Better Tradeoffs for Exact Distance Oracles in Planar Graphs

We present an $O(n^{1.5})$-space distance oracle for directed planar graphs that answers distance queries in $O(\log n)$ time. Our oracle both significantly simplifies and significantly improves the recent oracle of Cohen-Addad, Dahlgaard and Wulff-Nilsen [FOCS 2017], which uses $O(n^{5/3})$-space and answers queries in $O(\log n)$ time. We achieve this by designing an elegant and efficient point location data structure for Voronoi diagrams on planar graphs. We further show a smooth tradeoff between space and query-time. For any $S\in [n,n^2]$, we show an oracle of size $S$ that answers queries in $\tilde O(\max\{1,n^{1.5}/S\})$ time. This new tradeoff is currently the best (up to polylogarithmic factors) for the entire range of $S$ and improves by polynomial factors over all the previously known tradeoffs for the range $S \in [n,n^{5/3}]$

Spanners of Additively Weighted Point Sets

We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs $(p,r)$ where $p$ is a point in the plane and $r$ is a real number. The distance between two points $(p_i,r_i)$ and $(p_j,r_j)$ is defined as $|p_ip_j|-r_i-r_j$. We show that in the case where all $r_i$ are positive numbers and $|p_ip_j|\geq r_i+r_j$ for all $i,j$ (in which case the points can be seen as non-intersecting disks in the plane), a variant of the Yao graph is a $(1+\epsilon)$-spanner that has a linear number of edges. We also show that the Additively Weighted Delaunay graph (the face-dual of the Additively Weighted Voronoi diagram) has constant spanning ratio. The straight line embedding of the Additively Weighted Delaunay graph may not be a plane graph. We show how to compute a plane embedding that also has a constant spanning ratio

Districting Problems - New Geometrically Motivated Approaches

This thesis focuses on districting problems were the basic areas are represented by points or lines. In the context of points, it presents approaches that utilize the problem\u27s underlying geometrical information. For lines it introduces an algorithm combining features of geometric approaches, tabu search, and adaptive randomized neighborhood search that includes the routing distances explicitly. Moreover, this thesis summarizes, compares and enhances existing compactness measures

Shortest Paths in Geometric Intersection Graphs

This thesis studies shortest paths in geometric intersection graphs, which can model, among others, ad-hoc communication and transportation networks. First, we consider two classical problems in the field of algorithms, namely Single-Source Shortest Paths (SSSP) and All-Pairs Shortest Paths (APSP). In SSSP we want to compute the shortest paths from one vertex of a graph to all other vertices, while in APSP we aim to find the shortest path between every pair of vertices. Although there is a vast literature for these problems in many graph classes, the case of geometric intersection graphs has been only partially addressed. In unweighted unit-disk graphs, we show that we can solve SSSP in linear time, after presorting the disk centers with respect to their coordinates. Furthermore, we give the first (slightly) subquadratic-time APSP algorithm by using our new SSSP result, bit tricks, and a shifted-grid-based decomposition technique. In unweighted, undirected geometric intersection graphs, we present a simple and general technique that reduces APSP to static, offline intersection detection. Consequently, we give fast APSP algorithms for intersection graphs of arbitrary disks, axis-aligned line segments, arbitrary line segments, d-dimensional axis-aligned boxes, and d-dimensional axis-aligned unit hypercubes. We also provide a near-linear-time SSSP algorithm for intersection graphs of axis-aligned line segments by a reduction to dynamic orthogonal point location. Then, we study two problems that have received considerable attention lately. The first is that of computing the diameter of a graph, i.e., the longest shortest-path distance between any two vertices. In the second, we want to preprocess a graph into a data structure, called distance oracle, such that the shortest path (or its length) between any two query vertices can be found quickly. Since these problems are often too costly to solve exactly, we study their approximate versions. Following a long line of research, we employ Voronoi diagrams to compute a (1+epsilon)-approximation of the diameter of an undirected, non-negatively-weighted planar graph in time near linear in the input size and polynomial in 1/epsilon. The previously best solution had exponential dependency on the latter. Using similar techniques, we can also construct the first (1+epsilon)-approximate distance oracles with similar preprocessing time and space and only O(log(1/\epsilon)) query time. In weighted unit-disk graphs, we present the first near-linear-time (1+epsilon)-approximation algorithm for the diameter and for other related problems, such as the radius and the bichromatic closest pair. To do so, we combine techniques from computational geometry and planar graphs, namely well-separated pair decompositions and shortest-path separators. We also show how to extend our approach to obtain O(1)-query-time (1+epsilon)-approximate distance oracles with near linear preprocessing time and space. Then, we apply these oracles, along with additional ideas, to build a data structure for the (1+epsilon)-approximate All-Pairs Bounded-Leg Shortest Paths (apBLSP) problem in truly subcubic time

Optimization of urban mini-bus stop spacing: a case study of Shanghai (China)

