Showing papers on "Longest path problem published in 2010"

k-nearest neighbors in uncertain graphs

Abstract: Complex networks, such as biological, social, and communication networks, often entail uncertainty, and thus, can be modeled as probabilistic graphs. Similar to the problem of similarity search in standard graphs, a fundamental problem for probabilistic graphs is to efficiently answer k-nearest neighbor queries (k-NN), which is the problem of computing the k closest nodes to some specific node.In this paper we introduce a framework for processing k-NN queries in probabilistic graphs. We propose novel distance functions that extend well-known graph concepts, such as shortest paths. In order to compute them in probabilistic graphs, we design algorithms based on sampling. During k-NN query processing we efficiently prune the search space using novel techniques.Our experiments indicate that our distance functions outperform previously used alternatives in identifying true neighbors in real-world biological data. We also demonstrate that our algorithms scale for graphs with tens of millions of edges.

Penalized Likelihood Methods for Estimation of Sparse High Dimensional Directed Acyclic Graphs

Abstract: Directed acyclic graphs are commonly used to represent causal relationships among random variables in graphical models. Applications of these models arise in the study of physical and biological systems where directed edges between nodes represent the influence of components of the system on each other. Estimation of directed graphs from observational data is computationally NP-hard. In addition, directed graphs with the same structure may be indistinguishable based on observations alone. When the nodes exhibit a natural ordering, the problem of estimating directed graphs reduces to the problem of estimating the structure of the network. In this paper, we propose an efficient penalized likelihood method for estimation of the adjacency matrix of directed acyclic graphs, when variables inherit a natural ordering. We study variable selection consistency of lasso and adaptive lasso penalties in high-dimensional sparse settings, and propose an error-based choice for selecting the tuning parameter. We show that although the lasso is only variable selection consistent under stringent conditions, the adaptive lasso can consistently estimate the true graph under the usual regularity assumptions.

Fast and accurate estimation of shortest paths in large graphs

Abstract: Computing shortest paths between two given nodes is a fundamental operation over graphs, but known to be nontrivial over large disk-resident instances of graph data. While a number of techniques exist for answering reachability queries and approximating node distances efficiently, determining actual shortest paths (i.e. the sequence of nodes involved) is often neglected. However, in applications arising in massive online social networks, biological networks, and knowledge graphs it is often essential to find out many, if not all, shortest paths between two given nodes. In this paper, we address this problem and present a scalable sketch-based index structure that not only supports estimation of node distances, but also computes corresponding shortest paths themselves. Generating the actual path information allows for further improvements to the estimation accuracy of distances (and paths), leading to near-exact shortest-path approximations in real world graphs. We evaluate our techniques - implemented within a fully functional RDF graph database system - over large real-world social and biological networks of sizes ranging from tens of thousand to millions of nodes and edges. Experiments on several datasets show that we can achieve query response times providing several orders of magnitude speedup over traditional path computations while keeping the estimation errors between 0% and 1% on average.

A link-based day-to-day traffic assignment model

Abstract: Existing day-to-day traffic assignment models are all built upon path flow variables. This paper demonstrates two essential shortcomings of these path-based models. One is that their application requires a given initial path flow pattern, which is typically unidentifiable, i.e. mathematically nonunique and practically unobservable. In particular, we show that, for the path-based models, different initial path flow patterns constituting the same link flow pattern generally gives different day-to-day link flow evolutions. The other shortcoming of the path-based models is the path-overlapping problem. That is, the path-based models ignore the interdependence among paths and thus can give very unreasonable results for networks with paths overlapping with each other. These two path-based problems exist for most (if not all) deterministic day-to-day dynamics whose fixed points are the classic Wardrop user equilibrium. To avoid the two path-based problems, we propose a day-to-day traffic assignment model that directly deals with link flow variables. Our link-based model captures travelers’ cost-minimization behavior in their path finding as well as their inertia. The fixed point of our link-based dynamical system is the classic Wardrop user equilibrium.

