Showing papers on "Longest path problem published in 2002"

Frugal path mechanisms

Abstract: We consider the problem of selecting a low cost s --- t path in a graph, where the edge costs are a secret known only to the various economic agents who own them. To solve this problem, Nisan and Ronen applied the celebrated Vickrey-Clarke-Groves (VCG) mechanism, which pays a premium to induce edges to reveal their costs truthfully. We observe that this premium can be unacceptably high. There are simple instances where the mechanism pays Θ(k) times the actual cost of the path, even if there is alternate path available that costs only (1 + e) times as much. This inspires the frugal path problem, which is to design a mechanism that selects a path and induces truthful cost revelation without paying such a high premium.This paper contributes negative results on the frugal path problem. On two large classes of graphs, including ones having three node-disjoint s - t paths, we prove that no reasonable mechanism can always avoid paying a high premium to induce truthtelling. In particular, we introduce a general class of min function mechanisms, and show that all min function mechanisms can be forced to overpay just as badly VCG. On the other hand, we prove that (on two large classes of graphs) every truthful mechanism satisfying some reasonable properties is a min function mechanism.

Structure and value synopses for XML data graphs

Abstract: All existing proposals for querying XML (e.g., XQuery) rely on a pattern-specification language that allows (1) path navigation and branching through the label structure of the XML data graph, and (2) predicates on the values of specific path/branch nodes, in order to reach the desired data elements. Optimizing such queries depends crucially on the existence of concise synopsis structures that enable accurate compile-time selectivity estimates for complex path expressions over graph-structured XML data. In this paper, we extent our earlier work on structural XSKETCH synopses and we propose an (augmented) XSKETCH synopsis model that exploits localized stability and value-distribution summaries (e.g., histograms) to accurately capture the complex correlation patterns that can exist between and across path structure and element values in the data graph. We develop a systematic XSKETCH estimation framework for complex path expressions with value predicates and we propose an efficient heuristic algorithm based on greedy forward selection for building an effective XSKETCH for a given amount of space (which is, in general, an NP-hard optimization problem). Implementation results with both synthetic and real-life data sets verify the effectiveness of our approach.

Using Multi-level Graphs for Timetable Information in Railway Systems

Abstract: In many fields of application, shortest path finding problems in very large graphs arise. Scenarios where large numbers of on-line queries for shortest paths have to be processed in real-time appear for example in traffic information systems. In such systems, the techniques considered to speed up the shortest path computation are usually based on precomputed information. One approach proposed often in this context is a space reduction, where precomputed shortest paths are replaced by single edges with weight equal to the length of the corresponding shortest path. In this paper, we give a first systematic experimental study of such a space reduction approach. We introduce the concept of multi-level graph decomposition. For one specific application scenario from the field of timetable information in public transport, we perform a detailed analysis and experimental evaluation of shortest path computations based on multi-level graph decomposition.

Computing shortest paths for any number of hops

Abstract: In this paper, we introduce and investigate a "new" path optimization problem that we denote the all hops optimal path (AHOP) problem. The problem involves identifying, for all hop counts, the optimal, i.e., minimum weight, path(s) between a given source and destination(s). The AHOP problem arises naturally in the context of quality-of-service (QoS) routing in networks, where routes (paths) need to be computed that provide services guarantees, e.g., delay or bandwidth, at the minimum possible "cost" (amount of resources required) to the network. Because service guarantees are typically provided through some form of resource allocation on the path (links) computed for a new request, the hop count, which captures the number of links over which resources are allocated, is a commonly used cost measure. As a result, a standard approach for determining the cheapest path available that meets a desired level of service guarantees is to compute a minimum hop shortest (optimal) path. Furthermore, for efficiency purposes, it is desirable to precompute such optimal minimum hop paths for all possible service requests. Providing this information gives rise to solving the AHOP problem. The paper's contributions are to investigate the computational complexity of solving the AHOP problem for two of the most prevalent cost functions (path weights) in networks, namely, additive and bottleneck weights. In particular, we establish that a solution based on the Bellman-Ford algorithm is optimal for additive weights, but show that this does not hold for bottleneck weights for which a lower complexity solution exists.

Finding a small set of longest testable paths that cover every gate

Abstract: Testing the longest path passing through each gate is important to detect small localized delay defects at a gate, e.g. resistive opens or resistive shorts. In this paper we present ATPG techniques to automatically determine the longest testable path passing through a gate or wire in the circuit without first listing all long paths passing through it. This technique is based on a graph traversal algorithm that can traverse all paths of a given length in a weighted directed acyclic graph. Experimental results for ISCAS benchmarks are also presented.