U posljednjih se nekoliko godina služba mini-autobusa brzo razvija, a odgovarajući razmak između stajališta značajno poboljšava učinkovitost putovanja stanovnika. Kako bi mini-bus stajališta bila praktično locirana, u ovom se radu najprije predlaže matematički model temeljen na Voronoi dijagramima u svrhu smanjenja ukupnog vremena putovanja putnika. Za rješenje tog modela primijenjena je poboljšana tehnika Wilson-Han-Powell sekvencionalno kvadratno programiranje (SQP). Na temelju geografskih informacijskih sustava (GIS) korišten je Kartezijanski koordinatni sustav za izračunavanje pješačke udaljenosti između mini-bus stanica i početnih ili odredišta, koja se ne mogu izravno izmjeriti. Drugo, u skladu sa stvarnim stanjem, u radu se predlažu metode modifikacije za podešavanje stajališta mini-autobusa. Valjanost i korisnost metodologije ispitana je njenom primjenom na stvarnom slučaju u Šangaju (Kina). Rezultati proračuna pokazuje da se modelom dobro rješava problem određivanja razmaka između stajališta mini autobusa.In recent years, the mini-bus service has gained a rapid development, while the reasonable stop spacing is significant to improve residents’ travel efficiency. In order to locate the mini-bus stops practically, this paper firstly proposes a mathematical model based on Voronoi Diagrams to minimize residents’ total travel time. Improved Wilson-Han-Powell Sequential Quadratic Programming (SQP) technique is employed to solve the model. Based on Geographic Information Systems (GIS), the Cartesian coordinate system is used to calculate the walking distance between mini-bus stops and origins or destinations, which cannot be directly measured. Secondly, based on the actual situation, this paper proposes modification methods for adjusting mini-bus stops. The validity and usefulness of the methodology is tested through applying it to a real case in Shanghai (China). The calculation results suggest that the model deals well with the mini-bus stop spacing problem

Algorithms for Unit-Disk Graphs and Related Problems

In this dissertation, we study algorithms for several problems on unit-disk graphs and related problems. The unit-disk graph can be viewed as an intersection graph of a set of congruent disks. Unit-disk graphs have been extensively studied due to many of their applications, e.g., modeling the topology of wireless sensor networks. Lots of problems on unit-disk graphs have been considered in the literature, such as shortest paths, clique, independent set, distance oracle, diameter, etc. Specifically, we study the following problems in this dissertation: L1 shortest paths in unit-disk graphs, reverse shortest paths in unit-disk graphs, minimum bottleneck moving spanning tree, unit-disk range reporting, distance selection, etc. We develop efficient algorithms for these problems and our results are either first-known solutions or somehow improve the previous work. Given a set P of n points in the plane and a parameter r \u3e 0, a unit-disk graph G(P) can be defined using P as its vertex set and two points of P are connected by an edge if the distance between these two points is at most r. The weight of an edge is one in the unweighted case and is equal to the distance between the two endpoints in the weighted case. Note that the distance between two points can be measured by different metrics, e.g., L1 or L2 metric. In the first problem of L1 shortest paths in unit-disk graphs, we are given a point set P and a source point s ∈ P, the problem is to find all shortest paths from s to all other vertices in the L1 weighted unit-disk graph defined on set P. We present an O(n log n) time algorithm, which matches the Ω(n log n)-time lower bound. In the second problem, we are given a set P of n points, parameters r, λ \u3e 0, and two points s and t of P, the goal is to compute the smallest r such that the shortest path length between s and t in the unit-disk graph with respect to set P and parameter r is at most λ. This problem can be defined in both unweighted and weighted cases. We propose an algorithm of O(⌊λ⌋ · n log n) time and another algorithm of O(n5/4 log7/4 n) time for the unweighted case. We also given an O(n5/4 log5/2 n) time algorithm for the weighted case. In the third problem, we are given a set P of n points that are moving in the plane, the problem is to compute a spanning tree for these moving points that does not change its combinatorial structure during the point movement such that the bottleneck weight of the spanning tree (i.e., the largest Euclidean length of all edges) during the whole movement is minimized. We present an algorithm that runs in O(n4/3 log3 n) time. The fourth problem is unit-disk range reporting in which we are given a set P of n points in the plane and a value r, we need to construct a data structure so that given any query disk of radius r, all points of P in the disk can be reported efficiently. We build a data structure of O(n) space in O(n log n) time that can answer each query in O(k + log n) time, where k is the output size. The time complexity of our algorithm is the same as the previous result but our approach is much simpler. Finally, for the problem of distance selection, we are given a set P of n points in the plane and an integer 1 ≤ k ≤ (n2), the distance selection problem is to find the k-th smallest interpoint distance among all pairs of points of p. We propose an algorithm that runs in O(n4/3 log n) time. Our techniques yield two algorithmic frameworks for solving geometric optimization problems. Many algorithms and techniques developed in this dissertation are quite general and fundamental, and we believe they will find other applications in future