Generating multiple near-optimal solutions
Finding the k smallest spanning trees.
D. Eppstein.
2nd Scand. Worksh. Algorithm Theory, Bergen, Norway, 1990.
Springer, Lecture Notes in Comp. Sci. 447, 1990, pp. 38–47.
BIT 32: 237–248, 1992 (special issue for 2nd Scand. Worksh. Algorithm Theory).By removing edges not involved in some solution, and contracting edges involved in all solutions, we reduce the problem to one in a graph with O(k) edges and vertices. This simplification step transforms any time bound involving m or n to one involving min(m, k) or min(n, k) respectively. This paper also introduces the geometric version of the k smallest spanning trees problem (the graph version was long known) which it solves using order (k+1) Voronoi diagrams.
Iterated nearest neighbors and finding minimal polytopes.
D. Eppstein and J. Erickson.
Tech. Rep. 92-71, ICS, UCI, 1992.
4th ACM-SIAM Symp. Discrete Algorithms, Austin, 1993, pp. 64–73.
Disc. Comp. Geom. 11: 321–350, 1994.Showed that for various optimization criteria, the optimal polygon containing k of n points must be near one of the points, hence one can transform time bounds involving several factors of n to bounds linear in n but polynomial in k. Used as a subroutine are data structures for finding several nearest neighbors in rectilinear metric spaces, and algorithms for finding the deepest point in an arrangement of cubes or spheres.
Tree-weighted neighbors and geometric k smallest spanning trees.
D. Eppstein.
Tech. Rep. 92-77, ICS, UCI, 1992.
Int. J. Comp. Geom. & Appl. 4: 229–238, 1994."Finding the k smallest spanning trees" used higher order Voronoi diagrams to reduce the geometric k smallest spanning tree problem to the graph problem. Here I instead use nearest neighbors for a modified distance function where the bottleneck shortest path length is subtracted from the true distance between points. The result improves the planar time bounds and extends more easily to higher dimensions.
Ten algorithms for Egyptian fractions.
D. Eppstein.
Mathematica in Education and Research 4 (2): 5–15, 1995.Number theory. I survey and implement in Mathematica several methods for representing rational numbers as sums of distinct unit fractions. One of the methods involves searching for paths in a certain graph using a k shortest paths heuristic.
Sparsification—A technique for speeding up dynamic graph algorithms.
D. Eppstein, Z. Galil, G.F. Italiano, and A. Nissenzweig.
33rd IEEE Symp. Foundations of Comp. Sci., Pittsburgh, 1992, pp. 60–69.
Tech. Rep. RC 19272 (83907), IBM, 1993.
Tech. Rep. CS96-11, Univ. Ca' Foscari di Venezia, Oct. 1996.
J. ACM 44 (5): 669–696, 1997.Uses a divide and conquer on the edge set of a graph, together with the idea of replacing subgraphs by sparser certificates, to make various dynamic algorithms as fast on dense graphs as they are on sparse graphs. Applications include random generation of spanning trees as well as finding the \(k\) minimum weight spanning trees for a given parameter \(k\).
Improved sparsification.
D. Eppstein, Z. Galil, and G.F. Italiano.
Tech. Rep. 93-20, ICS, UCI, 1993.Saves a log factor over dynamic graph algorithms in "Sparsification" and their applications, by dividing vertices instead of edges. Merged into the journal version of "Sparsification".
Separator based sparsification for dynamic planar graph algorithms.
D. Eppstein, Z. Galil, G.F. Italiano, and T. Spencer.
25th ACM Symp. Theory of Computing, San Diego, 1993, pp. 208–217.Replaces portions of a hierarchical separator decomposition with smaller certificates to achieve fast update times for various dynamic planar graph problems. Applications include finding the k best spanning trees of a planar graph.
Separator based sparsification I: planarity testing and minimum spanning trees.
D. Eppstein, Z. Galil, G.F. Italiano, and T. Spencer.
J. Comp. Sys. Sci. 52: 3–27, 1996 (special issue for 25th STOC).First half of journal version of Separator based sparsification for dynamic planar graph algorithms.
Algorithms for proximity problems in higher dimensions.
