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[1708.08691] Extremal solutions to some art gallery and terminal-pairability problems
The chosen tool of this thesis is an extremal type approach. The lesson drawn by the theorems proved in the thesis is that surprisingly small compromise is necessary on the efficacy of the solutions to make the approach work. The problems studied have several connections to other subjects and practical applications.
The first part of the thesis is concerned with orthogonal art galleries. A sharp extremal bound is proved on partitioning orthogonal polygons into at most 8-vertex polygons using established techniques in the field of art gallery problems. This fills in the gap between already known results for partitioning into at most 6- and 10-vertex orthogonal polygons.
Next, these techniques are further developed to prove a new type of extremal art gallery result. The novelty provided by this approach is that it establishes a connection between mobile and stationary guards. This theorem has strong computational consequences, in fact, it provides the basis for an 83-approximation algorithm for guarding orthogonal polygons with rectangular vision.
In the second part, the graph theoretical concept of terminal-pairability is studied in complete and complete grid graphs. Once again, the extremal approach is conductive to discovering efficient methods to solve the problem.
In the case of a complete base graph, the new demonstrated lower bound on the maximum degree of realizable demand graphs is 4 times higher than previous best results. The techniques developed are then used to solve the classical extremal edge number problem for the terminal-pairability problem in complete base graphs.
The complete grid base graph lies on the other end of the spectrum in terms density amongst path-pairable graphs. It is shown that complete grid graphs are relatively efficient in routing edge-disjoint paths.
computational-geometry  combinatorics  plane-geometry  extreme-values  rather-interesting  to-write-about  nudge-targets  consider:approximation  consider:looking-to-see 
yesterday by Vaguery
[1706.01557] The minimum Manhattan distance and minimum jump of permutations
Let π be a permutation of {1,2,…,n}. If we identify a permutation with its graph, namely the set of n dots at positions (i,π(i)), it is natural to consider the minimum L1 (Manhattan) distance, d(π), between any pair of dots. The paper computes the expected value of d(π) when n→∞ and π is chosen uniformly, and settles a conjecture of Bevan, Homberger and Tenner (motivated by permutation patterns), showing that when d is fixed and n→∞, the probability that d(π)≥d+2 tends to e−d2−d.
The minimum jump 𝗆𝗃(π) of π, defined by 𝗆𝗃(π)=min1≤i≤n−1|π(i+1)−π(i)|, is another natural measure in this context. The paper computes the expected value of 𝗆𝗃(π), and the asymptotic probability that 𝗆𝗃(π)≥d+1 for any constant d.
combinatorics  probability-theory  permutations  rather-interesting  to-write-about  to-simulate  nudge-targets  consider:looking-to-see 
yesterday by Vaguery
[1610.06934] The K Shortest Paths Problem with Application to Routing
Due to the computational complexity of finding almost shortest simple paths, we propose that identifying a larger collection of (nonbacktracking) paths is more efficient than finding almost shortest simple paths on positively weighted real-world networks. First, we present an easy to implement O(mlogm+kL) solution for finding all (nonbacktracking) paths with bounded length D between two arbitrary nodes on a positively weighted graph, where L is an upperbound for the number of nodes in any of the k outputted paths. Subsequently, we illustrate that for undirected Chung-Lu random graphs, the ratio between the number of nonbacktracking and simple paths asymptotically approaches 1 with high probability for a wide range of parameters. We then consider an application to the almost shortest paths algorithm to measure path diversity for internet routing in a snapshot of the Autonomous System graph subject to an edge deletion process.