An optimization variant of multi-robot path planning is intractable

Abstract: An optimization variant of a problem of path planning for multiple robots is addressed in this work. The task is to find spatial-temporal path for each robot of a group of robots such that each robot can reach its destination by navigating through these paths. In the optimization variant of the problem, there is an additional requirement that the makespan of the solution must be as small as possible. A proof of the claim that optimal path planning for multiple robots is NP-complete is sketched in this short paper.

TEDI: efficient shortest path query answering on graphs

Abstract: Efficient shortest path query answering in large graphs is enjoying a growing number of applications, such as ranked keyword search in databases, social networks, ontology reasoning and bioinformatics. A shortest path query on a graph finds the shortest path for the given source and target vertices in the graph. Current techniques for efficient evaluation of such queries are based on the pre-computation of compressed Breadth First Search trees of the graph. However, they suffer from drawbacks of scalability. To address these problems, we propose TEDI, an indexing and query processing scheme for the shortest path query answering. TEDI is based on the tree decomposition methodology. The graph is first decomposed into a tree in which the node (a.k.a. bag) contains more than one vertex from the graph. The shortest paths are stored in such bags and these local paths together with the tree are the components of the index of the graph. Based on this index, a bottom-up operation can be executed to find the shortest path for any given source and target vertices. Our experimental results show that TEDI offers orders-of-magnitude performance improvement over existing approaches on the index construction time, the index size and the query answering.

Probabilistic path queries in road networks: traffic uncertainty aware path selection

Abstract: Path queries such as "finding the shortest path in travel time from my hotel to the airport" are heavily used in many applications of road networks. Currently, simple statistic aggregates such as the average travel time between two vertices are often used to answer path queries. However, such simple aggregates often cannot capture the uncertainty inherent in traffic. In this paper, we study how to take traffic uncertainty into account in answering path queries in road networks. To capture the uncertainty in traffic such as the travel time between two vertices, the weight of an edge is modeled as a random variable and is approximated by a set of samples. We propose three novel types of probabilistic path queries using basic probability principles: (1) a probabilistic path query like "what are the paths from my hotel to the airport whose travel time is at most 30 minutes with a probability of at least 90%?"; (2) a weight-threshold top-k path query like "what are the top-3 paths from my hotel to the airport with the highest probabilities to take at most 30 minutes?"; and (3) a probability-threshold top-k path query like "what are the top-3 shortest paths from my hotel to the airport whose travel time is guaranteed by a probability of at least 90%?" To evaluate probabilistic path queries efficiently, we develop three efficient probability calculation methods: an exact algorithm, a constant factor approximation method and a sampling based approach. Moreover, we devise the P* algorithm, a best-first search method based on a novel hierarchical partition tree index and three effective heuristic evaluation functions. An extensive empirical study using real road networks and synthetic data sets shows the effectiveness of the proposed path queries and the efficiency of the query evaluation methods.

A sketch-based distance oracle for web-scale graphs

Abstract: We study the fundamental problem of computing distances between nodes in large graphs such as the web graph and social networks. Our objective is to be able to answer distance queries between pairs of nodes in real time. Since the standard shortest path algorithms are expensive, our approach moves the time-consuming shortest-path computation offline, and at query time only looks up precomputed values and performs simple and fast computations on these precomputed values. More specifically, during the offline phase we compute and store a small "sketch" for each node in the graph, and at query-time we look up the sketches of the source and destination nodes and perform a simple computation using these two sketches to estimate the distance.

MSP algorithm: multi-robot patrolling based on territory allocation using balanced graph partitioning

Abstract: This article addresses the problem of efficient multi-robot patrolling in a known environment. The proposed approach assigns regions to each mobile agent. Every region is represented by a subgraph extracted from the topological representation of the global environment. A new algorithm is proposed in order to deal with the local patrolling task assigned for each robot, named Multilevel Subgraph Patrolling (MSP) Algorithm. It handles some major graph theory classic problems like graph partitioning, Hamilton cycles, non-Hamilton cycles and longest path searches. The flexible, scalable, robust and high performance nature of this approach is testified by simulation results.

