Dynamic Algorithms for the Shortest Path Routing Problem: Learning Automata-Based Solutions

Despite the numerous works related to the problem of finding the shortest paths (SP), attention to development of speed-efficient algorithms for constructing SP does not decrease. This is primarily due to the fact that in the overwhelming majority of cases such algorithms are often used to solve individual subtasks in many applications in various fields of natural science, and the time to solve the general optimization problem is largely determined by the time of constructing SP. Three groups of single-criterion algorithms are considered: network combinatorial algorithms; algebraic or matrix algorithms; algorithms based on methods for solving linear programming problems (LP-methods). The article presents an overview, analysis and classification of methods and algorithms for constructing the shortest paths on networks and graphs between given subsets of the network nodes (Single Source Shortest Path, SSSP) and between all pairs of nodes (Shortest Path Tree, SPT and All Pairs Shortest Paths, APSP). Estimates of the time complexity of the best known algorithms for solving SSST and APSP problems by combinatorial, matrix and LP methods for networks with non-negative arcs lengths and networks with negative arcs lengths and cycles of negative lengths are given. It is noted that for solving individual SSSP problems there are "almost optimal" algorithms in theory and practice, at the same time, for solving a wider class of problems, including APSP problem, there are prerequisites for improving existing algorithms. In recent years the evolution of methods for solving the problems of finding SP has been associated with development and further improvement of effective structures of abstract data types for representing objects of the problem and the creation of parallel algorithms for multiprocessor solving the problem. The main directions of further research on the development of effective methods and algorithms for solving the problems of finding the shortest paths are determined.

Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time, so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) revamping road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c) and (d) are restrictions to different graph classes. We show that (a) is intractable, even for relaxed variants of the problem. For (b), (c) and (d), we present efficient algorithms to solve the respective problems. We also generalise the problem to when at most k (for some k >= 2) contiguous vertices on a shortest path can be changed at a time.

Characterizing LR-visibility polygons and related problems

In order to solve the path planning problem of time-dependent road network(TDRN), an dynamic A* landmarks triangle algorithm(ALT) is proposed based on landmark-oriented technique and short-path tree(SPT). There are three main contributions: (1) constructing the shortest path tree in the preprocessing stage and calculating the distance between the landmark and other nodes; (2) using the dynamic shortest path tree to optimize the query in the point-to-point heuristic path planning process; (3) When the edge weight of the network changes, the shortest path tree is dynamically updated, and the structural characteristics of the tree are used to reduce the redundancy calculation. Experimental results indicate that the DALT algorithm not only outperforms the ALT implementation in point-to-point shortest path problem as the average query time is reduced by up to 51.71%, but also computes economically for updating shortest path tree compared with previous dynamic update algorithm as the average update times for increments are reduced by up to 9.90% with less modifications.

On the K shortest path trees problem

Voronoi-based splinegon decomposition and shortest-path tree computation

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.

This paper proposes a set of novel distributed algorithms on m-D mesh overlay configurations for short delay and low resource consumption application layer multicast. In contrast to previous approaches, our application layer multicast adopts two-layer tree architecture and the novelty and contribution are: (1) cluster formation algorithm assigns the closest group members into the same cluster that greatly decreases the multicast delay and resource consumption caused by the message transmission among the members with long distances; (2) optimal core selection algorithm seeks the cluster member who has the minimum sum of static delay distances to other cluster members as the optimal cores (i.e. cluster cores) that guarantees the short multicast delay; (3) weighted shortest path tree generation algorithm constructs a shortest path tree rooted at the optimal core for each cluster. The shortest path tree utilizes the minimum sum of links that are on the shortest paths among the cluster members; and (4) distributed multicast routing algorithm directs the multicast messages to be efficiently distributed along the two-layer multicast architecture in parallel without a global control. The extended simulation results indicate that the application layer multicast constructed by our algorithms is efficient in terms of short multicast delay and low network resource consumption as compared with other well-known existing multicast solutions.

