461 research outputs found
K-Dominance in Multidimensional Data: Theory and Applications
No full textWe study the problem of k-dominance in a set of d-dimensional vectors, prove bounds on the number of maxima (skyline vectors), under both worst-case and average-case models, perform experimental evaluation using synthetic and real-world data, and explore an application of k-dominant skyline for extracting a small set of top-ranked vectors in high dimensions where the full skylines can be unmanageably large
The Maximum Exposure Problem
No full textGiven a set of points P and axis-aligned rectangles R in the plane, a point p in P is called exposed if it lies outside all rectangles in R. In the max-exposure problem, given an integer parameter k, we want to delete k rectangles from R so as to maximize the number of exposed points. We show that the problem is NP-hard and assuming plausible complexity conjectures is also hard to approximate even when rectangles in R are translates of two fixed rectangles. However, if R only consists of translates of a single rectangle, we present a polynomial-time approximation scheme. For general rectangle range space, we present a simple O(k) bicriteria approximation algorithm; that is by deleting O(k^2) rectangles, we can expose at least Omega(1/k) of the optimal number of points
Shortest Paths in the Plane with Obstacle Violations
No full textWe study the problem of finding shortest paths in the plane among h convex obstacles, where the path is allowed to pass through (violate) up to k obstacles, for k <= h. Equivalently, the problem is to find shortest paths that become obstacle-free if k obstacles are removed from the input. Given a fixed source point s, we show how to construct a map, called a shortest k-path map, so that all destinations in the same region of the map have the same combinatorial shortest path passing through at most k obstacles. We prove a tight bound of Theta(kn) on the size of this map, and show that it can be computed in O(k^2 n log n) time, where n is the total number of obstacle vertices
07151 Abstracts Collection – Geometry in Sensor Networks
No full textFrom 9.4.2007 to 13.4.07, the Dagstuhl Seminar 07151 ``Geometry in Sensor
Networks'' was held in the International Conference and Research Center
(IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first
section describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Efficient Algorithms for Least Square Piecewise Polynomial Regression
No full textWe present approximation and exact algorithms for piecewise regression of univariate and bivariate data using fixed-degree polynomials. Specifically, given a set S of n data points (₁, y₁),… , (_n, y_n) ∈ ℝ^d × ℝ where d ∈ {1,2}, the goal is to segment _i’s into some (arbitrary) number of disjoint pieces P₁, … , P_k, where each piece P_j is associated with a fixed-degree polynomial f_j: ℝ^d → ℝ, to minimize the total loss function λ k + ∑_{i = 1}ⁿ (y_i - f(_i))², where λ ≥ 0 is a regularization term that penalizes model complexity (number of pieces) and f: ⨆_{j = 1}^k P_j → ℝ is the piecewise polynomial function defined as f|_{P_j} = f_j. The pieces P₁, … , P_k are disjoint intervals of ℝ in the case of univariate data and disjoint axis-aligned rectangles in the case of bivariate data. Our error approximation allows use of any fixed-degree polynomial, not just linear functions.
Our main results are the following. For univariate data, we present a (1 + ε)-approximation algorithm with time complexity O(n/(ε) log 1/(ε)), assuming that data is presented in sorted order of x_i’s. For bivariate data, we present three results: a sub-exponential exact algorithm with running time n^{O(√n)}; a polynomial-time constant-approximation algorithm; and a quasi-polynomial time approximation scheme (QPTAS). The bivariate case is believed to be NP-hard in the folklore but we could not find a published record in the literature, so in this paper we also present a hardness proof for completeness
Fault Tolerance in Euclidean Committee Selection
No full textIn the committee selection problem, the goal is to choose a subset of size k from a set of candidates C that collectively gives the best representation to a set of voters. We consider this problem in Euclidean d-space where each voter/candidate is a point and voters' preferences are implicitly represented by Euclidean distances to candidates. We explore fault-tolerance in committee selection and study the following three variants: (1) given a committee and a set of f failing candidates, find their optimal replacement; (2) compute the worst-case replacement score for a given committee under failure of f candidates; and (3) design a committee with the best replacement score under worst-case failures. The score of a committee is determined using the well-known (min-max) Chamberlin-Courant rule: minimize the maximum distance between any voter and its closest candidate in the committee. Our main results include the following: (1) in one dimension, all three problems can be solved in polynomial time; (2) in dimension d ≥ 2, all three problems are NP-hard; and (3) all three problems admit a constant-factor approximation in any fixed dimension, and the optimal committee problem has an FPT bicriterion approximation
Conflict-free Chromatic Art Gallery Coverage
No full textWe consider a chromatic variant of the art gallery problem, where each
guard is assigned one of k distinct colors. A placement of such colored guards is conflict-free if each point of the polygon is seen
by some guard whose color appears exactly once among the guards visible to that point. What is the smallest number k(n) of colors that
ensure a conflict-free covering of all n-vertex polygons? We call this
the conflict-free chromatic art gallery problem. The problem is motivated by applications in distributed robotics and wireless sensor
networks where colors indicate the wireless frequencies assigned to a
set of covering "landmarks" in the environment so that a mobile robot
can always communicate with at least one landmark in its line-of-sight
range without interference.
