15417765
The proof of a conjecture of Bouabdallah and Sotteau
Let G be a connected graph of order n. A routing in G is a set of n(n 1) fixed paths for all ordered pairs of vertices of G. The edge-forwarding index of G, (G), is the minimum of the maximum number of paths specified by a routing passing through any edge of G taken over all routings in G, and ,n is the minimum of (G) taken over all graphs of order n with maximum degree at most . To determine n 2p 1,n for 4p 2p/3 1 ≤ n ≤ 6p, A. Bouabdallah and D. Sotteau proposed the following conjecture in [On the edge forwarding index problem for small graphs, Networks 23 (1993), 249–255]. The set 3 {1, 2, . . . , (4p)/3} can be partitioned into 2p pairs plus singletons such that the set of differences of the pairs is the set 2 {1, 2, . . . , p}. This article gives a proof of this conjecture and determines that n 2p 1,n is equal to 5 if 4p 2p/3 1 ≤ n ≤ 6p and to 8 if 3p p/3 1 ≤ n ≤ 3p (3p)/5 for any p ≥ 2. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(4), 292–296 2004
10735574
Forwarding indices of folded n-cubes
For a given connected graph G of order n, a routing R is a set of n(n - 1) elementary paths specified for every ordered pair of vertices in G. The vertex (resp. edge) forwarding index of a graph is the maximum number of paths of R passing through any vertex (resp. edge) in the graph. In this paper, the authors determine the vertex and the edge forwarding indices of a folded n-cube as (n-1)2n-1 +1-((n+l)/2) (n ⌊n+1/2⌋) and 2n-(n,⌊n+1/2⌋),respectively.
297940
Connectivity of Cartesian product graphs
Use v"i,@k"i,@l"i,@d"i to denote order, connectivity, edge-connectivity and minimum degree of a graph G"i for i=1,2, respectively. For the connectivity and the edge-connectivity of the Cartesian product graph, up to now, the best results are @k(G"1xG"2)>=@k"1+@k"2 and @l(G"1xG"2)>=@l"1+@l"2. This paper improves these results by proving that @k(G"1xG"2)>=min{@k"1+@d"2,@k"2+@d"1} and @l(G"1xG"2)=min{@d"1+@d"2,@l"1v"2,@l"2v"1} if G"1 and G"2 are connected undirected graphs; @k(G"1xG"2)>=min{@k"1+@d"2,@k"2+@d"1,2@k"1+@k"2,2@k"2+@k"1} if G"1 and G"2 are strongly connected digraphs. These results are also generalized to the Cartesian products of n(>=3) connected graphs and n strongly connected digraphs, respectively.
30370540
On (d, 2)-dominating numbers of binary undirected de Bruijn graphs
Abstract In this paper, we show that: (i) For n-dimensional undirected binary de Bruijn graphs, UB(n), n⩾4, there is a vertex x=x 1 x 1 x 1 ⋯ x 1 x 1 (x1=1 or 0) such that for any other vertex t there exist at least two internally disjoint paths of length at most n−1 between x and t in UB(n), i.e., the (n−1,2)-dominating number of UB(n) is equal to one. (ii) For n⩾5, let S={100⋯01,011⋯10}. For any other vertex t there exist at least two internally disjoint paths of length at most n−2 between t and S in UB(n), i.e., the (n−2,2)-dominating number of UB(n) is no more than 2.
872168
The super connectivity of shuffle-cubes
The shuffle-cube SQn, where n ≡ 2 (mod 4), a new variation of hypercubes proposed by Li et al. [T.-K. Li, J.J.M. Tan, L.-H. Hsu, T.-Y. Sung, The shuffle-cubes and their generalization, Inform. Process. Lett. 77 (2001) 35-41], is an n-regular n-connected graph. This paper determines that the super connectivity of SQn is 2n - 4 and the super edge-connectivity is 2n - 2 for n ≥ 6.
7849638
Edge-fault-tolerant edge-bipancyclicity of hypercubes
In this paper, we consider the problem embedding a cycle into the hypercube Q"n with existence of faulty edges and show that for any edge subset F of Q"n with |F|==4 and all edges in F are not incident with the same vertex. This result improves some known results.