Graph indexing of road networks for shortest path queries with label restrictions

Abstract: The current widespread use of location-based services and GPS technologies has revived interest in very fast and scalable shortest path queries. We introduce a new shortest path query type in which dynamic constraints may be placed on the allowable set of edges that can appear on a valid shortest path (e.g., dynamically restricting the type of roads or modes of travel which may be considered in a multimodal transportation network). We formalize this problem as a specific variant of formal language constrained shortest path problems, which we call the Kleene Language Constrained Shortest Paths problem. To efficiently support this type of dynamically constrained shortest path query for large-scale datasets, we extend the hierarchical graph indexing technique known as Contraction Hierarchies. Our experimental evaluation using the North American road network dataset (with over 50 million edges) shows an average query speed and search space improvement of over 3 orders of magnitude compared to the naive adaptation of the standard Dijkstra's algorithm to support this query type. We also show an improvement of over 2 orders of magnitude compared to the only previously-existing indexing technique which could solve this problem without additional preprocessing.

Shortest paths in planar graphs with real lengths in O(n log 2 n/ log log n) time

Abstract: Given an n-vertex planar directed graphwith real edge lengths and with no negative cycles, we show how to compute single-source shortest path distances in the graph in O(n log2 n/ log log n) time with O(n) space. This improves on a recent O(n log2 n) time bound by Klein et al.

Path Problems in Networks

Abstract: The algebraic path problem is a generalization of the shortest path problem in graphs. Various instances of this abstract problem have appeared in the literature, and similar solutions have been independently discovered and rediscovered. The repeated appearance of a problem is evidence of its relevance. This book aims to help current and future researchers add this powerful tool to their arsenal, so that they can easily identify and use it in their own work. Path problems in networks can be conceptually divided into two parts: A distillation of the extensive theory behind the algebraic path problem, and an exposition of a broad range of applications. First of all, the shortest path problem is presented so as to fix terminology and concepts: existence and uniqueness of solutions, robustness to parameter changes, and centralized and distributed computation algorithms. Then, these concepts are generalized to the algebraic context of semirings. Methods for creating new semirings, useful for modeling new problems, are provided. A large part of the book is then devoted to numerous applications of the algebraic path problem, ranging from mobile network routing to BGP routing to social networks. These applications show what kind of problems can be modeled as algebraic path problems; they also serve as examples on how to go about modeling new problems. This monograph will be useful to network researchers, engineers, and graduate students. It can be used either as an introduction to the topic, or as a quick reference to the theoretical facts, algorithms, and application examples. The theoretical background assumed for the reader is that of a graduate or advanced undergraduate student in computer science or engineering. Some familiarity with algebra and algorithms is helpful, but not necessary. Algebra, in particular, is used as a convenient and concise language to describe problems that are essentially combinatorial. Table of Contents: Classical Shortest Path / The Algebraic Path Problem / Properties and Computation of Solutions / Applications / Related Areas / List of Semirings and Applications

Orion: shortest path estimation for large social graphs

Abstract: Through measurements, researchers continue to produce large social graphs that capture relationships, transactions, and social interactions between users. Efficient analysis of these graphs requires algorithms that scale well with graph size. We examine node distance computation, a critical primitive in graph problems such as computing node separation, centrality computation, mutual friend detection, and community detection. For large million-node social graphs, computing even a single shortest path using traditional breadth-first-search can take several seconds. In this paper, we propose a novel node distance estimation mechanism that effectively maps nodes in high dimensional graphs to positions in low-dimension Euclidean coordinate spaces, thus allowing constant time node distance computation. We describe Orion, a pro totype graph coordinate system, and explore critical decisions in its design. Finally, we evaluate the accuracy of Orion's node distance estimates, and show that it can produce accurate results in applications such as node separation, node centrality, and ranked social search.

