP Anatoly D. Plotnikov Department of Computer Systems and Networks,

Midterm Practice Problems

6.042/8.062J Mathematics for Computer Science October 2, 200 Tom Leighton, Marten van Dijk, and Brooke Cowan Midterm Practice Problems Problem. [0 points] In problem set you showed that the nand operator

HOMEWORK #3 SOLUTIONS - MATH 3260

HOMEWORK #3 SOLUTIONS - MATH 3260 ASSIGNED: FEBRUARY 26, 2003 DUE: MARCH 12, 2003 AT 2:30PM (1) Show either that each of the following graphs are planar by drawing them in a way that the vertices do not

THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER

THE 0/1-BORSUK CONJECTURE IS GENERICALLY TRUE FOR EACH FIXED DIAMETER JONATHAN P. MCCAMMOND AND GÜNTER ZIEGLER Abstract. In 1933 Karol Borsuk asked whether every compact subset of R d can be decomposed

10.1 Integer Programming and LP relaxation

CS787: Advanced Algorithms Lecture 10: LP Relaxation and Rounding In this lecture we will design approximation algorithms using linear programming. The key insight behind this approach is that the closely

Planar embeddability of the vertices of a graph using a fixed point set is NP-hard

Planar embeddability of the vertices of a graph using a fixed point set is NP-hard Sergio Cabello institute of information and computing sciences, utrecht university technical report UU-CS-2003-031 wwwcsuunl

Math 443/543 Graph Theory Notes 4: Connector Problems

Math 443/543 Graph Theory Notes 4: Connector Problems David Glickenstein September 19, 2012 1 Trees and the Minimal Connector Problem Here is the problem: Suppose we have a collection of cities which we

CS 598CSC: Combinatorial Optimization Lecture date: 2/4/2010

CS 598CSC: Combinatorial Optimization Lecture date: /4/010 Instructor: Chandra Chekuri Scribe: David Morrison Gomory-Hu Trees (The work in this section closely follows [3]) Let G = (V, E) be an undirected

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs

Approximated Distributed Minimum Vertex Cover Algorithms for Bounded Degree Graphs Yong Zhang 1.2, Francis Y.L. Chin 2, and Hing-Fung Ting 2 1 College of Mathematics and Computer Science, Hebei University,

2.3 Scheduling jobs on identical parallel machines

2.3 Scheduling jobs on identical parallel machines There are jobs to be processed, and there are identical machines (running in parallel) to which each job may be assigned Each job = 1,,, must be processed

M.S. Ibrahim 1*, N. Maculan 2 and M. Minoux 3

Pesquisa Operacional (2014) 34(1): 117-124 2014 Brazilian Operations Research Society Printed version ISSN 0101-7438 / Online version ISSN 1678-5142 www.scielo.br/pope A NOTE ON THE NP-HARDNESS OF THE

Mathematics for Algorithm and System Analysis

Mathematics for Algorithm and System Analysis for students of computer and computational science Edward A. Bender S. Gill Williamson c Edward A. Bender & S. Gill Williamson 2005. All rights reserved. Preface

A 2-factor in which each cycle has long length in claw-free graphs

A -factor in which each cycle has long length in claw-free graphs Roman Čada Shuya Chiba Kiyoshi Yoshimoto 3 Department of Mathematics University of West Bohemia and Institute of Theoretical Computer Science

Offline 1-Minesweeper is NP-complete

Offline 1-Minesweeper is NP-complete James D. Fix Brandon McPhail May 24 Abstract We use Minesweeper to illustrate NP-completeness proofs, arguments that establish the hardness of solving certain problems.

Lecture 16 : Relations and Functions DRAFT

CS/Math 240: Introduction to Discrete Mathematics 3/29/2011 Lecture 16 : Relations and Functions Instructor: Dieter van Melkebeek Scribe: Dalibor Zelený DRAFT In Lecture 3, we described a correspondence

Introduction to Relations

CHAPTER 7 Introduction to Relations 1. Relations and Their Properties 1.1. Definition of a Relation. Definition: A binary relation from a set A to a set B is a subset R A B. If (a, b) R we say a is related

On an anti-ramsey type result

On an anti-ramsey type result Noga Alon, Hanno Lefmann and Vojtĕch Rödl Abstract We consider anti-ramsey type results. For a given coloring of the k-element subsets of an n-element set X, where two k-element