The Shortest Path Problem with Forbidden Paths

Abstract: We consider a variant of the constrained shortest path problem, where the constraints come from a set of forbidden paths (arc sequences) that cannot be part of any feasible solution. Two solution approaches are proposed for this variant. The first uses Aho and Corasick's keyword matching algorithm to filter paths produced by a k-shortest paths algorithm. The second generalizes Martins' deviation path approach for the k-shortest paths problem by merging the original graph with a state graph derived from Aho and Corasick's algorithm. Like Martins' approach, the second method amounts to a polynomial reduction of the shortest path problem with forbidden paths to a classic shortest path problem. Its significant advantage over the first approach is that it allows considering forbidden paths in more general shortest path problems such as the shortest path problem with resource constraints.

Almost all graphs with average degree 4 are 3-colorable

Abstract: The technique of using differential equations to approximate the mean path of Markov chains has proved very useful in the average-case analysis of algorithms. Here, we significantly expand the range of this technique, by showing that it can be used to handle algorithms that favor high-degree vertices. In particular, we consider the problem of 3-coloring sparse random graphs and analyze a "smoothed" version of the Brelaz heuristic. This allows us to prove that i) almost all graphs with average degree d, i.e. G(n,p=d/n), are 3-colorable for d≤ 4.03, and that ii) almost all 4-regular graphs are 3-colorable. This improves over the previous lower bound of 3.847 for the G(n,p) 3-colorability threshold and gives the first non trivial result on the 3-colorability of random regular graphs.

Grid Resource Discovery Based on a Routing-Transferring Model

Abstract: The Grid technology emerges with the need of resource sharing and cooperation in wide area Compared with the traditional single computer system, effective resource locating in Grid is difficult because of huge amount and wide-area distribution of dynamical resources In this paper, we propose a Routing-Transferring resource discovery model, which includes three basic roles: the resource requester, the resource router and the resource provider The provider sends its resource information to a router, which maintains this information in routing tables When a router receives a resource request from a requester, it checks routing tables to choose a route for it and transfer it to another router or provider We give the formalization of this model and analyze the complexity of the SD-RT (Shortest Distance Routing-Transferring) algorithm The analysis shows that the resource discovery time depends on topology (the longest path in the graph) and distribution of resources When topology and distribution are definite, the SD-RT algorithm can finda resource in the shortest time Our experiments also show that when topology is definite, the performance is determined by resource distribution, which includes two important factors: resource frequency andresource location The testing result shows that high frequency and even location of resources can reduce the resource discovery time significantly

Efficient path conditions in dependence graphs

Abstract: Program slicing combined with constraint solving is a powerful tool for software analysis. Path conditions are generated for a slice or chop, which --- when solved for the input variables --- deliver compact "witnesses" for dependences or illegal influences between program points.In this contribution we show how to make path conditions work for large programs. Aggressive engineering, based on interval analysis and BDDs, is shown to overcome the potential combinatoric explosion. Case studies and empirical data will demonstrate the usefulness of path conditions for practical program analysis.

Approximability of dense and sparse instances of minimum 2-connectivity, TSP and path problems

Abstract: We study the approximability of dense and sparse instances of the following problems: the minimum 2-edge-connected (2-EC) and 2-vertex-connected (2-VC) spanning subgraph, metric TSP with distances 1 and 2 (TSP (1,2)), maximum path packing, and the longest path (cycle) problems. The approximability of dense instances of these problems was left open in Arora et al. [3]. We characterize the approximability of all these problems by proving tight upper (approximation algorithms) and lower bounds (inapproximability). We prove that 2-EC, 2-VC and TSP (1,2) are Max SNP-hard even on 3-regular graphs, and provide explicit hardness constants, under P ≠ NP. We also improve the approximation ratio for 2-EC and 2-VC on graphs with maximum degree 3. These are the first explicit hardness results on sparse and dense graphs for these problems. We apply our results to prove bounds on the integrality gaps of LP relaxations for dense and sparse 2-EC and TSP (1,2) problems, related to the famous metric TSP conjecture, due to Goemans [17].

Experimental Evaluation of a New Shortest Path Algorithm

Abstract: We evaluate the practical efficiency of a new shortest path algorithm for undirected graphs which was developed by the first two authors. This algorithm works on the fundamental comparison-addition model.Theoretically, this new algorithm out-performs Dijkstra's algorithm on sparse graphs for the all-pairs shortest path problem, and more generally, for the problem of computing single-source shortest paths from ?(1) different sources. Our extensive experimentalanal ysis demonstrates that this is also the case in practice. We present results which show the new algorithm to run faster than Dijkstra's on a variety of sparse graphs when the number of vertices ranges from a few thousand to a few million, and when computing single-source shortest paths from as few as three different sources.

