Showing papers on "Longest path problem published in 2003"
TL;DR: A rather simple algorithm for the single source shortest path problem, which the authors call Delta-stepping, can be implemented very efficiently in sequential and parallel setting for a large class of graphs and achieves significant speedup on real machines.
260 citations
TL;DR: In this article, a simple extension of Pearl's d-separation criterion, called mseparation, is applied to acycfic directed mixed graphs, in which directed edges and bi-directed edges may occur.
Abstract: We consider acycfic directed mixed graphs, in which directed edges (x->y) and bi-directed edges (x 4-+ y) may occur. A simple extension of Pearl's d-separation criterion, called m-separation, is applied to these graphs. We introduce a local Markov property which is equivalent to the global property resulting from the m-separation criterion for arbitrary distributions.
256 citations
09 Jun 2003
TL;DR: A fully dynamic algorithm for general directed graphs with non-negative real-valued edge weights that supports any sequence of operations in amortized time per update and unit worst-case time per distance query, where n is the number of vertices.
Abstract: We study novel combinatorial properties of graphs that allow us to devise a completely new approach to dynamic all pairs shortest paths problems. Our approach yields a fully dynamic algorithm for general directed graphs with non-negative real-valued edge weights that supports any sequence of operations in O(n2) amortized time per update and unit worst-case time per distance query, where n is the number of vertices. We can also report shortest paths in optimal worst-case time. These bounds improve substantially over previous results and solve a long-standing open problem. Our algorithm is deterministic and uses simple data structures.
189 citations
01 Jan 2003
TL;DR: Experimental results show that the path planning using the optimized path graph is an order of magnitude faster than the quadtree approach while the length of the path generated by the proposed method is almost the same as that of the course generated byThe quadtree.
Abstract: A fast path planning method by optimization of a path graph for both efficiency and accuracy is proposed. A conventional quadtree-based path planning approach is simple, robust, and efficient. However, it has two limitations. We propose a path graph optimization technique employing a compact mesh representation. A world space is triangulated into a base mesh and the base mesh is simplified to a compact mesh. The compact mesh representation is object-dependent; the positions of vertexes of the mesh are optimized according to the curvatures of the obstacles. The compact mesh represents the obstacles as accurately as the quadtree even though using much fewer vertexes than the quadtree. The compact mesh distributes vertexes in a free space in a balanced way by ensuring that the lengths of edges are below an edge length threshold. An optimized path graph is extracted from the compact mesh. An iterative vertex pushing method is proposed to include important obstacle boundary edges in the path graph. Dijkstra's shortest path searching algorithm is used to search the shortest path in the path graph. Experimental results show that the path planning using the optimized path graph is an order of magnitude faster than the quadtree approach while the length of the path generated by the proposed method is almost the same as that of the path generated by the quadtree.
103 citations
TL;DR: 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 efficiency the most.
102 citations
TL;DR: This paper adopts a probabilistic approach in which the state parameters (available bandwidth and delay) are characterized by random variables and provides efficient solutions for the most-probable bandwidth-delay-constrained path (MP-BDCP), and provides two complementary solutions for MP-DCP.
Abstract: One of the key issues in any quality-of-service (QoS) routing framework is how to compute a path that satisfies given QoS constraints. In this paper, we focus on the path computation problem subject to the bandwidth and delay constraints. This problem can be easily solved if the exact state information is available to the node performing the path computation function. In practice, however, nodes have only imprecise knowledge of the network state. The reliance on outdated information and treating this information as exact can significantly degrade the effectiveness of the path selection algorithm. To address this problem, we adopt a probabilistic approach in which the state parameters (available bandwidth and delay) are characterized by random variables. The goal is then to find the most-probable bandwidth-delay-constrained path (MP-BDCP). We provide efficient solutions for the MP-BDCP problem by decomposing it into two subproblems: the most-probable delay-constrained path (MP-DCP) problem and the most-probable bandwidth-constrained path (MP-BCP) problem. MP-DCP by itself is known to be NP-hard, necessitating the use of approximate solutions. By employing the central limit theorem and Lagrange relaxation techniques, we provide two complementary solutions for MP-DCP. These solutions are found to be highly efficient, requiring on average a few iterations of Dijkstra's shortest path algorithm. As for MP-BCP, it can be easily transformed into a variant of the shortest path problem. Our solutions for MP-DCP and MP-BCP are then combined to address the MP-BDCP problem by obtaining a set of near-nondominated paths. Decision makers can then select one or more of these paths based on a specific utility function. Extensive simulations are used to demonstrate the efficiency of the proposed algorithmic solutions and, more generally, to contrast the probabilistic path selection approach with the standard threshold-based triggered approach.
