Showing papers on "Longest path problem published in 2001"

Vickrey prices and shortest paths: what is an edge worth?

Abstract: We solve a shortest path problem that is motivated by recent interest in pricing networks or other computational resources. Informally, how much is an edge in a network worth to a user who wants to send data between two nodes along a shortest path? If the network is a decentralized entity, such as the Internet, in which multiple self-interested agents own different parts of the network, then auction-based pricing seems appropriate. A celebrated result from auction theory shows that the use of Vickrey pricing motivates the owners of the network resources to bid truthfully. In Vickrey's scheme, each agent is compensated in proportion to the marginal utility he brings to the auction. In the context of shortest path routing, an edge's utility is the value by which it lowers the length of the shortest path, i.e., the difference between the shortest path lengths with and without the edge. Our problem is to compute these marginal values for all the edges of the network efficiently. The naive method requires solving the single-source shortest path problem up to n times, for an n-node network. We show that the Vickrey prices for all the edges can be computed in the same asymptotic time complexity as one single-source shortest path problem. This solves an open problem posed by N. Nisan and A. Ronen (1999).

Models and approximation algorithms for channel assignment in radio networks

Abstract: We consider the frequency assignment (broadcast scheduling) problem for packet radio networks. Such networks are naturally modeled by graphs with a certain geometric structure. The problem of broadcast scheduling can be cast as a variant of the vertex coloring problem (called the distance-2 coloring problem) on the graph that models a given packet radio network. We present efficient approximation algorithms for the distance-2 coloring problem for various geometric graphs including those that naturally model a large class of packet radio networks. The class of graphs considered include (r, s)-civilized graphs, planar graphs, graphs with bounded genus, etc.

FAST-SP: a fast algorithm for block placement based on sequence pair

Abstract: In this paper we present FAST-SP which is a fast block placement algorithm based on the sequence-pair placement representation. FAST-SP has two significant improvements over previous sequence-pair based placement algorithms: (1) FAST-SP translates each sequence pair to its corresponding block placement in O(n log log n) time based on a fast longest common subsequence computation. This is much faster than the traditional O(n/sup 2/) method by first constructing horizontal and vertical constraint graphs and then performing longest path computations. As a result, FAST-SP can examine more sequence pairs and obtain a better placement solution in less runtime. (2) FAST-SP can handle placement constraints such as pre-placed constraint, range constraint, and boundary constraint. No previous sequence-pair based algorithms can handle range constraint and boundary constraint. Fast evaluation in O(n log log n) time is still valid in the presence of placement constraints and a novel cost function which unifies the evaluation of feasible and infeasible sequence pairs is used. We have implemented FAST-SP and obtained excellent experimental results. For all MCNC benchmark block placement problems, we have obtained the best results ever reported in the literature (including those reported by algorithms based on O-tree and B/sup */-tree) with significantly less runtime. For example, the best known result for ami49 (36.8 mm/sup 2/) was obtained by a B/sup */-tree based algorithm using 4752 seconds, and FAST-SP obtained a better result (36.5 mm/sup 2/) in 31 seconds.

Shortest viable path algorithm in multimodal networks

Abstract: We consider an approach using label correcting techniques to find the shortest viable path from an origin to a destination, in a multimodal transportation network. A path is called viable if its sequence of modes is feasible with respect to a set of constraints. We present an ad hoc modification of the Chronological Algorithm to solve the multimodal shortest viable path problem. We show the resulting paths of an application on a network, for different number of modal transfers. Since the results are a solution set, the choice of a path depends on the user's preferences with respect to cost and number of modal transfers.

Efficient computation of delay-sensitive routes from one source to all destinations

Abstract: In this paper we describe an efficient algorithm for the constrained shortest path problem which is defined as follows. Given a directed graph with two weights on each link e, a cost l/sub e/, and a delay t/sub e/, find the cheapest path from a source to all destinations such that the delay of each path is no more than a given threshold. The constrained shortest path problem arises in quality-of-service-sensitive routing in data networks and is of particular importance in real time services. The problem formulation and the algorithmic framework presented are quite general; they apply to IP, ATM, and optical networks. Unlike previous algorithms, our algorithm generates paths from one source to all destinations. Our algorithm is strongly polynomial, and is asymptotically faster than earlier algorithms. We corroborate our analysis by a preliminary simulation study.