planning  data-structures  algorithms  rather-interesting  operations-research  graph-theory  to-write-about  nudge-targets  consider:looking-to-see 
yesterday by Vaguery
[1711.03172] Curve Reconstruction via the Global Statistics of Natural Curves
Reconstructing the missing parts of a curve has been the subject of much computational research, with applications in image inpainting, object synthesis, etc. Different approaches for solving that problem are typically based on processes that seek visually pleasing or perceptually plausible completions. In this work we focus on reconstructing the underlying physically likely shape by utilizing the global statistics of natural curves. More specifically, we develop a reconstruction model that seeks the mean physical curve for a given inducer configuration. This simple model is both straightforward to compute and it is receptive to diverse additional information, but it requires enough samples for all curve configurations, a practical requirement that limits its effective utilization. To address this practical issue we explore and exploit statistical geometrical properties of natural curves, and in particular, we show that in many cases the mean curve is scale invariant and often times it is extensible. This, in turn, allows to boost the number of examples and thus the robustness of the statistics and its applicability. The reconstruction results are not only more physically plausible but they also lead to important insights on the reconstruction problem, including an elegant explanation why certain inducer configurations are more likely to yield consistent perceptual completions than others.
inference  computer-vision  rather-interesting  algorithms  nudge-targets  consider:looking-to-see  consider:representation  performance-measure  to-write-about 
yesterday by Vaguery
[1704.07422] Iterative Methods for Photoacoustic Tomography in Attenuating Acoustic Media
The development of efficient and accurate reconstruction methods is an important aspect of tomographic imaging. In this article, we address this issue for photoacoustic tomography. To this aim, we use models for acoustic wave propagation accounting for frequency dependent attenuation according to a wide class of attenuation laws that may include memory. We formulate the inverse problem of photoacoustic tomography in attenuating medium as an ill-posed operator equation in a Hilbert space framework that is tackled by iterative regularization methods. Our approach comes with a clear convergence analysis. For that purpose we derive explicit expressions for the adjoint problem that can efficiently be implemented. In contrast to time reversal, the employed adjoint wave equation is again damping and, thus has a stable solution. This stability property can be clearly seen in our numerical results. Moreover, the presented numerical results clearly demonstrate the Efficiency and accuracy of the derived iterative reconstruction algorithms in various situations including the limited view case.
tomography  numerical-methods  inverse-problems  rather-interesting  signal-processing  algorithms  nudge-targets  consider:benchmarking  consider:looking-to-see 
yesterday by Vaguery
[1705.01887] Streaming for Aibohphobes: Longest Palindrome with Mismatches
A palindrome is a string that reads the same as its reverse, such as "aibohphobia" (fear of palindromes). Given an integer d>0, a d-near-palindrome is a string of Hamming distance at most d from its reverse. We study the natural problem of identifying a longest d-near-palindrome in data streams. The problem is relevant to the analysis of DNA databases, and to the task of repairing recursive structures in documents such as XML and JSON. We present an algorithm that returns a d-near-palindrome whose length is within a multiplicative (1+ϵ)-factor of the longest d-near-palindrome. Our algorithm also returns the set of mismatched indices of the d-near-palindrome, using (dlog7nϵlog(1+ϵ)) bits of space, and (dlog6nϵlog(1+ϵ)) update time per arriving symbol. We show that Ω(dlogn) space is necessary for estimating the length of longest d-near-palindromes with high probability. We further obtain an additive-error approximation algorithm and a comparable lower bound, as well as an exact two-pass algorithm that solves the longest d-near-palindrome problem using (d2n‾√log6n) bits of space.
strings  computational-complexity  algorithms  probability-theory  inference  online-learning  rather-interesting  to-write-about  nudge-targets  consider:looking-to-see  consider:representation 
yesterday by Vaguery
[1609.07531] Popular Matchings with Multiple Partners
Our input is a bipartite graph G=(A∪B,E) where each vertex in A∪B has a preference list strictly ranking its neighbors. The vertices in A and in B are called students and courses, respectively. Each student a seeks to be matched to 𝖼𝖺𝗉(a)≥1 courses while each course b seeks 𝖼𝖺𝗉(b)≥1 many students to be matched to it. The Gale-Shapley algorithm computes a pairwise-stable matching (one with no blocking edge) in G in linear time. We consider the problem of computing a popular matching in G -- a matching M is popular if M cannot lose an election to any matching where vertices cast votes for one matching versus another. Our main contribution is to show that a max-size popular matching in G can be computed by the 2-level Gale-Shapley algorithm in linear time. This is an extension of the classical Gale-Shapley algorithm and we prove its correctness via linear programming.
