Abuaiadh and Kingston gave an efficient algorithm for the single source shortest path problem for a nearly acyclic graph with O(m+n log t) computing time, where m and n are the numbers of edges and vertices of the given directed graph and t is the number of delete-min operations in the priority queue manipulation. They use the Fibonacci heap for the priority queue. If the graph is acyclic, we have t = 0 and the time complexity becomes O(m + n) which is linear and optimal. They claim that if the graph...

Shortest path problems can be solved very efficiently when a directed graph is nearly acyclic. Earlier results defined a graph decomposition, now called the 1-dominator set, which consists of a unique collection of acyclic structures with each single acyclic structure dominated by a single associated trigger vertex. In this framework, a specialised shortest path algorithm only spends delete-min operations on trigger vertices, thereby making the computation of shortest paths through non-trigger vertices easier. A previously presented algorithm computed the 1-dominator set in O(mn) worst-case time, which allowed it to be integrated as part of an O(mn + nr log r) time all-pairs algorithm. Here m and n respectively denote the number of edges and vertices in the graph, while r denotes the number of trigger vertices. A new algorithm presented in this paper computes the 1-dominator set in just O(m) time. This can be integrated as part of the O(m+r log r) time spent solving single-source, improving on the value of r obtained by the earlier tree-decomposition single-source algorithm. In addition, a new bi-directional form of 1-dominator set is presented, which further improves the value of r by defining acyclic structures in both directions over edges in the graph. The bi-directional 1-dominator set can similarly be computed in O(m) time and included as part of the O(m + r log r) time spent computing single-source. This paper also presents a new all-pairs algorithm under the more general framework where r is defined as the size of any predetermined feedback vertex set of the graph, improving the previous all-pairs time complexity from O(mn + nr 2) to O(mn + r 3).

This paper presents new algorithms for computing shortest paths in a nearly acyclic directed graph G = (V, E). The new algorithms improve on the worst-case running time of previous algorithms. Such algorithms use the concept of a 1-dominator set. A 1-dominator set divides the graph into a unique collection of acyclic subgraphs, where each acyclic subgraph is dominated by a single associated trigger vertex. The previous time for computing a 1dominator set is improved from O(mn) to O(m), where m = |E| and n = |V|. Efficient shortest...

Abstract. If the given problem instance is partially solved, we want to minimize our effort to solve the problem using that information. In this paper we introduce the measure of entropy H(S) for uncertainty in partially solved input data S(X) = (X1,..., Xk), where X is the entire data set, and each Xi is already solved. We use the entropy measure to analyze three example problems, sorting, shortest paths and minimum spanning trees. For sorting Xi is an ascending run, and for shortest paths, Xi is an acyclic part in the given graph. For minimum spanning trees, Xi is interpreted as a partially obtained minimum spanning tree for a subgraph. The entropy measure, H(S), is defined by regarding pi = |Xi|/|X | as a probability measure, that is, H(S) = −nΣ k i=1pi log pi, where n = Σ k i=1|Xi|. Then we show that we can sort the input data S(X) in O(H(S)) time, and solve the shortest path problem in O(m + H(S)) time where m is the number of edges of the graph. Finally we show that the minimum spanning tree is computed in O(m + H(S)) time. Keywords:entropy, complexity, adaptive sort, minimal mergesort, ascending runs, shortest paths, nearly acyclic graphs, minimum spanning trees 1

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Abstract. If the given problem instance is partially solved, we want to minimize our effort to solve the problem using that information. In this paper we introduce the measure of entropy, H(S), for uncertainty in partially solved input data S(X) = (X1,..., Xk), where X is the entire data set, and each Xi is already solved. We propose a generic algorithm that merges Xi’s repeatedly, and finishes when k becomes 1. We use the entropy measure to analyze three example problems, sorting, shortest paths and minimum spanning trees. For sorting Xi is an ascending run, and for minimum spanning trees, Xi is interpreted as a partially obtained minimum spanning tree for a subgraph. For shortest paths, Xi is an acyclic part in the given graph. When k is small, the graph can be regarded as nearly acyclic. The entropy measure, H(S), is defined by regarding pi = |Xi|/|X | as a probability measure, that is, H(S) = −nΣ k i=1pi log pi, where n = Σ k i=1|Xi|. We show that we can sort the input data S(X) in O(H(S)) time, and that we can complete the minimum cost spanning tree in O(m + H(S)) time, where m in the number of edges. Then we solve the shortest path problem in O(m + H(S)) time. Finally we define dual entropy on the partitioning process, whereby we give the time bounds on a generic quicksort and the shortest path problem for another kind of nearly acyclic graphs.

Dijkstra’s algorithm solves the single-source shortest path problem on any directed graph in O(m + n log n) worst-case time when a Fibonacci heap is used as the frontier set data structure. Here n is the number of vertices and m is the number of edges in the graph. If the graph is nearly acyclic, then other algorithms can achieve a time complexity lower than that of Dijkstra’s algorithm. Abuaiadh and Kingston gave a single source shortest path algorithm for nearly acyclic graphs with O(m + n log t) worst-case time complexity, where the new parameter t is the number of delete-min operations performed in priority queue manipulation. For nearly acyclic graphs, the value of t is expected to be small, allowing the algorithm to outperform Dijkstra’s algorithm. Takaoka, using a different definition for acyclicity, gave an algorithm with O(m + n log k) worstcase time complexity. In this algorithm, the new parameter k is the maximum cardinality of the strongly connected components in the graph. This thesis presents several new shortest path algorithms that define trigger