100 citations
16 Sep 2003
TL;DR: This work shows that the edge-disjoint paths problem is W[1]-hard, and it is shown that the problem remains W[ 1]-hard even if the demand graph consists of two sets of parallel edges.
Abstract: Given a graph and pairs s i t i of terminals, the edge-disjoint paths problem is to determine whether there exist s i t i paths that do not share any edges. We consider this problem on acyclic digraphs. It is known to be NP-complete and solvable in time n O(k) where k is the number of paths. It has been a long-standing open question whether it is fixed-parameter tractable in k. We resolve this question in the negative: we show that the problem is W[1]-hard. In fact it remains W[1]-hard even if the demand graph consists of two sets of parallel edges.
65 citations
29 Sep 2003
TL;DR: Using the concepts of Constraint Programming, this work provides a theoretical study of cost-based filtering for shorter path constraints on acyclic, on undirected and on directed graphs that do not contain negative cycles.
Abstract: Many real world problems, e.g. in personnel scheduling and transportation planning, can be modeled naturally as Constrained Shortest Path Problems (CSPPs), i.e., as Shortest Path Problems with additional constraints. A well studied problem in this class is the Resource Constrained Shortest Path Problem. Reduction techniques are vital ingredients of solvers for the CSPP, that is frequently NP-hard, depending on the nature of the additional constraints. Viewed as heuristics, until today these techniques have not been studied theoretically with respect to their efficiency, i.e., with respect to the relation of filtering power and running time. Using the concepts of Constraint Programming, we provide a theoretical study of cost-based filtering for shorter path constraints on acyclic, on undirected and on directed graphs that do not contain negative cycles.
58 citations
Journal Article•
TL;DR: It is proved that superlinear lower bounds for some shortest path problems in directed graphs, where no such bounds were previously known, can be solved in near-linear time for undirected graphs.
Abstract: We prove super-linear lower bounds for some shortest path problems in directed graphs, where no such bounds were previously known. The central problem in our study is the replacement paths problem: Given a directed graph G with non-negative edge weights, and a shortest path P = {e 1 ,e 2 ,..., e p } between two nodes s and t, compute the shortest path distances from s to t in each of the p graphs obtained from G by deleting one of the edges e i . We show that the replacement paths problem requires Ω(mn) time in the worst case whenever m = O(nn). Our construction also implies a similar lower bound for the k shortest paths problem for a broad class of algorithms that includes all known algorithms for the problem. To put our lower bound in perspective, we note that both these problems (replacement paths and k shortest paths) can be solved in near linear time for undirected graphs.
56 citations
TL;DR: The k-path partition problem is a generalization of the Hamiltonian path problem and the problem of partitioning a graph into the minimum number of paths, and it is shown that it is NP-complete on the class of comparability graphs even for k=3.
53 citations
TL;DR: A new efficient algorithm for circular shortest path determination on a u-by-v trellis in O(u^{1.6} v)$ average time is presented, which uses a binary search tree on the set of path endpoints and a best-first Branch and Bound search technique to efficiently obtain the global minimum circular path.
TL;DR: It is proved that the problem of finding a maximum stable set can be solved in polynomial time in the class of (P5,Km,m)-free graphs for any fixed m.
TL;DR: A polynomial-time algorithm is presented that establishes the performance ratio O(n(log log n/log n)2) for the longest path problem, where n denotes the number of vertices in the graph.
Abstract: We consider the problem of finding a long, simple path in an undirected graph. We present a polynomial-time algorithm that finds a path of length $\Omega\bigl((\log L/\log\log L)^2\bigr)$, where L denotes the length of the longest simple path in the graph. This establishes the performance ratio O(n(log log n/log n)2) for the longest path problem, where n denotes the number of vertices in the graph.
TL;DR: A prediction formula can be used by a compiler to automatically determine the tiling parameters that minimizes the execution time of tiling loops, and the notion of rise is explained, a measure of the relationship between the shape of the tiles and theshape of the iteration space generated by the loop nest.