Approximate Distance Labeling Schemes

Abstract: We consider the problem of labeling the nodes of an n-node graph G with short labels in such a way that the distance between any two nodes u, v of G can be approximated efficiently (in constant time) by merely inspecting the labels of u and v, without using any other information. We develop such constant approximate distance labeling schemes for the classes of trees, bounded treewidth graphs, planar graphs, k-chordal graphs, and graphs with a dominating pair (including for instance interval, permutation, and AT-free graphs). We also establish lower bounds, and prove that most of our schemes are optimal in terms of the length of the labels generated and the quality of the approximation.

Planar graphs, negative weight edges, shortest paths, and near linear time

Abstract: The authors present an O(n log/sup 3/ n) time algorithm for finding shortest paths in a planar graph with real weights. This can be compared to the best previous strongly polynomial time algorithm developed by R. Lipton et al., (1978 )which ran in O(n/sup 3/2/) time, and the best polynomial algorithm developed by M. Henzinger et al. (1994) which ran in O/spl tilde/(n/sup 4/3/) time. We also present significantly improved algorithms for query and dynamic versions of the shortest path problems.

A neural network for shortest path computation

Abstract: This paper presents a new neural network to solve the shortest path problem for inter-network routing. The proposed solution extends the traditional single-layer recurrent Hopfield architecture introducing a two-layer architecture that automatically guarantees an entire set of constraints held by any valid solution to the shortest path problem. This new method addresses some of the limitations of previous solutions, in particular the lack of reliability in what concerns successful and valid convergence. Experimental results show that an improvement in successful convergence can be achieved in certain classes of graphs. Additionally, computation performance is also improved at the expense of slightly worse results.

Single-source shortest-paths on arbitrary directed graphs in linear average-case time

Abstract: The quest for a linear-time single-source shortest-path (SSSP) algorithm on directed graphs with positive edge weights is an ongoing hot research topic. While Thorup recently found an O(n + m) time RAM algorithm for undirected graphs with n nodes, m edges and integer edge weights in {0,…,2w - 1} where w denotes the word length, the currently best time bound for directed sparse graphs on a RAM is O(n + m · log log n).In the present paper we study the average-case complexity of SSSP. We give a simple algorithm for arbitrary directed graphs with random edge weights uniformly distributed in [0, 1] and show that it needs linear time O(n + m) with high probability.

Longest fault-free paths in star graphs with edge faults

Abstract: In this paper, we aim to embed longest fault-free paths in an n-dimensional star graph with edge faults. When n/spl ges/6 and there are n-3 edge faults, a longest fault-free path can be embedded between two arbitrary distinct vertices, exclusive of two exceptions in which at most two vertices are excluded. Since the star graph is regular of degree n-1, n-3 (edge faults) is maximal in the worst case. When n/spl ges/6 and there are n-4 edge faults, a longest fault-free path can be embedded between two arbitrary distinct vertices. The situation of n<6 is also discussed.

The Complexity of the Matching-Cut Problem

Abstract: Finding a cut or finding a matching in a graph are so simple problems that they are hardly considered problems at all. In this paper, by means of a reduction from the NAE3SAT problem, we prove that combining these two problems together, i.e., finding a cut whose split edges are a matching is an NP-complete problem. It remains intractable even if we impose the graph to be simple (no multiple edges allowed) or its maximum degree to be k, with k ? 4. On the contrary, we give a linear time algorithm that computes a matching-cut of a series-parallel graph. It's open whether the problem is tractable or not for planar graphs.

