Showing papers on "Longest path problem published in 2016"

Fast Memory-efficient Anomaly Detection in Streaming Heterogeneous Graphs

Abstract: Given a stream of heterogeneous graphs containing different types of nodes and edges, how can we spot anomalous ones in real-time while consuming bounded memory? This problem is motivated by and generalizes from its application in security to host-level advanced persistent threat (APT) detection. We propose StreamSpot, a clustering based anomaly detection approach that addresses challenges in two key fronts: (1) heterogeneity, and (2) streaming nature. We introduce a new similarity function for heterogeneous graphs that compares two graphs based on their relative frequency of local substructures, represented as short strings. This function lends itself to a vector representation of a graph, which is (a) fast to compute, and (b) amenable to a sketched version with bounded size that preserves similarity. StreamSpot exhibits desirable properties that a streaming application requires: it is (i) fully-streaming; processing the stream one edge at a time as it arrives, (ii) memory-efficient; requiring constant space for the sketches and the clustering, (iii) fast; taking constant time to update the graph sketches and the cluster summaries that can process over 100,000 edges per second, and (iv) online; scoring and flagging anomalies in real time. Experiments on datasets containing simulated system-call flow graphs from normal browser activity and various attack scenarios (ground truth) show that StreamSpot is high-performance; achieving above 95% detection accuracy with small delay, as well as competitive time and memory usage.

Efficient Algorithms for Temporal Path Computation

Abstract: Shortest path is a fundamental graph problem with numerous applications. However, the concept of classic shortest path is insufficient. In this paper, we study various concepts of “shortest” path in temporal graphs, called minimum temporal paths. Computing these minimum temporal paths is challenging as subpaths of a “shortest” path may not be “shortest” in a temporal graph. We propose efficient algorithms to compute minimum temporal paths and verified their efficiency using large real-world temporal graphs.

Indexing Variation Graphs

Abstract: Variation graphs, which represent genetic variation within a population, are replacing sequences as reference genomes. Path indexes are one of the most important tools for working with variation graphs. They generalize text indexes to graphs, allowing one to find the paths matching the query string. We propose using de Bruijn graphs as path indexes, compressing them by merging redundant subgraphs, and encoding them with the Burrows-Wheeler transform. The resulting fast, space-efficient, and versatile index is used in the variation graph toolkit vg.

Path cost distribution estimation using trajectory data

Abstract: With the growing volumes of vehicle trajectory data, it becomes increasingly possible to capture time-varying and uncertain travel costs in a road network, including travel time and fuel consumption. The current paradigm represents a road network as a weighted graph; it blasts trajectories into small fragments that fit the under-lying edges to assign weights to edges; and it then applies a routing algorithm to the resulting graph. We propose a new paradigm, the hybrid graph, that targets more accurate and more efficient path cost distribution estimation. The new paradigm avoids blasting trajectories into small fragments and instead assigns weights to paths rather than simply to the edges.We show how to compute path weights using trajectory data while taking into account the travel cost dependencies among the edges in the paths. Given a departure time and a query path, we show how to select an optimal set of weights with associated paths that cover the query path and such that the weights enable the most accurate joint cost distribution estimation for the query path. The cost distribution of the query path is then computed accurately using the joint distribution. Finally, we show how the resulting method for computing cost distributions of paths can be integrated into existing routing algorithms. Empirical studies with substantial trajectory data from two different cities offer insight into the design properties of the proposed method and confirm that the method is effective in real-world settings.

Shortest Path Problem under Bipolar Neutrosphic Setting

Abstract: This main purpose of this paper is to develop an algorithm to find the shortest path on a network in which the weights of the edges are represented by bipolar neutrosophic numbers. Finally, a numerical example has been provided for illustrating the proposed approach.

