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2004

Volume 17, Issue 4, pp. 521-685


Lower Bounds on the Broadcasting and Gossiping Time of Restricted Protocols

Michele Flammini and Stéphane Pérennès

SIAM J. Discrete Math. 17, pp. 521-540 (20 pages) | Cited 3 times

Online Publication Date: August 01, 2006

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In this paper we extend the technique provided in [M. Flammini and S. Pérennès, Inform. and Comput., to appear] to allow the determination of lower bounds on the broadcasting and gossiping time required by the so-called restricted protocols. Informally, a protocol is {\small $({\cal I}, {\cal O})$}-restricted if at every processor each outgoing activation of an arc depends on at most ${\cal I}$ previous incoming activations and any incoming activation influences at most ${\cal O}$ successive outgoing activations. Examples of restricted protocols are systolic ones and those running on bounded degree networks.
Thus, under the basic whispering model, we provide the first general lower bound on the gossiping time of d-bounded degree networks in the directed and half-duplex cases. Moreover, significantly improved broadcasting and gossiping lower bounds are obtained for well-known networks such as butterfly, de Bruijn, and Kautz graphs.
All the results are also extended to other communication models such as the c-port and/or postal one.

On Local Versus Global Satisfiability

Luca Trevisan

SIAM J. Discrete Math. 17, pp. 541-547 (7 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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We prove an extremal combinatorial result regarding the fraction of satisfiable clauses in Boolean conjunctive normal form (CNF) formulae enjoying a locally checkable property, thus solving a problem that has been open for several years.
We then generalize the problem to arbitrary constraint satisfaction problems. We prove a tight result even in the generalized case.

On the Maximal Codes of Length 3 with the 2-Identifiable Parent Property

Vu Dong Tô and Reihaneh Safavi-Naini

SIAM J. Discrete Math. 17, pp. 548-570 (23 pages) | Cited 3 times

Online Publication Date: August 01, 2006

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A $q$-ary code has identifiable parent property (IPP) if it allows one of the parents of a descendant word to be found. A 2-IPP code ensures that at least one parent of a pirate word constructed by a coalition of two users can be found. In this paper, we answer a question raised in [H. D. L. Hollmann et al., J. Combin. Theory Ser. A, 82 (1998), pp. 121--133] and show that F(q), the maximum number of codewords in a 2-IPP code of length 3, satisfies $|{\cal G}_0| \leq F(q) \leq |{\cal G}_0| +2$, where ${\cal G}_0$ is a well-defined graph. We also give an efficient algorithm (O(q3)) for finding maximal codes.

Radius Three Trees in Graphs with Large Chromatic Number

H. A. Kierstead and Yingxian Zhu

SIAM J. Discrete Math. 17, pp. 571-581 (11 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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A class $\Gamma$ of graphs is $\chi$-bounded if there exists a function $f$ such that $\chi \left(G\right) \leq f \left(\omega \left(G\right) \right)$ for all graphs $G \in \Gamma$, where $\chi$ denotes chromatic number and $\omega$ denotes clique number. Gyárfás and Sumner independently conjectured that, for any tree T, the class ${\rm Forb} \left(T\right)$, consisting of graphs that do not contain T as an induced subgraph, is $\chi$-bounded. The first author and Penrice showed that this conjecture is true for any radius two tree. Here we use the work of several authors to show that the conjecture is true for radius three trees obtained from radius two trees by making exactly one subdivision in every edge adjacent to the root. These are the only trees with radius greater than two, other than subdivided stars, for which the conjecture is known to be true.

Minimizing Wirelength in Zero and Bounded Skew Clock Trees

Moses Charikar, Jon Kleinberg, Ravi Kumar, Sridhar Rajagopalan, Amit Sahai, and Andrew Tomkins

SIAM J. Discrete Math. 17, pp. 582-595 (14 pages)

Online Publication Date: August 01, 2006

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An important problem in VLSI design is distributing a clock signal to synchronous elements in a VLSI circuit so that the signal arrives at all elements simultaneously. The signal is distributed by means of a clock routing tree rooted at a global clock source. The difference in length between the longest and shortest root-leaf path is called the skew of the tree. The problem is to construct a clock tree with zero skew (to achieve synchronicity) and minimal sum of edge lengths (so that circuit area and clock tree capacitance are minimized).
We give the first constant-factor approximation algorithms for this problem and its variants that arise in the VLSI context. For the zero skew problem in general metric spaces, we give an approximation algorithm with a performance guarantee of 2e. For the L1 version on the plane, we give an (8/ln 2)-approximation algorithm.

Encoding Fullerenes and Geodesic Domes

Jack E. Graver

SIAM J. Discrete Math. 17, pp. 596-614 (19 pages) | Cited 2 times

Online Publication Date: August 01, 2006

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Coxeter's classification of the highly symmetric geodesic domes (and, by duality, the highly symmetric fullerenes) is extended to a classification scheme for all geodesic domes and fullerenes. Each geodesic dome is characterized by its signature: a plane graph on twelve vertices with labeled angles and edges. In the case of the Coxeter geodesic domes, the plane graph is the icosahedron, all angles are labeled one, and all edges are labeled by the same pair of integers (p,q). Edges with these "Coxeter coordinates" correspond to straight line segments joining twovertices of $\Lambda$, the regular triangular tessellation of the plane, and the faces of the icosahedron are filled in with equilateral triangles from $\Lambda$ whose sides have coordinates (p,q).
We describe the construction of the signature for any geodesic dome. In turn, we describe how each geodesic dome may be reconstructed from its signature: the angle and edge labels around each face of the signature identify that face with a polygonal region of $\Lambda$ and, when the faces are filled by the corresponding regions, the geodesic dome is reconstituted. The signature of a fullerene is the signature of its dual. For each fullerene, the separation of its pentagons, the numbers of its vertices, faces, and edges, and its symmetry structure are easily computed directly from its signature. Also, it is easy to identify nanotubes by their signatures.

