Dynamic Single Source Shortest Path Research Articles

A Retroactive Approach for Dynamic Shortest Path Problem

Dynamic shortest path algorithms modify the existing shortest path tree or graph, taking into account changes in the underlying graph configuration. In the premise of this paper, the dynamic Dijkstra algorithm is specifically considered which is used for the solution of single source shortest path problem in dynamic graphs. Dynamic Single Source Shortest Path (DSSSP) problem is considered from a completely different perspective. For incorporating the dynamic changes into the solution, the retroactive data structure has been used. Dynamics of retroactive data structures provide a natural order for propagating the desired changes in the underlying graph configuration. DSSSP has been solved in efficient way with worst case complexity O(m log n) and also proved the correctness of our proposed algorithm.

Dynamizing Dijkstra: A solution to dynamic shortest path problem through retroactive priority queue

Dynamic shortest path algorithms are the ones which are used to accommodate the online sequence of update operations to the underlying graph topology and also facilitate the subsequent query operations. Many solutions exist for the different versions of the problem, all of which identify the set of vertices whose shortest paths may be affected by the changes and then update their shortest paths according to the update sequence. In this paper, we are dynamizing the Dijkstra algorithm which helps to efficiently solve the dynamic single source shortest path problem. Dynamization is achieved by using the retroactive priority queue data structure. Retroactive data structure identify the set of affected vertices step by step and thus help to accommodate the changes in least number of computations. So, with a suitable dynamic graph representation and the use of retroactive priority queue, we have proposed algorithm to dynamize Dijkstra algorithm giving solution of dynamic single source shortest path problem with complexity O(nlg m) for the update time. We have performed experimental analysis by comparing the performance of the proposed algorithm with other algorithms. Our experimental results indicate that the proposed algorithm has better performance in terms of time and memory usage.

EFS: Entropy First Search for Dynamic Shortest Path Problem in Large Sparse Graphs

Searching the dynamic shortest path is a hot topic recently. In this paper we proposed a new method to solve the dynamic single source shortest paths (SSSP) problem in sparse graphs. The main contributions are three: firstly, in the preprocessing stage, we use the unreachable and unstartable characteristics to avoid most of the non-solution path search. Secondly, Entropy first search (EFS) is introduced to speed up the search process in finding the possible shortest path and then the algorithm can converges as soon as possible. In addition, the update algorithm for dynamic shortest path search is proposed for practical use. The experiments in large random sparse graphs show the efficiency and benefits of the proposed method.

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

This paper presents the first Learning Automaton-based solution to the dynamic single source shortest path problem. It involves finding the shortest path in a single-source stochastic graph topology where there are continuous probabilistic updates in the edge-weights. The algorithm is significantly more efficient than the existing solutions, and can be used to find the "statistical" shortest path tree in the "average" graph topology. It converges to this solution irrespective of whether there are new changes in edge-weights taking place or not. In such random settings, the proposed learning automata solution converges to the set of shortest paths. On the other hand, the existing algorithms will fail to exhibit such a behavior, and would recalculate the affected shortest paths after each weight-change. The important contribution of the proposed algorithm is that all the edges in a stochastic graph are not probed, and even if they are, they are not all probed equally often. Indeed, the algorithm attempts to almost always probe only those edges that will be included in the shortest path graph, while probing the other edges minimally. This increases the performance of the proposed algorithm. All the algorithms were tested in environments where edge-weights change stochastically, and where the graph topologies undergo multiple simultaneous edge-weight updates. Its superiority in terms of the average number of processed nodes, scanned edges and the time per update operation, when compared with the existing algorithms, was experimentally established. The algorithm can be applicable in domains ranging from ground transportation to aerospace, from civilian applications to military, from spatial database applications to telecommunications networking.

Fully dynamic shortest paths in digraphs with arbitrary arc weights

We propose a new solution for the fully dynamic single source shortest paths problem in a directed graph G=( N, A) with arbitrary arc weights, that works for any digraph and has optimal space requirements and query time. If a negative-length cycle is introduced in the subgraph of G reachable from the source during an update operation, then it is detected by the algorithm. Zero-length cycles are explicitly handled. We evaluate the cost of the update operations as a function of a structural property of G and of the number of the output updates. We show that, if G has a k-bounded accounting function (as in the case of digraphs with genus, arboricity, degree, treewidth or page number bounded by k), then the update procedures require O(min{ m, k· n A }·log n) worst case time for weight-decrease operations, and O(min{ m·log n, k·( n A + n B )·log n+ n}) worst case time for weight-increase operations. Here, n=| N|, m=| A|, n A is the number of nodes affected by the input update, that is the nodes changing either the distance or the parent in the shortest paths tree, and n B is the number of nonaffected nodes considered by the algorithm that also belong to zero-length cycles. If zero-length cycles are not allowed, then n B is zero and the bound for weight-increase operations is O(min{ m·log n, k· n A ·log n+ n}). Similar amortized bounds hold if we perform also insertions and deletions of arcs.