Controllability of multiplex, multi-time-scale networks

Abstract: The paradigm of layered networks is used to describe many real-world systems, from biological networks to social organizations and transportation systems. While recently there has been much progress in understanding the general properties of multilayer networks, our understanding of how to control such systems remains limited. One fundamental aspect that makes this endeavor challenging is that each layer can operate at a different time scale; thus, we cannot directly apply standard ideas from structural control theory of individual networks. Here we address the problem of controlling multilayer and multi-time-scale networks focusing on two-layer multiplex networks with one-to-one interlayer coupling. We investigate the practically relevant case when the control signal is applied to the nodes of one layer. We develop a theory based on disjoint path covers to determine the minimum number of inputs (N_{i}) necessary for full control. We show that if both layers operate on the same time scale, then the network structure of both layers equally affect controllability. In the presence of time-scale separation, controllability is enhanced if the controller interacts with the faster layer: N_{i} decreases as the time-scale difference increases up to a critical time-scale difference, above which N_{i} remains constant and is completely determined by the faster layer. We show that the critical time-scale difference is large if layer I is easy and layer II is hard to control in isolation. In contrast, control becomes increasingly difficult if the controller interacts with the layer operating on the slower time scale and increasing time-scale separation leads to increased N_{i}, again up to a critical value, above which N_{i} still depends on the structure of both layers. This critical value is largely determined by the longest path in the faster layer that does not involve cycles. By identifying the underlying mechanisms that connect time-scale difference and controllability for a simplified model, we provide crucial insight into disentangling how our ability to control real interacting complex systems is affected by a variety of sources of complexity.

Shortest path problem under triangular fuzzy neutrosophic information

Abstract: In this paper, we develop a new approach to deal with neutrosphic shortest path problem in a network in which each edge weight (or length) is represented as triangular fuzzy neutrosophic number. The proposed algorithm also gives the shortest path length from source node to destination node using ranking function. Finally, an illustrative example is also included to demonstrate our proposed approach.

Interval Type 2 Fuzzy Set in Fuzzy Shortest Path Problem

Abstract: The shortest path problem (SPP) is one of the most important combinatorial optimization problems in graph theory due to its various applications. The uncertainty existing in the real world problems makes it difficult to determine the arc lengths exactly. The fuzzy set is one of the popular tools to represent and handle uncertainty in information due to incompleteness or inexactness. In most cases, the SPP in fuzzy graph, called the fuzzy shortest path problem (FSPP) uses type-1 fuzzy set (T1FS) as arc length. Uncertainty in the evaluation of membership degrees due to inexactness of human perception is not considered in T1FS. An interval type-2 fuzzy set (IT2FS) is able to tackle this uncertainty. In this paper, we use IT2FSs to represent the arc lengths of a fuzzy graph for FSPP. We call this problem an interval type-2 fuzzy shortest path problem (IT2FSPP). We describe the utility of IT2FSs as arc lengths and its application in different real world shortest path problems. Here, we propose an algorithm for IT2FSPP. In the proposed algorithm, we incorporate the uncertainty in Dijkstra’s algorithm for SPP using IT2FS as arc length. The path algebra corresponding to the proposed algorithm and the generalized algorithm based on the path algebra are also presented here. Numerical examples are used to illustrate the effectiveness of the proposed approach.

From Simplicity to Complexity Based on Consensus: A Case Study.

Abstract: Distributed consensus has been intensively studied in recent years as a means to mitigate state differences among dynamic nodes on a graph It has been successfully employed in various applications, eg, formation control of multi-robots, load balancing, clock synchronization However, almost all existing applications cast an impression of consensus as a simple process to iteratively reach agreement, without any clue on possibility to generate advanced complexity, say shortest path planning, which has been proved to be NP-hard Counter-intuitively, we show for the first time that the complexity of shortest path planning can emerge from a perturbed version of min-consensus protocol, which as a case study may shed lights to researchers in the field of distributed control to re-think the nature of complexity and the distance between control and intelligence Besides, we rigorously prove the convergence of graph dynamics and its equivalence to shortest path solutions An illustrative simulation on a small scale graph is provided to show the convergence of the biased min-consensus dynamics to shortest path solution over the graph To demonstrate the scalability to large scale problems, a graph with 43826 nodes, which corresponds to a map of a maze in 2D, is considered in the simulation study Apart from possible applications in robot path planning, the result is further extended to robot complete coverage, showing its potential in real practice such as cleaning robots