Search on high degree graphs

Abstract: We show that nodes of high degree tend to occur infrequently in random graphs but frequently in a wide variety of graphs associated with real world search problems. We then study some alternative models for randomly generating graphs which have been proposed to give more realistic topologies. For example, we show that Watts and Strogatz's small world model has a narrow distribution of node degree. On the other hand, Barabasi and Albert's power law model, gives graphs with both nodes of high degree and a small world topology. These graphs may therefore be useful for benchmarking. We then measure the impact of nodes of high degree and a small world topology on the cost of coloring graphs. The long tail in search costs observed with small world graphs disappears when these graphs are also constructed to contain nodes of high degree. We conjecture that this is a result of the small size of their "backbone", pairs of edges that are frozen to be the same color.

Solutions of Shortest Path Problems by Concentration Control

Abstract: In this paper, we present a concentration control method that may become a new framework of DNA computing. In this method, the concentration of each DNA is used as input and output data. By encoding the numeric data into concentrations of DNAs, a shortest path problem, which is a combinatorial optimization problem, can be solved. The method also enables local search among all candidate solutions instead of a exhaustive search. Furthermore, we can reduce the costs of some experimental operations in detecting process of DNA computing, because we have only to extract and analyze relatively intensive bands. Solutions of a shortest path problem by using a simulator and by laboratory experiments are presented to show the effectiveness of the concentration control method.

Online routing in geometric graphs

Abstract: This thesis considers the problem of finding a path from a source to a destination in a graph in which only local information is available This type of routing problem occurs regularly in robotics, parallel and distributed computing, mobile networks, and everyday life In particular, the research focuses on the case where the graph is geometric (nodes of the graph have locations in space) and planar (edges of the graph do not cross) The results in this thesis fall into four categories: (1) natural and intuitive algorithms that work on some well known and structured geometric graphs, (2) algorithms for special classes of graphs that find paths approximating shortest paths, (3) algorithms for arbitrary planar graphs, (4) algorithms for embedding graphs nicely so that simple algorithms can be used to find paths between vertices, and (5) simulation results that help to determine which routing algorithms work best in different settings In studying these problems we draw on a wide range of techniques from computer science and mathematics, improve some previous results, and report a number of open problems and directions for continuing research

Crosstalk mitigation method and system

Abstract: A method and system of crosstalk mitigation in integrated circuits employs delay change curves (DCCs) and uses targeted transistor sizing and/or buffer insertion. Based on a timing graph, a longest path capable of being shortened may be shortened by victim strengthening or aggressor weakening when a setup requirement time violation occurs and the path is capable of being shortened. The process is repeated based on an updated timing graph until the longest path is not capable of being further shortened, or there is no setup requirement time violation. Additionally, the path may be lengthened where a hold requirement time violation has occurred and the path is capable of being lengthened, by victim strengthening or aggressor weakening, until the path cannot be further lengthened or there is no hold requirement time violation. Victim strengthening is performed by altering the critical path, and aggressor weakening is performed by altering the non-critical path.

I/O-efficient algorithms for graphs of bounded treewidth

Abstract: We present I/O-efficient algorithms for the single source shortest path problem and NP-hard problems on graphs of bounded treewidth. The main step in these algorithms is a method to compute a tree-decomposition for the given graph I/O-efficiently.

Path Planning Using Intervals and Graphs

Abstract: In this paper, the problem of interest is to find a path with given endpoints such that the path lies inside a compact set S given by nonlinear inequalities. The proposed approach uses interval analysis for characterizing S by subpavings (union of boxes) and graph algorithms for finding short feasible paths. As an illustration, the problem of finding collision-free paths for a polygonal rigid object through a space that is cluttered with segment obstacles is considered.

Finding the Most Vital Node of a Shortest Path

Abstract: In an undirected, 2-node connected graph G= (V,E) with positive real edge lengths, the distance between anyt wo nodes r and s is the length of a shortest path between r and s in G. The removal of a node and its incident edges from G may increase the distance from r to s. A most vital node of a given shortest path from rto s is a node (other than r and s) whose removal from G results in the largest increase of the distance from r to s. In the past, the problem of finding a most vital node of a given shortest path has been studied because of its implications in network management, where it is important to know in advance which component failure will affect network efficiencythe most. In this paper, we show that this problem can be solved in O(m+ nlog n) time and O(m) space, where mand n denote the number of edges and the number of nodes in G.

