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David Eppstein

Publications and source records attributed to David Eppstein.

At least 253 records · Page 14Linked to original sources
arXiv2002

A steady state model for graph power laws

Power law distribution seems to be an important characteristic of web graphs. Several existing web graph models generate power law graphs by adding new vertices and non-uniform edge connectivities to existing graphs. Researchers have conjectured that preferential connectivity and incremental growth are both required for the power law distribution. In this paper, we propose a different web graph model with power law distribution that does not require incremental growth. We also provide a comparison of our model with several others in their ability to predict web graph clustering behavior.

cs.DM↗
arXiv2001

Vertex-Unfoldings of Simplicial Manifolds

We present an algorithm to unfold any triangulated 2-manifold (in particular, any simplicial polyhedron) into a non-overlapping, connected planar layout in linear time. The manifold is cut only along its edges. The resulting layout is connected, but it may have a disconnected interior; the triangles are connected at vertices, but not necessarily joined along edges. We extend our algorithm to establish a similar result for simplicial manifolds of arbitrary dimension.

cs.CG↗
arXiv2001

The Minimum Expectation Selection Problem

We define the min-min expectation selection problem (resp. max-min expectation selection problem) to be that of selecting k out of n given discrete probability distributions, to minimize (resp. maximize) the expectation of the minimum value resulting when independent random variables are drawn from the selected distributions. We assume each distribution has finitely many atoms. Let d be the number of distinct values in the support of the distributions. We show that if d is a constant greater than 2, the min-min expectation problem is NP-complete but admits a fully polynomial time approximation scheme. For d an arbitrary integer, it is NP-hard to approximate the min-min expectation problem with any constant approximation factor. The max-min expectation problem is polynomially solvable for constant d; we leave open its complexity for variable d. We also show similar results for binary selection problems in which we must choose one distribution from each of n pairs of distributions.

cs.DS↗
arXiv2001

Ununfoldable Polyhedra with Convex Faces

Unfolding a convex polyhedron into a simple planar polygon is a well-studied problem. In this paper, we study the limits of unfoldability by studying nonconvex polyhedra with the same combinatorial structure as convex polyhedra. In particular, we give two examples of polyhedra, one with 24 convex faces and one with 36 triangular faces, that cannot be unfolded by cutting along edges. We further show that such a polyhedron can indeed be unfolded if cuts are allowed to cross faces. Finally, we prove that ``open'' polyhedra with triangular faces may not be unfoldable no matter how they are cut.

cs.CG↗
arXiv2001

Phutball Endgames are Hard

We show that, in John Conway's board game Phutball (or Philosopher's Football), it is NP-complete to determine whether the current player has a move that immediately wins the game. In contrast, the similar problems of determining whether there is an immediately winning move in checkers, or a move that kings a man, are both solvable in polynomial time.

cs.CC↗
arXiv2001

Vertex-Unfoldings of Simplicial Polyhedra

We present two algorithms for unfolding the surface of any polyhedron, all of whose faces are triangles, to a nonoverlapping, connected planar layout. The surface is cut only along polyhedron edges. The layout is connected, but it may have a disconnected interior: the triangles are connected at vertices, but not necessarily joined along edges.

cs.CG↗
arXiv2001

Hinged Kite Mirror Dissection

Any two polygons of equal area can be partitioned into congruent sets of polygonal pieces, and in many cases one can connect the pieces by flexible hinges while still allowing the connected set to form both polygons. However it is open whether such a hinged dissection always exists. We solve a special case of this problem, by showing that any asymmetric polygon always has a hinged dissection to its mirror image. Our dissection forms a chain of kite-shaped pieces, found by a circle-packing algorithm for quadrilateral mesh generation. A hinged mirror dissection of a polygon with n sides can be formed with O(n) kites in O(n log n) time.

cs.CG↗
arXiv2001

Optimization Over Zonotopes and Training Support Vector Machines

We make a connection between classical polytopes called zonotopes and Support Vector Machine (SVM) classifiers. We combine this connection with the ellipsoid method to give some new theoretical results on training SVMs. We also describe some special properties of soft margin C-SVMs as parameter C goes to infinity.