Gabriel Triangulations and Angle-Monotone Graphs: Local Routing and Recognition

Abstract: A geometric graph is angle-monotone if every pair of vertices has a path between them that—after some rotation—is x- and y-monotone. Angle-monotone graphs are \(\sqrt{2}\)-spanners and they are increasing-chord graphs. Dehkordi, Frati, and Gudmundsson introduced angle-monotone graphs in 2014 and proved that Gabriel triangulations are angle-monotone graphs. We give a polynomial time algorithm to recognize angle-monotone geometric graphs. We prove that every point set has a plane geometric graph that is generalized angle-monotone—specifically, we prove that the half-\(\theta _6\)-graph is generalized angle-monotone. We give a local routing algorithm for Gabriel triangulations that finds a path from any vertex s to any vertex t whose length is within \(1 + \sqrt{2}\) times the Euclidean distance from s to t. Finally, we prove some lower bounds and limits on local routing algorithms on Gabriel triangulations.

Graph Indexing for Shortest-Path Finding over Dynamic Sub-Graphs

Abstract: A variety of applications spanning various domains, e.g., social networks, transportation, and bioinformatics, have graphs as first-class citizens. These applications share a vital operation, namely, finding the shortest path between two nodes. In many scenarios, users are interested in filtering the graph before finding the shortest path. For example, in social networks, one may need to compute the shortest path between two persons on a sub-graph containing only family relationships. This paper focuses on dynamic graphs with labeled edges, where the target is to find a shortest path after filtering some edges based on user-specified query labels. This problem is termed the Edge-Constrained Shortest Path query (or ECSP, for short). This paper introduces Edge-Disjoint Partitioning (EDP, for short), a new technique for efficiently answering ECSP queries over dynamic graphs. EDP has two main components: a dynamic index that is based on graph partitioning, and a traversal algorithm that exploits the regular patterns of the answers of ECSP queries. The main idea of EDP is to partition the graph based on the labels of the edges. On demand, EDP computes specific sub-paths within each partition and updates its index. The computed sub-paths act as pre-computations that can be leveraged by future queries. To answer an ECSP query, EDP connects sub-paths from different partitions using its efficient traversal algorithm. EDP can dynamically handle various types of graph updates, e.g., label, edge, and node updates. The index entries that are potentially affected by graph updates are invalidated and re-computed on demand. EDP is evaluated using real graph datasets from various domains. Experimental results demonstrate that EDP can achieve query performance gains of up to four orders of magnitude in comparison to state of the art techniques.

An ant colony optimization algorithm for partitioning graphs with supply and demand

Abstract: Graphical abstractDisplay Omitted HighlightsAnt colony optimization algorithm for the problem of partitioning graphs with supply and demand.Very effective method manages to find optimal solutions in more that 50% of the test instances.Average relative error of less than 0.5% when compared to known optimal solutions.Method analyzed on general graphs, Halin graphs, series-parallel graphs and trees. In this paper we focus on finding high quality solutions for the problem of maximum partitioning of graphs with supply and demand (MPGSD). There is a growing interest for the MPGSD due to its close connection to problems appearing in the field of electrical distribution systems, especially for the optimization of self-adequacy of interconnected microgrids. We propose an ant colony optimization algorithm for the problem. With the goal of further improving the algorithm we combine it with a previously developed correction procedure. In our computational experiments we evaluate the performance of the proposed algorithm on trees, 3-connected graphs, series-parallel graphs and general graphs. The tests show that the method manages to find optimal solutions for more than 50% of the problem instances, and has an average relative error of less than 0.5% when compared to known optimal solutions.

