Showing papers on "Longest path problem published in 2005"

Path slicing

Abstract: We present a new technique, path slicing, that takes as input a possibly infeasible path to a target location, and eliminates all the operations that are irrelevant towards the reachability of the target location. A path slice is a subsequence of the original path whose infeasibility guarantees the infeasibility of the original path, and whose feasibility guarantees the existence of some feasible variant of the given path that reaches the target location even though the given path may itself be infeasible. Our method combines the ability of program slicing to look at several program paths, with the precision that dynamic slicing enjoys by focusing on a single path. We have implemented Path Slicing to analyze possible counterexamples returned by the software model checker Blast. We show its effectiveness in drastically reducing the size of the counterexamples to less than 1% of their original size. This enables the precise verification of application programs (upto 100KLOC), by allowing the analysis to focus on the part of the counterexample that is relevant to the property being checked.

Worst-case update times for fully-dynamic all-pairs shortest paths

Abstract: We present here the first solution to the fully-dynamic all pairs shortest path problem where every update is faster than a recomputation from scratch in Ω(n3log ⁄n) time. This is for a directed graph with arbitrary non-negative edge weights. An update inserts or deletes a vertex with all incident edges. After each such vertex update, we update a complete distance matrix in O(n2.75) time.

Acceleration of shortest path and constrained shortest path computation

Abstract: We study acceleration methods for point-to-point shortest path and constrained shortest path computations in directed graphs, in particular in road and railroad networks. Our acceleration methods are allowed to use a preprocessing of the network data to create auxiliary information which is then used to speed-up shortest path queries. We focus on two methods based on Dijkstra's algorithm for shortest path computations and two methods based on a generalized version of Dijkstra for constrained shortest paths. The methods are compared with other acceleration techniques, most of them published only recently. We also look at appropriate combinations of different methods to find further improvements. For shortest path computations we investigate hierarchical multiway-separator and arc-flag approaches. The hierarchical multiway-separator approach divides the graph into regions along a multiway-separator and gathers information to improve the search for shortest paths that stretch over several regions. A new multiway-separator heuristic is presented which improves the hierarchical separator approach. The arc-flag approach divides the graph into regions and gathers information on whether an arc is on a shortest path into a given region. Both methods yield significant speed-ups of the plain Dijkstra's algorithm. The arc flag method combined with an appropriate partition and a bi-directed search achieves an average speed-up of up to 1,400 on large networks. This combination narrows down the search space of Dijkstra's algorithm to almost the size of the corresponding shortest path for long distance shortest path queries. For the constrained shortest path problem we show that goal-directed and bi-directed acceleration methods can be used both individually and in combination. The goal-directed search achieves the best speed-up factor of 110 for the constrained problem.

The robust shortest path problem with interval data via Benders decomposition

Abstract: Many real problems can be modelled as robust shortest path problems on digraphs with interval costs, where intervals represent uncertainty about real costs and a robust path is not too far from the shortest path for each possible configuration of the arc costs. In this paper we discuss the application of a Benders decomposition approach to this problem. Computational results confirm the efficiency of the new algorithm. It is able to clearly outperform state-of-the-art algorithms on many classes of networks. For the remaining classes we identify the most promising algorithm among the others, depending of the characteristics of the networks.

Resolver caching of a shortest path to a multihomed server as determined by a router

Abstract: A resolver queries a DNS server for any network addresses associated with the particular domain name. If the resolver detects a response for the DNS server with multiple network addresses for a particular domain name, then the resolver creates and sends a shortest path query to at least one router enabled to receive and respond to shortest path queries. The shortest path query indicates at least one source address and the multiple destination addresses returned by the DNS server for the particular domain name. The router detects a separate length for each path accessible between each source address and each of the multiple destination addresses. The router then orders the destination addresses from shortest path to longest path and returns the ordered destination addresses to the requesting resolver. The resolver caches the ordered network addresses in the local cache in association with the particular domain name, such that for future requests for the particular domain name, the resolver retrieves from local cache the shortest path network address as ordered by the router. In addition, the resolver, responsive to receiving the ordered destination addresses, selects the shortest path network address for a response to an application requesting the resolution of the particular domain name.

Connectedness preserving distributed coordination control over dynamic graphs

Abstract: This paper presents a solution to the limited information rendezvous problem over dynamic interaction graphs. In particular, we show how we, by adding appropriate weights to the edges in the graphs, can guarantee that the graph stays connected. In previous work on graph-based coordination, connectedness have been assumed, and this paper thus shows how to overcome this limitation even when the graphs are subject to dynamic changes.

