We consider single-source shortest path algorithms that perform a sequence of relaxation steps whose ordering depends only on the input graph structure and not on its weights or the results of prior steps. Each step examines one edge of the graph, and replaces the tentative distance to the endpoint of the edge by its minimum with the tentative distance to the start of the edge, plus the edge length. As we prove, among such algorithms, the Bellman-Ford algorithm has optimal complexity for dense graphs and near-optimal complexity for sparse graphs, as a function of the number of edges and vertices in the given graph. Our analysis holds both for deterministic algorithms and for randomized algorithms that find shortest path distances with high probability.

We give algorithms for finding the k shortest paths (not required to be simple) connecting a pair of vertices in a digraph. Our algorithms output an implicit representation of these paths in a digraph with n vertices and m edges, in time O(m + n log n + k). We can also find the k shortest paths from a given source s to each vertex in the graph, in total time O(m + n log n + kn). We describe applications to dynamic programming problems including the knapsack problem, sequence alignment, maximum inscribed polygons, and genealogical relationship discovery.

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Abstract. The degeneracy of an n-vertex graph G is the smallest number d such that every subgraph of G contains a vertex of degree at most d. We show that there exists a nearly-optimal fixed-parameter tractable algorithm for enumerating all maximal cliques, parametrized by degeneracy. To achieve this result, we modify the classic Bron-Kerbosch algorithm and show that it runs in time O(dn3 d/3 ). We also provide matching upper and lower bounds showing that the largest possible number of maximal cliques in an n-vertex graph with degeneracy d (when d is a multiple of 3 and n ≥ d + 3) is (n − d)3 d/3 . Therefore, our algorithm matches the) worst-case output size of the problem whenever n − d = Ω (n).

It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show that treewidth is bounded by a function of the diameter in a minor-closed family, if and only if some apex graph does not belong to the family. In particular, the O(D) bound above can be extended to bounded-genus graphs. As a consequence, we extend several approximation algorithms and exact subgraph isomorphism algorithms from planar graphs to other graph families.Comment: 15 pages, 12 figure

Natarajan reduced the problem of designing a certain type of mechanical parts orienter to that of finding reset sequences for monotonic deterministic finite automata. He gave algorithms that in polynomial time either find such sequences or prove that no such sequence exists. In this paper we present a new algorithm based on breadth first search that runs in faster asymptotic time than Natarajan's algorithms, and in addition finds the shortest possible reset sequence if such a sequence exists. We give tight bounds on the length of the minimum reset sequence. We further improve the time and space bounds of another algorithm given by Natarajan, which finds reset sequences for arbitrary deterministic finite automata when all states are initially possible.

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We provide data strutures that maintain a graph as edges are inserted and deleted, and keep track of the following properties with the following times: minimum spanning forests, graph connectivity, graph 2-edge connectivity, and bipartiteness in time O ( n 1/2 ) per change; 3-edge connectivity, in time O ( n 2/3 ) per change; 4-edge connectivity, in time O ( n α( n )) per change; k -edge connectivity for constant k , in time O ( n log n ) per change;2-vertex connectivity, and 3-vertex connectivity, in the O ( n ) per change; and 4-vertex connectivity, in time O ( n α( n )) per change. Further results speed up the insertion times to match the bounds of known partially dynamic algorithms. All our algorithms are based on a new technique that transforms an algorithm for sparse graphs into one that will work on any graph, which we call sparsification.