A method for finding a least-cost wide path in raster space

Abstract: Given a grid of cells each having an associated cost value, a raster version of the least-cost path problem seeks a sequence of cells connecting two specified cells such that its total accumulated cost is minimized. Identifying least-cost paths is one of the most basic functions of raster-based geographic information systems. Existing algorithms are useful if the path width is assumed to be zero or negligible compared to the cell size. This assumption, however, may not be valid in many real-world applications ranging from wildlife corridor planning to highway alignment. This paper presents a method to solve a raster-based least-cost path problem whose solution is a path having a specified width in terms of Euclidean distance rather than by number of cells. Assuming that all cell values are positive, it does so by transforming the given grid into a graph such that each node represents a neighborhood of a certain form determined by the specified path width, and each arc represents a possible transition from one neighborhood to another. An existing shortest path algorithm is then applied to the graph. This method is highly efficient, as the number of nodes in the transformed graph is not more than the number of cells in the given grid and decreases with the specified path width. However, a shortcoming of this method is the possibility of generating a self-intersecting path which occurs only when the given grid has an extremely skewed distribution of cost values.

PHAETON: A SAT-Based Framework for Timing-Aware Path Sensitization

Abstract: Knowledge about sensitizable paths through combinational logic is essential for numerous design tasks. We present the framework PHAETON which identifies sensitizable paths and generates test pairs to exercise these paths using Boolean satisfiability (SAT). PHAETON supports a large number of models and sensitization conditions and provides a generic interface that can be used by applications. It incorporates a novel application-specific unary representation of integer numbers to integrate timing information with logical conditions within the same monolithic SAT formula. Due to a number of further elaborate speed-up techniques, PHAETON scales to industrial circuits. Experimental results show the performance of PHAETON in classical K longest path generation tasks and in new post-silicon validation and characterization scenarios.

An Exact Algorithm for the Time-Constrained Traveling Salesman Problem

Abstract: The time-constrained traveling salesman problem is a variation of the familiar traveling salesman problem that includes time window constraints on the time a particular city, or cities, may be visited. This note presents a concise formulation of the time-constrained traveling salesman problem. The model assumes that the distances of the problem are symmetrical and that the triangle inequality holds. Additionally, the model allows the salesman to wait at a city, if necessary, for a time window to open. The dual of the formulation is shown to be a disjunctive graph model, which is well known from scheduling theory. A longest path algorithm is used to obtain bounding information for subproblems in a branch and bound solution procedure. Computational results are presented for several small to moderate size problems.

Solving Time Dependent Shortest Path Problems on Airway Networks Using Super-Optimal Wind

Abstract: We study the Flight Planning Problem for a single aircraft, which deals with finding a path of minimal travel time in an airway network. Flight time along arcs is affected by wind speed and direction, which are functions of time. We consider three variants of the problem, which can be modeled as, respectively, a classical shortest path problem in a metric space, a time-dependent shortest path problem with piecewise linear travel time functions, and a time-dependent shortest path problem with piecewise differentiable travel time functions. The shortest path problem and its time-dependent variant have been extensively studied, in particular, for road networks. Airway networks, however, have different characteristics: the average node degree is higher and shortest paths usually have only few arcs. We propose A* algorithms for each of the problem variants. In particular, for the third problem, we introduce an application-specific "super-optimal wind" potential function that overestimates optimal wind conditions on each arc, and establish a linear error bound. We compare the performance of our methods with the standard Dijkstra algorithm and the Contraction Hierarchies (CHs) algorithm. Our computational results on real world instances show that CHs do not perform as well as on road networks. On the other hand, A* guided by our potentials yields very good results. In particular, for the case of piecewise linear travel time functions, we achieve query times about 15 times shorter than CHs.

