Pairs Shortest Paths Research Articles

Subcubic Equivalences between Graph Centrality Problems, APSP, and Diameter

Measuring the importance of a node in a network is a major goal in the analysis of social networks, biological systems, transportation networks, and so forth. Different centrality measures have been proposed to capture the notion of node importance. For example, the center of a graph is a node that minimizes the maximum distance to any other node (the latter distance is the radius of the graph). The median of a graph is a node that minimizes the sum of the distances to all other nodes. Informally, the betweenness centrality of a node w measures the fraction of shortest paths that have w as an intermediate node. Finally, the reach centrality of a node w is the smallest distance r such that any s - t shortest path passing through w has either s or t in the ball of radius r around w . The fastest known algorithms to compute the center and the median of a graph and to compute the betweenness or reach centrality even of a single node take roughly cubic time in the number n of nodes in the input graph. It is open whether these problems admit truly subcubic algorithms, i.e., algorithms with running time Õ(n 3-δ ) for some constant δ > 0. 1 We relate the complexity of the mentioned centrality problems to two classical problems for which no truly subcubic algorithm is known, namely All Pairs Shortest Paths (APSP) and Diameter. We show that Radius, Median, and Betweenness Centrality are equivalent under subcubic reductions to APSP, i.e., that a truly subcubic algorithm for any of these problems implies a truly subcubic algorithm for all of them. We then show that Reach Centrality is equivalent to Diameter under subcubic reductions. The same holds for the problem of approximating Betweenness Centrality within any finite factor. Thus, the latter two centrality problems could potentially be solved in truly subcubic time, even if APSP required essentially cubic time. On the positive side, our reductions for Reach Centrality imply an improved Õ(Mn ω )-time algorithm for this problem in case of non-negative integer weights upper bounded by M , where ω is a fast matrix multiplication exponent.

Efficient construction of an approximate similarity graph for minimum spanning tree based clustering

Minimum spanning tree (MST) based unsupervised learning techniques are popular due to their ability to identify intrinsic clusters of heterogeneous structures. One of the important factors which affects their effectiveness is how to construct a sparse similarity graph which can effectively capture the local neighborhood information in sub quadratic time. In this paper, we propose a technique which efficiently uses the local nearest neighbors of data points to construct a similarity graph. The proposed approach consists of two steps. In the first step, the dataset is divided into groups using the dispersion level of data points and then all pair intra-partition edges are computed. In the second step, the boundary data points across the neighboring partitions are considered to produce inter-partition edges for increasing the accuracy. The resulting graph is generated by considering all intra- and inter-partition edges. Approximate MST of the similarity graph is constructed to show its efficacy. Experimental analyses demonstrate that the similarity graph captures shorter edges and discards the longest edges, based on graph diameter, all pair shortest path and weight error of MST. Moreover, the quality of the approximate MST is also validated by applying clustering technique on various synthetic and real data sets of different characteristics and cluster quality analyses demonstrate that it has a satisfying performance over other competing approximate MST construction techniques.

Matching Triangles and Basing Hardness on an Extremely Popular Conjecture

Due to the lack of unconditional polynomial lower bounds, it is now in fashion to prove conditional lower bounds in order to advance our understanding of the class P. The vast majority of these lower bounds are based on one of three famous hypotheses: the 3-SUM conjecture, the all pairs shortest paths (APSP) conjecture, and the Strong Exponential Time Hypothesis. Only circumstantial evidence is known in support of these hypotheses, and no formal relationship between them is known. In hopes of obtaining “less conditional" and therefore more reliable lower bounds, we consider the conjecture that at least one of the above three hypotheses is true. We design novel reductions from 3-SUM, APSP, and CNF-SAT, and derive interesting consequences of this very plausible conjecture, including tight $n^{3-o(1)}$ lower bounds for purely combinatorial problems about the triangles in unweighted graphs; new $n^{1-o(1)}$ lower bounds for the amortized update and query times of dynamic algorithms for Single-Source Reachability, Strongly Connected Components, and Max-Flow; new $n^{1.5-o(1)}$ lower bound for computing a set of $n$ $st$-maximum-flow values in a directed graph with $n$ nodes and $\tilde{O}(n)$ edges; and a hierarchy of natural graph problems on $n$ nodes with complexity $n^{c}$ for $c \in (2,3)$. Only slightly nontrivial consequences of this conjecture were known prior to our work. Along the way we also obtain new conditional lower bounds for the Single-Source Max-Flow problem.