operations-research  combinatorics  optimization  computational-complexity  algorithms  heuristics  nudge-targets  consider:looking-to-see  to-write-about  consider:comparing-to-random 
9 days ago by Vaguery
[1710.00104] Small Satellite Constellation Separation using Linear Programming based Differential Drag Commands
We study the optimal control of an arbitrarily large constellation of small satellites operating in low Earth orbit. Simulating the lack of on-board propulsion, we limit our actuation to the use of differential drag maneuvers to make in-plane changes to the satellite orbits. We propose an efficient method to separate a cluster of satellites into a desired constellation shape while respecting actuation constraints and maximizing the operational lifetime of the constellation. By posing the problem as a linear program, we solve for the optimal drag commands for each of the satellites on a daily basis with a shrinking-horizon model predictive control approach. We then apply this control strategy in a nonlinear orbital dynamics simulation with a simple, varying atmospheric density model. We demonstrate the ability to control a cluster of 100+ satellites starting at the same initial conditions in a circular low Earth orbit to form an equally spaced constellation (with a relative angular separation error tolerance of one-tenth a degree). The constellation separation task can be executed in 71 days, a time frame that is competitive for the state-of-the-practice. This method allows us to trade the time required to converge to the desired constellation with a sacrifice in the overall constellation lifetime, measured as the maximum altitude loss experienced by one of the satellites in the group after the separation maneuvers.
nonlinear-dynamics  control-theory  rather-interesting  simulation  nudge-targets  consider:coordination  consider:looking-to-see  to-write-about 
9 days ago by Vaguery
[1709.09130] Output Range Analysis for Deep Neural Networks
Deep neural networks (NN) are extensively used for machine learning tasks such as image classification, perception and control of autonomous systems. Increasingly, these deep NNs are also been deployed in high-assurance applications. Thus, there is a pressing need for developing techniques to verify neural networks to check whether certain user-expected properties are satisfied. In this paper, we study a specific verification problem of computing a guaranteed range for the output of a deep neural network given a set of inputs represented as a convex polyhedron. Range estimation is a key primitive for verifying deep NNs. We present an efficient range estimation algorithm that uses a combination of local search and linear programming problems to efficiently find the maximum and minimum values taken by the outputs of the NN over the given input set. In contrast to recently proposed "monolithic" optimization approaches, we use local gradient descent to repeatedly find and eliminate local minima of the function. The final global optimum is certified using a mixed integer programming instance. We implement our approach and compare it with Reluplex, a recently proposed solver for deep neural networks. We demonstrate the effectiveness of the proposed approach for verification of NNs used in automated control as well as those used in classification.
neural-networks  benchmarking  rather-interesting  feature-construction  performance-measure  modeling-is-not-mathematics  algorithms  nudge-targets  consider:looking-to-see  consider:generalizing 
9 days ago by Vaguery
[cs/0305016] The one-round Voronoi game replayed
We consider the one-round Voronoi game, where player one (``White'', called ``Wilma'') places a set of n points in a rectangular area of aspect ratio r <=1, followed by the second player (``Black'', called ``Barney''), who places the same number of points. Each player wins the fraction of the board closest to one of his points, and the goal is to win more than half of the total area. This problem has been studied by Cheong et al., who showed that for large enough n and r=1, Barney has a strategy that guarantees a fraction of 1/2+a, for some small fixed a.
