On the complexity of optimal homotopies

In this article, we provide new structural results and algorithms for the Homotopy Height problem. In broad terms, this problem quantifies how much a curve on a surface needs to be stretched to sweep continuously between two positions. More precisely, given two homotopic curves $\gamma_1$ and $\gamma_2$ on a combinatorial (say, triangulated) surface, we investigate the problem of computing a homotopy between $\gamma_1$ and $\gamma_2$ where the length of the longest intermediate curve is minimized. Such optimal homotopies are relevant for a wide range of purposes, from very theoretical questions in quantitative homotopy theory to more practical applications such as similarity measures on meshes and graph searching problems. We prove that Homotopy Height is in the complexity class NP, and the corresponding exponential algorithm is the best one known for this problem. This result builds on a structural theorem on monotonicity of optimal homotopies, which is proved in a companion paper. Then we show that this problem encompasses the Homotopic Fr\'echet distance problem which we therefore also establish to be in NP, answering a question which has previously been considered in several different settings. We also provide an O(log n)-approximation algorithm for Homotopy Height on surfaces by adapting an earlier algorithm of Har-Peled, Nayyeri, Salvatipour and Sidiropoulos in the planar setting

Minimum cycle and homology bases of surface embedded graphs

We study the problems of finding a minimum cycle basis (a minimum weight set of cycles that form a basis for the cycle space) and a minimum homology basis (a minimum weight set of cycles that generates the $1$-dimensional ($\mathbb{Z}_2$)-homology classes) of an undirected graph embedded on a surface. The problems are closely related, because the minimum cycle basis of a graph contains its minimum homology basis, and the minimum homology basis of the $1$-skeleton of any graph is exactly its minimum cycle basis. For the minimum cycle basis problem, we give a deterministic $O(n^\omega+2^{2g}n^2+m)$-time algorithm for graphs embedded on an orientable surface of genus $g$. The best known existing algorithms for surface embedded graphs are those for general graphs: an $O(m^\omega)$ time Monte Carlo algorithm and a deterministic $O(nm^2/\log n + n^2 m)$ time algorithm. For the minimum homology basis problem, we give a deterministic $O((g+b)^3 n \log n + m)$-time algorithm for graphs embedded on an orientable or non-orientable surface of genus $g$ with $b$ boundary components, assuming shortest paths are unique, improving on existing algorithms for many values of $g$ and $n$. The assumption of unique shortest paths can be avoided with high probability using randomization or deterministically by increasing the running time of the homology basis algorithm by a factor of $O(\log n)$.Comment: A preliminary version of this work was presented at the 32nd Annual International Symposium on Computational Geometr

Constructing monotone homotopies and sweepouts

This article investigates when homotopies can be converted to monotone homotopies without increasing the lengths of curves. A monotone homotopy is one which consists of curves which are simple or constant, and in which curves are pairwise disjoint. We show that, if the boundary of a Riemannian disc can be contracted through curves of length less than $L$, then it can also be contracted monotonously through curves of length less than $L$. This proves a conjecture of Chambers and Rotman. Additionally, any sweepout of a Riemannian $2$-sphere through curves of length less than $L$ can be replaced with a monotone sweepout through curves of length less than $L$. Applications of these results are also discussed.Comment: 16 pages, 6 figure

Recruitment, Preparation, Retention: A case study of computing culture at the University of Illinois at Urbana-Champaign

Computer science is seeing a decline in enrollment at all levels of education, including undergraduate and graduate study. This paper reports on the results of a study conducted at the University of Illinois at Urbana-Champaign which evaluated students attitudes regarding three areas which can contribute to improved enrollment in the Department of Computer Science: Recruitment, preparation and retention. The results of our study saw two themes. First, the department's tight research focus appears to draw significant attention from other activities -- such as teaching, service, and other community-building activities -- that are necessary for a department's excellence. Yet, as demonstrated by our second theme, one partial solution is to better promote such activities already employed by the department to its students and faculty. Based on our results, we make recommendations for improvements and enhancements based on the current state of practice at peer institutions.Comment: 37 pages, 13 figures. For better quality figures, please download the .pdf from http://www.cs.uiuc.edu/research/techreports.php?report=UIUCDCS-R-2007-281

A Family of Metrics from the Truncated Smoothing of Reeb Graphs

In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we "chop off" parts near local minima and maxima during the course of smoothing, where the amount cut is controlled by a parameter ?. After formalizing truncation as a functor, we show that when applied after the smoothing functor, this prevents extensive expansion of the range of the function, and yields particularly nice properties (such as maintaining connectivity) when combined with smoothing for 0 ? ? ? 2?, where ? is the smoothing parameter. Then, for the restriction of ? ? [0,?], we have additional structure which we can take advantage of to construct a categorical flow for any choice of slope m ? [0,1]. Using the infrastructure built for a category with a flow, this then gives an interleaving distance for every m ? [0,1], which is a generalization of the original interleaving distance, which is the case m = 0. While the resulting metrics are not stable, we show that any pair of these for m, m\u27 ? [0,1) are strongly equivalent metrics, which in turn gives stability of each metric up to a multiplicative constant. We conclude by discussing implications of this metric within the broader family of metrics for Reeb graphs

Algorithms for Contractibility of Compressed Curves on 3-Manifold Boundaries

In this paper we prove that the problem of deciding contractibility of an arbitrary closed curve on the boundary of a 3-manifold is in NP. We emphasize that the manifold and the curve are both inputs to the problem. Moreover, our algorithm also works if the curve is given as a compressed word. Previously, such an algorithm was known for simple (non-compressed) curves, and, in very limited cases, for curves with self-intersections. Furthermore, our algorithm is fixed-parameter tractable in the complexity of the input 3-manifold. As part of our proof, we obtain new polynomial-time algorithms for compressed curves on surfaces, which we believe are of independent interest. We provide a polynomial-time algorithm which, given an orientable surface and a compressed loop on the surface, computes a canonical form for the loop as a compressed word. In particular, contractibility of compressed curves on surfaces can be decided in polynomial time; prior published work considered only constant genus surfaces. More generally, we solve the following normal subgroup membership problem in polynomial time: given an arbitrary orientable surface, a compressed closed curve ?, and a collection of disjoint normal curves ?, there is a polynomial-time algorithm to decide if ? lies in the normal subgroup generated by components of ? in the fundamental group of the surface after attaching the curves to a basepoint