Utilization of OpenCL for Large Graph Problems on Graphics Processing Unit

During recent years graphics processing unit (GPU) has become an inexpensive high-performance parallel computing units and general purpose computation on graphics processing unit (GPGPU) become an alternative of conventional CPU computation, which provide massive parallel computation capability to a normal computer over a system which uses only CPU. This paper investigates the performance of the algorithm for the solution of single source shortest path (SSSP) problem, all pair shortest path (APSP) problem and graph coloring problem for large graph on GPU regardless of a specific vendor. The application programming interface (API) used for programming in graphic processing unit is open compute language (OpenCL), it is a specification for heterogeneous computing. The performance of solutions on GPU is compared with the solution on CPU in term of their execution time. The result indicates the significant improvement in the performance of the solutions on GPU over solutions on CPU.

Comparative Analysis of Floyd Warshall and Dijkstras Algorithm using Opencl

Shortest path algorithms finds applications in large real world domains. All pair shortest path (APSP) and single source shortest path (SSSP) both have their special applications domains. All though every SSSP can be applied for all vertices to calculate APSP. But APSP cant. In this paper heterogeneous implementation of Floyd warshalls algorithm and Dijkstra’s algorithm is compared on dense graphs have positive edge weights ranging from 1 to 10. It is found that Dijkstra’s algorithm is better than Floyd warshall algorithm in sequencial implementation. But as there is less parallelism identified in dijkstra algorithm as compared to parallel to parallel FW gives less execution time as compared to Dijkstra’s.

Distance closures on complex networks

AbstractTo expand the toolbox available to network science, we study the isomorphism between distance and Fuzzy (proximity or strength) graphs. Distinct transitive closures in Fuzzy graphs lead to closures of their isomorphic distance graphs with widely different structural properties. For instance, the All Pairs Shortest Paths (APSP) problem, based on the Dijkstra algorithm, is equivalent to a metric closure, which is only one of the possible ways to calculate shortest paths in weighted graphs. We show that different closures lead to different distortions of the original topology of weighted graphs. Therefore, complex network analyses that depend on the calculation of shortest paths on weighted graphs should take into account the closure choice and associated topological distortion. We characterize the isomorphism using the max-min and Dombi disjunction/conjunction pairs. This allows us to: (1) study alternative distance closures, such as those based on diffusion, metric, and ultra-metric distances; (2) identify the operators closest to the metric closure of distance graphs (the APSP), but which are logically consistent; and (3) propose a simple method to compute alternative path length measures and corresponding distance closures using existing algorithms for the APSP. In particular, we show that a specific diffusion distance is promising for community detection in complex networks, and is based on desirable axioms for logical inference or approximate reasoning on networks; it also provides a simple algebraic means to compute diffusion processes on networks. Based on these results, we argue that choosing different distance closures can lead to different conclusions about indirect associations on network data, as well as the structure of complex networks, and are thus important to consider.

Parallelization of Shortest Path Finder on GPU: Floyd-Warshall

The project deals with implementation of Floyd Warshall Algorithm i.e All Pair Shortest Path. This algorithm is implemented using parallel programming concept for faster solution. This is a research based project in which the serial and parallel computations are compared.Floyd Warshall algorithm has overcome the drawbacks of Dijkstra’s and Bellman Ford Algorithm. For parallel programming, the project is implemented using NVIDIA GPU(NVIDIA GeForce 820M , 410M) for which CUDA(CUDA Toolkit 6.0) is used . The purpose of developing this project is to find the shortest path between all the present nodes in a graph.This system is designed to work on a large dataset(set of 48 or 72 or 100 cities).This project can be implemented for Airline Systems,Transportation services, Courier Services, Networking.

Work efficient parallel algorithms for large graph exploration on emerging heterogeneous architectures

Graph algorithms play a prominent role in several fields of sciences and engineering. Notable among them are graph traversal, finding the connected components of a graph, and computing shortest paths. There are several efficient implementations of the above problems on a variety of modern multiprocessor architectures.It can be noticed in recent times that the size of the graphs that correspond to real world datasets has been increasing. Parallelism offers only a limited succor to this situation as current parallel architectures have severe short-comings when deployed for most graph algorithms. At the same time, these graphs are also getting very sparse in nature. This calls for particular solution strategies aimed at processing large, sparse graphs on modern parallel architectures.In this paper, we introduce graph pruning as a technique that aims to reduce the size of the graph. Certain elements of the graph can be pruned depending on the nature of the computation. Once a solution is obtained on the pruned graph, the solution is extended to the entire graph. Towards, this end we investigate pruning based on two strategies that justifies their use in current real world graphs.We apply the above technique on three fundamental graph algorithms: breadth first search (BFS), Connected Components (CC), and All Pairs Shortest Paths (APSP). For experimentations, we use three different sources for real world graphs. To validate our technique, we implement our algorithms on a heterogeneous platform consisting of a multicore CPU and a GPU. On this platform, we achieve an average of 35% improvement compared to state-of-the-art solutions. Such an improvement has the potential to speed up other applications reliant on these algorithms.