We resolve a number of open problems raised by that paper. In particular, we give a precise characterization of the outcome of the game for optimal play: We show that Barney has a winning strategy for n>2 and r>sqrt{2}/n, and for n=2 and r>sqrt{3}/2. Wilma wins in all remaining cases, i.e., for n>=3 and r<=sqrt{2}/n, for n=2 and r<=sqrt{3}/2, and for n=1. We also discuss complexity aspects of the game on more general boards, by proving that for a polygon with holes, it is NP-hard to maximize the area Barney can win against a given set of points by Wilma.
mathematical-recreations  game-theory  rather-interesting  planning  consider:looking-to-see  nudge-targets  to-write-about  to-simulate 
12 days ago by Vaguery
[1212.4771] Necklaces, Convolutions, and X+Y
We give subquadratic algorithms that, given two necklaces each with n beads at arbitrary positions, compute the optimal rotation of the necklaces to best align the beads. Here alignment is measured according to the p norm of the vector of distances between pairs of beads from opposite necklaces in the best perfect matching. We show surprisingly different results for p = 1, p even, and p = \infty. For p even, we reduce the problem to standard convolution, while for p = \infty and p = 1, we reduce the problem to (min, +) convolution and (median, +) convolution. Then we solve the latter two convolution problems in subquadratic time, which are interesting results in their own right. These results shed some light on the classic sorting X + Y problem, because the convolutions can be viewed as computing order statistics on the antidiagonals of the X + Y matrix. All of our algorithms run in o(n^2) time, whereas the obvious algorithms for these problems run in \Theta(n^2) time.
optimization  operations-research  combinatorics  distance  nudge-targets  consider:looking-to-see  consider:representation 
12 days ago by Vaguery
[1308.5550] Fitting Voronoi Diagrams to Planar Tesselations
Given a tesselation of the plane, defined by a planar straight-line graph G, we want to find a minimal set S of points in the plane, such that the Voronoi diagram associated with S "fits" \ G. This is the Generalized Inverse Voronoi Problem (GIVP), defined in \cite{Trin07} and rediscovered recently in \cite{Baner12}. Here we give an algorithm that solves this problem with a number of points that is linear in the size of G, assuming that the smallest angle in G is constant.
inverse-problems  computational-geometry  rather-interesting  nudge-targets  consider:looking-to-see  to-write-about 
12 days ago by Vaguery
[1705.04715] A catalogue of 4-regular and (2,4)-regular matchstick graphs
The first part (page 1 - 6) of this article presents the currently known examples of 4-regular matchstick graphs with 63 - 70 vertices. The second part (page 7 - 13) presents the currently known examples of (2,4)-regular matchstick graphs with less than 42 vertices which contain only two vertices of degree 2.
stamp-collecting  combinatorics  graph-theory  computational-geometry  rather-interesting  to-w  anomaly-detection  consider:looking-to-see 
12 days ago by Vaguery
[1205.2425] Flip Distance Between Two Triangulations of a Point-Set is NP-complete
Given two triangulations of a convex polygon, computing the minimum number of flips required to transform one to the other is a long-standing open problem. It is not known whether the problem is in P or NP-complete. We prove that two natural generalizations of the problem are NP-complete, namely computing the minimum number of flips between two triangulations of (1) a polygon with holes; (2) a set of points in the plane.
computational-complexity  computational-geometry  algorithms  planning  optimization  rather-interesting  plane-geometry  nudge-targets  consider:looking-to-see 
12 days ago by Vaguery
[1611.08035] Automating the Last-Mile for High Performance Dense Linear Algebra
High performance dense linear algebra (DLA) libraries often rely on a general matrix multiply (Gemm) kernel that is implemented using assembly or with vector intrinsics. In particular, the real-valued Gemm kernels provide the overwhelming fraction of performance for the complex-valued Gemm kernels, along with the entire level-3 BLAS and many of the real and complex LAPACK routines. Thus,achieving high performance for the Gemm kernel translates into a high performance linear algebra stack above this kernel. However, it is a monumental task for a domain expert to manually implement the kernel for every library-supported architecture. This leads to the belief that the craft of a Gemm kernel is more dark art than science. It is this premise that drives the popularity of autotuning with code generation in the domain of DLA.
