**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 – Martins implementation – Cliff OpenStreetMap demo)

**Geometric lower bounds for parametric matroid optimization**.

D. Eppstein.

Tech. Rep. 95-11, ICS, UCI, 1995.

*27th ACM Symp. Theory of Computing,*Las Vegas, 1995, pp. 662–671.

*Disc. Comp. Geom.*20: 463–476, 1998.Considers graphs in which edge weights are linear functions of time. Shows nonlinear lower bounds on the number of different minimum spanning trees appearing over time by translation from geometric problem of lower envelopes of line segments. A matroid generalization has a better lower bound coming from many faces in line arrangements, and the uniform matroid problem is equivalent to the geometric

*k*-set problem.**On triangulating three-dimensional polygons**.

G. Barequet, M. Dickerson, and D. Eppstein.

*12th ACM Symp. Comp. Geom.,*Philadelphia, 1996, pp. 38–47.

*Comp. Geom. Theory & Applications*10: 155–170, 1998.It is NP-complete, given a simple polygon in 3-space, to find a triangulated simply-connected surface (without extra vertices) spanning that polygon. If extra vertices are allowed, or the surface may be curved, such a surface exists if and only if the polygon is unknotted; the complexity of testing knottedness remains open. Snoeyink has shown that exponentially many extra vertices may be required for a triangulated spanning disk.

(SCG paper – Full paper)

**The crust and the beta-skeleton: combinatorial curve reconstruction**.

N. Amenta, M. Bern, and D. Eppstein.

*Graphical Models & Image Processing*60/2 (2): 125–135, 1998.We consider the problem of "connect the dots": if we have an unknown smooth curve from which sample points have been selected, we would like to find a curve through the sample points that approximates the unknown curve. We show that if the local sample density is sufficiently high, a simple algorithm suffices: form the Delaunay triangulation of the sample points together with their Voronoi vertices, and keep only those Delaunay edges connecting original sample points. There have been many follow-up papers suggesting alternative methods, generalizing the problem to the reconstruction of curves with sharp corners or to curves and surfaces in higher dimensions, etc.

**Fast hierarchical clustering and other applications of dynamic closest pairs**.

D. Eppstein.

*9th ACM-SIAM Symp. Discrete Algorithms,*San Francisco, 1998, pp. 619–628.

arXiv:cs.DS/9912014.

*J. Experimental Algorithmics*5 (1): 1–23, 2000.This paper shows how to use my dynamic closest pair data structure from "Dynamic Euclidean minimum spanning trees" for some non-geometric problems including hierarchical clustering, greedy matching, and TSP heuristics. Experiments show variants of my data structures to be faster than previously used heuristics.

(Source code and experimental data – SODA paper – JEA home page)

**Raising roofs, crashing cycles, and playing pool: applications of a data structure for finding pairwise interactions**.

D. Eppstein and J. Erickson.

*14th ACM Symp. Comp. Geom.,*Minneapolis, 1998, pp. 58–67.

*Disc. Comp. Geom.*22 (4): 569–592, 1999 (special issue for SCG 1998).We use my dynamic closest pair data structure from "Dynamic Euclidean minimum spanning trees" to detect collisions among a collection of moving objects in sublinear time per collision. As one application, we can construct the straight skeleton of Aichholzer et al (and the mitered offset curves from which it is defined) in subquadratic time.

(Jeff's publications page and copy of the journal version)

**Geometric thickness of complete graphs**.

M. Dillencourt, D. Eppstein, and D. S. Hirschberg.

*6th Int. Symp. Graph Drawing,*Montreal, August 1998.

Springer,*Lecture Notes in Comp. Sci.*1547, 1998, pp. 102–110.

arXiv:math.CO/9910185.

*J. Graph Algorithms and Applications*4 (3): 5–17, 2000 (special issue for GD98).We define a notion of geometric thickness, intermediate between the previously studied concepts of graph thickness and book thickness: a graph has geometric thickness T if its vertices can be embedded in the plane, and its edges partitioned into T subsets, so that each subset forms a planar straight line graph. We then give upper and lower bounds on the geometric thickness of complete graphs.

**A disk-packing algorithm for an origami magic trick**.

M. Bern, E. Demaine, D. Eppstein, and B. Hayes.

*Int. Conf. Fun with Algorithms*, Elba, June 1998.

Proceedings in Informatics 4, Carleton Scientific, Waterloo, Canada, 1999, pp. 32–42.

*Origami*, A K Peters, 2002, pp. 17–28.^{3}: Proc. 3rd Int. Mtg. Origami Science, Math, and Education (Asilomar, California, 2001)We apply techniques from "Quadrilateral meshing by circle packing" to a magic trick of Houdini: fold a piece of paper so that with one straight cut, you can form your favorite polygon.

**Parametric and kinetic minimum spanning trees**.

P. K. Agarwal, D. Eppstein, L. J. Guibas, and M. R. Henzinger.

*39th IEEE Symp. Foundations of Comp. Sci.*, 1998, pp. 596–605..We describe algorithms for maintaining the minimum spanning tree in a graph in which the edge weights are piecewise linear functions of time that may change unpredictably. We solve the problem in time O(n

^{2/3}polylog n) per combinatorial change to the tree for general graphs, and in time O(n^{1/4}polylog n) per combinatorial change to the tree for planar graphs.**Incremental and decremental maintenance of planar width**.

D. Eppstein.

arXiv:cs.CG/9809038.

*10th ACM-SIAM Symp. Discrete Algorithms,*Baltimore, 1999, pp. S899-S900.

*J. Algorithms*37 (2): 570–577, 2000.We show how to maintain the width of a planar point set, subject to insertions or deletions (but not both) in time O(

*n*^{c}) per update for any*c*> 0. The idea is to apply our dynamic closest pair data structure to an appropriate measure of distance between pairs of convex hull features.**Regression depth and center points**.

N. Amenta, M. Bern, D. Eppstein, and S.-H. Teng.

arXiv:cs.CG/9809037.

*3rd CGC Worksh. Computational Geometry*, Brown Univ., 1998.

*Disc. Comp. Geom.*23 (3): 305–323, 2000.We show that, for any set of n points in d dimensions, there exists a hyperplane with regression depth at least ceiling(n/(d+1)). as had been conjectured by Rousseeuw and Hubert. Dually, for any arrangement of n hyperplanes in d dimensions there exists a point that cannot escape to infinity without crossing at least ceiling(n/(d+1)) hyperplanes. We also apply our approach to related questions on the existence of partitions of the data into subsets such that a common plane has nonzero regression depth in each subset, and to the computational complexity of regression depth problems.

**Guest editor's forward to special issue on dynamic graph algorithms**.

D. Eppstein.

*Algorithmica*22 (3): 233–234, 1998.

Years – Publications – David Eppstein – Theory Group – Inf. & Comp. Sci. – UC Irvine

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