Pattern matching for permutations

Given a permutation T of 1 to n, and a permutation P of 1 to k, for k ≤ n, we wish to find a k-element subsequence of T whose elements are ordered according to the permutation P. For example, if P is (1,2,..., k), then we wish to find an increasing subsequence of length k in T; this special case was done in time O(n log log n) by Chang and Wang. We prove that the general problem is NP-complete. We give a polynomial time algorithm for the decision problem, and the corresponding counting problem, in the case that P is separable - i.e., contains neither the subpattern (3,1,4,2) nor its reverse, the subpattern (2,4,1,3)

Flipping edge-labelled triangulations

Flips in triangulations have received a lot of attention over the past decades. However, the problem of tracking where particular edges go during the flipping process has not been addressed. We examine this question by attaching unique labels to the triangulation edges. We introduce the concept of the orbit of an edge e, which is the set of all edges reachable from e via flips. We establish the first upper and lower bounds on the diameter of the flip graph in this setting. Specifically, we prove tight Θ(nlog⁡n) bounds for edge-labelled triangulations of n-vertex convex polygons and combinatorial triangulations, contrasting with the Θ(n) bounds in their respective unlabelled settings. The Ω(nlog⁡n) lower bound for the convex polygon setting might be of independent interest, as it generalizes lower bounds on certain sorting models. When simultaneous flips are allowed, the upper bound for convex polygons decreases to O(log2⁡n), although we no longer have a matching lower bound. Moving beyond conve

Efficient Algorithms for Petersen's Matching Theorem

Petersen's theorem is a classic result in matching theory from 1891, stating that every 3-regular bridgeless graph has a perfect matching. Our work explores efficient algorithms for finding perfect matchings in such graphs. Previously, the only relevant matching algorithms were for general graphs, and the fastest algorithm ran in O(n3/2) time for 3-regular graphs. We have developed an O(n log4 n)-time algorithm for perfect matching in a 3-regular bridgeless graph. When the graph is also planar, we have as the main result of our paper an optimal O(n)-time algorithm. We present three applications of this result: terrain guarding, adaptive mesh refinement, and quadrangulation

Approximation algorithms for shortest descending paths in terrains

A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. No efficient algorithm is known to find a shortest descending path (SDP) from s to t in a polyhedral terrain. We present two approximation algorithms that solve the SDP problem on general terrains. We also introduce a generalization of the shortest descending path problem, called the shortest gently descending path (SGDP) problem, where a path descends, but not too steeply. The additional constraint to disallow a very steep descent makes the paths more realistic in practice. We present two approximation algorithms to solve the SGDP problem on general terrains. All of our algorithms are simple, robust and easy to implement

The floodlight problem

Given three angles summing to 2π, given n points in the plane and a tripartition κ1 + k2 + κ3 = n, we can tripartition the plane into three wedges of the given angles so that the i-th wedge contains κi of the points. This new result on dissecting point sets is used to prove that lights of specified angles not exceeding π can be placed at n fixed points in the plane to illuminate the entire plane if and only if the angles sum to at least 2. We give O(n log n) algorithms for both these problems

On simultaneous planar graph embeddings

AbstractWe consider the problem of simultaneous embedding of planar graphs. There are two variants of this problem, one in which the mapping between the vertices of the two graphs is given and another in which the mapping is not given. We present positive and negative results for the two versions of the problem. Among the positive results with given mapping, we show that we can embed two paths on an n×n grid, and two caterpillar graphs on a 3n×3n grid. Among the negative results with given mapping, we show that it is not always possible to simultaneously embed three paths or two general planar graphs. If the mapping is not given, we show that any number of outerplanar graphs can be embedded simultaneously on an O(n)×O(n) grid, and an outerplanar and general planar graph can be embedded simultaneously on an O(n2)×O(n2) grid

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On a visibility representation for graphs in 3D

This paper proposes a 3-dimensional visibility representation of graphs G = (V,E) in which vertices are mapped to rectangles floating in R 3 parallel to the x,y-plane, with edges represented by vertical lines of sight. We apply an extension of the Erdös-Szekeres Theorem in a geometric setting to obtain an upper bound of n = 56 for the largest representable complete graph Kn. On the other hand, we show by construction that n>=22. These are the best existing bounds. We also note that planar graphs and complete bipartite graphs Km,n are representable, but that the family of representable graphs is not closed under graph minors