Balanced partitions of 3-colored geometric sets in the plane

Let SS be a finite set of geometric objects partitioned into classes or colors . A subset S'¿SS'¿S is said to be balanced if S'S' contains the same amount of elements of SS from each of the colors. We study several problems on partitioning 33-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are 2m2m lines of each color, there is a segment intercepting mm lines of each color. (b) Given nn red points, nn blue points and nn green points on any closed Jordan curve ¿¿, we show that for every integer kk with 0=k=n0=k=n there is a pair of disjoint intervals on ¿¿ whose union contains exactly kk points of each color. (c) Given a set SS of nn red points, nn blue points and nn green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of SS.Peer ReviewedPostprint (published version

A Generalized Approach To Partitioning Weighted Points In A Plane

A theorem commonly known as the Discrete Pancake Theorem states that for two finite disjoint sets S and T of points in a plane where the union of S and T contains no three collinear points, there exists a line that simultaneously bisects |S| and |T| within an error of at most one. This thesis considers a more general situation in which each point is assigned two non-negative weights and, instead of simply bisecting the plane to obtain a balance in the number of points, we prove there exists a line that simultaneously balances weight one and weight two accumulations within a prescribed tolerance. The Discrete Pancake Theorem is shown to be a special case of this Dual-Balanced Theorem, and a computational implementation of this generalization is applied to various examples

Geometric partitioning algorithms for fair division of geographic resources

University of Minnesota Ph.D. dissertation. July 2014. Major: Industrial and Systems Engineering. Advisor: John Gunnar Carlsson. 1 computer file (PDF): vi, 140 pages, appendices p. 129-140.This dissertation focuses on a fundamental but under-researched problem: how does one divide a piece of territory into smaller pieces in an efficient way? In particular, we are interested in \emph{map segmentation problem} of partitioning a geographic region into smaller subregions for allocating resources or distributing a workload among multiple agents. This work would result in useful solutions for a variety of fundamental problems, ranging from congressional districting, facility location, and supply chain management to air traffic control and vehicle routing. In a typical map segmentation problem, we are given a geographic region $R$, a probability density function defined on $R$ (representing, say population density, distribution of a natural resource, or locations of clients) and a set of points in $R$ (representing, say service facilities or vehicle depots). We seek a \emph{partition} of $R$ that is a collection of disjoint sub-regions $\{R_1, . . . , R_n\}$ such that $\bigcup_i R_i = R$, that optimizes some objective function while satisfying a shape condition. As examples of shape conditions, we may require that all sub-regions be compact, convex, star convex, simply connected (not having holes), connected, or merely measurable.Such problems are difficult because the search space is infinite-dimensional (since we are designing boundaries between sub-regions) and because the shape conditions are generally difficult to enforce using standard optimization methods. There are also many interesting variants and extensions to this problem. It is often the case that the optimal partition for a problem changes over time as new information about the region is collected. In that case, we have an \emph{online} problem and we must re-draw the sub-region boundaries as time progresses. In addition, we often prefer to construct these sub-regions in a \emph{decentralized} fashion: that is, the sub-region assigned to agent $i$ should be computable using only local information to agent $i$ (such as nearby neighbors or information about its surroundings), and the optimal boundary between two sub-regions should be computable using only knowledge available to those two agents.This dissertation is an attempt to design geometric algorithms aiming to solve the above mentioned problems keeping in view the various design constraints. We describe the drawbacks of the current approach to solving map segmentation problems, its ineffectiveness to impose geometric shape conditions and its limited utility in solving the online version of the problem. Using an intrinsically interdisciplinary approach, combining elements from variational calculus, computational geometry, geometric probability theory, and vector space optimization, we present an approach where we formulate the problems geometrically and then use a fast geometric algorithm to solve them. We demonstrate our success by solving problems having a particular choice of objective function and enforcing certain shape conditions. In fact, it turns out that such methods actually give useful insights and algorithms into classical location problems such as the continuous $k$-medians problem, where the aim is to find optimal locations for facilities. We use a map segmentation technique to present a constant factor approximation algorithm to solve the continuous $k$-medians problem in a convex polygon. We conclude this thesis by describing how we intend to build on this success and develop algorithms to solve larger classes of these problems

A spatial decision support system for autodistricting collection units for the taking of the Canadian census