Processing directed acyclic graphs with recursive neural networks

Abstract: Recursive neural networks are conceived for processing graphs and extend the well-known recurrent model for processing sequences. In Frasconi et al. (1998), recursive neural networks can deal only with directed ordered acyclic graphs (DOAGs), in which the children of any given node are ordered. While this assumption is reasonable in some applications, it introduces unnecessary constraints in others. In this paper, it is shown that the constraint on the ordering can be relaxed by using an appropriate weight sharing, that guarantees the independence of the network output with respect to the permutations of the arcs leaving from each node. The method can be used with graphs having low connectivity and, in particular, few outcoming arcs. Some theoretical properties of the proposed architecture are given. They guarantee that the approximation capabilities are maintained, despite the weight sharing.

An Efficient Algorithm for Finding a Maximum Weight k-Independent Set on Trapezoid Graphs

Abstract: The maximum weight k-independent set problem has applications in many practical problems like k-machines job scheduling problem, k-colourable subgraph problem, VLSI design layout and routing problem. Based on DAG (Directed Acyclic Graph) approach, an O(kn2) time sequential algorithm is designed in this paper to solve the maximum weight k-independent set problem on weighted trapezoid graphs. The weights considered here are all non-negative and associated with each of the n vertices of the graph.

Implementing parallel shortest path for parallel transportation applications

Abstract: Shortest path algorithms are required by several transportation applications; furthermore, the shortest path computation in these applications can account for a large percentage of the total execution time. Since these algorithms are very computationally intense, parallel processing can provide the compute power and memory required to solve large problems quickly. Therefore, good parallel shortest algorithms are critical for efficient parallel implementations of transportation applications. The experimental work related to parallel shortest path algorithms has focused on the development of parallel algorithms; however, very little work has been done with analyzing and understanding the performance impact of various implementation issues. In this paper, we conduct a thorough experimental analysis of parallel shortest path algorithms for sparse networks, concentrating on three implementation issues: (1) choice of shortest path algorithm, (2) termination detection and (3) network decomposition. The paper focuses on the choice of shortest path algorithm and network decomposition since the work on termination detection was published previously. We determine that all three issues affect the communication and convergence of the shortest path algorithm. Furthermore, we find that communicating the most information at a time results in the best convergence; this is contrary to most scientific applications where it is optimal to minimize communication.

Pseudo-geodesics on three-dimensional surfaces and pseudo-geodesic meshes

Abstract: We present two numerical methods to approximate the shortest path or a geodesic between two points on a three-dimensional parametric surface. The first one consists of minimizing the path length, working in the parameter domain, where the approximation class is composed of Bezier curves. In the second approach, we consider Bezier surfaces and their control net. The numerical implementation is based on finding the shortest path on the successive control net subdivisions. The convergence property of the Bezier net to the surface gives an approximation of the required shortest path. These approximations, also called pseudo-geodesics, are then applied to the creation of pseudo-geodesic meshes. Experimental results are also provided.

The Maximum Acyclic Subgraph Problem and Degree-3 Graphs

Abstract: We study the problem of finding a maximum acyclic subgraph of a given directed graph in which the maximum total degree (in plus out) is 3. For these graphs, we present: (i) a simple combinatorial algorithm that achieves an 11/12-approximation (the previous best factor was 2/3 [1]), (ii) a lower bound of 125/126 on approximability, and (iii) an approximation-preserving reduction from the general case: if for any Ɛ > 0, there exists a (17/18 + Ɛ)-approximation algorithm for the maximum acyclic subgraph problem in graphs with maximum degree 3, then there is a (1/2+δ)-approximation algorithm for general graphs for some δ > 0. The problem of finding a better-than-half approximation for general graphs is open.