M. T. Dickerson and D. Eppstein.
Comp. Geom. Theory & Applications 5: 277–291, 1996.Combines a method from "Provably good mesh generation" for finding sparse high-dimensional Delaunay triangulations, a method of Dickerson, Drysdale, and Sack ["Simple algorithms for enumerating interpoint distances", IJCGA 1992] for using Delaunay triangulations to search for nearest neighbors, and a method of Frederickson for speeding up tree-based searches. The results are fast algorithms for several proximity problems such as finding the k nearest neighbors to each point in a given point set.
Finding the k shortest paths.
D. Eppstein.
35th IEEE Symp. Foundations of Comp. Sci., Santa Fe, 1994, pp. 154–165.
Tech. Rep. 94-26, ICS, UCI, 1994.
SIAM J. Computing 28 (2): 652–673, 1998.This paper presents an algorithm that finds multiple short paths connecting two terminals in a graph (allowing repeated vertices and edges in the paths) in constant time per path after a preprocessing stage dominated by a single-source shortest path computation. The paths it finds are the k shortest in the graph, where k is a parameter given as input to the algorithm.
The k shortest paths problem has many important applications for finding alternative solutions to geographic path planning problems, network routing, hypothesis generation in computational linguistics, and sequence alignment and metabolic pathway finding in bioinformatics. Although there have been many papers on the k shortest paths problem before and after this one, it has become frequently cited in those application areas. Additionally, it marks a boundary in the theoretical study of the problem: prior theoretical work largely concerned how quickly the problem could be solved, a line of research that was closed off by the optimal time bounds of this paper. Subsequent work has focused instead on devising efficient algorithms for more complex alternative formulations of the problem that avoid the repeated vertices and other shortcomings of the alternative paths produced by this formulation.
The journal version also includes material from a separate 1995 technical report, "Finding common ancestors and disjoint paths in DAGs".
(Full paper – Graehl implementation – Jiménez-Marzal implementations – Shibuya implementation – Cliff OpenStreetMap demo)
Representing all minimum spanning trees with applications to counting and generation.
D. Eppstein.
Tech. Rep. 95-50, ICS, UCI, 1995.Shows how to find for any edge weighted graph G an equivalent graph EG such that the minimum spanning trees of G correspond one-for-one with the spanning trees of EG. The equivalent graph can be constructed in time O(m+n log n) given a single minimum spanning tree of G. As a consequence one can find fast algorithms for counting, listing, and randomly generating MSTs. Also discusses similar equivalent graph constructions for shortest paths, minimum cost flows, and bipartite matching.
Finding common ancestors and disjoint paths in DAGs.
D. Eppstein.
Tech. Rep. 95-52, ICS, UCI, 1995.This paper describes algorithms for finding pairs of vertex-disjoint paths in a DAG, either connecting two given nodes to a common ancestor, or connecting two given pairs of terminals. The main results were merged into the journal version of "Finding the k shortest paths".
All maximal independent sets and dynamic dominance for sparse graphs.
D. Eppstein.
arXiv:cs.DS/0407036.
16th ACM-SIAM Symp. Discrete Algorithms, Vancouver, 2005, pp. 451–459.
ACM Trans. Algorithms 5(4):A38, 2009.We show how to apply reverse search to list all maximal independent sets in bounded-degree graphs in constant time per set, in graphs from minor closed families in linear time per set, and in sparse graphs in subquadratic time per set. The latter two results rely on new data structures for maintaining a dynamic vertex set in a graph and quickly testing whether the set dominates all other vertices.
k-best enumeration.
D. Eppstein.
Encyclopedia of Algorithms (Ming-Yang Kao, ed.), Springer, added 2014.
arXiv:1412.5075.
Bull. EATCS 115, 2015.A brief survey of algorithms for finding the k shortest paths and related k-best enumeration problems. The arXiv/EATCS version is significantly longer and with more references than the Springer version.
K-best solutions of MSO problems on tree-decomposable graphs.
D. Eppstein and D. Kurz.
arXiv:1703.02784.
Proc. 12th International Symposium on Parameterized and Exact Computation (IPEC 2017), Vienna, Austria, 2017.
Leibniz International Proceedings in Informatics (LIPIcs) 89, pp. 16.1–16.13We show that, on graphs of bounded treewidth, for any optimization problem definable in monadic second-order logic, we can find the k best solutions in logarithmic time per solution.