Approximate Shortest Paths Avoiding a Failed Vertex: Near Optimal Data Structures for Undirected Unweighted Graphs

Let G=(V,E) be an undirected unweighted graph. A path between any two vertices u,v∈V is said to be t-approximate shortest path if its length is at most t times the length of the shortest path between u and v. We address the problem of building a compact data structure which can efficiently answer the following query for any u,v,x∈V and t>1: Report t-approximate shortest path between u and v when vertex x fails. We present data structures for the single source as well as all-pairs versions of this problem. The query time guaranteed by our data structures is optimal up to a constant factor. Moreover, the size of each of them nearly matches the size of the corresponding data structure with no failures.     

Improved algorithms for the k simple shortest paths and the replacement paths problems

Given a directed, non-negatively weighted graph G=(V,E) and s,t∈V, we consider two problems. In the k simple shortest paths problem, we want to find the k simple paths from s to t with the k smallest weights. In the replacement paths problem, we want the shortest path from s to t that avoids e, for every edge e in the original shortest path from s to t. The best known algorithm for the k simple shortest paths problem has a running of O(k(mn+n²logn)). For the replacement paths problem the best known result is the trivial one running in time O(mn+n²logn).In this paper we present two simple algorithms for the replacement paths problem and the k simple shortest paths problem in weighted directed graphs (using a solution of the All Pairs Shortest Paths problem). The running time of our algorithm for the replacement paths problem is O(mn+n²loglogn). For the k simple shortest paths we will perform O(k) iterations of the second simple shortest path (each in O(mn+n²loglogn) running time) using a useful property of Roditty and Zwick [L. Roditty, U. Zwick, Replacement paths and k simple shortest paths in unweighted directed graphs, in: Proc. of International Conference on Automata, Languages and Programming (ICALP), 2005, pp. 249-260]. These running times immediately improve the best known results for both problems over sparse graphs.Moreover, we prove that both the replacement paths and the k simple shortest paths (for constant k) problems are not harder than APSP (All Pairs Shortest Paths) in weighted directed graphs.     

基于SPB树的公路网络最短路径查询

针对在线地图服务和路程安排等领域中的点对点最短路径查询方法,提出一种新的数据结构——最短路径B+树(SPB树),以有效存储预先计算好的点空间信息和与之对应的最短路径信息.实验结果证明,利用SPB树在公路网络上进行最短路径查询比经典的Dijkstra算法最高快出3个数量级.     

Parallel Algorithm for Shortest Pairs of Edge-Disjoint Paths

LetG= (V,E) be a directed graph having a nonnegative cost associated with each edge. LetsVbe a special vertex called the source andWVbe a set of other vertices called sinks inG. In this paper, a parallel algorithm is proposed for finding a pair of edge-disjoint paths fromsto each possible sinktWsuch that the sum of the costs of the two paths is minimized. This algorithm has processor and time complexities same as those needed to find shortest paths fromsto all sinkstW, i.e.,n³/lognprocessors andO(log²n) time.     

Maximum flow and critical cutset as descriptors of multi-state systems with randomly capacitated components

Let G = (V, E, s, t) denote a directed network with node set V, arc set E = {1,…, n}, source node s and sink node t. Let Γ denote the set of all minimal s−t cutsets and b₁(τ), …, B_n(τ), the random arc capacities at time τ with known joint probability distribution function. Let Λ(τ) denote the maximum s−t flow at time τ and D(τ), the corresponding critical minimal s−t cutset. Let Ω denote a set of minimal s−t cutsets. This paper describes a comprehensive Monte Carlo sampling plan for efficiently estimating the probability that D(τ)εΩ-Γ and x<λ(τ)y at time τ and the probability that D(τ) Ω given that x < Λ(τ) y at time τ. The proposed method makes use of a readily obtainable upper bound on the probability that Λ(τ) > x to gain its computational advantage. Techniques are described for computing confidence intervals and credibility measures for assessing that specified accuracies have been achieved. The paper includes an algorithm for performing the Monte Carlo sampling experiment, an example to illustrate the technique and a listing of all steps needed for implementation.     

Reduction approaches for robust shortest path problems

We investigate the uncertain versions of two classical combinatorial optimization problems, namely the Single-Pair Shortest Path Problem (SP-SPP) and the Single-Source Shortest Path Problem (SS-SPP). The former consists of finding a path of minimum length connecting two specific nodes in a finite directed graph G; the latter consists of finding the shortest paths from a fixed node to the remaining nodes of G. When considering the uncertain versions of both problems we assume that cycles may occur in G and that arc lengths are (possibly degenerating) nonnegative intervals. We provide sufficient conditions for a node and an arc to be always or never in an optimal solution of the Minimax regret Single-Pair Shortest Path Problem (MSP-SPP). Similarly, we provide sufficient conditions for an arc to be always or never in an optimal solution of the Minimax regret Single-Source Shortest Path Problem (MSS-SPP). We exploit such results to develop pegging tests useful to reduce the overall running time necessary to exactly solve both problems.     