Flight path planning of UAV based on heuristically search and genetic algorithms

Abstract: Flight path planning of UAV is a complicated optimum problem. The research in this field is usually classified as two directions: optimal path planning without considering the computation cost and real-time suboptimal path planning. Aimed at the two problems, this paper presents two methods of path planning of UAV. One method is based on heuristically search. In this method, we search the threat net by using A-star algorithm. A shortest suboptimum path is obtained, which is composed of the lines on the Voronoi diagram, and then considering the UAV's turn constraint, the path can be smoothed by geometry method. The other method is based on genetic algorithm and potential fields technology. Because potential fields, however popular, have shortcomings, viz, trap situations due to local minima, the genetic algorithm is introduced. In this method, Firstly, construct the threats' Delaunay triangle net based on the principle of nearest neighborhood and a safe planning path goes across the lines of the triangles. Then designate each point (position of a threat) on the left of the path a index 0 and on the right 1 and only designate the points of line which is not passed the same symbol. Thus, based on the directions of passing the lines of Delaunay triangle, a kind of encoding is designed on the principle of "Left 0, Right 1". At last, a global planning path can obtained by the genetic algorithm and potential fields technology. At the same time, simulation results of the two path planning methods are given.

Combining speed-up techniques for shortest-path computations

Abstract: In practice, computing a shortest path from one node to another in a directed graph is a very common task. This problem is classically solved by Dijkstra's algorithm. Many techniques are known to speed up this algorithm heuristically, while optimality of the solution can still be guaranteed. In most studies, such techniques are considered individually. The focus of our work is combination of speed-up techniques for Dijkstra's algorithm. We consider all possible combinations of four known techniques, namely, goal-directed search, bidirectional search, multilevel approach, and shortest-path containers, and show how these can be implemented. In an extensive experimental study, we compare the performance of the various combinations and analyze how the techniques harmonize when jointly applied. Several real-world graphs from road maps and public transport and three types of generated random graphs are taken into account.

Polynomial Time Approximation Schemes for MAX-BISECTION on Planar and Geometric Graphs

Abstract: The max-bisection and min-bisection problems are to find a partition of the vertices of a graph into two equal size subsets that, respectively, maximizes or minimizes the number of edges with endpoints in both subsets. We design the first polynomial time approximation scheme for the max-bisection problem on arbitrary planar graphs solving a long-standing open problem. The method of solution involves designing exact polynomial time algorithms for computing optimal partitions of bounded treewidth graphs, in particular max- and min-bisection, which could be of independent interest. Using a similar method we design also the first polynomial time approximation scheme for max-bisection on unit disk graphs (which could also be easily extended to other geometrically defined graphs).

Longest-path selection for delay test under process variation

Abstract: Under manufacturing process variation, a path through a net is called longest if there exists a process condition under which the path has the maximum delay among all paths through the net. There are often multiple longest paths for each net, due to different process conditions. In addition, a local defect, such as resistive open or a resistive bridge, increases the delay of the affected net. To detect delay faults due to local defects and process variation, it is necessary to test all longest paths through each net. Previous approaches to this problem were inefficient because of the large number of paths that are not longest. This paper presents an efficient method to generate the set of longest paths for delay test under process variation. To capture both structural and process correlation between path delays, we use linear delay functions to express path delays under process variation. A novel technique is proposed to prune paths that are not longest, resulting in a significant reduction in the number of paths. In experiments on International Symposium on Circuits and Systems (ISCAS) circuits, our number of longest paths is 1-6% of the previous best approach, with 300/spl times/ less running time.

A shortest path algorithm for real road network based on path overlap

Abstract: Existing k-shortest path algorithms has some weaknesses such as path similarity among determined paths and network expansion for describing turn prohibitions. Path similarity represents that many of the alternative paths derived from the k-shortest path algorithm are likely to share a lots of links, so they could not represent heterogeneity. The turning restrictions popularly adopted in real road may violate Bellman's principle of optimality in searching shortest path, so network expansion technique is widely used to avoid such difficulty. But, this method needs additional works to expand the network. This paper proposes a link-based shortest path algorithm to generate dissimilar paths for the travel information in real road network where exists turn prohibitions. The main merit of proposed model is to provide efficient alternative paths under consideration of overlaps among paths to alleviate the path similarity. Another merit is that it does not require extra nodes and links for expanding the network. Thus it is possible to save the time of network modification and of computer running. The algorithm is tested with some example networks and then will be expanded to a dynamic case.

Maintenance event planning and scheduling for gas turbines

Abstract: A method for scheduling a project such as the inspection and maintenance of a gas turbine utilizes a branch and bound technique for arriving at a solution The branch and bound technique is improved by using an all-pair longest path algorithm in preprocessing to tighten the set of possible start times of the tasks That set is further tightened by considering two-forbidden-task pairs; ie, pairs of tasks that cannot execute at the same time due to conflicting resource needs A hard lower bound of a branch is determined by using all-pair longest path update and two-forbidden-task pair update, reducing the need to recalculate

Replacement paths and k simple shortest paths in unweighted directed graphs

Abstract: Let G=(V,E) be a directed graph and let P be a shortest path from s to t in G. In the replacement paths problem we are required to find, for every edge e on P, a shortest path from s to t in G that avoids e. We present the first non-trivial algorithm for computing replacement paths in unweighted directed graphs (and in graphs with small integer weights). Our algorithm is Monte-Carlo and its running time is ${\tilde O}(m\sqrt{n})$. Using the improved algorithm for the replacement paths problem we get an improved algorithm for finding the ksimple shortest paths between two given vertices.