17736236
Algorithms for congruent sphere packing and applications
The problem of packing congruent spheres (i.e., copies of the same sph ere) in a bounded domain arises in many applications. In this paper, we present a new pack-and-shake scheme for packing congruent spheres in various bounded 2-D domains. Our packing scheme is based on a number of interesting ideas, such as a trimming and packing approach, optimal lattice packing under translation and/or rotation, shaking procedures, etc. Our packing algorithms have fairly low time complexities. In certain cases, they even run in nearly linear time. Our techniques can be easily generalized to congruent packing of other shapes of objects, and are readily extended to higher dimensional spaces. Applications of our packing algorithms to treatment planning of radiosurgery are discussed. Experimental results suggest that our algorithms produce reasonably dense packings.
5147011
Wide Diameters of Cartesian Product Graphs and Digraphs
In graph theory and study of fault tolerance and transmission delay of networks, connectivity and diameter of a graph are two very important parameters and have been deeply studied by many authors. Wide diameter combining connectivity with diameter is a more important parameter to measure fault tolerance and efficiency of parallel processing computer networks and has received much attention in the recent years. Diameter with width k of a graph G is defined as the minimum integer d for which between any two distinct vertices in G there exist at least k internally disjoint paths of length at most d. In the present paper, the tight upper bounds of wide diameter of the Cartesian product graphs are obtained. Some known results can be deduced or improved from ours.
2583878
A data streaming algorithm for estimating subpopulation flow size distribution
Statistical information about the flow sizes in the traffic passing through a network link helps a network operator to characterize network resource usage, infer traffic demands, detect traffic anomalies, and improve network performance through traffic engineering. Previous work on estimating the flow size distribution for the complete population of flows has produced techniques that either make inferences from sampled network traffic, or use data streaming approaches. In this work, we identify and solve a more challenging problem of estimating the size distribution and other statistical information about arbitrary subpopulations of flows. Inferring subpopulation flow statistics is more challenging than the complete population counterpart, since subpopulations of interest are often specified a posteriori (i.e., after the data collection is done), making it impossible for the data collection module to "plan in advance".Our solution consists of a novel mechanism that combines data streaming with traditional packet sampling to provide highly accurate estimates of subpopulation flow statistics. The algorithm employs two data collection modules operating in parallel --- a NetFlow-like packet sampler and a streaming data structure made up of an array of counters. Combining the data collected by these two modules, our estimation algorithm uses a statistical estimation procedure that correlates and decodes the outputs (observations) from both data collection modules to obtain flow statistics for any arbitrary subpopulation. Evaluations of this algorithm on real-world Internet traffic traces demonstrate its high measurement accuracy.
11532584
Sequentially t-Diagnosable Systems: A Characterization and Its Applications
In the system level fault diagnosis area, the fundamental problem of characterizing sequentially t-diagnosable systems in the PMC model has remained open for more than two decades. We resolve this problem by providing a complete characterization of such systems. Our solution to the characterization problem leads to the correct identification of optimal sequentially t-diagnosable D/sub /spl delta/,k/ systems. Given a set of n units where n=2t+1, an optimal D/sub /spl delta/,k/ system can be constructed with just n([(t+2)/3]) tests, rather than n([t/2]+1) tests-a previously misjudged bound. An efficient algorithm for identifying the set of faulty units in a sequentially t-diagnosable D/sub /spl delta/,k/ system is given along the line of the proposed characterization, which is linear with respect to the number of tests in the system. >
2254306
An efficient direct approach for computing shortest rectilinear paths among obstacles in a two-layer interconnection model
In this paper, we present a direct approach for routing a shortest rectilinear path between two points among a set of rectilinear obstacles in a two-layer interconnection model that is used for VLSI routing applications. The previously best known direct approach for this problem takes O(nlog^2n) time and O(nlogn) space, where n is the total number of obstacle edges. By using integer data structures and an implicit graph representation scheme (i.e., a generalization of the distance table method), we improve the time bound to O(nlog^3^/^2n) while still maintaining the O(nlogn) space bound. Comparing with the indirect approach for this problem, our algorithm is simpler to implement and is probably faster for a quite large range of input sizes.