A Fine Tuned Hybrid Implementation for Solving Shortest Path Problems using Bellman Ford

this paper a hybrid implementation for Bellman-Ford to solve shortest path problems is proposed using OpenCL. Here first parallel implementation for Bellman-Ford for single source shortest path (SSSP) problem and all pair shortest path (APSP) are analyzed on CPU and GPU and based on this analysis work is divided among CPU and GPU and hybrid implementation is done. As proper resource utilization is done here we have termed it a fine tuned implementation. We have got considerable speedup of 2.88x over parallel implementation on GPU for SSSP and 3.3x over parallel implementation of Bellman-Ford for APSP on GPU. Keywordspath problem , OpenCL , Graphical processing unit(GPU).

Sharing information for the all pairs shortest path problem

We improve the time complexity of the all pairs shortest path (APSP) problem for special classes of directed graphs. One is a nearly acyclic directed graph and the other is a directed graph with limited edge costs. The common idea for speed-up is information sharing by n single source shortest path (SSSP) problems that are solved for APSP. We measure the degree of acyclicity by the size, r, of a given set of feedback vertices. If r is small, the given graph can be considered to be nearly acyclic. We consider this parameter, r, in addition to the traditional parameters of the number of vertices, n, and that of edges, m. In the first part we improve the existing time complexity of O(mn+r3) for the all pairs shortest path problem to O(mn+rnlogn). This complexity is equal to or better than the previous one for all values of r under a reasonable assumption of m⩾n. In the second part, we deal with a directed graph with non-negative integer edge costs bounded by c. We show the all pairs shortest path (APSP) problem can be solved in O(mn+n2log(c/n)) time with the data structure of cascading bucket system. We use the traditional computational model such that comparison–addition operations on distance data and random access with O(logn) bits can be done in O(1) time. Also the graph is not separated, meaning m⩾n−1.

Energy and Throughput Analysis of Multicast Routing Algorithm for 2D Mesh Network on Chip

In Multi-Core Technology epoch, Network on Chip (NoC) architectures has been acknowledged as a solution to solve the design challenges of System on Chips (SoCs). The communication issue is played major role in design of effective NoCs. In order to achieve better communication among the multiple cores in NoC, an efficient routing algorithm is required. To evaluate the performance of NoCs, the performance parameters like throughput, energy and path length are focused by applying different routing algorithms. In this paper, we have analyzed network partitioning based on routing path and implemented two different routing algorithms named as Path Based Shortest Path (PBSP) and All Pair Shortest Path (APSP) algorithms for multicast messaging. These algorithms are developed and evaluated using C/C++ and Network Simulator for 2D Mesh NoC respectively. And finally we compare and analyze these two algorithms in terms of throughput and energy

Approximate shortest paths in weighted graphs

We present an approximation algorithm for the all pairs shortest paths (APSP) problem in weighed graphs. Our algorithm solves the APSP problem for weighted directed graphs, with real (positive or negative) weights, up to an additive error of ϵ. For any pair of vertices u , v , the algorithm finds a path whose length is at most δ ( u , v ) + ϵ . The algorithm is randomized and runs in O ˜ ( n ( ω + 3 ) / 2 ) < O ( n 2.688 ) time, where n is the number of vertices and ω is the matrix multiplication exponent. The absolute weights are allowed to mildly depend upon n, being at most n o ( 1 ) (we note that even if the weights are constants, δ ( u , v ) can be linear in n, while the error requirement ϵ is a small constant independent of n). Clearly, ϵ-additive approximations generalize exact algorithms for integer weighted instances. Hence, if ω = 2 + o ( 1 ) , this algorithm is as fast as any algorithm known for integer APSP in directed graphs, and is more general.

Computing the Degrees of All Cofactors in Mixed Polynomial Matrices

A mixed polynomial matrix is a polynomial matrix which has two kinds of nonzero coefficients: fixed constants that account for conservation laws and independent parameters that represent physical characteristics. This paper presents an algorithm for computing the degrees of all cofactors simultaneously in a regular mixed polynomial matrix. The algorithm is based on the valuated matroid intersection and all pair shortest paths. The technique is also used for improving the running time of the algorithm for minimizing the index of the differential-algebraic equation in the hybrid analysis for circuit simulation.

