Computer Science

We study approximate-near-neighbor data structures for time series under the continuous Fréchet distance. For an attainable approximation factor c>1 and a query radius r , an approximate-near-neighbor data structure can be used to preprocess n curves in R (aka time series), each of complexity m , to answer queries with a curve of complexity k by either returning a curve that lies within Fréchet distance cr , or answering that there exists no curve in the input within distance r . In both cases, the answer is correct. Our first data structure achieves a (5+ϵ) approximation factor, uses space in n⋅O ( ϵ −1 ) k +O(nm) and has query time in O(k) . Our second data structure achieves a (2+ϵ) approximation factor, uses space in n⋅O ( m kϵ ) k +O(nm) and has query time in O(k⋅ 2 k ) . Our third positive result is a probabilistic data structure based on locality-sensitive hashing, which achieves space in O(nlogn+nm) and query time in O(klogn) , and which answers queries with an approximation factor in O(k) . All of our data structures make use of the concept of signatures, which were originally introduced for the problem of clustering time series under the Fréchet distance. In addition, we show lower bounds for this problem. Consider any data structure which achieves an approximation factor less than 2 and which supports curves of arclength up to L and answers the query using only a constant number of probes. We show that under reasonable assumptions on the word size any such data structure needs space in L Ω(k) .

A 1-bend boundary labelling problem consists of an axis-aligned rectangle B , n points (called sites) in the interior, and n points (called ports) on the labels along the boundary of B . The goal is to find a set of n axis-aligned curves (called leaders), each having at most one bend and connecting one site to one port, such that the leaders are pairwise disjoint. A 1-bend boundary labelling problem is k -sided ( 1≤k≤4 ) if the ports appear on k different sides of B . Kindermann et al. ["Multi-Sided Boundary Labeling", Algorithmica, 76(1): 225-258, 2016] showed that the 1-bend three-sided and four-sided boundary labelling problems can be solved in O( n 4 ) and O( n 9 ) time, respectively. Bose et al. [SWAT, 12:1-12:14, 2018] improved the latter running time to O( n 6 ) by reducing the problem to computing maximum independent set in an outerstring graph. In this paper, we improve both previous results by giving new algorithms with running times O( n 3 logn) and O( n 5 ) to solve the 1-bend three-sided and four-sided boundary labelling problems, respectively.

An important problem in terrain analysis is modeling how water flows across a terrain creating floods by forming channels and filling depressions. In this paper we study a number of \emph{flow-query} related problems: Given a terrain Σ , represented as a triangulated xy -monotone surface with n vertices, a rain distribution R which may vary over time, determine how much water is flowing over a given edge as a function of time. We develop internal-memory as well as I/O-efficient algorithms for flow queries. This paper contains four main results: (i) We present an internal-memory algorithm that preprocesses Σ into a linear-size data structure that for a (possibly time varying) rain distribution R can return the flow-rate functions of all edges of Σ in O(ρk+|ϕ|logn) time, where ρ is the number of sinks in Σ , k is the number of times the rain distribution changes, and |ϕ| is the total complexity of the flow-rate functions that have non-zero values; (ii) We also present an I/O-efficient algorithm for preprocessing Σ into a linear-size data structure so that for a rain distribution R , it can compute the flow-rate function of all edges using O(Sort(|ϕ|)) I/Os and O(|ϕ|log|ϕ|) internal computation time. (iii) Σ can be preprocessed into a linear-size data structure so that for a given rain distribution R , the flow-rate function of an edge (q,r)∈Σ under the single-flow direction (SFD) model can be computed more efficiently. (iv) We present an algorithm for computing the two-dimensional channel along which water flows using Manning's equation; a widely used empirical equation that relates the flow-rate of water in an open channel to the geometry of the channel along with the height of water in the channel.

Fractional cascading is one of the influential techniques in data structures, as it provides a general framework for solving the important iterative search problem. In the problem, the input is a graph G with constant degree and a set of values for every vertex of G . The goal is to preprocess G such that when given a query value q , and a connected subgraph π of G , we can find the predecessor of q in all the sets associated with the vertices of π . The fundamental result of fractional cascading is that there exists a data structure that uses linear space and it can answer queries in O(logn+|π|) time [Chazelle and Guibas, 1986]. While this technique has received plenty of attention in the past decades, an almost quadratic space lower bound for "2D fractional cascading" [Chazelle and Liu, 2001] has convinced the researchers that fractional cascading is fundamentally a 1D technique. In 2D fractional cascading, the input includes a planar subdivision for every vertex of G and the query is a point q and a subgraph π and the goal is to locate the cell containing q in all the subdivisions associated with the vertices of π . In this paper, we show that it is possible to circumvent the lower bound of Chazelle and Liu for axis-aligned planar subdivisions. We present a number of upper and lower bounds which reveal that in 2D, the problem has a much richer structure. When G is a tree and π is a path, then queries can be answered in O(logn+|π|+min{|π| logn − − − − √ ,α(n) |π| − − √ logn}) time using linear space where α is an inverse Ackermann function; surprisingly, we show both branches of this bound are tight, up to the inverse Ackermann factor. When G is a general graph or when π is a general subgraph, then the query bound becomes O(logn+|π| logn − − − − √ ) and this bound is once again tight in both cases.