cs.CG↗
arXiv2001

Optimal Moebius Transformations for Information Visualization and Meshing

We give linear-time quasiconvex programming algorithms for finding a Moebius transformation of a set of spheres in a unit ball or on the surface of a unit sphere that maximizes the minimum size of a transformed sphere. We can also use similar methods to maximize the minimum distance among a set of pairs of input points. We apply these results to vertex separation and symmetry display in spherical graph drawing, viewpoint selection in hyperbolic browsing, element size control in conformal structured mesh generation, and brain flat mapping.

cs.CG↗
arXiv2000

Small Maximal Independent Sets and Faster Exact Graph Coloring

We show that, for any n-vertex graph G and integer parameter k, there are at most 3^{4k-n}4^{n-3k} maximal independent sets I \subset G with |I| <= k, and that all such sets can be listed in time O(3^{4k-n} 4^{n-3k}). These bounds are tight when n/4 <= k <= n/3. As a consequence, we show how to compute the exact chromatic number of a graph in time O((4/3 + 3^{4/3}/4)^n) ~= 2.4150^n, improving a previous O((1+3^{1/3})^n) ~= 2.4422^n algorithm of Lawler (1976).

cs.DS↗
arXiv2000

Internet Packet Filter Management and Rectangle Geometry

We consider rule sets for internet packet routing and filtering, where each rule consists of a range of source addresses, a range of destination addresses, a priority, and an action. A given packet should be handled by the action from the maximum priority rule that matches its source and destination. We describe new data structures for quickly finding the rule matching an incoming packet, in near-linear space, and a new algorithm for determining whether a rule set contains any conflicts, in time O(n^{3/2}).

cs.CG↗
arXiv2000

Computing the Depth of a Flat

We give algorithms for computing the regression depth of a k-flat for a set of n points in R^d. The running time is O(n^(d-2) + n log n) when 0 < k < d-1, faster than the best time bound for hyperplane regression or for data depth.

cs.CG↗
arXiv2000

Fast Approximation of Centrality

Social studies researchers use graphs to model group activities in social networks. An important property in this context is the centrality of a vertex: the inverse of the average distance to each other vertex. We describe a randomized approximation algorithm for centrality in weighted graphs. For graphs exhibiting the small world phenomenon, our method estimates the centrality of all vertices with high probability within a (1+epsilon) factor in near-linear time.

cs.DS↗
arXiv2000

Improved Algorithms for 3-Coloring, 3-Edge-Coloring, and Constraint Satisfaction

We consider worst case time bounds for NP-complete problems including 3-SAT, 3-coloring, 3-edge-coloring, and 3-list-coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems; 3-SAT is equivalent to (2,3)-CSP while the other problems above are special cases of (3,2)-CSP. We give a fast algorithm for (3,2)-CSP and use it to improve the time bounds for solving the other problems listed above. Our techniques involve a mixture of Davis-Putnam-style backtracking with more sophisticated matching and network flow based ideas.

cs.DS↗
arXiv2000

One-Dimensional Peg Solitaire, and Duotaire

We solve the problem of one-dimensional Peg Solitaire. In particular, we show that the set of configurations that can be reduced to a single peg forms a regular language, and that a linear-time algorithm exists for reducing any configuration to the minimum number of pegs. We then look at the impartial two-player game, proposed by Ravikumar, where two players take turns making peg moves, and whichever player is left without a move loses. We calculate some simple nim-values and discuss when the game separates into a disjunctive sum of smaller games. In the version where a series of hops can be made in a single move, we show that neither the P-positions nor the N-positions (i.e. wins for the previous or next player) are described by a regular or context-free language.

math.CO↗
arXiv2000

One-Dimensional Peg Solitaire

We solve the problem of one-dimensional peg solitaire. In particular, we show that the set of configurations that can be reduced to a single peg forms a regular language, and that a linear-time algorithm exists for reducing any configuration to the minimum number of pegs.

math.CO↗
arXiv2000

3-Coloring in Time O(1.3289^n)

We consider worst case time bounds for NP-complete problems including 3-SAT, 3-coloring, 3-edge-coloring, and 3-list-coloring. Our algorithms are based on a constraint satisfaction (CSP) formulation of these problems. 3-SAT is equivalent to (2,3)-CSP while the other problems above are special cases of (3,2)-CSP; there is also a natural duality transformation from (a,b)-CSP to (b,a)-CSP. We give a fast algorithm for (3,2)-CSP and use it to improve the time bounds for solving the other problems listed above. Our techniques involve a mixture of Davis-Putnam-style backtracking with more sophisticated matching and network flow based ideas.

cs.DS↗