A fast algorithm for Steiner trees
IBM1
TL;DR: The heuristic algorithm has a worst case time complexity of O(¦S¦¦V¦2) on a random access computer and it guarantees to output a tree that spans S with total distance on its edges no more than 2(1−1/l) times that of the optimal tree.
Abstract: Given an undirected distance graph G=(V, E, d) and a set S, where V is the set of vertices in G, E is the set of edges in G, d is a distance function which maps E into the set of nonnegative numbers and S?V is a subset of the vertices of V, the Steiner tree problem is to find a tree of G that spans S with minimal total distance on its edges. In this paper, we analyze a heuristic algorithm for the Steiner tree problem. The heuristic algorithm has a worst case time complexity of O(¦S¦¦V¦ 2) on a random access computer and it guarantees to output a tree that spans S with total distance on its edges no more than 2(1?1/l) times that of the optimal tree, where l is the number of leaves in the optimal tree.
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02 Jul 2001TL;DR: Covering the basic techniques used in the latest research work, the author consolidates progress made so far, including some very recent and promising results, and conveys the beauty and excitement of work in the field.
Abstract: Covering the basic techniques used in the latest research work, the author consolidates progress made so far, including some very recent and promising results, and conveys the beauty and excitement of work in the field. He gives clear, lucid explanations of key results and ideas, with intuitive proofs, and provides critical examples and numerous illustrations to help elucidate the algorithms. Many of the results presented have been simplified and new insights provided. Of interest to theoretical computer scientists, operations researchers, and discrete mathematicians.
4,290 citations
TL;DR: In this article, a weighted greedy algorithm is proposed for a version of the dynamic Steiner tree problem, which allows endpoints to come and go during the life of a connection.
Abstract: The author addresses the problem of routing connections in a large-scale packet-switched network supporting multipoint communications. He gives a formal definition of several versions of the multipoint problem, including both static and dynamic versions. He looks at the Steiner tree problem as an example of the static problem and considers the experimental performance of two approximation algorithms for this problem. A weighted greedy algorithm is considered for a version of the dynamic problem which allows endpoints to come and go during the life of a connection. One of the static algorithms serves as a reference to measure the performance of the proposed weighted greedy algorithm in a series of experiments. >
2,866 citations
TL;DR: An overview of the QoS routing problem as well as the existing solutions is given, the strengths and weaknesses of different routing strategies, and the challenges are outlined.
Abstract: The upcoming gigabit-per-second high-speed networks are expected to support a wide range of communication-intensive real-time multimedia applications. The requirement for timely delivery of digitized audio-visual information raises new challenges for next-generation integrated services broadband networks. One of the key issues is QoS routing. It selects network routes with sufficient resources for the requested QoS parameters. The goal of routing solutions is twofold: (1) satisfying the QoS requirements for every admitted connection, and (2) achieving global efficiency in resource utilization. Many unicast/multicast QoS routing algorithms have been published, and they work with a variety of QoS requirements and resource constraints. Overall, they can be partitioned into three broad classes: (1) source routing, (2) distributed routing, and (3) hierarchical routing algorithms. We give an overview of the QoS routing problem as well as the existing solutions. We present the strengths and weaknesses of different routing strategies, and outline the challenges. We also discuss the basic algorithms in each class, classify and compare them, and point out possible future directions in the QoS routing area.
936 citations
TL;DR: This article outlined some of the key elements of an integrated network technology, capable of supporting voice, data, and video communication and having the flexibility to accommodate new services as the need arose, and identified Integrated switching architectures, generalized connection signaling protocols, quality-of-service, multicast switching, and routing were all identified as key challenges to be addressed.