Maximum flows and parametric shortest paths in planar graphs

Abstract: We observe that the classical maximum flow problem in any directed planar graph G can be reformulated as a parametric shortest path problem in the oriented dual graph G*. This reformulation immediately suggests an algorithm to compute maximum flows, which runs in O(n log n) time. As we continuously increase the parameter, each change in the shortest path tree can be effected in O(log n) time using standard dynamic tree data structures, and the special structure of the parametrization implies that each directed edge enters the evolving shortest path tree at most once. The resulting maximum-flow algorithm is identical to the recent algorithm of Borradaile and Klein [J. ACM 2009], but our new formulation allows a simpler presentation and analysis. On the other hand, we demonstrate that for a similarly structured parametric shortest path problem on the torus, the shortest path tree can change Ω(n2) times in the worst case, suggesting that a different method may be required to efficiently compute maximum flows in higher-genus graphs.

Efficiently answering probability threshold-based shortest path queries over uncertain graphs

Abstract: Efficiently processing shortest path (SP) queries over stochastic networks attracted a lot of research attention as such queries are very popular in the emerging real world applications such as Intelligent Transportation Systems and communication networks whose edge weights can be modeled as a random variable. Some pervious works aim at finding the most likely SP (the path with largest probability to be SP), and others search the least-expected-weight path. In all these works, the definitions of the shortest path query are based on simple probabilistic models which can be converted into the multi-objective optimal issues on a weighted graph. However, these simple definitions miss important information about the internal structure of the probabilistic paths and the interplay among all the uncertain paths. Thus, in this paper, we propose a new SP definition based on the possible world semantics that has been widely adopted for probabilistic data management, and develop efficient methods to find threshold-based SP path queries over an uncertain graph. Extensive experiments based on real data sets verified the effectiveness of the proposed methods.

Network Simplification with Minimal Loss of Connectivity

Abstract: We propose a novel problem to simplify weighted graphs by pruning least important edges from them. Simplified graphs can be used to improve visualization of a network, to extract its main structure, or as a pre-processing step for other data mining algorithms. We define a graph connectivity function based on the best paths between all pairs of nodes. Given the number of edges to be pruned, the problem is then to select a subset of edges that best maintains the overall graph connectivity. Our model is applicable to a wide range of settings, including probabilistic graphs, flow graphs and distance graphs, since the path quality function that is used to find best paths can be defined by the user. We analyze the problem, and give lower bounds for the effect of individual edge removal in the case where the path quality function has a natural recursive property. We then propose a range of algorithms and report on experimental results on real networks derived from public biological databases. The results show that a large fraction of edges can be removed quite fast and with minimal effect on the overall graph connectivity. A rough semantic analysis of the removed edges indicates that few important edges were removed, and that the proposed approach could be a valuable tool in aiding users to view or explore weighted graphs.

A case for time-dependent shortest path computation in spatial networks

Abstract: The problem of point-to-point shortest path computation in spatial networks is extensively studied with many approaches proposed to speed-up the computation. Most of the existing approaches make the simplifying assumption that weights (e.g., travel-time) of the network edges are constant. However, with real-world spatial networks the edge travel-times are time-dependent, where the arrival-time to an edge determines the actual travel-time of the edge. With this paper, we study the applicability of existing shortest path algorithms to real-world large time-dependent spatial networks. In addition, we evaluate the importance of considering time-dependent edge travel-times for route planning in spatial networks. We show that time-dependent shortest path computation can reduce the travel-time by 36% on average as compared to the static shortest path computation that assumes constant edge travel-times.

