30320002
A Short Proof of a Result of Pollak on Steiner Minimal Trees
The long-standing conjecture of Gilbert and Pollak states that for any set of n given points in the euclidean plane, the ratio of the length of a Steiner minimal tree and the length of a minimal (spanning) tree is at least 32. This conjecture was shown to be true for n = 3 by Gilbert and Pollak, and for n = 4 by Pollak. However, the proof for n = 4 by Pollak is sufficiently complicated that no generalization to any other value of n has been found. We use a different approach to give a very short proof for the n = 4 case. This approach also allows us to attack the n = 5 case, though the proof is no longer short (to be reported in a subsequent paper).
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.
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.
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).
9933311
The k-Steiner Ratio in the Rectilinear Plane
A Steiner minimum tree (SMT) in the rectilinear plane is the shortest length tree interconnecting a set of points, called the regular points, possibly using additional vertices. Ak-size Steiner minimum tree (kSMT) is one that can be split into components where all regular points are leaves and all components have at mostkleaves. Thek-Steiner ratio in the rectilinear plane, ?k, is the infimum of the ratios SMT/kSMT over all finite sets of regular points. Thek-Steiner ratio is used to determine the performance ratio of several recent polynomial-time approximations for Steiner minimum trees. Previously it was known that in the rectilinear plane, ?2=2/3, ?3=4/5, and (2k?2)/(2k?1)??k(L1)?(2k?1)/(2k) fork?4. In 1991, P. Berman and V. Ramaiyer conjectured that in fact ?k=(2k?1)/(2k) fork?4. In this paper we prove their conjecture.
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
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.
7236819
Creating a virtual network laboratory
Networking courses are traditionally conceptual in nature, with little opportunity for students to apply what they have learned in order to strengthen their knowledge. To support these courses, the University of Minnesota established a hands-on network laboratory. Educational and operational concerns have surfaced throughout the years due to the physical laboratory constraints and difficulty of satisfying the high demand. To address these issues, and to overcome the shortcomings of current approaches, we are developing a virtual network laboratory environment to eliminate most of the constraints, though it is not necessarily replacing all of the physical settings at once. The kernel consists of a collection of online multimedia learning modules which integrate the key laboratory devices with instructional material. The modules are designed to allow students to carry out hands-on networking in a virtual environment or through remote access. This virtual laboratory is easily accessible over the Internet.
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.