Fast All-Pairs Shortest Paths Algorithm in Large Sparse Graph

Wireless Sensor Network (WSN) has many applications in day to day life. One of the applications of WSN is target tracking that uses sensors to track and localize the targets. This makes sensor node deployment in a network necessary and highly important. But the main challenge faced by sensor networks is the power consumption by sensor nodes. Hence an energy optimization technique must be in place that ultimately reduces the actual power consumption of the sensor nodes. Here, the shortest path algorithms are used in routing and it helps in reducing the energy consumption. The shortest path algorithms used in wireless sensor network are Dijkstra's algorithm and Floyd-Warshall's algorithm. They are also used for calculating throughput, packet delivery ratio and energy consumption for given number of nodes. The Floyd-Warshall's algorithm is not scalable and efficient for routing in WSN. Hence, we use modified Floyd-Warshall's algorithm that obtains better throughput, packet delivery ratio and minimized energy consumption. The new Modified Floyd-Warshall's algorithm calculates the shortest path from node to node by adding an acknowledgment mechanism. We have done a comparative analysis of Dijkstra's algorithm, Floyd-Warshall's algorithm, and modified Floyd-Warshall's algorithm for parameters like packet deliver ratio, throughput, and energy optimization for multiple nodes. From simulation results it is observed that the modified Floyd-Warshall's algorithm works better than Dijkstra's and Floyd Warshall's algorithm in terms of Packet Delivery ratio, throughput and energy consumption.

The all-pairs shortest path problem is a classic problem to study characteristics of the given graphs. Though many efficient all-pairs shortest path algorithms have been published, it is still a very expensive computing task, especially with large graph datasets. In this paper, we propose an efficient parallel all-pairs shortest path algorithm based on Peng et al.'s fast sequential algorithm on shared-memory parallel environments to achieve faster and more efficient calculation for large-scale real-world networks. Peng et al.'s algorithm needs to sort vertices with respect to their degrees. However, it turns out the original algorithm uses less efficient sorting method, which is a significant portion of parallel overhead. Therefore, we also propose an efficient parallel method to sort data within a fixed range, in order to minimize the parallel overhead in our parallel algorithm. The optimized efficient sorting method can be used for general sorting purposes. Our experimental analysis shows that our proposed parallel algorithm achieves very high parallel speedup, even hyper-linear speedup, with real-world test datasets on two different shared-memory multi-core systems.

This paper explores techniques to quickly solve the maximum weight clique problem (MWCP) in very large scale sparse graphs. Due to their size, and the hardness of MWCP, it is infeasible to solve many of these graphs with exact algorithms. Although recent heuristic algorithms make progress in solving MWCP in large graphs, they still need considerable time to get a high-quality solution. In this work, we focus on solving MWCP for large sparse graphs within a short time limit. We propose a new method for MWCP which interleaves clique finding with data reduction rules. We propose novel ideas to make this process efficient, and develop an algorithm called FastWClq. Experiments on a broad range of large sparse graphs show that FastWClq finds better solutions than state-of-the-art algorithms while the running time of FastWClq is much shorter than the competitors for most instances. Further, FastWClq proves the optimality of its solutions for roughly half of the graphs, all with at least 105 vertices, with an average time of 21 seconds.

The main result of this paper is building a new parallel algorithm based on Floyd-Warshall algorithm to find the Shortest Path for all-pair. The problem of finding All Pair Shortest Path (APSP). APSP is performed on many structures such as: MPI, OpenMP, Cuda and others. When the input data is large, we need to find ways to improve computing power to reduce calculation time for the APSP algorithm. We present a contribution for the APSP in the MapReduce architectures. The idea of this algorithm is to multi Workers to work in parallel by Floyd-Warshall algorithm. There is one Master assigning Map and Reduce. The mapper simultaneously executes their work and sends their data to the reducer until the job is finished. The MapReduce algorithm has two functions: Map and Reduce. They receive key/value pairs based on an adjacency list. The algorithm performs in MapReduce and our results prove that the proposed approach improved Computing Performance for Algorithm Finding the Shortest Path for all-pair. Some fundamental results are systematized and proved.