On edges crossing few other edges in simple topological complete graphs

The critical problem for multicasting is to find optimal multicast tree. Most popular multicast algorithms over WAN are usually based on either Dijkstra’s shortest path tree or Steiner tree, which assume that sender can send a message to the next destination node right away. This can lead to a non-optimal multicast tree as it ignores gap parameter in LogP model. From our preliminary study, we found that gap is very significant especially in WAN. In this paper, we take gap into consideration and propose a minimization technique, which apply gap parameters to Dijkstra’s shortest path algorithm to construct minimized shortest path trees. We compare the multicast latency of normal Dijkstra’s shortest path multicast tree and our minimized multicast tree with simulation. The results show that multicast tree based on our proposed algorithm can clearly out-perform the multicast tree based on Dijkstra’s algorithm. Our results further indicate that our proposed algorithm can perform very well when message size is large with lots of nodes.

Motivated by the increasing need for fast distributed processing of large-scale graphs such as the Web graph and various social networks, we study a number of fundamental graph problems in the message-passing model, where we have k machines that jointly perform computation on an arbitrary n-node (typically, n > k) input graph. The graph is assumed to be randomly partitioned among the k ≥ 2 machines (a common implementation in many real world systems). The communication is point-to-point, and the goal is to minimize the time complexity i.e., the number of communication rounds, of solving various fundamental graph problems.We present lower bounds that quantify the fundamental time limitations of distributively solving graph problems. We first show a lower bound of Ω(n/k) rounds for computing a spanning tree (ST) of the input graph. This result also implies the same bound for other fundamental problems such as computing a minimum spanning tree (MST), breadth-first tree (BFS), and shortest paths tree (SPT). We also show an Ω(n/k2) lower bound for connectivity ST verification and other related problems. Our lower bounds develop and use new bounds in random-partition communication complexity.To complement our lower bounds, we also give algorithms for various fundamental graph problems, e.g., PageRank, MST, connectivity, ST verification, shortest paths, cuts, spanners, covering problems, densest subgraph, subgraph isomorphism, finding triangles, etc. We show that problems such as PageRank, MST, connectivity, and graph covering can be solved in O(n/k) time (the notation O hides polylog(n) factors and an additive polylog(n) term); this shows that one can achieve almost linear (in k) speedup, whereas for shortest paths, we present algorithms that run in O(n/Ok) time (for (1 + e)-factor approximation) and in O(n/k) time (for O(log n)-factor approximation) respectively.Our results are a step towards understanding the complexity of distributively solving large-scale graph problems.

Efficient algorithm for finding k shortest paths based on re-optimization technique

摘要: 针对无线传感器网络数据汇集应用中, 由于数据流量大, 相邻路径之间容易发生串扰、信道竞争和冲突, 造成拥塞问题, 提出了基于拥塞控制的无线传感器网络数据汇集树生成算法(Data gather tree algorithm based on congestion control, DGT-CC). DGT-CC算法通过层次发现、邻居发现、启发式搜索和流量均衡策略构造一棵最短路径最小拥塞权值树. 理论分析证明DGT-CC算法收敛, 并能够构造一棵最短路径最小拥塞权值树, 仿真实验表明DGT-CC算法在丢包率、网络吞吐量和时延方面都较普通的最短路径树具有更好的性能. 关键词: 无线传感器网络 / 数据汇集 / 拥塞控制 / 路由树

Computing shortest paths is a fundamental operation in processing graph data. In many real-world applications, discovering shortest paths between two vertices empowers us to make full use of the underlying structure to understand how vertices are related in a graph, e.g. the strength of social ties between individuals in a social network. In this paper, we study the shortest-path-graph problem that aims to efficiently compute a shortest path graph containing exactly all shortest paths between any arbitrary pair of vertices on complex networks. Our goal is to design an exact solution that can scale to graphs with millions or billions of vertices and edges. To achieve high scalability, we propose a novel method, Query-by-Sketch (QbS), which efficiently leverages offline labelling (i.e., precomputed labels) to guide online searching through a fast sketching process that summarizes the important structural aspects of shortest paths in answering shortest-path-graph queries. We theoretically prove the correctness of this method and analyze its computational complexity. To empirically verify the efficiency of QbS, we conduct experiments on 12 real-world datasets, among which the largest dataset has 1.7 billion vertices and 7.8 billion edges. The experimental results show that QbS can answer shortest-path graph queries in microseconds for million-scale graphs and less than half a second for billion-scale graphs.

In the dynamic Single-Source Shortest Paths (SSSP) problem, we are given a graph G=(V,E) subject to edge insertions and deletions and a source vertex s∈ V, and the goal is to maintain the distance d(s,t) for all t∈ V.