Abstract: hat a difference 25 years makes! When this article was written the telecommunications industry was still a heavily regulated, non-competitive but highly integrated place; the Internet was the private preserve of congnoscenti at a handful of research universities; integrated circuits with a feature size of 2 micron were state-of-the-art; and the room-sized minicomputers of the day could send and receive maybe a thousand packets per second, on a good day. This article articulated an observation that was gaining recognition throughout the early 1980s: first, that the fragmentation of the communications infrastructure into several distinct application-specific networks was wasteful and unnecessary; and second, that by combining concepts from packet switching with the implementation practices of circuit switching (highly parallel, hardware switching systems), one could develop systems with unprecedented flexibility and the performance levels and reliability needed to support large-scale, ubiquitous deployment. By the mid to late 1980s, these ideas were being pursued most vigorously in the development of ATM standards and technology, which was widely expected to play a leading role in the next generation of the public telecommunications infrastructure. The article outlined some of the key elements of such an integrated network technology, capable of supporting voice, data, and video communication and having the flexibility to accommodate new services as the need arose. Integrated switching architectures, generalized connection signaling protocols, quality-of-service, multicast switching, and routing were all identified as key challenges to be addressed. Subsequent years have seen tremendous progress on all these fronts in both the ATM context and in the now all-important Internet context. The changes in packet switching technology over the last two decades are particularly striking. While advanced switching architectures were first developed in the ATM context, these concepts have been applied with striking success in Ethernet switches and IP routers in recent years. Emerging IP router products with aggregate capacities exceeding a terabit per second are becoming available now, and gigabit Ethernet switches are available for just hundreds of dollars per port. Progress on other fronts has been more disappointing. While practical and effective solutions for connection signaling, quality-of-service, congestion control, and multicast have been developed and demonstrated, there has been little progress toward widespread commercial deployment. We still operate separate networks for voice, data, and video, and while the Internet promises to play a growing role in voice and video communication, it cannot achieve that promise without major technical improvements. Fortunately, the potential value of such improvements is becoming more and more clear, as the Internet’s role in our evolving information society continues to develop. While challenges remain, there seems little doubt that the necessary changes can and will be made, and that the full realization of the vision outlined in this article 25 years ago is now within our grasp. Jonathan S. Turner
918 citations
TL;DR: The protocol independent multicast (PIM) architecture maintains the traditional IP multicast service model of receiver-initiated membership, supports both shared and source-specific (shortest-path) distribution trees, and uses soft-state mechanisms to adapt to underlying network conditions and group dynamics.
Abstract: The purpose of multicast routing is to reduce the communication costs for applications that send the same data to multiple recipients. Existing multicast routing mechanisms were intended for use within regions where a group is widely represented or bandwidth is universally plentiful. When group members, and senders to those group members, are distributed sparsely across a wide area, these schemes are not efficient; data packets or membership report information are occasionally sent over many links that do not lead to receivers or senders, respectively. We have developed a multicast routing architecture that efficiently establishes distribution trees across wide area internets, where many groups will be sparsely represented. Efficiency is measured in terms of the router state, control message processing, and data packet processing, required across the entire network in order to deliver data packets to the members of the group. Our protocol independent multicast (PIM) architecture: (a) maintains the traditional IP multicast service model of receiver-initiated membership, (b) supports both shared and source-specific (shortest-path) distribution trees, (c) is not dependent on a specific unicast routing protocol, and (d) uses soft-state mechanisms to adapt to underlying network conditions and group dynamics. The robustness, flexibility, and scaling properties of this architecture make it well-suited to large heterogeneous internetworks.
863 citations
References
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TL;DR: A tree is a graph with one and only one path between every two nodes, where at least one path exists between any two nodes and the length of each branch is given.
Abstract: We consider n points (nodes), some or all pairs of which are connected by a branch; the length of each branch is given. We restrict ourselves to the case where at least one path exists between any two nodes. We now consider two problems. Problem 1. Constrnct the tree of minimum total length between the n nodes. (A tree is a graph with one and only one path between every two nodes.) In the course of the construction that we present here, the branches are subdivided into three sets: I. the branches definitely assignec~ to the tree under construction (they will form a subtree) ; II. the branches from which the next branch to be added to set I, will be selected ; III. the remaining branches (rejected or not yet considered). The nodes are subdivided into two sets: A. the nodes connected by the branches of set I, B. the remaining nodes (one and only one branch of set II will lead to each of these nodes), We start the construction by choosing an arbitrary node as the only member of set A, and by placing all branches that end in this node in set II. To start with, set I is empty. From then onwards we perform the following two steps repeatedly. Step 1. The shortest branch of set II is removed from this set and added to
22,704 citations
"A fast algorithm for Steiner trees" refers methods in this paper
...The readers are referred to the literatures with regard to the algorithms for constructing the shortest path as required in Step 1 [ 3-5 ] and for finding a minimal spanning tree as required in Step 2 and Step 3 [3, 6, 7]. We would only mention that, as far as computational complexity is concerned, using algorithms as mentioned above, in the worst case, Step 1 could be done in O(]SI IVI 2) time, Step 2 could be done in O(ISI 2) time, Step 3 ......
[...]