Algorithms for algebraic path properties in concurrent systems of constant treewidth components

Abstract: We study algorithmic questions for concurrent systems where the transitions are labeled from a complete, closed semiring, and path properties are algebraic with semiring operations. The algebraic path properties can model dataflow analysis problems, the shortest path problem, and many other natural problems that arise in program analysis. We consider that each component of the concurrent system is a graph with constant treewidth, a property satisfied by the controlflow graphs of most programs. We allow for multiple possible queries, which arise naturally in demand driven dataflow analysis. The study of multiple queries allows us to consider the tradeoff between the resource usage of the one-time preprocessing and for each individual query. The traditional approach constructs the product graph of all components and applies the best-known graph algorithm on the product. In this approach, even the answer to a single query requires the transitive closure (i.e., the results of all possible queries), which provides no room for tradeoff between preprocessing and query time. Our main contributions are algorithms that significantly improve the worst-case running time of the traditional approach, and provide various tradeoffs depending on the number of queries. For example, in a concurrent system of two components, the traditional approach requires hexic time in the worst case for answering one query as well as computing the transitive closure, whereas we show that with one-time preprocessing in almost cubic time, each subsequent query can be answered in at most linear time, and even the transitive closure can be computed in almost quartic time. Furthermore, we establish conditional optimality results showing that the worst-case running time of our algorithms cannot be improved without achieving major breakthroughs in graph algorithms (i.e., improving the worst-case bound for the shortest path problem in general graphs). Preliminary experimental results show that our algorithms perform favorably on several benchmarks.

Proxies for Shortest Path and Distance Queries

Abstract: Computing shortest paths and distances is one of the fundamental problems on graphs, and it remains a challenging task today. This article investigates a light-weight data reduction technique for speeding-up shortest path and distance queries on large graphs. To do this, we propose a notion of routing proxies (or simply proxies), each of which represents a small subgraph, referred to as deterministic routing areas ( dra s). We first show that routing proxies hold good properties for speeding-up shortest path and distance queries. Then, we design a linear-time algorithm to compute routing proxies and their corresponding dra s. Finally, we experimentally verify that our solution is a general technique for reducing graph sizes and speeding-up shortest path and distance queries, using real-life large graphs.

On dynamic approximate shortest paths for planar graphs with worst-case costs

Abstract: Given a base weighted planar graph Ginput on n nodes and parameters M, e we present a dynamic distance oracle with 1 + e stretch and worst case update and query costs of e--3M4 · poly-log(n). We allow arbitrary edge weight updates as long as the shortest path metric induced by the updated graph has stretch of at most M relative to the shortest path metric of the base graph Ginput.For example, on a planar road network, we can support fast queries and dynamic traffic updates as long as the shortest path from any source to any target (including using arbitrary detours) is between, say, 80 and 3 miles-per-hour.As a warm-up we also prove that graphs of bounded treewidth have exact distance oracles in the dynamic edge model.To the best of our knowledge, this is the first dynamic distance oracle for a non-trivial family of dynamic changes to planar graphs with worst case costs of o(n1/2) both for query and for update operations.

Efficient fast recovery mechanism in Software-Defined Networks: Multipath routing approach

Abstract: Although Software-Defined Networking (SDN) is a mature paradigm, failure recovery and management in SDN still need much research attention. OpenFlow, as an implementation of SDN, provides flexible and abstracted approach to configure SDN networks. This Paper focuses on failure recovery mechanism using OpenFlow. The proposed mechanism is divided into: (a) computing the working paths proactively between each source-destination pair and return a list of paths ordered by-path latency from shortest to longest path, (b)implement per-Link Bidirectional Forwarding Detection(BFD) for failure detection, to enable fast detection time, thus fast recovery time, (d) configure OpenFlow Fast Failover Group for restoration such that, the highest-priority bucket to be linked to the shortest path, the second highest-priority bucket linked to the second shortest path and so on. Upon failure, the switch will revert to the second fastest path, (e) to achieve high resource utilization, OpenFlow Select Group used to split the flow among the working paths. The proposed architecture of this work provides high utilization of network resources in addition to efficient fast detection and recovery time. As the evaluation shows, the usage of multipath routing with fast failover scheme provide much resource utilization and fast recovery time.