Computing Approximate Shortest Paths on Convex Polytopes

Abstract: The algorithms for computing a shortest path on a polyhedral surface are slow, complicated, and numerically unstable. We have developed and implemented a robust and efficient algorithm for computing approximate shortest paths on a convex polyhedral surface. Given a convex polyhedral surface P in \reals3 , two points s, t ź P , and a parameter \eps > 0 , it computes a path between s and t on P whose length is at most (1+\eps) times the length of the shortest path between those points. It constructs in time O(n/\sqrt \eps ) a graph of size O(1/\eps4) , computes a shortest path on this graph, and projects the path onto the surface in O(n/\eps) time, where n is the number of vertices of P . In the postprocessing step we have added a heuristic that considerably improves the quality of the resulting path.

What can we learn from the path equations?: Identifiability, constraints, equivalence

Abstract: Procedures are given for determining identified parameters, finding constraints on the covariances, and checking equivalence, in acyclic (recursive) linear path models with correlated error terms (disturbances), by inspection of the path equations, aided by simple recursions. This provides a useful and general alternative to the employment of directed acyclic graph theory for such purposes.

An optimal algorithm for solving all-pairs shortest paths on trapezoid graphs

Abstract: The shortest-paths problem is an important problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The Shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one node to another often gives the best way to route message between the nodes. This paper presents an O(n2) time algorithm for solving all pairs shortest path problems on trapezoid graphs which are extensions of interval graphs and permutation graphs. The space complexity of this algorithm is of O(n2). This problem has been solved by constructing n breadth-first search (BFS) trees with each of the n vertices as root. As the lower bound of time complexity for computing the all pairs shortest paths is known to be of O(n2), this proposed algorithm is optimal.

Choosing good paths for fast distributed reconfiguration of hexagonal metamorphic robots

Abstract: The problem addressed is the distributed reconfiguration of a metamorphic robot system composed of any number of two dimensional robots (modules) front specific initial to specific goal configurations. The initial configuration we consider is a straight chain of modules, while the goal configuration satisfies a simple admissibility condition. Reconfiguration of the modules depends on finding a contiguous path of cells, called a substrate path, that spans the goal configuration. Modules fill in this substrate path and then move along the path to fill in the remainder of the goal without collision or deadlock. In this paper, we examine the problem of finding the substrate path most likely to result in fast parallel reconfiguration, drawing on results from our previous papers (2000, 2001). Admissible goal configurations are represented as directed acyclic graphs (DAGs). We present a combination graph traversal-weighting algorithm that traverses all paths in the rooted DAG and use this algorithm to determine the best substrate path. We extend our definition of admissible substrate paths to consider admissible obstacle surfaces for reconfiguration when obstacles are present in the environment.

Optimal Hamiltonian completions and path covers for trees, and a reduction to maximum flow

Abstract: A minimum Hamiltonian completion of a graph G is a minimum-size set of edges that, when added to G, guarantee a Hamiltonian path. Finding a Hamiltonian completion has applications to frequency assignment as well as distributed computing. If the new edges are deleted from the Hamiltonian path, one is left with a minimum path cover, a minimum-size set of vertex-disjoint paths that cover the vertices of G. For arbitrary graphs, constructing a minimum Hamiltonian completion or path cover is clearly NP-hard, but there exists a linear-time algorithm for trees. In this paper we first give a description and proof of correctness for this linear-time algorithm that is simpler and more intuitive than those given previously. We show that the algorithm extends also to unicyclic graphs. We then give a new method for finding an optimal path cover or Hamiltonian completion for a tree that uses a reduction to a maximum flow problem. In addition, we show how to extend the reduction to construct, if possible, a covering of the vertices of a bipartite graph with vertex-disjoint cycles, that is, a 2-factor.

Cost allocation in shortest path games

Abstract: A class of cooperative games arising from shortest path problems is defined. These shortest path games are totally balanced and allow a population-monotonic allocation scheme. Possible methods for obtaining core elements are indicated; first, by relating to the allocation rules in taxation and bankruptcy problems, second, by constructing an explicit rule that takes opportunity costs into account by considering the costs of the second best alternative and that rewards players who are crucial to the construction of the shortest path. The core and the bargaining sets of Davis-Maschler and Mas-Colell are shown to coincide. Finally, noncooperative games arising from shortest path problems are introduced, in which players make bids or claims on paths. The core allocations of the cooperative shortest path game coincide with the payoff vectors in the strong Nash equilibria of the associated noncooperative shortest path game.