...The readers are referred to the literatures with regard to the algorithms for constructing the shortest path as required in Step 1 [3-5] and for finding a minimal spanning tree as required in Step 2 and Step 3 [ 3 , 6, 7]. We would only mention that, as far as computational complexity is concerned, using algorithms as mentioned above, in the worst case, Step 1 could be done in O(]SI IVI 2) time, Step 2 could be done in O(ISI 2) time, Step 3 ......
[...]
TL;DR: The work of Dantzig, Fulkerson, Hoffman, Edmonds, Lawler and other pioneers on network flows, matching and matroids acquainted me with the elegant and efficient algorithms that were sometimes possible.
Abstract: Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held. These experiences made me aware that seemingly simple discrete optimization problems could hold the seeds of combinatorial explosions. The work of Dantzig, Fulkerson, Hoffman, Edmonds, Lawler and other pioneers on network flows, matching and matroids acquainted me with the elegant and efficient algorithms that were sometimes possible. Jack Edmonds’ papers and a few key discussions with him drew my attention to the crucial distinction between polynomial-time and superpolynomial-time solvability. I was also influenced by Jack’s emphasis on min-max theorems as a tool for fast verification of optimal solutions, which foreshadowed Steve Cook’s definition of the complexity class NP. Another influence was George Dantzig’s suggestion that integer programming could serve as a universal format for combinatorial optimization problems.
8,644 citations
01 Jan 1972
TL;DR: Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held, which made me aware of the importance of distinction between polynomial-time and superpolynomial-time solvability.
Abstract: Throughout the 1960s I worked on combinatorial optimization problems including logic circuit design with Paul Roth and assembly line balancing and the traveling salesman problem with Mike Held. These experiences made me aware that seemingly simple discrete optimization problems could hold the seeds of combinatorial explosions. The work of Dantzig, Fulkerson, Hoffman, Edmonds, Lawler and other pioneers on network flows, matching and matroids acquainted me with the elegant and efficient algorithms that were sometimes possible. Jack Edmonds’ papers and a few key discussions with him drew my attention to the crucial distinction between polynomial-time and superpolynomial-time solvability. I was also influenced by Jack’s emphasis on min-max theorems as a tool for fast verification of optimal solutions, which foreshadowed Steve Cook’s definition of the complexity class NP. Another influence was George Dantzig’s suggestion that integer programming could serve as a universal format for combinatorial optimization problems.
7,714 citations
TL;DR: The procedure was originally programmed in FORTRAN for the Control Data 160 desk-size computer and was limited to te t ra t ion because subroutine recursiveness in CONTROL Data 160 FORTRan has been held down to four levels in the interests of economy.
Abstract: procedure ari thmetic (a, b, c, op); in t eger a, b, c, op; ¢ o n l m e n t This procedure will perform different order ar i thmetic operations with b and c, put t ing the result in a. The order of the operation is given by op. For op = 1 addit ion is performed. For op = 2 multiplicaLion, repeated addition, is done. Beyond these the operations are non-commutat ive. For op = 3 exponentiat ion, repeated multiplication, is done, raising b to the power c. Beyond these the question of grouping is important . The innermost implied parentheses are at the right. The hyper-exponent is always c. For op = 4 te t ra t ion, repeated exponentiat ion, is done. For op = 5, 6, 7, etc., the procedure performs pentat ion, hexation, heptat ion, etc., respectively. The routine was originally programmed in FORTRAN for the Control Data 160 desk-size computer. The original program was limited to te t ra t ion because subroutine recursiveness in Control Data 160 FORTRAN has been held down to four levels in the interests of economy. The input parameter , b, c, and op, must be positive integers, not zero; b e g i n own i n t e g e r d, e, f, drop; i f o p = 1 t h e n b e g i n a := h-4c; go t o l e n d i f o p = 2 t h e n d := 0; else d := 1; e := c; drop := op 1; for f := I s t e p 1 u n t i l e do b e g i n ari thmetic (a, b, d, drop);
3,848 citations
"A fast algorithm for Steiner trees" refers methods in this paper
...The readers are referred to the literatures with regard to the algorithms for constructing the shortest path as required in Step 1 [3-5] and for finding a minimal spanning tree as required in Step 2 and Step 3 [3, 6, 7]....
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TL;DR: A Steiner minimal tree for given points in the plane is a tree which interconnects these points using lines of shortest possible total length as mentioned in this paper, where the length of the shortest possible line is chosen.
Abstract: A Steiner minimal tree for given points $A_1 , \cdots ,A_n $ in the plane is a tree which interconnects these points using lines of shortest possible total length. In order to achieve minimum lengt...
946 citations