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SIAM J. on Matrix Analysis and Applications

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1980

Volume 1, Issue 4, pp. 363-462


Dynamic-Programming Algorithms for Recognizing Small-Bandwidth Graphs in Polynomial Time

James B. Saxe

SIAM. J. on Algebraic and Discrete Methods 1, pp. 363-369 (7 pages) | Cited 17 times

Online Publication Date: August 02, 2006

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In this paper we investigate the problem of testing the bandwidth of a graph: Given a graph, $G$, can the vertices of $G$ be mapped to distinct positive integers so that no edge of $G$ has its endpoints mapped to integers which differ by more than some fixed constant, $k$? We exhibit an algorithm to solve this problem in $O ( f ( k )N^{k + 1} )$ time, where $N$ is the number of vertices of $G$ and $f ( k )$ depends only on $k$. This result implies that the “Bandwidth $\overset{?}{\leqq} k$” problem is not NP-complete (unless P = NP) for any fixed $k$, answering an open question of Garey, Graham, Johnson, and Knuth. We also show how the algorithm can be modified to solve some other problems closely related to the “Bandwidth $\overset{?}{\leqq} k$” problem.

An $O ( ( n\log p )^2 )$ Algorithm for the Continuous $p$-Center Problem on a Tree

R. Chandrasekaran and A. Tamir

SIAM. J. on Algebraic and Discrete Methods 1, pp. 370-375 (6 pages) | Cited 7 times

Online Publication Date: August 02, 2006

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This paper considers the problem of locating $p$ facilities on a tree network in order to minimize the maximum of the distances of the points on the network to their respective nearest facilities. An $O ( ( n\log p )^2 )$ algorithm for a tree network with $n$ nodes is presented.

The Erdös-Ko-Rado Theorem for Integer Sequences

Peter Frankl and Zoltán Füredi

SIAM. J. on Algebraic and Discrete Methods 1, pp. 376-381 (6 pages) | Cited 4 times

Online Publication Date: August 02, 2006

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For positive integers $n,k,t$ we investigate the problem how many integer sequences $( a_1 ,a_2 , \cdots ,a_n )$ we can take, such that $1\leqq a_i \leqq k$ for $1\leqq i\leqq n$, and any two sequences agree in at least $t$ positions. This problem was solved by Kleitman (J. Combin. Theory, 1 (1966), pp. 209–214) for $k = 2$, and by Berge (in Hypergraph Seminar, Columbus, Ohio (1972), Springer-Verlag, New York, 1974) for $t = 1$. We prove that for $t\geqq 15$ the maximum number of such sequences is $k^{n - t} $ if and only if $k\geqq t + 1$.

On Additive Bases and Harmonious Graphs

R. L. Graham and N. J. A. Sloane

SIAM. J. on Algebraic and Discrete Methods 1, pp. 382-404 (23 pages) | Cited 18 times

Online Publication Date: August 02, 2006

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This paper first considers several types of additive bases. A typical problem is to find $n_\gamma ( k )$, the largest $n$ for which there exists a set $\{ 0 = a_1 < a_2 < \cdots < a_k \}$ of distinct integers modulo $n$ such that each $r$ in the range $0\leqq r \leqq n - 1$ can be written at least once as $r \equiv a_i + a_j $ (modulo $n$) with $i < j$. For example, $n_\gamma ( 8 ) = 24$, as illustrated by the set {0, 1, 2, 4, 8, 13, 18, 22}. The other problems arise if at least is changed to at most, or $i < j$ to $i\leqq j$, or if the words modulo $n$ are omitted. Tables and bounds are given for each of these problems. Then a closely related graph labeling problem is studied. A connected graph with $n$ edges is called harmonious if it is possible to label the vertices with distinct numbers (modulo $n$) in such a way that the edge sums are also distinct (modulo $n$). Some infinite families of graphs (odd cycles, ladders, wheels, $ \cdots $) are shown to be harmonious while others (even cycles, most complete or complete bipartite graphs, $ \cdots $) are not. In fact most graphs are not harmonious. The function $n_\gamma ( k )$ is the size of the largest harmonious subgraph of the complete graph on $k$ vertices.

On Unimodality for Linear Extensions of Partial Orders

F. R. K. Chung, P. C. Fishburn, and R. L. Graham

SIAM. J. on Algebraic and Discrete Methods 1, pp. 405-410 (6 pages)

Online Publication Date: August 02, 2006

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R. Rivest has recently proposed the following intriguing conjecture: Let $x^* $ denote an arbitrary fixed element in an $n$-element partially ordered set $P$, and for each $k$ in $\{ 1,2, \cdots ,n \}$ let $N_k $ be the number of order-preserving maps from $P$ onto $\{ 1,2, \cdots ,n \}$ that map $x^* $ into $k$. Then the sequence $N_1 , \cdots ,N_n $ is unimodal. This note proves the conjecture for the special case in which $P$ can be covered by two linear orders. It also generalizes this result for $P$ that have disjoint components, one of which can be covered by two linear orders.