I/o-efficient algorithms for shortest path related problems

Abstract: In this thesis, we study I/O-efficient algorithms for problems related to computing shortest paths in outerplanar and planar graphs and in spanner graphs for point sets in d-dimensional space and sets of obstacles in the plane. In particular, we show in the first part of the thesis that the following problems can be solved in sorting complexity or even in a linear number of I/Os: outerplanarity testing, outerplanar embedding, planarity testing, planar embedding, computing optimal e-separators of planar and outerplanar graphs, breadth-first search; depth-first search, and single source shortest paths on planar and outerplanar graphs. In the second part of the thesis, we show that the well-separated pair decomposition of [37] can be computed in sorting complexity. We use this decomposition to construct two types of Euclidean spanners of linear size for point sets in d dimensions. The first spanner is derived in a natural manner from the well-separated pairs in the decomposition. The second spanner is a supergraph of the first spanner. The particular structure of this spanner makes it possible to construct a data structure which can be used to report spanner paths in an I/O-efficient manner. For sets of polygonal obstacles in the plane, we use a subdivision derived from the fair split tree of a point set to compute a planar Steiner spanner of the set of obstacles. Given the results from the first part of the thesis and results from [2, 100, 177]; the planarity of the graph can be exploited to compute spanner paths I/O-efficiently and to preprocess the graph so that shortest path queries can be answered and paths in the graph can be traversed in an I/O-efficient manner. As part of the results in Part I of the thesis; we present improved and much simplified algorithms for computing maximal matchings and maximal independent sets of general undirected graphs. As an additional application of the well-separated pair decomposition, which is at the core of the algorithms presented in Part II, we obtain nearly I/O-optimal algorithms for solving the K-nearest neighbor and K-closest pair problems for point sets in d dimensions.

Reversibility of the time-dependent shortest path problem

Abstract: Time-dependent shortest path problems arise in a variety of applications; e.g., dynamic traffic assignment (DTA), network control, automobile driver guidance, ship routing and airplane dispatching. In the majority of cases one seeks the cheapest (least generalized cost) or quickest (least time) route between an origin and a destination for a given time of departure. This is the “forward” shortest path problem. In some applications, however, e.g., when dispatching airplanes from airports and in DTA versions of the “morning commute problem”, one seeks the cheapest or quickest routes for a given arrival time. This is the “backward” shortest path problem. It is shown that an algorithm that solves the forward quickest path problem on a network with first-in-first-out (FIFO) links also solves the backward quickest path problem on the same network. More generally, any algorithm that solves forward (or backward) problems of a particular type is shown also to solve backward (forward) problems of a conjugate type.

Shortest Path Algorithm in Time-Dependent Networks

Abstract: The time dependent network shortest path problems are generalized from and more realistic than the classical shortest path problem, and are applicable in a wide range of fields Many transportation and communication systems can be represented by networks with travel times that are time dependent, which has lead to a need for extensive research on path planning in time dependent networks When the arc length of a system model is time dependent, the problem becomes considerably more difficult Standard shortest path algorithms, such as the Dijkstra algorithm, are not valid in this circumstance, since travel times are now time dependent variables Early researchers have found the incorrectness of the traditional shortest path algorithms by presenting examples in the time dependent networks and have afterwards given conditions to make the classical shortest path algorithms valid in the time dependent networks In this paper, based on the theoretical foundations of traditional algorithmic approaches for solving shortest path problems, we theoretically prove that the traditional shortest algorithms, such as the label setting algorithms and the famous Dijkstra algorithm, can not find the least time path in time dependent networks; and by establishing the theoretical foundations and the algorithms we solve the shortest path problems in the time dependent networks without giving any conditions We construct time dependent network models, describe shortest path problems in time dependent networks, give the properties of shortest path problems, and present theoretical foundations and the optimality conditions for solving the least time path problems in time dependent networks Next, we present the SPTDN algorithm for solving least time path problems in various scenarios, and theoretically prove that the SPTDN algorithm is valid in time dependent networks The SPTDN algorithm solves the shortest path problem with computational complexity O(nmM 2C) in time dependent networks We then implement the SPTDN algorithm with micro computer and obtain the correct experimental results Finally, we illustrate how the algorithm can be applied to solve real world problems with time dependent properties

A New Distributed Learning Automata Based Algorithm For Solving Stochastic Shortest Path Problem.

Abstract: A new distributed learning automata (DLA) based algorithm for solving stochastic shortest path problem is presented. The objective is to use DLA to nd a policy that determines a path from a source node to a destination node with minimal expected cost. It has been shown that the proposed algorithm nds the shortest path in a stochastic graph with probability as close as to unity.