Geographic information systems related to the position of objects in the earth's surface such as the position of the attractions that are connected with roads. Streets that are interconnected to form a complex network. Usually people use our manual to find the shortest route of an object toward a particular object but it is less than optimal because it requires precision and a long time. The shortest route search problem has been solved by several algorithms such as Dijkstra's algorithm, Astar, Floyd Warshall, Ant Colony and others. Based on this final task is to discuss the implementation of the shortest route to the geographic information system of tourism in a city using the distance parameter. City to be addressed is Sawahlunto has 12 attractions that are connected with road network. Shortest path search algorithm used is Dijkstra's and Astar algorithm. Geographic information system of tourism is displayed in a web form, with the programming language PHP, database PosgreSQL and support PostGIS and pgRouting. Based on the testing of these applications can provide the shortest route along the travel distance to and from 12 tourist sites in Sawahlunto. Astar algorithm is faster in the process of finding the shortest route instead of Dijkstra's algorithm with an average gap of 40 ms. Application also provides information supporting the tourism profile information, attractions, culture, and contacts. Keyword : shortest path, Dijkstra algorithm, Astar algorithm, geographic information system

Graph algorithms play an important role in several fields of sciences and engineering. Prominent among them are the All-Pairs-Shortest-Paths (APSP) and related problems. Indeed there are several efficient implementations for such problems on a variety of modern multi- and many-core architectures. It can be noticed that for several graph problems, parallelism offers only a limited success as current parallel architectures have severe short-comings when deployed for most graph algorithms. At the same time, some of these graphs exhibit clear structural properties due to their sparsity. This calls for particular solution strategies aimed at scalable processing of large, sparse graphs on modern parallel architectures. In this paper, we study the applicability of an ear decomposition of graphs to problems such as all-pairs-shortestpaths and minimum cost cycle basis. Through experimentation, we show that the resulting solutions are scalable in terms of both memory usage and also their speedup over best known current implementations. We believe that our techniques have the potential to be relevant for designing scalable solutions for other computations on large sparse graphs.

The multiple pairs shortest path problem (MPSP) arises in many applications where the shortest paths and distances between only some specific pairs of origin-destination (OD) nodes in a network are desired. The traditional repeated single-source shortest path (SSSP) and all pairs shortest paths (APSP) algorithms often do unnecessary computation to solve the MPSP problem. We propose a new shortest path algorithm to save computational work when solving the MPSP problem. Our method is especially suitable for applications with fixed network topology but changeable arc lengths and desired OD pairs. Preliminary computational experiments demonstrate our algorithm’s superiority on airline network problems over other APSP and SSSP algorithms.

In this paper we consider two families of competing algorithms for finding the shortest paths between all pairs of vertices (APSP) in directed weighted large graphs with different edge densities: Dijkstra and Floyd-Warshall. For comparison, we have taken Dijkstra's algorithm with dynamically varying binary heap, which solves the APSP prob-lem purely in parallel by repeatedly executing on all vertices of the graph considered as source vertices, and we have taken blocked Floyd-Warshall algorithm, which is also well-parallelizable. It is known that in terms of computational complexity, the first algorithm is preferable on sparse graphs and the second algorithm is preferable on dense graphs. At the same time, it is not clear what are the ranges of graph densities at which the first algorithm will consume less CPU time than the second algorithm. This paper describes multithreaded implementations of parallel algorithms on multicore processors that make different usage of synchronization primitives such as mutex, conditional variable, lock-ing, and atomic operation. By conducting computational experiments on an 8-core Intel(R) Core(TM) i7-10700 CPU @ 2.90GHz, we found that each algorithm has a preferred graph density. In the case of multi-threaded parallel imple-mentation, the blocked Floyd-Warshall algorithm has lower running time than Dijkstra's algorithm if the graph densi-ty is greater than 0.5. Otherwise, Dijkstra's algorithm runs faster. In the case of single-threaded implementation, the split point is 0.43.

