39689433
On 1-rate Wide-sense Nonblocking for 3-stage Clos Networks
Abstract This paper considers the multirate environment in which each network link has a unity capacity and each call is associated with a rate w, 1 ⩾ w>0. The multirate environment is reduced to the 1 -rate environment if all rates are identical. Circuit switching is a special case of 1 -rate in which the constant rate is w = 1. We establish some general relations between wide-sense nonblocking in circuit switching and 1-rate wide-sense nonblocking for the 3-stage Clos network. We also give necessary and sufficient conditions for a 3-stage Clos network with two input switches and two output switches to be wide-sense nonblocking, and give a necessary condition when there are three input switches and three output switches. We conjecture that 1-rate wide-sense nonblocking implies circuit switching wide-sense nonblocking and give some partial evidence.
30354625
A class of full Steiner minimal trees
A Steiner minimal tree for a given set of points in the plane is a tree which interconnects these points using lines of shortest possible total length. We construct an infinite class of trees which are the unique full Steiner minimal trees for their sets of endpoints (vertices of degree one).
2453836
On connectivity of consecutive-d digraphs
The concept of a consecutive-d digraph was proposed by Du, Hsu and Hwang as a generalization of de Bruijn digraphs, Kautz digraphs, and their generalizations given by Imase and Itoh and Reddy, Pradhan and Kuhl. In this paper, we determine the connectivity of consecutive-d digraphs and study how to modify consecutive-d digraphs to reach maximum connectivity. Our results will generalize and improve several existing results on the connectivity of de Bruijn digraphs, Kautz digraphs and Imase-Itoh digraphs.
14358235
New bounds on a hypercube coloring problem
In studying the scalability of optical networks, one problem which arises involves coloring the vertices of the n-cube with as few colors as possible such that any two vertices whose Hamming distance is at most k are colored differently. Determining the exact value of χ??(n), the minimum number of colors needed, appears to be a difficult problem. In this note, we improve the known lower and upper bounds of χ??(n) and indicate the connection of this coloring problem to linear codes.
43203779
The k-Steiner ratio in graphs
A Steiner minimum tree, SMT, is the shortest length tree in a metric space interconnecting a set of points, called the regular points, possibly using additional vertices. A k-size Steiner minimum tree, kSMT, is one that can be split into components where all regular points are leaves and all components have at most k leaves. The k-$lteiner ratio, pk, is the infimum of the ratios SMT/kSMT over all sets of regular points in all possible metric spaces, where the distances are given by a complete graph. Previously only p2 and p3 in graphs had been known exactly, and some bounds where know for the others. In this paper we determine pk exactly for all k.
36038781
A proof of the Gilbert-Pollak conjecture on the Steiner ratio
LetP be a set ofn points on the euclidean plane. LetLs(P) andLm(P) denote the lengths of the Steiner minimum tree and the minimum spanning tree onP, respectively. In 1968, Gilbert and Pollak conjectured that for anyP,Ls(P)≥(√3/2)Lm(P). We provide a proof for their conjecture in this paper.
17041567
On-line scheduling of small open shops
We investigate the problem of on-line scheduling open shops of two and three machines with an objective of minimizing the schedule makespan. We first propose a 1.848-competitive permutation algorithm for the non-preemptive scheduling problem of two machines and show that no permutation algorithm can be better than 1.754-competitive. Secondly, we develop a (27/19)-competitive algorithm for the preemptive scheduling problem of three machines, which is most competitive.
18860643
The multiroute maximum flow problem revisited
We are given a directed network G = (V, A, u) with vertex setV , arc setA, a source vertexs ∈ V , a destination vertex t ∈ V , a finite capacity vector u = {uij}ij∈A, and a positive integerm ∈ Z+. The multiroute maximum flow problem ( m-MFP) generalizes the ordinary maximum flow problem by seeking a maximum flow from s to t subject to not only the regular flow conservation constraints at the vertices (except s andt) and the flow capacity constraints at the arcs, but also the extra constraints that any flow must be routed alon g m arc-disjoints-t paths. In this paper, we devise two new combinatorial algorithms for m-MFP. One is based on Newton’s method and another is based on augmenting-path technique. We also show how the Newton-based algorithm unifies two existing algorithms, and how the augmen ting-path algorithm is strongly polynomial for case m = 2.
35793386
An improved algorithm for decomposing arc flows into multipath flows
We consider a multipath maximum flow problem introduced by Kishimoto (Networks 27(4)(1996)279-291). The focus is on efficient transformation from arc flows into multipath flows, where a multipath flow is a nonnegative combination of multipaths. A new algorithm that is more efficient than existing ones is proposed for the transformation.
16666020
On-line Network Synthesis
We consider on-line network synthesis problems. Let N = {1, , n} be a set of n sites. Trafﬁc ﬂow requirements between pairs of sites are revealed one by one. Whenever a new request rij = rji (i < j) between sites i and j is revealed, an on-line algorithm must install the additional necessary capacity without decreasing the existing network capacity such that all the trafﬁc requirements are met. The objective is to minimize the total capacity installed by the algorithm. The performance of an on-line algorithm is measured by the competitive ratio, deﬁned to be the worst-case ratio between the total capacity by the on-line algorithm and the total optimal (off-line) capacity assuming we have priori information on all the requirements initially. We distinguish between two on-line versions of the problem depending on whether the entire set of sites is known a prior or not. For the ﬁrst version where the entire set of site is unknown, we present a best possible algorithm along with a matching lower bound. For the second version where the entire set of sites is known a priori, we present a best possible algorithm for n ≤ 6. 
 
Key words: On-line algorithm; Approximation algorithm; Network synthesis; Competitive analysis
8600977
Optimal preemptive semi-online scheduling on two uniform processors
We investigate a preemptive semi-online scheduling problem. Jobs with sizes within a certain range [1, r] (r≥1 ) arrive one by one to be scheduled on two uniform parallel processors with speed 1 and s≥1, respectively. The objective is to minimize makespan. We characterize the optimal competitive ratio as a function of both s and r by devising a deterministic on-line scheduling algorithm along with a matching lower bound, which also holds for randomized algorithms.