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

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

Exploring Membership in Black Greek-Letter Sororities and the Influence on Career Advancement for Black Women in Higher Education

The aim of this study was to explore membership in Black Greek-letter sororities and the influence on career advancement for Black women in higher education. Research has neglected to account for the role that Black Greek-letter organizations play in the development of Black women beyond their undergraduate experience. This research is motivated by two research questions: (1) How do Black women perceive that membership in Black Greek-letter sororities prepared them for career advancement in higher education?; and (2) How do Black women perceive that membership in Black Greek-letter sororities influenced their professional success in higher education? To examine these questions, the study explored the perceptions of 12 Black women holding a membership in one of four Black Greek-letter sororities on the influence these memberships had on career advancement using interpretative phenomenological analysis. The findings from the research show that the impact of Black-Greek letter sorority membership on the career advancement for Black women in higher education is more complex than previously thought. The results, implications for institutions of higher education and Black Greek-letter sororities, and future research are discussed

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

Design of a Passive Intensity Modulation Device for Bolus Electron Conformal Therapy

Purpose: To develop a process that can design an island block matrix that produces an intensity distribution (70-100%), which acceptably matches one planned for an intensity modulated (IM) bolus electron conformal therapy (ECT) patient. The intensity modulator concept is that electrons laterally scatter behind hexagonally-packed, small-diameter island blocks such that local intensity equals the fractional unblocked area. Methods: A pencil beam algorithm (PBA) was used to calculate the modulated electron intensity created by varying diameter (d) of island blocks in a hexagonal array (separation r). Accuracy of the PBA model was assessed by comparing with Monte Carlo (MC) calculations. PBA calculations determined acceptability of (r, d) values for achieving clinical intensity reduction factors (70% Results: PBA and MC calculations agreed within &177;5%. At 103 cm SSD PBA results showed r \u3c 0.5 cm and 0.75 cm at 0.5 cm and 2.0 cm depths, respectively, acceptable for 7-20 MeV electrons; larger r values were acceptable for lower energies. Although larger r require fewer blocks, smaller r decreased the distance to transition to the desired IRF, helping achieve intensity distributions with sharp gradients. The Modulator Generator required \u3c5sec (r=0.5cm) to produce clinically-acceptable distributions for the buccal mucosa patient (\u3e99% of points within 3% of planned intensity). Conclusions: The PBA model was sufficient to study the impact of island block parameters (r, d) on achieving desired IRFs for differing conditions (energy, SSD, depth); however, PBA-MC agreement should be improved for patient use. The primary objective was achieved; electron intensity modulators comprised of island blocks of variable diameter can be designed to deliver a desired intensity distribution of clinical complexity (70-100%) with an accuracy of &177;3% for 95% of modulated points

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

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum

Homotopy Measures for Representative Trajectories

An important task in trajectory analysis is defining a meaningful representative for a cluster of similar trajectories. Formally defining and computing such a representative r is a challenging problem. We propose and discuss two new definitions, both of which use only the geometry of the input trajectories. The definitions are based on the homotopy area as a measure of similarity between two curves, which is a minimum area swept by all possible deformations of one curve into the other. In the first definition we wish to minimize the maximum homotopy area between r and any input trajectory, whereas in the second definition we wish to minimize the sum of the homotopy areas between r and the input trajectories. For both definitions computing an optimal representative is NP-hard. However, for the case of minimizing the sum of the homotopy areas, an optimal representative can be found efficiently in a natural class of restricted inputs, namely, when the arrangement of trajectories forms a directed acyclic graph