Obtaining Specified Irreducible Polynomials over Finite Fields

Solomon W. Golomb

SIAM. J. on Algebraic and Discrete Methods 1, pp. 411-418 (8 pages) | Cited 1 time

Online Publication Date: August 02, 2006

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In numerous applications, it is necessary to find an irreducible polynomial $f( x )$ of degree $n$ over $GF( q )$ whose roots are primitive $d$th roots of unity. (Here $d$ must divide $q^n - 1$.) Let $\alpha$ be one such root. A direct method is to write \[ f ( x ) = \prod\limits_{i = 0}^{n - 1} {\left( x - \alpha^{q^i } \right)} = \sum\limits_{j = 0}^n {( - 1 )} ^j C_j x^{n - j} ,\] where $C_0 = 1$ and all $C_j $ are in $GF ( q )$. Explicitly, $C_j $ is the sum of all powers of $\alpha $ whose exponents, written as $n$-digit numbers in base $q$, look like binary numbers of weight $j$. Formulas for the number of such polynomials $f ( x )$ are given, several computational shortcuts exploiting properties of cyclotomic polynomials are noted, and numerous illustrative examples are presented.

Totally Nonnegative, $M$-, and Jacobi Matrices

Mordechai Lewin

SIAM. J. on Algebraic and Discrete Methods 1, pp. 419-421 (3 pages) | Cited 7 times

Online Publication Date: August 02, 2006

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It is shown among other results that a nonsingular $M$-matrix is a Jacobi matrix if and only if its inverse is totally nonnegative and it is a normal Jacobi matrix if and only if its inverse is oscillatory.
This is an extension of a previous result of Markham [Proc. Amer. Math. Soc., 161 (1912), pp. 326–330].

The Elimination Matrix: Some Lemmas and Applications

Jan R. Magnus and H. Neudecker

SIAM. J. on Algebraic and Discrete Methods 1, pp. 422-449 (28 pages) | Cited 36 times

Online Publication Date: August 02, 2006

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Two transformation matrices are introduced, $L$ and $D$, which contain zero and unit elements only. If $A$ is an arbitrary $( n,n )$ matrix, $L$ eliminates from vec$A$ the supradiagonal elements of $A$, while $D$ performs the inverse transformation for symmetric$A$. Many properties of $L$ and $D$ are derived, in particular in relation to Kronecker products. The usefulness of the two matrices is demonstrated in three areas of mathematical statistics and matrix algebra: maximum likelihood estimation of the multivariate normal distribution, the evaluation of Jacobians of transformations with symmetric or lower triangular matrix arguments, and the solution of matrix equations.

Decomposing a Permutation into Two Large Cycles: An Enumeration

Edward A. Bertram and Victor K. Wei

SIAM. J. on Algebraic and Discrete Methods 1, pp. 450-461 (12 pages)

Online Publication Date: August 02, 2006

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Let $c_{l,m,\tau }^{( n )} $ denote the number of ways a permutation $\tau $ can be expressed as the product of an $l$-cycle and an $m$-cycle, all in the symmetric group on $n$ symbols. In 1972, the first author gave a necessary and sufficient condition on $l$ such that $c_{l,i,\tau }^{( n )} > 0$ for every even permutation $\tau $. In 1978, G. Boccara gave a necessary and sufficient condition on $l,m,$ and $\tau $ such that $c_{l,m,\tau }^{( n )} > 0$. More recently, D. W. Walkup developed a recursion for $c_{n,n,\tau }^{( n )} $. In this paper, we show how to recursively calculate the values of $c_{n,n - i,\tau }^{( n )} $. Theorem 1 states that $c_{n,n - 1,\tau }^{( n )} = 2 \cdot ( n - 2 )$! for every odd $\tau $. Theorem 2 exhibits $c_{n + 1,n - i,\sigma}^{( n + 1 )} $ as a linear, combination (with easily obtained integral coefficients) of a specified set of $c_{n,n - i,\tau }^{( n )} $. Applications include a method to evaluate, by inverting an integral triangular matrix, all values in {$c_{n,n,\tau }^{( n )} :\tau $ has exactly $k$ disjoint cycles}, for arbitrary $k\leqq n$.

Acknowledgment of Priority: On the Order of Random Channel Networks

A. Meir, J. W. Moon, and J. R. Pounder

SIAM. J. on Algebraic and Discrete Methods 1, pp. 462-462 (1 page)

Online Publication Date: August 02, 2006

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