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.
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).
40136335
On Rearrangeability of Multirate Clos Networks
Chung and Ross [SIAM J. Comput., 20 (1991), pp. 726--736] conjectured that the multirate three-stage Clos network C(n,2n-1,r) is rearrangeable in the general discrete bandwidth case; i.e., each connection has a weight chosen from a given finite set {p1, p2,. . .,pk} where $1 \geq p_1 > p_2 > \cdots > p_k > 0$ and pi is an integer multiple of pi, denoted by $p_k \mid p_i$, for $1 \leq i \leq k-1$. In this paper, we prove that multirate three-stage Clos network C(n,2n-1,r) is rearrangeable when each connection has a weight chosen from a given finite set {p1, p2,. . .,pk} where $1 \geq p_1 > p_2 > \cdots > p_{h} > 1/2 \geq p_{h+1} > \cdots > p_k > 0$ and ph+2 | ph+1,ph+3 |ph+2,. . . ,pk | ph+1. We also prove that C(n,2n-1,r) is two-rate rearrangeable and $C(n, \lceil \frac{7n}{3} \rceil, r)$ is three-rate rearrangeable.
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.
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.
6724306
Line Digraph Iterations and Connectivity Analysis of de Bruijn and Kautz Graphs
A graph has spread (m, k, l) if for any m+1 distinct nodes x, y/sub 1/, . . ., y/sub m/ and m positive integers r/sub 1/, . . ., r/sub m/, such that Sigma /sub i/r/sub i/=k, there exist k node-disjoint paths of length at most 1 from x to the y/sub i/, where r/sub i/ of them end at y/sub i/. This concept contains, and is related to many important concepts used in communications and graph theory. The authors prove an optimal general theorem about the spreads of digraphs generated by line digraph iterations. Useful graphs, like the de Bruijn and Kautz digraphs, can be thus generated. The theorem is applied to the de Bruijn and Kautz digraphs to derive optimal bounds on their spreads, which implies previous results and resolves open questions on their connectivity, diameter, k-diameter, vulnerability, and some other measures related to length-bound disjoint paths. >
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.
2477251
PCR-assist CBR for delivering pre-recorded MPEG-2 transport streams
Transmitting compressed variable bit-rate (VBR) video with a traditional constant bit-rate (CBR) service for continuous playback requires a large buffer on the viewer's side. In this paper, we study a new CBR transmission scheme which employs the program clock references (PCR) that are embedded in the MPEG-2 transport streams to reduce the buffer requirement. The scheme is called "PCR-assist CBR" (PCBR). The PCBR scheme uses PCRs to examine its transmission regularly. It holds up the transmission if it is ahead of schedule. Based on experimental results with video traces, the PCBR scheme reduces the buffer requirement at the cost of a higher transmission rate.
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
11088545
Supporting Random Access on Real-Time Retrieval of Digital Continuous Media
In addition to the large data size requirement and real-time constraint in continuous media, future video applications such as video editing demand a random access capability at the video-frame level. This paper introduces our study on effective buffering control for the real-time retrieval of jitter-free digital video medium. We adopt a video-frame level approach to maintaining the flexibility on placement and analysing the efficiency of the buffering schemes. An integrated solution which offers efficient buffering schemes and flexible storage placement to support random access is our goal. We present two buffering schemes: the two-buffer scheme and the k-buffer compensation scheme. The two-buffer scheme requires that all the frames in a block are stored consecutively, while providing random access between blocks. However, this intuitive buffering scheme potentially requires a large block size and buffer space. The k-buffer compensation scheme is proposed to resolve this large buffer space requirement, by using more than two buffers and requiring a minimal number of blocks randomly placed in each cylinder. This scheme differs from the contiguous placement scheme because individual blocks can be stored anywhere in each cylinder. Compared to the two-buffer scheme, the k-buffer compensation scheme requires less buffer space, has higher disk utilization and finer granularity on disk data transfer. The placement requirements are more flexible and implementable than the contiguous and storage pattern placement schemes. Experimental measurement results reveal the significant improvements on the buffer-size reduction and placement flexibility by using the k-buffer compensation scheme. Extensions of the k-buffer compensation scheme to support multiple streams are also addressed.
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.