Deterministic computations in time-varying graphs: Broadcasting under unstructured mobility

Abstract: Most highly dynamic infrastructure-less networks have in common that the assumption of connectivity does not necessarily hold at a given instant. Still, communication routes can be available between any pair of nodes over time and space. These networks (variously called delay-tolerant, disruptive-tolerant, challenged) are naturally modeled as time-varying graphs (or evolving graphs), where the existence of an edge is a function of time. In this paper we study deterministic computations under unstructured mobility, that is when the edges of the graph appear infinitely often but without any (known) pattern. In particular, we focus on the problem of broadcasting with termination detection. We explore the problem with respect to three possible metrics: the date of message arrival (foremost), the time spent doing the broadcast (fastest), and the number of hops used by the broadcast (shortest). We prove that the solvability and complexity of this problem vary with the metric considered, as well as with the type of knowledge a priori available to the entities. These results draw a complete computability map for this problem when mobility is unstructured.

The Online Loop-free Stochastic Shortest-Path Problem.

Abstract: We consider a stochastic extension of the loop-free shortest path problem with adversarial rewards. In this episodicMarkov decision problem an agent traverses through an acyclic graph with random transitions: at each step of an episode the agent chooses an action, receives some reward, and arrives at a random next state, where the reward and the distribution of the next state depend on the actual state and the chosen action. We consider the bandit situation when only the reward of the just visited state-action pair is revealed to the agent. For this problem we develop algorithms that perform asymptotically as well as the best stationary policy in hindsight. Assuming that all states are reachable with probability a>0 under all policies, we give an algorithm and prove that its regret is O(L^2 sqrt(T|A|)/a), where T is the number of episodes, A denotes the (finite) set of actions, and L is the length of the longest path in the graph. Variants of the algorithm are given that improve the dependence on the transition probabilities under specific conditions. The results are also extended to variations of the problem, including the case when the agent competes with time varying policies.

Edge-based representation beats vertex-based representation in shortest path problems

Abstract: In this paper, we present a new representation for individuals in the single-source shortest path problem. Contrary to previous approaches, it has the natural property that different vertex degrees do not induce unfairness in the mutation step. In particular, at any time each edge has roughly the same probability of being added to or removed from the current individual.This turns out to be a crucial property. Mainly based on this, we prove superior bounds for the optimization time two evolutionary algorithms for the single-source shortest path problem. For both the multi-criteria formulation of the problem (introduced by Scharnow, Tinnefeld and Wegener (2002, 2004)) and the single-criteria one (regarded in Baswana et al. (2009)), we improve the existing bounds by a factor of n2/m, where m denotes the number of edges and n the number of vertices of the underlying graph. Given that most graphs found in practical applications are sparse, this is a considerable gain.

A Simple Polynomial Algorithm for the Longest Path Problem on Cocomparability Graphs

Abstract: Given a graph $G$, the longest path problem asks to compute a simple path of $G$ with the largest number of vertices. This problem is the most natural optimization version of the well known and well studied Hamiltonian path problem, and thus it is NP-hard on general graphs. However, in contrast to the Hamiltonian path problem, there are only few restricted graph families such as trees and some small graph classes where polynomial algorithms for the longest path problem have been found. Recently it has been shown that this problem can be solved in polynomial time on interval graphs by applying dynamic programming to a characterizing ordering of the vertices of the given graph \cite{longest-int-algo}, thus answering an open question. In the present paper, we provide the first polynomial algorithm for the longest path problem on a much greater class, namely on cocomparability graphs. Our algorithm uses a similar - but essentially simpler - dynamic programming approach, which is applied to a Lexicographic Depth First Search (LDFS) characterizing ordering of the vertices of a cocomparability graph. Therefore, our results provide evidence that this general dynamic programming approach can be used in a more general setting, leading to efficient algorithms for the longest path problem on greater classes of graphs. LDFS has recently been introduced in \cite{Corneil-LDFS08}. Since then, a similar phenomenon of extending an existing interval graph algorithm to cocomparability graphs by using an LDFS preprocessing step has also been observed for the minimum path cover problem \cite{Corneil-MPC}. Therefore, more interestingly, our results also provide evidence that cocomparability graphs present an interval graph structure when they are considered using an LDFS ordering of their vertices, which may lead to other new and more efficient combinatorial algorithms.