CSL851: Algorithmic Graph Theory Semester I Lecture 1: July 24

CSL851: Algorithmic Graph Theory Semester I 2013-2014 Lecture 1: July 24 Lecturer: Naveen Garg Scribes: Suyash Roongta Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer: These notes have

Complexity and Completeness of Finding Another Solution and Its Application to Puzzles

yato@is.s.u-tokyo.ac.jp seta@is.s.u-tokyo.ac.jp Π (ASP) Π x s x s ASP Ueda Nagao n n-asp parsimonious ASP ASP NP Complexity and Completeness of Finding Another Solution and Its Application to Puzzles Takayuki

Notes on Matrix Multiplication and the Transitive Closure

ICS 6D Due: Wednesday, February 25, 2015 Instructor: Sandy Irani Notes on Matrix Multiplication and the Transitive Closure An n m matrix over a set S is an array of elements from S with n rows and m columns.

JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS. Received December May 12, 2003; revised February 5, 2004

Scientiae Mathematicae Japonicae Online, Vol. 10, (2004), 431 437 431 JUST-IN-TIME SCHEDULING WITH PERIODIC TIME SLOTS Ondřej Čepeka and Shao Chin Sung b Received December May 12, 2003; revised February

5. GRAPHS ON SURFACES

. GRPHS ON SURCS.. Graphs graph, by itself, is a combinatorial object rather than a topological one. But when we relate a graph to a surface through the process of embedding we move into the realm of topology.

COLORED GRAPHS AND THEIR PROPERTIES

COLORED GRAPHS AND THEIR PROPERTIES BEN STEVENS 1. Introduction This paper is concerned with the upper bound on the chromatic number for graphs of maximum vertex degree under three different sets of coloring

TRAINING A 3-NODE NEURAL NETWORK IS NP-COMPLETE

494 TRAINING A 3-NODE NEURAL NETWORK IS NP-COMPLETE Avrim Blum'" MIT Lab. for Computer Science Cambridge, Mass. 02139 USA Ronald L. Rivest t MIT Lab. for Computer Science Cambridge, Mass. 02139 USA ABSTRACT

On Integer Additive Set-Indexers of Graphs

On Integer Additive Set-Indexers of Graphs arxiv:1312.7672v4 [math.co] 2 Mar 2014 N K Sudev and K A Germina Abstract A set-indexer of a graph G is an injective set-valued function f : V (G) 2 X such that

Large induced subgraphs with all degrees odd

Large induced subgraphs with all degrees odd A.D. Scott Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, England Abstract: We prove that every connected graph of order

Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs

CSE599s: Extremal Combinatorics November 21, 2011 Lecture 15 An Arithmetic Circuit Lowerbound and Flows in Graphs Lecturer: Anup Rao 1 An Arithmetic Circuit Lower Bound An arithmetic circuit is just like

On the Union of Arithmetic Progressions

On the Union of Arithmetic Progressions Shoni Gilboa Rom Pinchasi August, 04 Abstract We show that for any integer n and real ɛ > 0, the union of n arithmetic progressions with pairwise distinct differences,

Graph Theory Problems and Solutions

raph Theory Problems and Solutions Tom Davis tomrdavis@earthlink.net http://www.geometer.org/mathcircles November, 005 Problems. Prove that the sum of the degrees of the vertices of any finite graph is

Planar embeddability of the vertices of a graph using a fixed point set is NP-hard

Planar embeddability of the vertices of a graph using a fixed point set is NP-hard Sergio Cabello institute of information and computing sciences, utrecht university technical report UU-CS-2003-031 wwwcsuunl

Euler Paths and Euler Circuits

Euler Paths and Euler Circuits An Euler path is a path that uses every edge of a graph exactly once. An Euler circuit is a circuit that uses every edge of a graph exactly once. An Euler path starts and

Clique coloring B 1 -EPG graphs

Clique coloring B 1 -EPG graphs Flavia Bonomo a,c, María Pía Mazzoleni b,c, and Maya Stein d a Departamento de Computación, FCEN-UBA, Buenos Aires, Argentina. b Departamento de Matemática, FCE-UNLP, La

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold

Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various

Class One: Degree Sequences

Class One: Degree Sequences For our purposes a graph is a just a bunch of points, called vertices, together with lines or curves, called edges, joining certain pairs of vertices. Three small examples of

(Vertex) Colorings. We can properly color W 6 with. colors and no fewer. Of interest: What is the fewest colors necessary to properly color G?

