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Register a new userComputing a Minimum-Width Cubic and Hypercubic Shell
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Sang Won Bae
Publication date
2019
fields
Informatics Engineering
and research's language is
English
Authors
Sang Won Bae
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In this paper, we study the problem of computing a minimum-width axis-aligned cubic shell that encloses a given set of $n$ points in a three-dimensional space. A cubic shell is a closed volume between two concentric and face-parallel cubes. Prior to this work, there was no known algorithm for this problem in the literature. We present the first nontrivial algorithm whose running time is $O(n log^2 n)$. Our approach easily extends to higher dimension, resulting in an $O(n^{lfloor d/2 rfloor} log^{d-1} n)$-time algorithm for the hypercubic shell problem in $dgeq 3$ dimension.
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In this paper, we study the problem of computing a minimum-width double-strip or parallelogram annulus that encloses a given set of $n$ points in the plane. A double-strip is a closed region in the plane whose boundary consists of four parallel lines and a parallelogram annulus is a closed region between two edge-parallel parallelograms. We present several first algorithms for these problems. Among them are $O(n^2)$ and $O(n^3 log n)$-time algorithms that compute a minimum-width double-strip and parallelogram annulus, respectively, when their orientations can be freely chosen.
We provide exact and approximation methods for solving a geometric relaxation of the Traveling Salesman Problem (TSP) that occurs in curve reconstruction: for a given set of vertices in the plane, the problem Minimum Perimeter Polygon (MPP) asks for a (not necessarily simply connected) polygon with shortest possible boundary length. Even though the closely related problem of finding a minimum cycle cover is polynomially solvable by matching techniques, we prove how the topological structure of a polygon leads to NP-hardness of the MPP. On the positive side, we show how to achieve a constant-factor approximation. When trying to solve MPP instances to provable optimality by means of integer programming, an additional difficulty compared to the TSP is the fact that only a subset of subtour constraints is valid, depending not on combinatorics, but on geometry. We overcome this difficulty by establishing and exploiting additional geometric properties. This allows us to reliably solve a wide range of benchmark instances with up to 600 vertices within reasonable time on a standard machine. We also show that using a natural geometry-based sparsification yields results that are on average within 0.5% of the optimum.
In a geometric network G = (S, E), the graph distance between two vertices u, v in S is the length of the shortest path in G connecting u to v. The dilation of G is the maximum factor by which the graph distance of a pair of vertices differs from their Euclidean distance. We show that given a set S of n points with integer coordinates in the plane and a rational dilation delta > 1, it is NP-hard to determine whether a spanning tree of S with dilation at most delta exists.
We present time-space trade-offs for computing the Euclidean minimum spanning tree of a set $S$ of $n$ point-sites in the plane. More precisely, we assume that $S$ resides in a random-access memory that can only be read. The edges of the Euclidean minimum spanning tree $text{EMST}(S)$ have to be reported sequentially, and they cannot be accessed or modified afterwards. There is a parameter $s in {1, dots, n}$ so that the algorithm may use $O(s)$ cells of read-write memory (called the workspace) for its computations. Our goal is to find an algorithm that has the best possible running time for any given $s$ between $1$ and $n$. We show how to compute $text{EMST}(S)$ in $Obig((n^3/s^2)log s big)$ time with $O(s)$ cells of workspace, giving a smooth trade-off between the two best known bounds $O(n^3)$ for $s = 1$ and $O(n log n)$ for $s = n$. For this, we run Kruskals algorithm on the relative neighborhood graph (RNG) of $S$. It is a classic fact that the minimum spanning tree of $text{RNG}(S)$ is exactly $text{EMST}(S)$. To implement Kruskals algorithm with $O(s)$ cells of workspace, we define $s$-nets, a compact representation of planar graphs. This allows us to efficiently maintain and update the components of the current minimum spanning forest as the edges are being inserted.
We study the problem of finding a minimum homology basis, that is, a shortest set of cycles that generates the $1$-dimensional homology classes with $mathbb{Z}_2$ coefficients in a given simplicial complex $K$. This problem has been extensively studied in the last few years. For general complexes, the current best deterministic algorithm, by Dey et al., runs in $O(N^omega + N^2 g)$ time, where $N$ denotes the number of simplices in $K$, $g$ denotes the rank of the $1$-homology group of $K$, and $omega$ denotes the exponent of matrix multiplication. In this paper, we present two conceptually simple randomized algorithms that compute a minimum homology basis of a general simplicial complex $K$. The first algorithm runs in $tilde{O}(m^omega)$ time, where $m$ denotes the number of edges in $K$, whereas the second algorithm runs in $O(m^omega + N m^{omega-1})$ time. We also study the problem of finding a minimum cycle basis in an undirected graph $G$ with $n$ vertices and $m$ edges. The best known algorithm for this problem runs in $O(m^omega)$ time. Our algorithm, which has a simpler high-level description, but is slightly more expensive, runs in $tilde{O}(m^omega)$ time.
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