Semigroup generation and stabilization by A-bounded feedback perturbations

Let A be a generator of a strongly continuous semigroup of operators, and assume that C and H are operators such that A + CH generates a strongly continuous semigroup S_H(t) on X. Let λ₀ be a real number in the resolvent set of A, and let ε [−1, 1]. Then there are some fairly unrestrictive conditions under which A+(λ₀ − A)CH(λ₀ − A)⁻ also generates a strongly continuous semigroup S_K(t) on X which has the same exponential growth rate as S_H(t). Given an input operator B, we can use this to identify a class of feedback perturbations K such that A + BK generates a strongly continuous semigroup. We can also use this result to identify classes of feedbacks which can and cannot uniformly stabilize a system. For example, we show that if the control on a cantilever beam in the state space H₀²[0, 1] × L²[0, 1] is a moment force on the free end, then we cannot stabilize the beam with an A^−1/2-bounded feedback, but we can find an A^−1/4-bounded feedback, for any > 0, which does stabilize the beam.     

A Simpler and More Efficient Algorithm for the Next-to-Shortest Path Problem

Given an undirected graph G=(V,E) with positive edge lengths and two vertices s and t, the next-to-shortest path problem is to find an st-path which length is minimum amongst all st-paths strictly longer than the shortest path length. In this paper we show that the problem can be solved in linear time if the distances from s and t to all other vertices are given. Particularly our new algorithm runs in O(|V|log|V|+|E|) time for general graphs, which improves the previous result of O(|V|²) time and takes only linear time for unweighted graphs, planar graphs, and graphs with positive integer edge lengths.     

An Algorithm for Computing a Convex and Simple Path of Bounded Curvature in a Simple Polygon

Abstract. In this paper we study the collision-free path planning problem for a point robot, whose path is of bounded curvature (i.e., constrained to have curvature at most 1), moving in the plane inside an n -sided simple polygon P . Given two points s and t inside P and two directions of travel, one at s and one at t , the problem is to compute a convex and simple path of bounded curvature inside P from s to t consisting of straight-line segments and circular arcs such that (i) the radius of each circular arc is at least 1, (ii) each segment on the path is the tangent between the two consecutive circular arcs on the path, (iii) the given initial direction at s is tangent to the path at s and (iv) the given final direction at t is tangent to the path at t . We propose an O(n ⁴ ) time algorithm for this problem. Using the notion of complete visibility, P is pruned to another polygon P' such that any convex and simple path from s to t inside P also remains inside P' . Then our algorithm constructs the locus of center of a circle of unit radius translating along the boundary of P' and, using this locus, the algorithm constructs a convex and simple path of bounded curvature. Our algorithm is based on the relationship between the Euclidean shortest path, link paths and paths of bounded curvature, and uses several properties derived here on convex and simple paths of bounded curvature. We also show that the path computed by our algorithm can be transformed in O(n) time to a minimal convex and simple path of bounded curvature. Using this transformation and two necessary conditions proposed here for the shortest convex and simple path of bounded curvature, a minimal bounded curvature path is located in O(n ⁴ ) time whose length, except in special situations involving arcs of length greater than π , is at most twice the length of a shortest convex and simple path of bounded curvature.     

On Approximate Jumbled Pattern Matching in Strings

Given a string s, the Parikh vector of s, denoted p(s), counts the multiplicity of each character in s. Searching for a match of a Parikh vector q in the text s requires finding a substring t of s with p(t)=q. This can be viewed as the task of finding a jumbled (permuted) version of a query pattern, hence the term Jumbled Pattern Matching. We present several algorithms for the approximate version of the problem: Given a string s and two Parikh vectors u,v (the query bounds), find all maximal occurrences in s of some Parikh vector q such that u≤q≤v. This definition encompasses several natural versions of approximate Parikh vector search. We present an algorithm solving this problem in sub-linear expected time using a wavelet tree of s, which can be computed in time O(n) in a preprocessing phase. We then discuss a Scrabble-like variation of the problem, in which a weight function on the letters of s is given and one has to find all occurrences in s of a substring t with maximum weight having Parikh vector p(t)≤v. For the case of a binary alphabet, we present an algorithm which solves the decision version of the Approximate Jumbled Pattern Matching problem in constant time, by indexing the string in subquadratic time.     

Shortest hop multipath algorithm for wireless sensor networks

Shortest hop or distance path is one of the most common methods used for relaying messages in a wide variety of networks. It provides an efficient message relaying to destination in terms of energy and time. There are many algorithms for constructing shortest hop or distance path. However, according to our knowledge, no algorithm for constructing a shortest hop multipath for wireless sensor networks (WSNs) has yet been proposed in the literature. In this paper, we propose a novel distributed shortest hop multipath algorithm for WSNs in order to generate energy efficient paths for data dissemination or routing. The proposed algorithm generates shortest hop braided multipath to be used for fault-tolerance or load-balancing. It guarantees the BFS tree and generates near optimal paths in O(V.D+V) message complexity and O(D²) time complexity regarding the communication costs towards the sink after termination of algorithm.     