This paper, instead, focuses on an analytical approach to code generation of the Gemm kernel for different architecture, in order to shed light on the details or voo-doo required for implementing a high performance Gemm kernel. We distill the implementation of the kernel into an even smaller kernel, an outer-product, and analytically determine how available SIMD instructions can be used to compute the outer-product efficiently. We codify this approach into a system to automatically generate a high performance SIMD implementation of the Gemm kernel. Experimental results demonstrate that our approach yields generated kernels with performance that is competitive with kernels implemented manually or using empirical search.
matrices  algorithms  rather-interesting  optimization  code-generation  nudge-targets  consider:looking-to-see  consider:performance-measures 
13 days ago by Vaguery
[1603.02597] Prediction of Infinite Words with Automata
In the classic problem of sequence prediction, a predictor receives a sequence of values from an emitter and tries to guess the next value before it appears. The predictor masters the emitter if there is a point after which all of the predictor's guesses are correct. In this paper we consider the case in which the predictor is an automaton and the emitted values are drawn from a finite set; i.e., the emitted sequence is an infinite word. We examine the predictive capabilities of finite automata, pushdown automata, stack automata (a generalization of pushdown automata), and multihead finite automata. We relate our predicting automata to purely periodic words, ultimately periodic words, and multilinear words, describing novel prediction algorithms for mastering these sequences.
formal-languages  automata  consider:looking-to-see  prediction  modeling  to-write-about 
13 days ago by Vaguery
[1707.00219] Angle-monotone Paths in Non-obtuse Triangulations
We reprove a result of Dehkordi, Frati, and Gudmundsson: every two vertices in a non-obtuse triangulation of a point set are connected by an angle-monotone path--an xy-monotone path in an appropriately rotated coordinate system. We show that this result cannot be extended to angle-monotone spanning trees, but can be extended to boundary-rooted spanning forests. The latter leads to a conjectural edge-unfolding of sufficiently shallow polyhedral convex caps.
computational-geometry  plane-geometry  rather-interesting  proof  nudge-targets  consider:looking-to-see  consider:algorithms 
13 days ago by Vaguery
[1702.02017] Pushing the Bounds for Matrix-Matrix Multiplication
A tight lower bound for required I/O when computing a matrix-matrix multiplication on a processor with two layers of memory is established. Prior work obtained weaker lower bounds by reasoning about the number of \textit{phases} needed to perform C:=AB, where each phase is a series of operations involving S reads and writes to and from fast memory, and S is the size of fast memory. A lower bound on the number of phases was then determined by obtaining an upper bound on the number of scalar multiplications performed per phase. This paper follows the same high level approach, but improves the lower bound by considering C:=AB+C instead of C:=AB, and obtains the maximum number of scalar fused multiply-adds (FMAs) per phase instead of scalar additions. Key to obtaining the new result is the decoupling of the per-phase I/O from the size of fast memory. The new lower bound is 2mnk/S‾√−2S. The constant for the leading term is an improvement of a factor 42‾√. A theoretical algorithm that attains the lower bound is given, and how the state-of-the-art Goto's algorithm also in some sense meets the lower bound is discussed.
computational-complexity  algorithms  matrices  rather-interesting  nudge-targets  consider:looking-to-see  consider:representation  consider:performance-measures 
13 days ago by Vaguery
[1001.0695] The Geodesic Diameter of Polygonal Domains
This paper studies the geodesic diameter of polygonal domains having h holes and n corners. For simple polygons (i.e., h = 0), the geodesic diameter is determined by a pair of corners of a given polygon and can be computed in linear time, as known by Hershberger and Suri. For general polygonal domains with h >= 1, however, no algorithm for computing the geodesic diameter was known prior to this paper. In this paper, we present the first algorithms that compute the geodesic diameter of a given polygonal domain in worst-case time O(n^7.73) or O(n^7 (log n + h)). The main difficulty unlike the simple polygon case relies on the following observation revealed in this paper: two interior points can determine the geodesic diameter and in that case there exist at least five distinct shortest paths between the two.
computational-complexity  computational-geometry  distance  algorithms  rather-interesting  plane-geometry  optimization  nudge-targets  consider:looking-to-see 
14 days ago by Vaguery

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