A Multiple Pairs Shortest Path Algorithm

A comparison of algorithms for origin–destination matrix generation on real road networks and an approximation approach

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

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.

In various applications where the problem domain can be modeled into graphs, the shortest path computation in the graph is an indispensable challenge. In applications like online social networks and shortest route computation problems, the size of the graph is so large; the number of nodes have become close to hundreds of billions. Shortest path graph algorithms like SSSP (Single Source Shortest Path) and APSP (All Pairs Shortest Path) have low arithmetic intensity and irregular memory access patterns. The GPU implementation of many arithmetic and logical problems exceed the performance of the CPU system implementations. There is an increasing need for faster computation of shortest path in graphs for various applications. The objective of the work is to demonstrate that GPUs can efficiently perform shortest path computations on undirected weighted graphs considering space efficiency. For the implementation, GPUs supporting CUDA (Compute Unified Device Architecture) programming have been used. Additionally a space efficient approach called Triangular Matrix Method has been used.

Let G = (V, E, w) be a directed graph, where w : V ← R is an arbitrary weight function defined on its vertices. The bottleneck weight, or the capacity, of a path is the smallest weight of a vertex on the path. For two vertices u, v the bottleneck weight, or the capacity, from u to v, denoted c(u, v), is the maximum bottleneck weight of a path from u to v. In the All-Pairs Bottleneck Paths (APBP) problem we have to find the bottleneck weights for all ordered pairs of vertices. Our main result is an O(n2.575) time algorithm for the APBP problem. The exponent is derived from the exponent of fast matrix multiplication. Our algorithm is the first sub-cubic algorithm for this problem. Unlike the sub-cubic algorithm for the all-pairs shortest paths (APSP) problem, that only applies to bounded (or relatively small) integer edge or vertex weights, the algorithm presented for APBP problem works for arbitrary large vertex weights.The APBP problem has numerous applications, and several interesting problems that have recently attracted attention can be reduced to it, with no asymptotic loss in the running times of the known algorithms for these problems. Some examples are a result of Vassilevska and Williams [STOC 2006] on finding a triangle of maximum weight, a result of Bender et al. [SODA 2001] on computing least common ancestors in DAGs and a result of Kowaluk and Lingas [ICALP 2005] on finding maximum witnesses for boolean matrix multiplication. Thus, the APBP problem provides a uniform framework for these applications. For some of these problems, we can in fact show that their complexity is equivalent to that of the APBP problem.A slight modification of our algorithm enables us to compute shortest paths of maximum bottleneck weight. Let d(u, v) denote the (unweighted) distance from u to v, and let sc(u, v) denote the maximum bottleneck weight of a path from u to v having length d(u, v). The All-Pairs Bottleneck Shortest Paths (APBSP) problem is to compute sc(u, v) for all ordered pairs of vertices. We present an algorithm for the APBSP problem whose running time is O(n2.86).

Finding the All-Pairs Shortest Paths (APSP) in a graph is the key for various domains. Motivated by the graphs are sparse in most real-world applications, we store the whole graph as a compressed storage format in each process of the distributed computing clusters and combine the Floyd algorithm with the Dijkstra algorithm to solve the APSP problem in this work, which leads to the novel Fast APSP algorithm. In contrast to the state-of-the-art Part APSP algorithm, our algorithm adds some memory overhead to store the original sparse graph and uses local Floyd and global Dijkstra algorithms simultaneously. The payoff is the circumvention of expensive global communication, reducing one local FW operation, simplifying the Minplus function, and making its data access continuous. Furthermore, we propose a parallel framework to solve the problem of mismatch between the number of GPUs and the number of divisible blocks of a graph. The Fast APSP algorithm exhibits an average speedup of 16.97x compared to the CPU Dijkstra algorithm, 7.09x compared to the GPU Dijkstra algorithm, 7.09x compared to the Part APSP algorithm, and 4.6x compared to the decentralized Part APSP algorithm. It also shows good scalability in our experiments. It takes about 12.45 minutes to solve the APSP problem for the graph with 11,548,845 vertices by engaging 2048 GPUs.

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.

We propose a simple and effective heuristic that allows fast generation of a large set of shortest path alternatives in weighted directed graphs. The heuristic is based on existing deviation path algorithms for exact k shortest paths. It precalculates a backward shortest path tree and thus avoids doing many shortest path computations, but as a result it does not necessarily find the exact set of k shortest paths. Computational results on real-world road networks are reported. Our tests show that the quality of the paths produced by the heuristic is most satisfactory: typically, the kth path found by the heuristic is less than 1% longer than the exact kth shortest path, for values of k up to 10,000. Moreover, the heuristic runs very fast. We also show how the heuristic can be enhanced to an exact k shortest paths algorithm, which performs well in comparison with the existing exact k shortest path algorithms.