GIS中最短路径的求取及三维可视化

最短路径是GIS网络分析的主要问题之一,而经典的Dijkstra算法是目前解决这一问题的理论基础。论文在Dijkstra算法的基础上,根据Shape矢量地图的自身特点,对算法的存储结构和算法过程进行了相应的设计,完成了最短路径的显示。并且最终分别利用一种求交和插值算法,结合OpenGL实现了最短路径在三维地形(基于规则格网)中的可视化,从而为用户提供了一个更加真实沉浸的可视化环境。     

An Efficient Scaling Algorithm for the Minimum Weight Bibranching Problem

Let G=(VG,AG) be a digraph and let S ⊔ T be a bipartition of VG. A bibranching is a subset B⊆AG such that for each node s∈S there exists a directed s–T path in B and, vice versa, for each node t∈T there exists a directed S–t path in B.     

<Emphasis Type="Italic">f</Emphasis>-Sensitivity Distance Oracles and Routing Schemes

An f-sensitivity distance oracle for a weighted undirected graph G(V,E) is a data structure capable of answering restricted distance queries between vertex pairs, i.e., calculating distances on a subgraph avoiding some forbidden edges. This paper presents an efficiently constructible f-sensitivity distance oracle that given a triplet (s,t,F), where s and t are vertices and F is a set of forbidden edges such that |F|≤f, returns an estimate of the distance between s and t in G(V,E∖F). For an integer parameter k≥1, the size of the data structure is O(fkn ^1+1/k log (nW)), where W is the heaviest edge in G, the stretch (approximation ratio) of the returned distance is (8k−2)(f+1), and the query time is O(|F|⋅log ² n⋅log log n⋅log log d), where d is the distance between s and t in G(V,E∖F).     

A new algorithm for shortest paths among obstacles in the plane

We introduce a new algorithm for computing Euclidean shortest paths in the plane in the presence of polygonal obstacles. In particular, for a given start points, we build a planar subdivision (ashortest path map) that supports efficient queries for shortest paths froms to any destination pointt. The worst-case time complexity of our algorithm isO(kn log² n), wheren is the number of vertices describing the polygonal obstacles, andk is a parameter we call the illumination depth of the obstacle space. Our algorithm usesO(n) space, avoiding the possibly quadratic space complexity of methods that rely on visibility graphs. The quantityk is frequently significantly smaller thann, especially in some of the cases in which the visibility graph has quadratic size. In particular,k is bounded above by the number of different obstacles that touch any shortest path froms.Partially supported by NSF Grants IRI-8710858 and ECSE-8857642 and by a grant from Hughes Research Laboratories, Malibu, CA.     

Cost-based Filtering for Shorter Path Constraints

Many real world problems, e.g. 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, 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. We then show empirically how reasoning about path-substructures in combination with CP-based Lagrangian relaxation can help to improve significantly over previously developed problem-tailored filtering algorithms for the resource constrained shortest path problem and investigate the impact of required-edge detection, undirected versus directed filtering, and the choice of the algorithm optimizing the Lagrangian dual.     

Weighted path queries on semistructured databases

Path queries have been extensively used to query semistructured data, such as the Web and XML documents. In this paper we introduce weighted path queries, an extension of path queries enabling several classes of optimization problems (such as the computation of shortest paths) to be easily expressed. Weighted path queries are based on the notion of weighted regular expression, i.e., a regular expression whose symbols are associated to a weight. We characterize the problem of answering weighted path queries and provide an algorithm for computing their answer. We also show how weighted path queries can be effectively embedded into query languages for XML data to express in a simple and compact form several meaningful research problems.     

Minimum-link paths among obstacles in the plane

Given a set of nonintersecting polygonal obstacles in the plane, thelink distance between two pointss andt is the minimum number of edges required to form a polygonal path connectings tot that avoids all obstacles. We present an algorithm that computes the link distance (and a corresponding minimum-link path) between two points in timeO(Eα(n) log² n) (and spaceO(E)), wheren is the total number of edges of the obstacles,E is the size of the visibility graph, and α(n) denotes the extremely slowly growing inverse of Ackermann's function. We show how to extend our method to allow computation of a tree (rooted ats) of minimum-link paths froms to all obstacle vertices. This leads to a method of solving the query version of our problem (for query pointst).     

Computing Approximate Shortest Paths on Convex Polytopes

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 \reals ³ , two points s, t ∈ 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/\eps ⁴ ) , 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. Received July 25, 2000; revised June 6, 2001.