Hybrid Bellman–Ford–Dijkstra algorithm

Abstract

We consider the single-source cheapest paths problem in a digraph with negative edge costs allowed. A hybrid of Bellman-Ford and Dijkstra algorithms is suggested, improving the running time bound upon Bellman-Ford for graphs with a sparse distribution of negative cost edges. The algorithm iterates Dijkstra several times without reinitializing the tentative value d(v) at vertices. At most k + 2 executions of Dijkstra solve the problem, if for any vertex reachable from the source, there exists a cheapest path to it with at most k negative cost edges.In addition, a new, straightforward proof is suggested that the Bellman-Ford algorithm results in a cheapest path tree from the source.

Setting an Optimization Problem of Distributing Traffic Across Multiple Independent Paths

“…This problem is a special case of the multi-path routing problem considered in the literature [7][8][9][10][11][12][13][14][15][16][17]. The algorithm for solving such a generalized problem has a large computational complexity and the time of solution of such a problem cannot meet the requirements for the means of multipath routing.…”

“…, where N is the number of vertices of the graph [13][14]. Then, assuming that the optimal number of independent shortest paths is equal to the connectivity of the graph of the network S (the worst case), the computation volume of this part of the problem can be estimated by the following formula:…”

“…It is shown in [12] that in the general case the linear programming problem is NPcomplete. However, in [11][12][13][14][15][16][17] it is shown that when solving applied problems, such as finding the maximum flow, etc., the simplex method has a polynomial complexity. The computational efficiency of the simplex method according to [11] can be estimated using the following two parameters: number of iterations (admissible basic solutions needed to achieve the optimal solution of the problem) and the total machine time required to solve the problem.…”

“…The paper presents the optimization procedure providing traffic distribution in a critical mode of network operation based on [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. In [2][3][4], network coding technique was applied to obtain the optimum transmission algorithm based on a minimum number of transmitted packets and hence obtaining the shortest path to be shorter than the traditional optimization algorithm.…”

Cache Friendly Bellman-Ford algorithm using OpenCL

“…Bellman-Ford algorithm [12] is applicable on graphs with negative weights and can also detect negative cycles where majority of algorithms fail. Bellman-Ford is also used in wireless sensor networks and other ad hoc networks as distributed Bellman Ford [7] can be used there. Distributed Bellman-Ford is also used as first ARPANET routing algorithm in 1969 [14].…”