Abstract: Many computationally-intensive programs, such as those for differential equations, spatial interpolation, and dynamic programming, spend a large portion of their execution time in multiply-nested loops that have a regular stencil of data dependences. Tiling is a well-known compiler optimization that improves performance on such loops, particularly for computers with a multilevel hierarchy of parallelism and memory. Most previous work on tiling is limited in at least one of the following ways: they only handle nested loops of depth two, orthogonal tiling, or rectangular tiles. In our work, we tile loop nests of arbitrary depth using polyhedral tiles. We derive a prediction formula for the execution time of such tiled loops, which can be used by a compiler to automatically determine the tiling parameters that minimizes the execution time. We also explain the notion of rise, a measure of the relationship between the shape of the tiles and the shape of the iteration space generated by the loop nest. The rise is a powerful tool in predicting the execution time of a tiled loop. It allows us to reason about how the tiling affects the length of the longest path of dependent tiles, which is a measure of the execution time of a tiling. We use a model of the tiled iteration space that allows us to determine the length of the longest path of dependent tiles using linear programming. Using the rise, we derive a simple formula for the length of the longest path of dependent tiles in rectilinear iteration spaces, a subclass of the convex iteration spaces, and show how to choose the optimal tile shape.
09 Jun 2003
TL;DR: This work presents a new algorithm for answering short path queries in planar graphs, which significantly improves the previous result of D. Eppstein and can be applied to compute the girth of a planar graph and a corresponding shortest cycle in O(|V|) time provided that the constant bound on theGirth is known.
Abstract: We present a new algorithm for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G=(V,E) one can build in O(|V|) time a data structure, which allows to check in O(1) time whether two given vertices are distant by at most k in G and if so a shortest path between them is returned. This significantly improves the previous result of D. Eppstein [5] where after a linear preprocessing the queries are answered in O(log |V|) time. Our approach can be applied to compute the girth of a planar graph and a corresponding shortest cycle in O(|V|) time provided that the constant bound on the girth is known.Our results can be easily generalized to other wide classes of graphs~--~for instance we can take graphs embeddable in a surface of bounded genus or graphs of bounded tree-width.
TL;DR: The aim is to estimate the entropy, Hausdorff dimension and complexity for a path in a general sub-Riemannian manifold and states that complexity and entropy are equivalent for generic paths.
Abstract: We characterize the geometry of a path in a sub-Riemannian manifold using two metric invariants, the entropy and the complexity. The entropy of a subset A of a metric space is the minimum number of balls of a given radius ? needed to cover A. It allows one to compute the Hausdorff dimension in some cases and to bound it from above in general. We define the complexity of a path in a sub-Riemannian manifold as the infimum of the lengths of all trajectories contained in an ?-neighborhood of the path, having the same extremities as the path. The concept of complexity for paths was first developed to model the algorithmic complexity of the nonholonomic motion planning problem in robotics. In this paper, our aim is to estimate the entropy, Hausdorff dimension and complexity for a path in a general sub-Riemannian manifold. We construct first a norm || * ||? on the tangent space that depends on a parameter ? > 0. Our main result states then that the entropy of a path is equivalent to the integral of this ?-norm along the path. As a corollary we obtain upper and lower bounds for the Hausdorff dimension of a path. Our second main result is that complexity and entropy are equivalent for generic paths. We give also a computable sufficient condition on the path for this equivalence to happen. © EDP Sciences, SMAI 2003.
Journal Article•
TL;DR: In this paper, the authors extend Borovkov's renovation theory to obtain criteria for coup- ling-convergence of stochastic processes that do not necessarily obey recchastic recursions.
Abstract: We extend Borovkov's renovation theory to obtain criteria for coup- ling-convergence of stochastic processes that do not necessarily obey stochastic recursions. The results are applied to an "infinite bin model", a particular system that is an abstraction of a stochastic ordered graph, i.e., a graph on the integers that has (i,j), i < j, as an edge, with probability p, independently from edge to edge. A question of interest is an estimate of the length Ln of a longest path between two vertices at distance n. We give sharp bounds on C = limn!1(Ln/n). This is done by first constructing the unique stationary version of the infinite bin model, using extended renovation theory. We also prove a functional law of large numbers and a functional central limit theorem for the infinite bin model. Finally, we discuss perfect simulation, in connection to extended renovation theory, and as a means for simulating the particular stochastic models considered in this paper.