Vertex Coloring 2.1 33 (Vertex) Colorings Definition: A coloring of a graph G is a labeling of the vertices of G with colors. [Technically, it is a function f : V (G) {1, 2,...,l}.] Definition: A proper

Graphical degree sequences and realizations

swap Graphical and realizations Péter L. Erdös Alfréd Rényi Institute of Mathematics Hungarian Academy of Sciences MAPCON 12 MPIPKS - Dresden, May 15, 2012 swap Graphical and realizations Péter L. Erdös

Connectivity and cuts

Math 104, Graph Theory February 19, 2013 Measure of connectivity How connected are each of these graphs? > increasing connectivity > I G 1 is a tree, so it is a connected graph w/minimum # of edges. Every

On Data Recovery in Distributed Databases

On Data Recovery in Distributed Databases Sergei L. Bezrukov 1,UweLeck 1, and Victor P. Piotrowski 2 1 Dept. of Mathematics and Computer Science, University of Wisconsin-Superior {sbezruko,uleck}@uwsuper.edu

6.896 Probability and Computation February 14, Lecture 4

6.896 Probability and Computation February 14, 2011 Lecture 4 Lecturer: Constantinos Daskalakis Scribe: Georgios Papadopoulos NOTE: The content of these notes has not been formally reviewed by the lecturer.

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH. 1. Introduction

COMBINATORIAL PROPERTIES OF THE HIGMAN-SIMS GRAPH ZACHARY ABEL 1. Introduction In this survey we discuss properties of the Higman-Sims graph, which has 100 vertices, 1100 edges, and is 22 regular. In fact

8.1 Min Degree Spanning Tree

CS880: Approximations Algorithms Scribe: Siddharth Barman Lecturer: Shuchi Chawla Topic: Min Degree Spanning Tree Date: 02/15/07 In this lecture we give a local search based algorithm for the Min Degree

1 Introduction. Dr. T. Srinivas Department of Mathematics Kakatiya University Warangal 506009, AP, INDIA tsrinivasku@gmail.com

A New Allgoriitthm for Miiniimum Costt Liinkiing M. Sreenivas Alluri Institute of Management Sciences Hanamkonda 506001, AP, INDIA allurimaster@gmail.com Dr. T. Srinivas Department of Mathematics Kakatiya

Math 55: Discrete Mathematics

Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What

RAMSEY FOR COMPLETE GRAPHS WITH DROPPED CLIQUES

RAMSEY FOR COMPLETE GRAPHS WITH DROPPED CLIQUES JONATHAN CHAPPELON, LUIS PEDRO MONTEJANO, AND JORGE LUIS RAMÍREZ ALFONSÍN Abstract Let K [k,t] be the complete graph on k vertices from which a set of edges,

Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 20

CS 70 Discrete Mathematics and Probability Theory Fall 009 Satish Rao, David Tse Note 0 Infinity and Countability Consider a function (or mapping) f that maps elements of a set A (called the domain of

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs

Discrete Mathematics & Mathematical Reasoning Chapter 10: Graphs Kousha Etessami U. of Edinburgh, UK Kousha Etessami (U. of Edinburgh, UK) Discrete Mathematics (Chapter 6) 1 / 13 Overview Graphs and Graph

ON THE COMPLEXITY OF THE GAME OF SET. {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu

ON THE COMPLEXITY OF THE GAME OF SET KAMALIKA CHAUDHURI, BRIGHTEN GODFREY, DAVID RATAJCZAK, AND HOETECK WEE {kamalika,pbg,dratajcz,hoeteck}@cs.berkeley.edu ABSTRACT. Set R is a card game played with a

Finite Sets. Theorem 5.1. Two non-empty finite sets have the same cardinality if and only if they are equivalent.

MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we

Generalizing the Ramsey Problem through Diameter

Generalizing the Ramsey Problem through Diameter Dhruv Mubayi Submitted: January 8, 001; Accepted: November 13, 001. MR Subject Classifications: 05C1, 05C15, 05C35, 05C55 Abstract Given a graph G and positive

Rational exponents in extremal graph theory

Rational exponents in extremal graph theory Boris Bukh David Conlon Abstract Given a family of graphs H, the extremal number ex(n, H) is the largest m for which there exists a graph with n vertices and

Partitioning edge-coloured complete graphs into monochromatic cycles and paths

arxiv:1205.5492v1 [math.co] 24 May 2012 Partitioning edge-coloured complete graphs into monochromatic cycles and paths Alexey Pokrovskiy Departement of Mathematics, London School of Economics and Political

The Open University s repository of research publications and other research outputs

Open Research Online The Open University s repository of research publications and other research outputs The degree-diameter problem for circulant graphs of degree 8 and 9 Journal Article How to cite:

A Note on Maximum Independent Sets in Rectangle Intersection Graphs

A Note on Maximum Independent Sets in Rectangle Intersection Graphs Timothy M. Chan School of Computer Science University of Waterloo Waterloo, Ontario N2L 3G1, Canada tmchan@uwaterloo.ca September 12,

How Many Lions Can One Man Avoid?