We present the buffer heap , a cache-oblivious priority queue that supports Delete-Min , Delete , and a hybrid Insert / Decrease-Key operation in O (1/ B log 2 N / M ) amortized block transfers from main memory, where M and B are the (unknown) cache size and block size, respectively, and N is the number of elements in the queue. We introduce the notion of a slim data structure that captures the situation when only a limited portion of the cache, which we call a slim cache , is available to the data structure to retain data between data structural operations. We show that a buffer heap automatically adapts to such an environment and supports all operations in O (1/λ + 1/ B log 2 N /λ) amortized block transfers each when the size of the slim cache is λ. Our results provide substantial improvements over known trivial cache performance bounds for cache-oblivious priority queues with Decrease-Keys . Using the buffer heap, we present cache-oblivious implementations of Dijkstra’s algorithm for undirected and directed single-source shortest path (SSSP) problems for graphs with non-negative real edge-weights. On a graph with n vertices and m edges, our algorithm for the undirected case performs O ( n + m / B log 2 n / M ) block transfers and for the directed case performs O (( n + m / B ) ċ log 2 n / B ) block transfers. These results give the first non-trivial cache-oblivious bounds for the SSSP problem on general graphs. For the all-pairs shortest path (APSP) problem on weighted undirected graphs, we incorporate slim buffer heaps into multi-buffer-buffer-heaps and use these to improve the cache-aware cache complexity. We also present a simple cache-oblivious APSP algorithm for unweighted undirected graphs that performs O ( mn / B log M / B n / B ) block transfers. This matches the cache-aware bound and is a substantial improvement over the previous cache-oblivious bound for the problem.

This paper presents a blocked algorithm for the all-pairs shortest paths (APSP) problem for a hybrid CPU-GPU system. In the blocked APSP algorithm, the amount of data communication between CPU (host) memory and GPU memory is minimized. When a problem size (the number of vertices in a graph) is large enough compared with a blocking factor, the blocked algorithm virtually requires CPU⇋GPU exchanging of two block matrices for a block computation on the GPU. We also estimate a required memory/communication bandwidth to utilize the GPU efficiently. On a system containing an Intel West mere CPU (Core i7 970) and an AMD Cypress GPU (Radeon HD 5870), our implementation of the blocked APSP algorithm achieves the performance up to 1 TFlop/s in single precision.

The Single-Source Shortest Path (SSSP) problem is to compute the shortest distances in a weighted graph from a source vertex to every other vertex. This paper focuses on parallel efficiency and scalability of SSSP computations on large-scale graphs. We propose an edge-fencing strategy to customize a SSSP algorithm's schedule for every SSSP computation, and devise a path-centric SSSP algorithm with this strategy. This strategy aims at reducing both the relaxed edges and relaxations repeated on each edge. It exploits a few fence values to select the relaxed edges and schedule edge relaxations according to lengths of the created paths. The path-centric algorithm works on a hierarchical graph model, and exploits the edge-fencing strategy to schedule edge relaxations in parallel settings. The hierarchical graph model quantifies the length distribution of shortest paths in large-scale graphs, provides appropriate fence values for every SSSP computation. The algorithm was evaluated on a wide range of synthetic graphs and real-world graphs. The experimental results suggest that our algorithm is efficient and scalable for graphs with skewed degree distributions, and its performance is relatively insensitive to the hierarchical graph model's accuracy.

Finding shortest distances and paths have always been a crucial area of study. In this article, we have proposed a linear time pre-processing method for optimization of shortest path and distance algorithms, for an undirected weighted graph with non-negative edges. In this approach, certain parts of a graph are grouped together as subgraphs termed as optimized deterministic routing areas (ODRAs). Each ODRA has a unique representative element called optimized proxy. The input graph is reduced by replacing ODRAs with optimized proxies and the new reduced graph is used for the shortest path and distance queries. This is a linear time pre-processing stage which helps in significant decrease in computation time for finding shortest distances and paths in graphs. The lower bound and upper bound of time complexity of the proposed algorithm is O(n/2) and O(n) respectively.

We present a study of multithreaded implementations of Thorup's algorithm for solving the Single Source Shortest Path (SSSP) problemfor undirected graphs. Our implementations leverage thefledgling MultiThreaded Graph Library (MTGL) to perform operations such as finding connected components and extracting induced subgraphs. To achieve good parallel performance from this algorithm, we deviate from several theoretically optimal algorithmic steps. In this paper; we present simplifications that perform better in practice, and we describe details of the multithreaded implementation that were necessary for scalability. We study synthetic graphs that model unstructured networks, such as social networks and economic transaction networks. Most of the recent progress in shortest path algorithms relies on structure that these networks do not have. In this work, we take a step back and explore the synergy between an elegant theoretical algorithm and an elegant computer architecture. Finally, we conclude with a prediction that this work will become relevant to shortest path computation on structured networks.

Faster Shortest-Path Algorithms for Planar Graphs

The reserved delivery service can help the information service providers to provide more consistent performance to their customers by the provisioning of reserved bandwidth on a delivery subnetwork. However, the configuration problem of a reserved delivery subnetwork is a hard optimization problem with no efficient exact algorithm besides exhaust searches. In this paper, we introduce a reserved delivery subnetwork configuration algorithm based on an idea of the maximum sharing shortest path tree (MSSPT). The proposed algorithm is motivated by the observation that the path sharing of multiple flows reduce the cost in reserved delivery subnetworks. Thus, a solution close to the optimal could occur in a subnetwork with the maximum degree of flow sharing. The maximum sharing shortest path tree problem can be categorized as a multicriteria shortest path problem. Using an algorithm based on the shortest path network (SPN, a unique subnetwork in which every path s → u is a shortest path in the original graph), we develop an efficient algorithm for the maximum sharing shortest path problem. The proposed algorithm is an approximation algorithm in nature because it takes the MSSPT as the approximation solution to the reserved delivery subnetwork configuration problem. Our experimental results show that the proposed algorithm has good performance against an easily computed lower bound, but has time complexity comparable to a single source shortest path algorithm.