The 1-Fixed-Endpoint Path Cover Problem is Polynomial on Interval Graphs

Abstract: We consider a variant of the path cover problem, namely, the k-fixed-endpoint path cover problem, or kPC for short, on interval graphs. Given a graph G and a subset \(\mathcal{T}\) of k vertices of V(G), a k-fixed-endpoint path cover of G with respect to \(\mathcal{T}\) is a set of vertex-disjoint paths ℘ that covers the vertices of G such that the k vertices of \(\mathcal{T}\) are all endpoints of the paths in ℘. The kPC problem is to find a k-fixed-endpoint path cover of G of minimum cardinality; note that, if \(\mathcal{T}\) is empty the stated problem coincides with the classical path cover problem. In this paper, we study the 1-fixed-endpoint path cover problem on interval graphs, or 1PC for short, generalizing the 1HP problem which has been proved to be NP-complete even for small classes of graphs. Motivated by a work of Damaschke (Discrete Math. 112:49–64, 1993), where he left both 1HP and 2HP problems open for the class of interval graphs, we show that the 1PC problem can be solved in polynomial time on the class of interval graphs. We propose a polynomial-time algorithm for the problem, which also enables us to solve the 1HP problem on interval graphs within the same time and space complexity.

Maximum Flux Transition Paths of Conformational Change

Abstract: Given two metastable states A and B of a biomolecular system, the problem is to calculate the likely paths of the transition from A to B. Such a calculation is more informative and more manageable if done for a reduced set of collective variables chosen so that paths cluster in collective variable space. The computational task becomes that of computing the “center” of such a cluster. A good way to define the center employs the concept of a committor, whose value at a point in collective variable space is the probability that a trajectory at that point will reach B before A. The committor “foliates” the transition region into a set of isocommittors. The maximum flux transition path is defined as a path that crosses each isocommittor at a point which (locally) has the highest crossing rate of distinct reactive trajectories. This path is based on the same principle as the minimum resistance path of Berkowitz et al (1983), but it has two advantages: (i) the path is invariant with respect to a change of coordinates in collective variable space and (ii) the differential equations that define the path are simpler. It is argued that such a path is nearer to an ideal path than others that have been proposed with the possible exception of the finite-temperature string method path. To make the calculation tractable, three approximations are introduced, yielding a path that is the solution of a nonsingular two-point boundary-value problem. For such a problem, one can construct a simple and robust algorithm. One such algorithm and its performance is discussed.

Approximate shortest path and distance queries in networks

Abstract: Computing shortest paths in graphs is one of the most fundamental and well-studied problems in combinatorial optimization. Numerous real-world applications have stimulated research investigations for more than 50 years. Finding routes in road and public transportation networks is a classical application motivating the study of the shortest path problem. Shortest paths are also sought by routing schemes for computer networks: the transmission time of messages is less when they are sent through a short sequence of routers. The problem is also relevant for social networks: one may more likely obtain a favor from a stranger by establishing contact through personal connections. This thesis investigates the problem of efficiently computing exact and approximate shortest paths in graphs, with the main focus being on shortest path query processing. Strategies for computing answers to shortest path queries may involve the use of pre-computed data structures (often called distance oracles) in order to improve the query time. Designing a shortest path query processing method raises questions such as: How can these data structures be computed efficiently? What amount of storage is necessary? How much improvement of the query time is possible? How good is the approximation quality of the query result? What are the tradeoffs between precomputation time, storage, query time, and approximation quality? For distance oracles applicable to general graphs, the quantitative tradeoff between the storage requirement and the approximation quality is known up to constant factors. For distance oracles that take advantage of the properties of certain classes of graphs, however, the tradeoff is less well understood: for some classes of sparse graphs such as planar graphs, there are data structures that enable query algorithms to efficiently compute distance estimates of much higher precision than what the tradeoff for general graphs would predict. The first main contribution of this thesis is a proof that such data structures cannot exist for all sparse graphs. We prove a space lower bound implying that distance oracles with good precision and very low query costs require large amounts of space. A second contribution consists of spaceand time-efficient data structures for a large family of complex networks. We prove that exploiting well-connected nodes yields efficient distance oracles for scale-free graphs. A third contribution is a practical method to compute approximate shortest paths. By means of random sampling and graph Voronoi duals, our method successfully accommodates both highly structured graphs stemming from transportation networks and less structured graphs stemming from complex networks such as social networks.

