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.
26138426
2-diameter of de Bruijn graphs
Binary strings are used as vertex labels or components of vertex labels. We will abbreviate “the vertex with label X” as “vertex x.” The binarji direcred de Briiijn graph of dimension n , denoted B ( n ) , has 2 ” vertices which are labeled with the binary strings of length n . There is an arc from any vertex x, x2 xn to the vertices x2x3 x,O and x2x3 x,, 1. We say that the ith coordinate of ,Y is x, . Any .u, is equal toOor l , andZ, = 1 x i . The undirected binary de Bruijn graph UB( n ) is obtained from B( n) by deleting the orientation of the arcs and omitting multiple edges and loops. It is well known that UB( n ) is 2-connected and that its diameter (maximum of the distances between all pairs of vertices) is equal to n . Due to their bounded maximum degree equal to 4 and their low diameter, de Bruijn graphs have been proposed as a possible good interconnection network for a parallel architecture [ 11. They have also been shown to solve a wide class of problems [ 21. For a k-connected graph, Hsu [ 4 ] introduced the notion of the kdiameter as follows: Given any two vertices x and y , a k-container between x and y is a set of k vertex disjoint paths between xand y . The length ofa k-container is the length of the longest of the k paths. The kdistance d k ( x , y ) between x and y is the minimum of the lengths of all the k-containers. Then, the k-diameter of a graph G is the maximum of d k ( x , y) for all pairs of vertices x and y . General results on the k-diameters of k-regular kconnected graphs can be found in [ 41 or [ 51. Results for some particular classes ofgraphs and many open problems can also be found in [ 41. It was shown by Pradhan and Reddy [ 7 ] that the 2diameter of U B ( n ) is less than or equal to 2n. The 2diameter is also obviously at least n. The aim of this paper was to prove that it is exactly n. In other words, between any two vertices of UB( n ) , there exist two internally vertex disjoint paths of length at most n. This gives a good measure of the fault tolerance of the graph [ 61. Let us first introduce some notation and recall some properties of the de Bruijn digraph B( n ) : Given two vertices x and y , we will denote f i x , y ] as a shortest path P from x to y . The length of this path, denoted by I PIX, y ] 1, is the number of edges in the path and is also the distance d ( x , y ) from x to y . P [ x , y ] also represents the set of vertices of the path, including its extremities.
15426263
Super connectivity of line graphs
The super connectivity κ' and the super edge-connectivity λ' are more refined network reliability indices than connectivity κ and edge-connectivity λ. This paper shows that for a connected graph G with order at least four rather than a star and its line graph L(G), κ'(L(G)) = λ'(G) if and only if G is not super-λ'. As a consequence, we obtain the result of Hellwig et al. [Note on the connectivity of line graphs, Inform. Process. Lett. 91 (2004) 7] that κ(L(G)) = λ'(G). Furthermore, the authors show that the line graph of a super-λ' graph is super-λ if the minimum degree is at least three.
7639532
Fault diameter of Cartesian product graphs
The (k-1)-fault diameter D"k(G) of a k-connected graph G is the maximum diameter of G-F for any F@?V(G) with |F|=1 for i=1,2.
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.
1351954
Cost-Sensitive Learning of SVM for Ranking
In this paper, we propose a new method for learning to rank. ‘Ranking SVM' is a method for performing the task. It formulizes the problem as that of binary classification on instance pairs and performs the classification by means of Support Vector Machines (SVM). In Ranking SVM, the losses for incorrect classifications of instance pairs between different rank pairs are defined as the same. We note that in many applications such as information retrieval the negative effects of making errors between higher ranks and lower ranks are larger than making errors among lower ranks. Therefore, it is natural to bring in the idea of cost-sensitive learning to learning to rank, or more precisely, to set up different losses for misclassification of instance pairs between different rank pairs. Given a cost-sensitive loss function we can construct a Ranking SVM model on the basis of the loss function. Simulation results show that our method works better than Ranking SVM in practical settings of ranking. Experimental results also indicate that our method can outperform existing methods including Ranking SVM on real information retrieval tasks such as document search and definition search.
33339608
Almost Optimal Solutions for Bin Coloring Problems
In this paper we study two interesting bin coloring problems: Minimum Bin Coloring Problem (MinBC) and Online Maximum Bin Coloring Problem (OMaxBC), motivated from several applications in networking. For the MinBC problem, we first show that it is NP-complete, and then present two near linear time approximation algorithms to achieve almost optimal solutions, i.e., no more than OPT+2 and OPT+1 respectively, where OPT is the optimal solution. For the OMaxBC problem, we first introduce a deterministic 2-competitive greedy algorithm, and then give lower bounds for any deterministic and randomized (against adaptive offline adversary) online algorithms. The lower bounds show that our deterministic algorithm achieves the best possible competitive ratio.
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.
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.
9688726
Motion Editing in 3D Video Database
As the next generation of media, 3D video is attracting increased attention. 3D video is a sequence of three dimensional mesh models, captured and generated for a real dynamic object. In this paper, we present a simple framework of motion editing in 3D video database to re-use 3D video data. Our system is composed of two modules. In the first module, a motion database is automatically set up from original 3D video sequences off-line by analyzing the feature vectors of each frame. It is observed that our original 3D video sequences have a two-level temporal structure. A fine-to-coarse method is proposed to extract such a structure. 3D video is segmented into the fine-level structure by a three reference frame strategy and then is clustered into the coarse-level structure. In the second module, users can synthesize the motions to edit a new 3D video sequence online. A cost function is optimized to transit between two motions with the users' requirements. All the algorithms in the system are based on the analysis in feature vector space and the edited 3D video sequence is played using OpenGL.