Our main result shows that k(n) is O(log n) for orthogonal and for
monotone polygons, and O(log^2 n) for arbitrary simple polygons. By
contrast, if all guards visible from each point must have distinct
colors, then k(n)is Omega(n) for arbitrary simple polygons and Omega(sqrt(n)) for orthogonal polygons, as shown by Erickson and LaValle [Proc. of RSS 2011]
Embedding Graphs as Euclidean kNN-Graphs
No full textLet G = (V,E) be a directed graph on n vertices where each vertex has out-degree k. We say that G is kNN-realizable in d-dimensional Euclidean space if there exists a point set P = {p_1, p_2, …, p_n} in ℝ^d along with a one-to-one mapping ϕ: V → P such that for any u,v ∈ V, u is an out-neighbor of v in G if and only if ϕ(u) is one of the k nearest neighbors of ϕ(v); we call the map ϕ a kNN-realization of G in ℝ^d. The kNN-realization problem, which aims to compute a kNN-realization of an input graph in ℝ^d, is known to be NP-hard already for d = 2 and k = 1 [Eades and Whitesides, Theoretical Computer Science, 1996], and to the best of our knowledge has not been studied in dimension d = 1. The main results of this paper are the following:
- For any fixed dimension d ≥ 2, we can efficiently compute an embedding realizing at least a 1 - ε fraction of G’s edges, or conclude that G is not kNN-realizable in ℝ^d.
- For d = 1, we can decide in O(kn) time whether G is kNN-realizable and, if so, compute a realization in O(n^{2.5} poly(log n)) time
Most Likely Voronoi Diagrams in Higher Dimensions
No full textThe Most Likely Voronoi Diagram is a generalization of the well known Voronoi Diagrams to a stochastic setting, where a stochastic point is a point associated with a given probability of existence, and the cell for such a point is the set of points which would classify the given point as its most likely nearest neighbor. We investigate the complexity of this subdivision of space in d dimensions. We show that in the general case, the complexity of such a subdivision is Omega(n^{2d}) where n is the number of points. This settles an open question raised in a recent (ISAAC 2014) paper of Suri and Verbeek, which first defined the Most Likely Voronoi Diagram. We also show that when the probabilities are assigned using a random permutation of a fixed set of values, in expectation the complexity is only ~O(n^{ceil{d/2}}) where the ~O(*) means that logarithmic factors are suppressed. In the worst case, this bound is tight up to polylog factors
An ETH-Tight Algorithm for Multi-Team Formation
No full textIn the Multi-Team Formation problem, we are given a ground set C of n candidates, each of which is characterized by a d-dimensional attribute vector in ℝ^d, and two positive integers α and β satisfying α β ≤ n. The goal is to form α disjoint teams T₁,...,T_α ⊆ C, each of which consists of β candidates in C, such that the total score of the teams is maximized, where the score of a team T is the sum of the h_j maximum values of the j-th attributes of the candidates in T, for all j ∈ {1,...,d}. Our main result is an 2^{2^O(d)} n^O(1)-time algorithm for Multi-Team Formation. This bound is ETH-tight since a 2^{2^{d/c}} n^O(1)-time algorithm for any constant c > 12 can be shown to violate the Exponential Time Hypothesis (ETH). Our algorithm runs in polynomial time for all dimensions up to d = clog log n for a sufficiently small constant c > 0. Prior to our work, the existence of a polynomial time algorithm was an open problem even for d = 3
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