Bounded-angle (minimum) spanning trees were first introduced in the context of wireless networks with directional antennas. They are reminiscent of bounded-degree spanning trees, which have received significant attention. Let P={ p 1 ,…, p n } be a set of n points in the plane, let Π be the polygonal path ( p 1 ,…, p n ) , and let 0<α<2π be an angle. An α -spanning tree ( α -ST) of P is a spanning tree of the complete Euclidean graph over P , with the following property: For each vertex p i ∈P , the (smallest) angle that is spanned by all the edges incident to p i is at most α . An α -minimum spanning tree ( α -MST) is an α -ST of P of minimum weight, where the weight of an α -ST is the sum of the lengths of its edges. In this paper, we consider the problem of computing an α -MST, for the important case where α= 2π 3 . We present a simple 4-approximation algorithm, thus improving upon the previous results of Aschner and Katz and Biniaz et al., who presented algorithms with approximation ratios 6 and 16 3 , respectively. In order to obtain this result, we devise a simple O(n) -time algorithm for constructing a 2π 3 -ST\, T of P , such that T 's weight is at most twice that of Π and, moreover, T is a 3-hop spanner of Π . This latter result is optimal in the sense that for any ε>0 there exists a polygonal path for which every 2π 3 -ST has weight greater than 2−ε times the weight of the path.

Additive manufacturing (AM) enables enormous freedom for design of complex structures. However, the process-dependent limitations that result in discrepancies between as-designed and as-manufactured shapes are not fully understood. The tradeoffs between infinitely many different ways to approximate a design by a manufacturable replica are even harder to characterize. To support design for AM (DfAM), one has to quantify local discrepancies introduced by AM processes, identify the detrimental deviations (if any) to the original design intent, and prescribe modifications to the design and/or process parameters to countervail their effects. Our focus in this work will be on topological analysis. There is ample evidence in many applications that preserving local topology (e.g., connectivity of beams in a lattice) is important even when slight geometric deviations can be tolerated. We first present a generic method to characterize local topological discrepancies due to material under- and over-deposition in AM, and show how it captures various types of defects in the as-manufactured structures. We use this information to systematically modify the as-manufactured outcomes within the limitations of available 3D printer resolution(s), which often comes at the expense of introducing more geometric deviations (e.g., thickening a beam to avoid disconnection). We validate the effectiveness of the method on 3D examples with nontrivial topologies such as lattice structures and foams.

We give a complete description of all convex polyhedra whose surface can be constructed from several congruent regular pentagons by folding and gluing them edge to edge. Our method of determining the graph structure of the polyhedra from a gluing is of independent interest and can be used in other similar settings.

Given two points s and t in the plane and a set of obstacles defined by closed curves, what is the minimum number of obstacles touched by a path connecting s and t? This is a fundamental and well-studied problem arising naturally in computational geometry, graph theory (under the names Min-Color Path and Minimum Label Path), wireless sensor networks (Barrier Resilience) and motion planning (Minimum Constraint Removal). It remains NP-hard even for very simple-shaped obstacles such as unit-length line segments. In this paper we give the first constant factor approximation algorithm for this problem, resolving an open problem of [Chan and Kirkpatrick, TCS, 2014] and [Bandyapadhyay et al., CGTA, 2020]. We also obtain a constant factor approximation for the Minimum Color Prize Collecting Steiner Forest where the goal is to connect multiple request pairs (s1, t1), . . . ,(sk, tk) while minimizing the number of obstacles touched by any (si, ti) path plus a fixed cost of wi for each pair (si, ti) left disconnected. This generalizes the classic Steiner Forest and Prize-Collecting Steiner Forest problems on planar graphs, for which intricate PTASes are known. In contrast, no PTAS is possible for Min-Color Path even on planar graphs since the problem is known to be APXhard [Eiben and Kanj, TALG, 2020]. Additionally, we show that generalizations of the problem to disconnected obstacles

The problem of vertex guarding a simple polygon was first studied by Subir K. Ghosh (1987), who presented a polynomial-time O(logn) -approximation algorithm for placing as few guards as possible at vertices of a simple n -gon P , such that every point in P is visible to at least one of the guards. Ghosh also conjectured that this problem admits a polynomial-time algorithm with constant approximation ratio. Due to the centrality of guarding problems in the field of computational geometry, much effort has been invested throughout the years in trying to resolve this conjecture. Despite some progress (surveyed below), the conjecture remains unresolved to date. In this paper, we confirm the conjecture for the important case of weakly visible polygons, by presenting a (2+ε) -approximation algorithm for guarding such a polygon using vertex guards. A simple polygon P is weakly visible if it has an edge e , such that every point in P is visible from some point on e . We also present a (2+ε) -approximation algorithm for guarding a weakly visible polygon P , where guards may be placed anywhere on P 's boundary (except in the interior of the edge e ). Finally, we present a 3c -approximation algorithm for vertex guarding a polygon P that is weakly visible from a chord, given a subset G of P 's vertices that guards P 's boundary whose size is bounded by c times the size of a minimum such subset. Our algorithms are based on an in-depth analysis of the geometric properties of the regions that remain unguarded after placing guards at the vertices to guard the polygon's boundary. It is plausible that our results will enable Bhattacharya et al. to complete their grand attempt to prove the original conjecture, as their approach is based on partitioning the underlying simple polygon into a hierarchy of weakly visible polygons.

Generated Jacobian Equations have been introduced by Trudinger [Disc. cont. dyn. sys (2014), pp. 1663-1681] as a generalization of Monge-Amp{è}re equations arising in optimal transport. In this paper, we introduce and study a damped Newton algorithm for solving these equations in the semi-discrete setting, meaning that one of the two measures involved in the problem is finitely supported and the other one is absolutely continuous. We also present a numerical application of this algorithm to the near-field parallel refractor problem arising in non-imaging problems.