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).
206027538
A coloring problem on the n-cube
Abstract In this paper, we consider a coloring problem on the n-cube that arises in the study of scalability of optical networks. Let χ k (n) be the minimum number of colors needed to color the vertices of the n-cube so that every two vertices with Hamming distance ⩽k have different colors. We show that for k=3, 2n⩽χ 3 (n)⩽2 ⌈ log 2 n⌉+1 . We also provide upper and lower bounds on χ k (n) for general k.
42394808
on Component-size Bounded Steiner Trees
A Steiner tree is a tree interconnecting a given set of points in a metric space such that all leaves are given points. A (full) component of a Steiner tree is a subtree which results from splitting the Steiner tree at some given points. A k-size Steiner tree is a Steiner tree in which every component has at most k given points. The k-Steiner ratio is the largest lower bound for the ratio between lengths of a minimum Steiner tree and a minimum k-size Steiner tree for the same set of points. In this paper, we determine the 3-Steiner ratio in weighted graphs.
20341554
On greedy heuristics for steiner minimum trees
We disprove a conjecture of Shor and Smith on a greedy heuristic for the Steiner minimum tree by showing that the length ratio between the Steiner minimum tree and the greedy tree constructed by their method for the same set of points can be arbitrarily close to√3/2. We also propose a new conjecture.
44914304
Steiner minimal trees on sets of four points
LetS = {A, B, C, D} consist of the four corner points of a convex quadrilateral where diagonals [A, C] and [B, D] intersect at the pointO. There are two possible full Steiner trees forS, theAB-CD tree hasA andB adjacent to one Steiner point, andC andD to another; theAD-BC tree hasA andD adjacent to one Steiner point, andB andC to another. Pollak proved that if both full Steiner trees exist, then theAB-CD (AD-BC) tree is the Steiner minimal tree ifAOD>3 (<) 90°, and both are Steiner minimal trees ifAOD=90°. While the theorem has been crucially used in obtaining results on Steiner minimal trees in general, its applicability is sometimes restricted because of the condition that both full Steiner trees must exist. In this paper we remove this obstacle by showing: (i) Necessary and sufficient conditions for the existence of either full Steiner tree forS. (ii) IfAOD≥90°, then theAB-CD tree is the SMT even if theAD-BC tree does not exist. (iii) IfAOD<90° but theAD-BC tree does not exist, then theAB-CD tree cannot be ruled out as a Steiner minimal tree, though under certain broad conditions it can.
24107558
Lower Bounds on the Minus Domination and k-Subdomination Numbers
A three-valued function f defined on the vertex set of a graph G = (V,E), f : V → {-1, 0, 1} is a minus dominating function if the sum of its function values over any closed neighborhood is at least one. That is, for every v ∈ V, f(N[v]) ≥ 1, where N[v] consists of v and all vertices adjacent to v. The weight of a minus function is f(V) = Σv∈V f(v). The minus domination number of a graph G, denoted by γ-(G), equals the minimum weight of a minus dominating function of G. In this paper, sharp lower bounds on minus domination of a bipartite graph are given. Thus, we prove a conjecture proposed by J. Dunbar etc. (Discrete Math. 199(1999) 35-47), and we give a lower bound on γks(G) of a graph G.
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.
12802794
A systematic approach to design the network-based learning environment for home and office
An effective network-based learning environment provides synchronous and asynchronous education to learners who are physically, remote from or opt to telecommute to the base location of instruction. The environment provides content creation, user registration, course material storage, multiple delivery means, and end-user viewing tools. An intriguing aspect of the environment is how to model a lecture in terms of pre-, in-, and post-class information. We regard a lecture model collectively as an activity consisting of three elements: lecture materials, lecture presentation, and lecture interaction. We derive the synchronization specification for the media objects and define strong-weak networked hyperlink (SWNH) to represent the lecture structure and inter/intra media object reference in the distributed network environment. The specification and representation exhibit an object-oriented modular design. The presentation of the lecture embody both the event-driven and timeline-based presentation synchronization where the browsing and authoring tools developed earlier fit right into the learning environment, e.g. FLIPS and HQT by the authors, or are immediately available through other providers, e.g. Web browsers.
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
16471624
Preemptive on-line scheduling for two uniform processors
This paper considers the problem of preemptive on-line scheduling for two uniform processors in which one of the processors has speed 1 and the other has speed s>=1. The objective is to minimize the makespan. A best possible algorithm with competitive ratio of (1+s)^2/(1+s+s^2) is proposed for this problem.