Speeding up A* search on visibility graphs defined over quadtrees to enable long distance path planning for unmanned surface vehicles

Abstract: We introduce an algorithm for long distance path planning in complex marine environments. The available free space in marine environments changes over time as a result of tides, environmental restrictions, and weather. As a result of these considerations, the free space region in marine environments needs to be dynamically generated and updated. The approach presented in this paper demonstrates that it is feasible to compute optimal paths using A* search on visibility graphs defined over quadtrees. Our algorithm exploits quadtree data structures for efficiently computing tangent edges in visibility graphs. We have developed an admissible heuristic that accounts for large islands while estimating the cost-to-go and provides a better lower bound than the Euclidean distance-based heuristic. During the search over visibility graphs, the branching factor of A* can be large due to the large size of the region. We introduce the idea of focusing the search by limiting the child nodes to be in certain regions of the workspace. Our results show that focusing the search significantly improves the computational efficiency without any noticeable degradation in path quality. We have also developed a method to estimate bounds on how far the computed path can be from the optimal path when methods for focusing the search are utilized for speeding up the computation.

Parametric search for the bi-attribute concave shortest path problem

Abstract: A bi-attribute concave shortest path (BC-SP) problem seeks to find an optimal path in a bi-attribute network that minimizes a linear combination of two path costs, one of which is evaluated by a nondecreasing concave function. Due to the nonadditivity of its objective function, Bellman’s principle of optimality does not hold. This paper proposes a parametric search method to solve the BC-SP problem, which only needs to solve a series of shortest path problems, i.e., the parameterized subproblems (PSPs). Several techniques are developed to reduce both the number of PSPs and the computation time for these PSPs. Specifically, we first identify two properties of the BC-SP problem to guide the parametric search using the gradient and concavity of its objective function. Based on the properties, a monotonic descent search (MDS) and an intersection point search (IPS) are proposed. Second, we design a speedup label correcting (LC) algorithm, which uses optimal solutions of previously solved PSPs to reduce the number of labeling operations for subsequent PSPs. The MDS, IPS and speedup LC techniques are embedded into a branch-and-bound based interval search to guarantee optimality. The performance of the proposed method is tested on the mean-standard deviation shortest path problem and the route choice problem with a quadratic disutility function. Experiments on both real transportation networks and grid networks show that the proposed method reduces the computation time of existing algorithms by one to two orders of magnitude.

Routing under balance

Abstract: We introduce the notion of balance for directed graphs: a weighted directed graph is α-balanced if for every cut S ⊆ V, the total weight of edges going from S to V∖ S is within factor α of the total weight of edges going from V∖ S to S. Several important families of graphs are nearly balanced, in particular, Eulerian graphs (with α = 1) and residual graphs of (1+є)-approximate undirected maximum flows (with α=O(1/є)). We use the notion of balance to give a more fine-grained understanding of several well-studied routing questions that are considerably harder in directed graphs. We first revisit oblivious routings in directed graphs. Our main algorithmic result is an oblivious routing scheme for single-source instances that achieve an O(α · log3 n / loglogn) competitive ratio. In the process, we make several technical contributions which may be of independent interest. In particular, we give an efficient algorithm for computing low-radius decompositions of directed graphs parameterized by balance. We also define and construct low-stretch arborescences, a generalization of low-stretch spanning trees to directed graphs. On the negative side, we present new lower bounds for oblivious routing problems on directed graphs. We show that the competitive ratio of oblivious routing algorithms for directed graphs is Ω(n) in general; this result improves upon the long-standing best known lower bound of Ω(√n) by Hajiaghayi et al. We also show that our restriction to single-source instances is necessary by showing an Ω(√n) lower bound for multiple-source oblivious routing in Eulerian graphs. We also study the maximum flow problem in balanced directed graphs with arbitrary capacities. We develop an efficient algorithm that finds an (1+є)-approximate maximum flows in α-balanced graphs in time O(m α2 / є2). We show that, using our approximate maximum flow algorithm, we can efficiently determine whether a given directed graph is α-balanced. Additionally, we give an application to the directed sparsest cut problem.