27 Feb 2003
TL;DR: It is shown that the replacement paths problem requires ?(m?n) time in the worst case whenever m = O(n?n), which implies a similar lower bound for the k shortest paths problem for a broad class of algorithms that includes all known algorithms for the problem.
Abstract: We prove super-linear lower bounds for some shortest path problems in directed graphs, where no such bounds were previously known. The central problem in our study is the replacement paths problem: Given a directed graph G with non-negative edge weights, and a shortest path P = {e1, e2, ..., ep} between two nodes s and t, compute the shortest path distances from s to t in each of the p graphs obtained from G by deleting one of the edges ei. We show that the replacement paths problem requires ?(m?n) time in the worst case whenever m = O(n?n). Our construction also implies a similar lower bound for the k shortest paths problem for a broad class of algorithms that includes all known algorithms for the problem. To put our lower bound in perspective, we note that both these problems (replacement paths and k shortest paths) can be solved in near linear time for undirected graphs.
09 Nov 2003
TL;DR: The study shows that with a single longest path, the delay produced by these algorithms is from 10% to 18% longer than the optima on the average, and from 34% to 53% longer in the worst case.
Abstract: This work studies the optimality and stability of timing-driven placement algorithms. The contributions of this work include two parts: 1) We develop an algorithm for generating synthetic examples with known optimal delay for timing driven placement (T-PEKO). The examples generated by our algorithm can closely match the characteristics of real circuits. 2) Using these synthetic examples with known optimal solutions, we studied the optimality of several timing-driven placement algorithms for FPGAs by comparing their solutions with the optimal solutions, and their stability by varying the number of longest paths in the examples. Our study shows that with a single longest path, the delay produced by these algorithms is from 10% to 18% longer than the optima on the average, and from 34% to 53% longer in the worst case. Furthermore, their solution quality deteriorates as the number of longest paths increases. For examples with more than 5 longest paths, their delay is from 23% to 35% longer than the optima on the average, and is from 41% to 48% longer in the worst case.
TL;DR: This paper introduces a new kind of network called on–off time-switch network in which an arc is associated with a sequence of windows with status “on” or “off” analogous to “go”or “wait”, and shows that both networks have the same mathematical structure in the sense that a path in one network corresponds to a route in the other one.
TL;DR: A symbolic approach to the shortest path problem is proposed using an interpretation of a shortest path algorithm as an algebraic expression and an algorithm generating expressions of O(n2) length for Fibonacci graphs is proposed.
Abstract: The paper investigates relationship between algebraic expressions and graphs. Through out the paper we consider two kinds of digraphs: series-parallel graphs and Fibonacci graphs (which give a generic example of non-series-parallel graphs). Motivated by the fact that the most compact expressions of series-parallel graphs are read-once formulae, and, thus, of O(n) length, we propose an algorithm generating expressions of O(n2) length for Fibonacci graphs. A serious effort was made to prove that this algorithm yields expressions with a minimum number of terms. Using an interpretation of a shortest path algorithm as an algebraic expression, a symbolic approach to the shortest path problem is proposed.
07 Dec 2003
TL;DR: This dual network provides an alternative proof of the reversibility property of tandem queueing networks under communication blocking and extends to other systems.
Abstract: An important class of discrete event systems, tandem queueing networks, are considered and formulated as mathematical programming problems where the constraints represent the system dynamics. The dual of the mathematical programming formulation is a network flow problem where the longest path equals the makespan of n jobs. This dual network provides an alternative proof of the reversibility property of tandem queueing networks under communication blocking. The approach extends to other systems.
TL;DR: It is shown that an optimal value function is a unique solution to an optimality equation and find an optimal stationary policy and a value iteration method is given.
Abstract: We consider a minimizing risk model in a stochastic shortest path problem in which for each node of a graph we select a probability distribution over the set of successor nodes so as to reach a given target node with minimum threshold probability. We formulate such a problem as undiscounted finite Markov decision processes. We show that an optimal value function is a unique solution to an optimality equation and find an optimal stationary policy. A value iteration method is also given.
TL;DR: In this paper, for the general problem, the NP-completeness is shown, and for the cases of trees and the case of single source-terminal, polynomial-time algorithms are presented.