Rheinische Friedrich-Wilhelms-Universität Bonn Institut für Informatik I Florian Berger Ansgar Grüne Rolf Klein How Many Lions Can One Man Avoid? Technical Report 006 November 007 Abstract A pride of

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS

COUNTING INDEPENDENT SETS IN SOME CLASSES OF (ALMOST) REGULAR GRAPHS Alexander Burstein Department of Mathematics Howard University Washington, DC 259, USA aburstein@howard.edu Sergey Kitaev Mathematics

Mathematical Induction

Mathematical Induction Victor Adamchik Fall of 2005 Lecture 2 (out of three) Plan 1. Strong Induction 2. Faulty Inductions 3. Induction and the Least Element Principal Strong Induction Fibonacci Numbers

About the inverse football pool problem for 9 games 1

Seventh International Workshop on Optimal Codes and Related Topics September 6-1, 013, Albena, Bulgaria pp. 15-133 About the inverse football pool problem for 9 games 1 Emil Kolev Tsonka Baicheva Institute

The number of generalized balanced lines

The number of generalized balanced lines David Orden Pedro Ramos Gelasio Salazar Abstract Let S be a set of r red points and b = r + 2δ blue points in general position in the plane, with δ 0. A line l

(Knight) 3 : A Graphical Perspective of the Knight's Tour on a Multi-Layered Chess Board

Bridgewater State University Virtual Commons - Bridgewater State University Honors Program Theses and Projects Undergraduate Honors Program 5-2-2016 (Knight) 3 : A Graphical Perspective of the Knight's

Random trees. Jean-François Le Gall. Université Paris-Sud Orsay and Institut universitaire de France. IMS Annual Meeting, Göteborg, August 2010

Random trees Jean-François Le Gall Université Paris-Sud Orsay and Institut universitaire de France IMS Annual Meeting, Göteborg, August 2010 Jean-François Le Gall (Université Paris-Sud) Random trees Göteborg

Some Results on 2-Lifts of Graphs

Some Results on -Lifts of Graphs Carsten Peterson Advised by: Anup Rao August 8, 014 1 Ramanujan Graphs Let G be a d-regular graph. Every d-regular graph has d as an eigenvalue (with multiplicity equal

Odd induced subgraphs in graphs of maximum degree three

Odd induced subgraphs in graphs of maximum degree three David M. Berman, Hong Wang, and Larry Wargo Department of Mathematics University of New Orleans New Orleans, Louisiana, USA 70148 Abstract A long-standing

vertex, 369 disjoint pairwise, 395 disjoint sets, 236 disjunction, 33, 36 distributive laws

Index absolute value, 135 141 additive identity, 254 additive inverse, 254 aleph, 466 algebra of sets, 245, 278 antisymmetric relation, 387 arcsine function, 349 arithmetic sequence, 208 arrow diagram,

On Total Domination in Graphs

University of Houston - Downtown Senior Project - Fall 2012 On Total Domination in Graphs Author: David Amos Advisor: Dr. Ermelinda DeLaViña Senior Project Committee: Dr. Sergiy Koshkin Dr. Ryan Pepper

Nan Kong, Andrew J. Schaefer. Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA

A Factor 1 2 Approximation Algorithm for Two-Stage Stochastic Matching Problems Nan Kong, Andrew J. Schaefer Department of Industrial Engineering, Univeristy of Pittsburgh, PA 15261, USA Abstract We introduce

UNIVERSAL JUGGLING CYCLES. Fan Chung 1 Department of Mathematics, University of California, San Diego, La Jolla, CA Ron Graham 2.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 7(2) (2007), #A08 UNIVERSAL JUGGLING CYCLES Fan Chung 1 Department of Mathematics, University of California, San Diego, La Jolla, CA 92093 Ron

Max-Min Representation of Piecewise Linear Functions

Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 1, 297-302. Max-Min Representation of Piecewise Linear Functions Sergei Ovchinnikov Mathematics Department,

