Improved Shortest Path Algorithms by Dynamic Graph Decomposition

Abstract

In this thesis, we introduce three new approaches for solving the single source shortest path (SSSP) problem in nearly acyclic directed graphs, and algorithms based on these approaches. In the first approach, we extend a technique of strongly connected components (sc-components) decomposition by Takaoka [23], and the generalized decomposition approach is called a higher-order decomposition. According to Takaoka's definition of acyclicity, the degree of cyclicity of a graph G, cyc(G), is defined by the maximum cardinality of the strongly connected components of G. Based on the higher-order decomposition, we give a generalization of Takaoka's definition of acyclicity. That is, the degree of cyclicity cych(G) is the maximum cardinality of the hth order strongly connected components of G, where h is the number of times that the graph has been decomposed. Then, the original definition introduced by Takaoka [23] can be presented as: The degree of cyclicity cyc(G) is the maximum cardinality of the 1th order strongly connected components of G. The second approach presents a new method for measuring acyclicity based on modifications to two existing methods. In the new method, we decompose the given graph into a 1-dominator set, which is a set of acyclic subgraphs, where each sub-graph is dominated by one trigger vertex. Meanwhile we compute sc-components of a degenerated graph derived from triggers. Using this preprocessing, we can efficiently compute the single source shortest paths (SSSPs) for nearly acyclic graphs in O(m + r logl ) time, where r is the size of the 1-dominator set, and l is the size of the largest sc-component. In the third approach, we modify the concept of a 1-dominator set to that of a 1-2-dominator set, and achieve O(m + r²) time to compute a 1- 2-dominator set in a graph. Each of acyclic sub-graphs obtained by the 1-2-dominator set are dominated by one or two trigger vertices cooperatively. Such sub-graphs are potentially larger than those decomposed by the 1-dominator set. Thus fewer trigger vertices are needed to cover the graph, that is, rʹ ≤ r, where rʹ is the number of triggers in the 1-2-dominator set. When rʹ is much smaller than r, we can efficiently compute SSSPs in O(m + rʹlogrʹ) time.