This dissertation documents the districting requirements for collection units for taking the Canadian Census and provides a spatial decision support system for their automatic creation. In the context of the literature on autodistricting, this problem falls under the general category of creating districts for monitoring, surveillance and inventory applications since the Census is essentially an spatial inventory exercise. The basic requirement is to create an area-based categorical coverage such that the workload is equitably distributed amongst Census Representatives within the limits of a large number of constraints and conditions.A new omnibus automated districting process that combines a 3-stage cascading selection procedure for identifying sub-blockface, blockface and block level collection units with a 4- stage heuristic solution procedure for grouping blocks (termed 'assigns', 'annexes', 're-assigns' and 'adjusts') is contributed by this research to provide a systematic response to varying districting situations.The resulting spatial decision support system for autodistricting has been tested on test data sets and on one of the larger urban population centres of Canada. The set of test pattern sites mimicking typical settlement patterns was generated to ensure that the various alternative assignment or block grouping methods (i.e., unidirectional and bidirectional tessellations based on circular and rectangular grids and regular, random and 'extrema-based' seeds) performed as designed and specified. The Census Subdivision of Laval (in the Census Metropolitan Area of Montreal) was selected as the test site for comparing the performance of the autodistricting capacity to the actual, manually created, results from the 1986 Census.To permit the comparison of results from classical manual and automated processes, a set of satisficing evaluation functions that vary in accordance with data availability was implemented in the context of a competing set of districting objectives. The most sophisticated of these evaluation functions incorporates a composite index that combines the distribution and a measure of the 'density' of the dwellings with the length of the route that must be followed to complete the collection activity (including travel time to the start of the route and between route parts).To assess the continued acceptability of the districting from the previous Census, and/or to select between alternative results generated by computer-assisted approaches, a set of objective functions is provided that vary depending upon the available amount of geographic, cartographic or statistical data

Empirical Analysis and Modelling of Information and Communications Technology in Agriculture for Southern Ontario, Canada

Information and communications technology (ICT) represents an important enabling technology for on-farm operations that helps to maximise yield and minimise on-farm inputs. This study investigates the adoption factors and coverage characteristics of ICT in Southern Ontario. A set of eight site and situation adoption factors were identified explaining 57% of the variation in agricultural high-speed Internet utilisation for Southern Ontario. ICT coverage was assessed through service carrier and band factors, and their presence in rural settlements. Findings of the research indicate that there exists a digital divide among settlements in Southern Ontario and recommendations for targeted policy and investment in infrastructure are proposed to bridge the gap

New Applications of the Nearest-Neighbor Chain Algorithm

The nearest-neighbor chain algorithm was proposed in the eighties as a way to speed up certain hierarchical clustering algorithms. In the first part of the dissertation, we show that its application is not limited to clustering. We apply it to a variety of geometric and combinatorial problems. In each case, we show that the nearest-neighbor chain algorithm finds the same solution as a preexistent greedy algorithm, but often with an improved runtime. We obtain speedups over greedy algorithms for Euclidean TSP, Steiner TSP in planar graphs, straight skeletons, a geometric coverage problem, and three stable matching models. In the second part, we study the stable-matching Voronoi diagram, a type of plane partition which combines properties of stable matchings and Voronoi diagrams. We propose political redistricting as an application. We also show that it is impossible to compute this diagram in an algebraic model of computation, and give three algorithmic approaches to overcome this obstacle. One of them is based on the nearest-neighbor chain algorithm, linking the two parts together

Minimax and Maximin Fitting of Geometric Objects to Sets of Points

This thesis addresses several problems in the facility location sub-area of computational geometry. Let S be a set of n points in the plane. We derive algorithms for approximating S by a step function curve of size k \u3c n, i.e., by an x-monotone orthogonal polyline ℜ with k \u3c n horizontal segments. We use the vertical distance to measure the quality of the approximation, i.e., the maximum distance from a point in S to the horizontal segment directly above or below it. We consider two types of problems: min-ε, where the goal is to minimize the error for a given number of horizontal segments k and min-#, where the goal is to minimize the number of segments for a given allowed error ε. After O(n) preprocessing time, we solve instances of the latter in O(min{k log n, n}) time per instance. We can then solve the former problem in O(min{n2, nk log n}) time. Both algorithms require O(n) space. The second contribution is a heuristic for the min-ε problem that computes a solution within a factor of 3 of the optimal error for k segments, or with at most the same error as the k-optimal but using 2k - 1 segments. Furthermore, experiments on real data show even better results than what is guaranteed by the theoretical bounds. Both approximations run in O(n log n) time and O(n) space. Then, we present an exact algorithm for the weighted version of this problem that runs in O(n2) time and generalize the heuristic to handle weights at the expense of an additional log n factor. At this point, a randomized algorithm that runs in O(n log2 n) expected time for the unweighted version is presented. It easily generalizes to the weighted case, though at the expense of an additional log n factor. Finally, we treat the maximin problem and present an O(n3 log n) solution to the problem of finding the furthest separating line through a set of weighted points. We conclude with solutions to the obnoxious wedge problem: an O(n2 log n) algorithm for the general case of a wedge with its apex on the boundary of the convex hull of S and an O(n2) algorithm for the case of the apex of a wedge coming from the input set S