Graphs without proper subgraphs of minimum degree 3 and short cycles

Graphs without proper subgraphs of minimum degree 3 and short cycles Lothar Narins, Alexey Pokrovskiy, Tibor Szabó Department of Mathematics, Freie Universität, Berlin, Germany. August 22, 2014 Abstract

CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992

CS268: Geometric Algorithms Handout #5 Design and Analysis Original Handout #15 Stanford University Tuesday, 25 February 1992 Original Lecture #6: 28 January 1991 Topics: Triangulating Simple Polygons

Approximating the entropy of a 2-dimensional shift of finite type

Approximating the entropy of a -dimensional shift of finite type Tirasan Khandhawit c 4 July 006 Abstract. In this paper, we extend the method used to compute entropy of -dimensional subshift and the technique

Applied Algorithm Design Lecture 5

Applied Algorithm Design Lecture 5 Pietro Michiardi Eurecom Pietro Michiardi (Eurecom) Applied Algorithm Design Lecture 5 1 / 86 Approximation Algorithms Pietro Michiardi (Eurecom) Applied Algorithm Design

Week 5 Integral Polyhedra

Week 5 Integral Polyhedra We have seen some examples 1 of linear programming formulation that are integral, meaning that every basic feasible solution is an integral vector. This week we develop a theory

Completely Positive Cone and its Dual

On the Computational Complexity of Membership Problems for the Completely Positive Cone and its Dual Peter J.C. Dickinson Luuk Gijben July 3, 2012 Abstract Copositive programming has become a useful tool

1.3 Induction and Other Proof Techniques

4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.

. 0 1 10 2 100 11 1000 3 20 1 2 3 4 5 6 7 8 9

Introduction The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive integer We say d is a

SETS, RELATIONS, AND FUNCTIONS

September 27, 2009 and notations Common Universal Subset and Power Set Cardinality Operations A set is a collection or group of objects or elements or members (Cantor 1895). the collection of the four

A Lower Bound for Area{Universal Graphs. K. Mehlhorn { Abstract. We establish a lower bound on the eciency of area{universal circuits.

A Lower Bound for Area{Universal Graphs Gianfranco Bilardi y Shiva Chaudhuri z Devdatt Dubhashi x K. Mehlhorn { Abstract We establish a lower bound on the eciency of area{universal circuits. The area A

Cycles and clique-minors in expanders

Cycles and clique-minors in expanders Benny Sudakov UCLA and Princeton University Expanders Definition: The vertex boundary of a subset X of a graph G: X = { all vertices in G\X with at least one neighbor

ON THE TREE GRAPH OF A CONNECTED GRAPH

Discussiones Mathematicae Graph Theory 28 (2008 ) 501 510 ON THE TREE GRAPH OF A CONNECTED GRAPH Ana Paulina Figueroa Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria,

On the crossing number of K m,n

On the crossing number of K m,n Nagi H. Nahas nnahas@acm.org Submitted: Mar 15, 001; Accepted: Aug 10, 00; Published: Aug 1, 00 MR Subject Classifications: 05C10, 05C5 Abstract The best lower bound known

Discrete Mathematics Problems

Discrete Mathematics Problems William F. Klostermeyer School of Computing University of North Florida Jacksonville, FL 32224 E-mail: wkloster@unf.edu Contents 0 Preface 3 1 Logic 5 1.1 Basics...............................

Steiner Tree NP-completeness Proof

Steiner Tree NP-completeness Proof Alessandro Santuari May 7, 2003 Abstract This document is an eercise for the Computational Compleity course taken at the University of Trento. We propose an NP-completeness

Lecture 7: NP-Complete Problems

IAS/PCMI Summer Session 2000 Clay Mathematics Undergraduate Program Basic Course on Computational Complexity Lecture 7: NP-Complete Problems David Mix Barrington and Alexis Maciel July 25, 2000 1. Circuit

NP-Completeness. CptS 223 Advanced Data Structures. Larry Holder School of Electrical Engineering and Computer Science Washington State University

NP-Completeness CptS 223 Advanced Data Structures Larry Holder School of Electrical Engineering and Computer Science Washington State University 1 Hard Graph Problems Hard means no known solutions with

Boulder Dash is NP hard

Boulder Dash is NP hard Marzio De Biasi marziodebiasi [at] gmail [dot] com December 2011 Version 0.01:... now the difficult part: is it NP? Abstract Boulder Dash is a videogame created by Peter Liepa and