Replacement Paths via Fast Matrix Multiplication

Abstract: Let G be a directed edge-weighted graph and let P be a shortest path from s to t in G. The replacement paths problem asks to compute, for every edge e on P, the shortest s-to-t path that avoids e. Apart from approximation algorithms and algorithms for special graph classes, the naive solution to this problem – removing each edge e on P one at a time and computing the shortest s-to-t path each time – is surprisingly the only known solution for directed weighted graphs, even when the weights are integrals. In particular, although the related shortest paths problem has benefited from fast matrix multiplication, the replacement paths problem has not, and still required cubic time. For an n-vertex graph with integral edge-lengths between -M and M, we give a randomized algorithm that uses fast matrix multiplication and is sub-cubic for appropriate values of M. We also show how to construct a distance sensitivity oracle in the same time bounds. A query (u,v,e) to this oracle requires sub-quadratic time and returns the length of the shortest u-to-v path that avoids the edge e. In fact, for any constant number of edge failures, we construct a data structure in sub-cubic time, that answer queries in sub-quadratic time. Our results also apply for avoiding vertices rather than edges.

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

Abstract: Let $G=(V,E)$ be any undirected graph on $V$ vertices and $E$ edges. A path $\textbf{P}$ between any two vertices $u,v\in V$ 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 consider the problem of building a compact data structure for a given graph $G$ which is capable of answering the following query for any $u,v,z\in V$ and $t>1$. \centerline{\em report $t$-approximate shortest path between $u$ and $v$ when vertex $z$ fails} We present data structures for the single source as well all-pairs versions of this problem. Our data structures guarantee optimal query time. Most impressive feature of our data structures is that their size {\em nearly} match the size of their best static counterparts.

Fast top-k simple shortest paths discovery in graphs

Abstract: With the wide applications of large scale graph data such as social networks, the problem of finding the top-k shortest paths attracts increasing attention. This paper focuses on the discovery of the top-k simple shortest paths (paths without loops). The well known algorithm for this problem is due to Yen, and the provided worstcase bound O(kn(m + nlogn)), which comes from O(n) times single-source shortest path discovery for each of k shortest paths, remains unbeaten for 30 years, where n is the number of nodes and m is the number of edges. In this paper, we observe that there are shared sub-paths among O(kn) single-source shortest paths. The basic idea behind our method is to pre-compute the shortest paths to the target node, and utilize them to reduce the discovery cost at running time. Specifically, we transform the original graph by encoding the pre-computed paths, and prove that the shortest path discovered over the transformed graph is equivalent to that in the original graph. Most importantly, the path discovery over the transformed graph can be terminated much earlier than before. In addition, two optimization strategies are presented. One is to reduce the total iteration times for shortest path discovery, and the other is to prune the search space in each iteration with an adaptively-determined threshold. Although the worst-case complexity cannot be lowered, our method is proven to be much more efficient in a general case. The final extensive experimental results (on both real and synthetic graphs) also show that our method offers a significant performance improvement over the existing ones.