Independent Sets in Regular Hypergraphs and Multidimensional Runlength-Limited Constraints

Erik Ordentlich and Ron M. Roth

SIAM J. Discrete Math. 17, pp. 615-623 (9 pages) | Cited 1 time

Online Publication Date: August 01, 2006

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Let G be a t-uniform s-regular linear hypergraph with r vertices. It is shown that the number of independent sets $\IS(\hgraph)$ in $\hgraph$ satisfies \[ \log_2 \IS(\hgraph) \le \frac{r}{t} \left( 1 + O \biggl( \frac{\log^2(ts)}{s} \biggr) \right) . \] This leads to an improvement of a previous bound by Alon obtained for t = 2 (i.e., for regular ordinary graphs). It is also shown that for the Hamming graph $\Hamming(n,q)$ (with vertices consisting of all n-tuples over an alphabet of size q and edges connecting pairs of vertices with Hamming distance $1$), \[ \frac{\log_2 \IS(\Hamming(n,q))}{q^n} = \frac{1}{q} + O \biggl(\frac{\log^2 (q n)}{q n} \biggr). \] The latter result is then applied to show that the Shannon capacity of the n-dimensional $(d,\infty)$-runlength-limited (RLL) constraint converges to 1/(d+1) as n goes to infinity.

On Distributions Computable by Random Walks on Graphs

Guy Kindler and Dan Romik

SIAM J. Discrete Math. 17, pp. 624-633 (10 pages)

Online Publication Date: August 01, 2006

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We answer a question raised by Donald E. Knuth and Andrew C. Yao, concerning the class of polynomials on [0,1] that can be realized as the distribution function of a random variable, whose binary expansion is the output of a finite state automaton driven by unbiased coin tosses. The polynomial distribution functions which can be obtained in this way are precisely those with rational coefficients, whose derivative has no irrational roots on [0,1].
We also show, strengthening a result of Knuth and Yao, that all smooth distribution functions which can be obtained by such automata are polynomials.

Listen to Your Neighbors: How (Not) to Reach a Consensus

Nabil H. Mustafa and Aleksandar Pekec

SIAM J. Discrete Math. 17, pp. 634-660 (27 pages) | Cited 3 times

Online Publication Date: August 01, 2006

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We study the following rather generic communication\slash coordination\slash computation problem: In a finite network of agents, each initially having one of the two possible states, can the majority initial state be computed and agreed upon by means of local computation only? We study an iterative synchronous application of the local majority rule and describe the architecture of networks that are always capable of reaching the consensus on the majority initial state of its agents. In particular, we show that, for any truly local network of agents, there are instances in which the network is not capable of reaching such a consensus. Thus, every truly local computational approach that requires reaching a consensus is not failure-free.

Fractional Packing of T-Joins

Francisco Barahona

SIAM J. Discrete Math. 17, pp. 661-669 (9 pages) | Cited 1 time

Online Publication Date: August 01, 2006

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Given a graph with nonnegative capacities on its edges, it is well known that the capacity of a minimum T-cut is equal to the value of a maximum fractional packing of T-joins. The Padberg--Rao algorithm finds a minimum capacity T-cut, but it does not produce a T-join packing. We present a polynomial combinatorial algorithm for finding an optimal T-join packing.

Exact Formulae for the Lovász Theta Function of Sparse Circulant Graphs

Valentino Crespi

SIAM J. Discrete Math. 17, pp. 670-674 (5 pages) | Cited 1 time

Online Publication Date: August 01, 2006

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The Lovász theta function has attracted much attention for its connection with diverse issues such as communicating without errors and computing large cliques in graphs. Indeed, this function enjoys the remarkable property of being computable in polynomial time despite being sandwiched between clique and chromatic numbers, two well-known, hard to compute quantities.
In this paper I provide a closed formula for the Lovász function of all the circulant graphs of degree 4 with even displacement, thus generalizing Lovász results on cycle graphs (circulant graphs of degree 2).

Counting Strings with Given Elementary Symmetric Function Evaluations I: Strings over \boldmath$\mathbbZ_p$ with p Prime

C. R. Miers and F. Ruskey

SIAM J. Discrete Math. 17, pp. 675-685 (11 pages) | Cited 1 time

Online Publication Date: August 01, 2006

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Let $\alpha$ be a string over $\mathbb{Z}_p$ with $p$ prime. The $j$th elementary symmetric function evaluated at $\alpha$ is denoted $T_j(\alpha)$. We study the cardinalities $S_p(n;\tau_1,\tau_2,\ldots,\tau_t)$ of the set of length $n$ strings for which $T_i(\alpha) = \tau_i$. The \emph{profile} $\langle k_0,k_1,\ldots,k_{p-1} \rangle$ of a string $\alpha$ is the sequence of frequencies with which each letter occurs. The profile of $\alpha$ determines $T_j(\alpha)$, and hence $S_p$. Let $f_n : \mathbb{Z}_{p^n}^{p-1} \mapsto\nobreak \mathbb{Z}_p^{p^n-1}$ be the map that takes $\langle k_0,k_1,\ldots,k_{p-1} \rangle \bmod {p^n}$ to $(T_1,T_2,\ldots,T_{p^n-1}) \bmod p$. We show that $f_n$ is well defined and injective and show how to efficiently determine its range. These results are used to efficiently compute $S_p(n;\tau_1,\tau_2,\ldots,\tau_t)$.
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