Given a bipartite graph, the maximum balanced biclique (MBB) problem, discovering a mutually connected while disjoint sets of equal size with the maximum cardinality, plays a significant role for mining the bipartite graph and has numerous applications. Despite the NP-hardness of the MBB problem, in this paper, we show that an exact MBB can be discovered extremely fast in bipartite graphs for real applications. We propose two exact algorithms dedicated for small dense and large sparse bipartite graphs respectively. For dense bipartite graphs, an O*(1.3803n) algorithm is proposed. This algorithm in fact can find an MBB very fast for small dense bipartite graphs that are common for applications such as VLSI design. This is because, using our proposed novel techniques, the search can fast converge to sufficiently dense bipartite graphs which we prove to be polynomial-time solvable. For large sparse bipartite graphs typical for applications such as biological data analysis, an O*(1.3803 δ) algorithm is proposed, where δ is only a few hundred for large sparse bipartite graphs with millions of vertices. The indispensible optimization that leads to this time complexity is: we transform a large sparse bipartite graph into a limited number of dense subgraphs such that each of the dense subgraphs has up to δ vertices and then apply our proposed algorithm for dense bipartite graphs on each of the subgraphs. To further speed up this algorithm, tighter upper bounds, faster heuristics and more effective reductions are proposed, allowing an MBB to be discovered within a few seconds for bipartite graphs with millions of vertices. Extensive experiments are conducted on synthetic and real large bipartite graphs to demonstrate the efficiency and effectiveness of our proposed algorithms and techniques.

Solving all-pairs shortest path by single-source computations: Theory and practice

External matrix multiplication and all-pairs shortest path

We implement a new algorithm for listing all maximal cliques in sparse graphs due to Eppstein, Löffler, and Strash (ISAAC 2010) and analyze its performance on a large corpus of real-world graphs. Our analysis shows that this algorithm is the first to offer a practical solution to listing all maximal cliques in large sparse graphs. All other theoretically-fast algorithms for sparse graphs have been shown to be significantly slower than the algorithm of Tomita et al. (Theoretical Computer Science, 2006) in practice. However, the algorithm of Tomita et al. uses an adjacency matrix, which requires too much space for large sparse graphs. Our new algorithm opens the door for fast analysis of large sparse graphs whose adjacency matrix will not fit into working memory.Keywordsmaximal clique listingBron–Kerbosch algorithmsparse graphs d-degenerate graphs

We propose a blocked version of Floyd's all-pairs shortestpaths algorithm. The blocked algorithm makes better utilization of cache than does Floyd's original algorithm. Experiments indicate that the blocked algorithm delivers a speedup (relative to the unblocked Floyd's algorithm) between 1.6 and 1.9 on a Sun Ultra Enterprise 4000/5000 for graphs that have between 480 and 3200 vertices. The measured speedup on an SGI O2 for graphs with between 240 and 1200 vertices is between 1.6 and 2.

Solving the all pairs shortest path (APSP) problem is crucial for graph theory and various practical applications, but remains challenging for large graphs with an enormous number of nodes and edges when the computing resources is limited. In this work, based on an effective graph partition method, we further propose a divisive approach to obtain the APSP for large graphs. A large graph is firstly divided into smaller subgraphs that may not be of an equal size, and then getting the APSP of each subgraph separately. By concatenating the APSPs of the subgraphs, the APSP of the whole graph can be obtained. Experimental results confirm the effectiveness and correctness of our approach for obtaining APSP on large graphs with limited computing resources.

Zwick's $(1+\varepsilon)$-approximation algorithm for the All Pairs Shortest Path (APSP) problem runs in time $\widetilde{O}(\frac{n^\omega}{\varepsilon} \log{W})$, where $\omega \le 2.373$ is the exponent of matrix multiplication and $W$ denotes the largest weight. This can be used to approximate several graph characteristics including the diameter, radius, median, minimum-weight triangle, and minimum-weight cycle in the same time bound. Since Zwick's algorithm uses the scaling technique, it has a factor $\log W$ in the running time. In this paper, we study whether APSP and related problems admit approximation schemes avoiding the scaling technique. That is, the number of arithmetic operations should be independent of $W$; this is called strongly polynomial. Our main results are as follows. - We design approximation schemes in strongly polynomial time $O(\frac{n^\omega}{\varepsilon} \text{polylog}(\frac{n}{\varepsilon}))$ for APSP on undirected graphs as well as for the graph characteristics diameter, radius, median, minimum-weight triangle, and minimum-weight cycle on directed or undirected graphs. - For APSP on directed graphs we design an approximation scheme in strongly polynomial time $O(n^{\frac{\omega + 3}{2}} \varepsilon^{-1} \text{polylog}(\frac{n}{\varepsilon}))$. This is significantly faster than the best exact algorithm. - We explain why our approximation scheme for APSP on directed graphs has a worse exponent than $\omega$: Any improvement over our exponent $\frac{\omega + 3}{2}$ would improve the best known algorithm for Min-Max Product In fact, we prove that approximating directed APSP and exactly computing the Min-Max Product are equivalent.