David Eppstein

Finding the k Shortest Paths

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.

Finding the k shortest paths

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, and maximum inscribed polygons.<<ETX>>

Provably good mesh generation

Several versions of the problem of generating triangular meshes for finite-element methods are studied. It is shown how to triangulate a planar point set or a polygonally bounded domain with triangles of bounded aspect ratio, how to triangulate a planar point set with triangles having no obtuse angles, how to triangulate a point set in arbitrary dimension with simplices of bounded aspect ratio, and how to produce a linear-size Delaunay triangulation of a multidimensional point set by adding a linear number of extra points. All the triangulations have size within a constant factor of optimal and run in optimal time O(n log n+k) with input of size n and output of size k. No previous work on mesh generation simultaneously guarantees well-shaped elements and small total size. >

Chapter 9 – Spanning Trees and Spanners

We survey results in geometric network design theory, including algorithms for constructing minimum spanning trees and low-dilation graphs.

Sparsification—a technique for speeding up dynamic graph algorithms

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<italic>O</italic>(<italic>n</italic><supscrpt>1/2</supscrpt>) per change; 3-edge connectivity, in time <italic>O</italic>(<italic>n</italic><supscrpt>2/3</supscrpt>) per change; 4-edge connectivity, in time <italic>O</italic>(<italic>n</italic>α(<italic>n</italic>)) per change; <italic>k</italic>-edge connectivity for constant <italic>k</italic>, in time <italic>O</italic>(<italic>n</italic>log<italic>n</italic>) per change;2-vertex connectivity, and 3-vertex connectivity, in the <italic>O</italic>(<italic>n</italic>) per change; and 4-vertex connectivity, in time <italic>O</italic>(<italic>n</italic>α(<italic>n</italic>)) 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 <italic>sparsification.</italic>

Subgraph Isomorphism in Planar Graphs and Related Problems

We solve the subgraph isomorphism problem in planar graphs in linear time, for any pattern of constant size. Our results are based on a technique of partitioning the planar graph into pieces of small tree-width, and applying dynamic programming within each piece. The same methods can be used to solve other planar graph problems including connectivity, diameter, girth, induced subgraph isomorphism, and shortest paths.

Diameter and Treewidth in Minor-Closed Graph Families

Abstract. 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.

Reset sequences for monotonic automata

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 a new algorithm based on breadth-first search is presented that runs in faster asymptotic time than Natarajan’s algorithms, and in addition finds the shortest possible reset sequence if such a sequence exists. Tight bounds on the length of the minimum reset sequence are given. The time and space bounds of another algorithm given by Natarajan are further improved.That algorithm finds reset sequences for arbitrary deterministicfinite automata when all states are initially possible.

Dynamic graph algorithms

In dynamic graph algorithms the following provide-or-boundproblem has to be solved quickly: Given a set S containing a subset R and a way of generating random elements fromS testing for membership inR, either (i) provide an element of R or (ii) give a (small) upper bound on the size of R that holds with high probability. We give an optimal algorithm for this problem. This algorithm improves the time per operation for various dyamic graph algorithms by a factor ofO(logn). For example, it improves the time per update for fully dynamic connectivity fromO(log n) toO(log n).

Optimal point placement for mesh smoothing

We study the problem of moving a vertex in an unstructured mesh of triangular, quadrilateral, or tetrahedral elements to optimize the shapes of adjacent elements. We show that many such problems can be solved in linear time using generalized linear programming. We also give efficient algorithms for some mesh smoothing problems that do not fit into the generalized linear programming paradigm.