Sensitivity analysis for shortest path problems and maximum capacity path problems in undirected graphs

Abstract: This paper addresses sensitivity analysis questions concerning the shortest path problem and the maximum capacity path problem in an undirected network. For both problems, we determine the maximum and minimum weights that each edge can have so that a given path remains optimal. For both problems, we show how to determine these maximum and minimum values for all edges in O(m + K log K) time, where m is the number of edges in the network, and K is the number of edges on the given optimal path.

The directed planar reachability problem

Abstract: We investigate the s-t-connectivity problem for directed planar graphs, which is hard for L and is contained in NL but is not known to be complete We show that this problem is logspace-reducible to its complement, and we show that the problem of searching graphs of genus 1 reduces to the planar case We also consider a previously-studied subclass of planar graphs known as grid graphs We show that the directed planar s-t-connectivity problem reduces to the reachability problem for directed grid graphs A special case of the grid-graph reachability problem where no edges are directed from right to left is known as the “layered grid graph reachability problem” We show that this problem lies in the complexity class UL

Simultaneous timing-driven placement and duplication

Abstract: Logic duplication is an effective method for improving circuit performance. In this paper we present an algorithm named SPD that performs simultaneous placement and duplication to minimize the longest path delay. We introduce the notion of feasible region and super feasible region to improve the critical path monotonicity from a global perspective. We introduce a constrained gain graph to perform optimal incremental legalization under complex constraints. We also formulate a timing-constrained global redundancy removal problem and propose a heuristic solution. Our SPD algorithm outperforms the state-of-the-art FPGA placement flow (T-VPack + VPR) with an average reduction of up to 27% in longest path estimate delay and 18% in routed delay. The increase in overall runtime is less than 2% and the increase in area is less than 1%.

Influence of the link weight structure on the shortest path

Abstract: The shortest path tree rooted at a source to all other nodes is investigated in a graph with polynomial link weights tunable by the power exponent alpha. By varying alpha, different types of shortest path trees, in short alpha trees, appear. Especially, the alpha --> 0 regime that corresponds to heavily fluctuating link weights possesses a peculiar type of tree. The most important properties of this alpha --> 0 tree are derived in the asymptotic limit for large N. The application of the theoretical insights to real networks (such as the Internet) are discussed: steering flow by adjusting link weights (traffic engineering), sensitivity of link weights and modeling of the network by alpha trees.

Constrained shortest path computation

Abstract: This paper proposes and solves a-autonomy and k-stops shortest path problems in large spatial databases. Given a source s and a destination d, an a-autonomy query retrieves a sequence of data points connecting s and d, such that the distance between any two consecutive points in the path is not greater than a. A k-stops query retrieves a sequence that contains exactly k intermediate data points. In both cases our aim is to compute the shortest path subject to these constraints. Assuming that the dataset is indexed by a data-partitioning method, the proposed techniques initially compute a sub-optimal path by utilizing the Euclidean distance information provided by the index. The length of the retrieved path is used to prune the search space, filtering out large parts of the input dataset. In a final step, the optimal (a-autonomy or k-stops) path is computed (using only the non-eliminated data points) by an exact algorithm. We discuss several processing methods for both problems, and evaluate their efficiency through extensive experiments.

Maintaining Longest Paths Incrementally

Abstract: Modeling and programming tools for neighborhood search often support invariants, i.e., data structures specified declaratively and automatically maintained incrementally under changes. This paper considers invariants for longest paths in directed acyclic graphs, a fundamental abstraction for many applications. It presents bounded incremental algorithms for arc insertion and deletion which run in O(??? + |?| log|?|) time and O(???) time respectively, where |?| and ??? are measures of the change in the input and output. The paper also shows how to generalize the algorithm to various classes of multiple insertions/deletions encountered in scheduling applications. Preliminary experimental results show that the algorithms behave well in practice.

Fast generation of random connected graphs with prescribed degrees

Abstract: We address here the problem of generating random graphs uniformly from the set of simple connected graphs having a prescribed degree sequence. Our goal is to provide an algorithm designed for practical use both because of its ability to generate very large graphs (efficiency) and because it is easy to implement (simplicity). We focus on a family of heuristics for which we prove optimality conditions, and show how this optimality can be reached in practice. We then propose a different approach, specifically designed for typical real-world degree distributions, which outperforms the first one. Assuming a conjecture which we state and argue rigorously, we finally obtain an log-linear algorithm, which, in spite of being very simple, improves the best known complexity.

PathStack ¬ : a holistic path join algorithm for path query with not-predicates on XML data

Abstract: The evaluation of path queries forms the basis of complex XML query processing which has attracted a lot of research attention. However, none of these works have examined the processing of more complex queries that contain not-predicates. In this paper, we present the first study on evaluating path queries with not-predicates. We propose an efficient holistic path join algorithm, PathStack¬, which has the following advantages: (1) it requires only one scan of the relevant data to evaluate path queries with not-predicates; (2) it does not generate any intermediate results; and (3) its memory space requirement is bounded by the longest path in the input XML document. We also present an improved variant of PathStack¬ that further minimizes unnecessary computations.