All-pairs bottleneck paths and max-min matrix products in truly subcubic time †

In the all pairs bottleneck paths (APBP) problem, one is given a directed graph with real weights on its edges. Viewing the weights as capacities, one is asked to determine, for all pairs (s,t) of vertices, the maximum amount of flow that can be routed along a single path from s to t. The APBP problem was first studied in operations research, shortly after the introduction of maximum flows and all pairs shortest paths. We present the first truly subcubic algorithm for APBP in general dense graphs. In par- ticular, we give a procedure for computing the (max,min)-product of two arbitrary matrices over R ( {¥, ¥} in O(n 2+w/3 ) O(n 2.792 ) time, where n is the number of vertices and w is the exponent for multiplication over rings. Max-min products can be used to compute the maximum bottleneck values for all pairs of vertices together with a successor matrix from which one can extract an explicit maximum bottleneck path for any pair of vertices in time linear in the length of the path.

An O(n 3(log log n/log n)5/4) Time Algorithm for All Pairs Shortest Path

We present an O(n 3(log log n/log n)5/4) time algorithm for all pairs shortest paths. This algorithm improves on the best previous result of O(n 3/log n) time.

Implications from a Network-Based Topological Analysis of Ubiquitin Unfolding Simulations

BackgroundThe architectural organization of protein structures has been the focus of intense research since it can hopefully lead to an understanding of how proteins fold. In earlier works we had attempted to identify the inherent structural organization in proteins through a study of protein topology. We obtained a modular partitioning of protein structures with the modules correlating well with experimental evidence of early folding units or “foldons”. Residues that connect different modules were shown to be those that were protected during the transition phase of folding.Methodology/Principal FindingsIn this work, we follow the topological path of ubiquitin through molecular dynamics unfolding simulations. We observed that the use of recurrence quantification analysis (RQA) could lead to the identification of the transition state during unfolding. Additionally, our earlier contention that the modules uncovered through our graph partitioning approach correlated well with early folding units was vindicated through our simulations. Moreover, residues identified from native structure as connector hubs and which had been shown to be those that were protected during the transition phase of folding were indeed more stable (less flexible) well beyond the transition state. Further analysis of the topological pathway suggests that the all pairs shortest path in a protein is minimized during folding.ConclusionsWe observed that treating a protein native structure as a network by having amino acid residues as nodes and the non-covalent interactions among them as links allows for the rationalization of many aspects of the folding process. The possibility to derive this information directly from 3D structure opens the way to the prediction of important residues in proteins, while the confirmation of the minimization of APSP for folding allows for the establishment of a potentially useful proxy for kinetic optimality in the validation of sequence-structure predictions.

Graph Theory and Economic Models: from Small to Large Size Applications

Abstract This empirical study explores the structure of macroeconomic models using major concepts and algorithms of the graph theory. Different sizes of applications with dynamic effects are considered. We will firstly examine the matching problem when assigning the equations to the variables. We'll also propose a simple method for improving the regular circular embedding of graphs on the basis of one of the longest circuit and adequate permutations. The determination of the maximal list of edge-disjoint circuits also produces an useful insight into the structure. A typology of the interdependent variables is proposed using the all- pairs shortest paths matrix. This classification is based on both the emissions of nodes towards the rest of the directed graph and the perturbations that the rest of the graph exerts on these nodes. The computations have been done using the softwares MATHEMATICA® 5.1, LINDO 6.1 and our own programs in Fortran 77L and C++.

Estimating all pairs shortest paths in restricted graph families

In this paper we show that a very simple and efficient approach can be used to solve the all pairs almost shortest path problem on the class of weakly chordal graphs and its different subclasses. M...

An optimal parallel algorithm for solving all-pairs shortest paths problem on circular-arc graphs

The shortest-paths problem is a fundamental problem in graph theory and finds diverse applications in various fields. This is why shortest path algorithms have been designed more thoroughly than any other algorithm in graph theory. A large number of optimization problems are mathematically equivalent to the problem of finding shortest paths in a graph. The shortest-path between a pair of vertices is defined as the path with shortest length between the pair of vertices. The shortest path from one vertex to another often gives the best way to route a message between the vertices. This paper presents anO(n2) time sequential algorithm and anO(n2/p+logn) time parallel algorithm on EREW PRAM model for solving all pairs shortest paths problem on circular-arc graphs, wherep andn represent respectively the number of processors and the number of vertices of the circular-arc graph.

NC Algorithms for the Single Most Vital Edge Problem with Respect to All Pairs Shortest Paths

NC Algorithms for the Single Most Vital Edge Problem with Respect to All Pairs Shortest Paths