Abstract: The shortest-path problem in a network is to find shortest paths between some specified sources and terminals when the lengths of edges are given. This paper studies a reverse problem: how to shorten the lengths of edges with as less cost as possible such that the distances between specified sources and terminals are reduced to the required bounds. This can be regarded as a routing speed-up model in transportation networks. In this paper, for the general problem, the NP-completeness is shown, and for the case of trees and the case of single source-terminal, polynomial-time algorithms are presented.
TL;DR: The main result is that the stable set problem in the class of P5-free graphs is polynomially equivalent to the problem of finding augmenting graphs containing a P4, and is applied to detect a new infinite family of graph classes where the problem has a polynomial time solution.
TL;DR: In this paper, it was shown that the problem of finding and reporting the smallest number of vertex-disjoint paths that cover the vertices of a graph can be solved time and work-optimally for cographs.
TL;DR: Based on a Directed Acyclic Graph approach, an O(kn 2) time sequential algorithm is presented to solve the maximum weight k-independent set problem on weighted-permutation graphs.
Abstract: Based on a Directed Acyclic Graph approach, an O(kn 2) time sequential algorithm is presented to solve the maximum weight k-independent set problem on weighted-permutation graphs. The weights considered here are all non-negative and associated with each of the n vertices of the graph. This problem has many applications in practical problems like k-machines job scheduling problem, k-colourable subgraph problem, VLSI design and routing problem.
17 Nov 2003
TL;DR: Two techniques are presented that add scheduling steps to the branch of a conditional construct with fewer scheduling steps that 'balances' or equalises the number of scheduling steps in the conditional branches and increases the scope for application of speculative code motions.
Abstract: The quality of high-level synthesis results for designs with complex and nested conditionals and loops can be improved significantly by employing speculative code motions. Two techniques are presented that add scheduling steps to the branch of a conditional construct with fewer scheduling steps. This 'balances' or equalises the number of scheduling steps in the conditional branches and increases the scope for application of speculative code motions. These branch balancing techniques have been applied 'dynamically' during scheduling. The authors have implemented algorithms for dynamic branch balancing techniques in a C-to-VHDL high-level synthesis framework called Spark. The utility of these techniques is demonstrated by experimental results on four designs derived from two moderately complex applications, namely, MPEG-1 and the GIMP image processing tool. These results show that the two branch balancing techniques can reduce the cycles on the longest path through the design by up to 38% and the number of states in the controller by up to 37%.
09 Feb 2003
TL;DR: A new all-pairs algorithm with a time complexity of O(mn + nr log r), where r is the number of acyclic parts resulting when the graph is decomposed into acyClic parts, is presented.
Abstract: Dijkstra's algorithm solves the single-source shortest path problem on any directed graph in O(m + n log n) time when a Fibonacci heap is used as the frontier set data structure. Here n is the number of vertices and m is the number of edges in the graph. If the graph is nearly acyclic, other algorithms can achieve a time complexity lower than that of Dijkstra's algorithm. Abuaiadh and Kingston gave a single-source shortest path algorithm for nearly acyclic graphs with O(m + n log t) time complexity, where the new parameter, t, is the number of delete-min operations performed in priority queue manipulation. If the graph is nearly acyclic, then t is expected to be small, and the algorithm out-performs Dijkstra's algorithm. Takaoka, using a different definition for acyclicity, gave an algorithm with O(m + n log k) time complexity. In this algorithm, the new parameter, k, is the maximum cardinality of the strongly connected components in the graph.The generalised single-source (GSS) problem allows an initial distance to be defined at each vertex in the graph. Decomposing a graph into r trees allows the GSS problem to be solved within O(m + r logr) time. This paper presents a new all-pairs algorithm with a time complexity of O(mn + nr log r), where r is the number of acyclic parts resulting when the graph is decomposed into acyclic parts. The acyclic decomposition used is setwise unique and can be computed in O(mn) time. If the decomposition has been pre-calculated, then GSS can be solved within O(m + r log r) time whenever edge-costs in the graph change. A second new all-pairs algorithm is presented, with O(mn + nr2) worst-case time complexity, where r is the number of vertices in a pre-calculated feedback vertex set for the nearly acyclic graph. For certain graphs, these new algorithms offer an improvement on the time complexity of the previous algorithms.
TL;DR: This paper first proposes a characterization of all connected P5-free augmenting graphs, and then uses this characterization to detect families of subclasses of P 5-free graphs where the maximum stable set problem has a polynomial time solution.