Near-Optimal Lower Bounds for ε-Nets for Half-Spaces and Low Complexity Set Systems

The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

Discrete & Computational Geometry ◽

10.1007/s00454-021-00318-z ◽

2021 ◽

Author(s):

Anne Driemel ◽

André Nusser ◽

Jeff M. Phillips ◽

Ioannis Psarros

Keyword(s):

Density Estimation ◽

Lower Bounds ◽

Upper And Lower Bounds ◽

Similarity Metrics ◽

Vc Dimension ◽

Set Systems ◽

Polygonal Curves ◽

The Individual ◽

Curve Similarity ◽

Metric Balls

AbstractThe Vapnik–Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set X is a set of polygonal curves in $$\mathbb {R}^d$$ R d and the sets $$\mathcal {R}$$ R are metric balls defined by curve similarity metrics, such as the Fréchet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper and lower bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

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Context-bounded verification of thread pools

Proceedings of the ACM on Programming Languages ◽

10.1145/3498678 ◽

2022 ◽

Vol 6 (POPL) ◽

pp. 1-28

Author(s):

Pascal Baumann ◽

Rupak Majumdar ◽

Ramanathan S. Thinniyam ◽

Georg Zetzsche

Keyword(s):

Lower Bounds ◽

Finite Automata ◽

Low Complexity ◽

Safety Verification ◽

Nondeterministic Finite Automata ◽

Thread Pool ◽

Fixed Set ◽

Verification Problem ◽

Downward Closure ◽

Two Parameters

Thread pooling is a common programming idiom in which a fixed set of worker threads are maintained to execute tasks concurrently. The workers repeatedly pick tasks and execute them to completion. Each task is sequential, with possibly recursive code, and tasks communicate over shared memory. Executing a task can lead to more new tasks being spawned. We consider the safety verification problem for thread-pooled programs. We parameterize the problem with two parameters: the size of the thread pool as well as the number of context switches for each task. The size of the thread pool determines the number of workers running concurrently. The number of context switches determines how many times a worker can be swapped out while executing a single task---like many verification problems for multithreaded recursive programs, the context bounding is important for decidability. We show that the safety verification problem for thread-pooled, context-bounded, Boolean programs is EXPSPACE-complete, even if the size of the thread pool and the context bound are given in binary. Our main result, the EXPSPACE upper bound, is derived using a sequence of new succinct encoding techniques of independent language-theoretic interest. In particular, we show a polynomial-time construction of downward closures of languages accepted by succinct pushdown automata as doubly succinct nondeterministic finite automata. While there are explicit doubly exponential lower bounds on the size of nondeterministic finite automata accepting the downward closure, our result shows these automata can be compressed. We show that thread pooling significantly reduces computational power: in contrast, if only the context bound is provided in binary, but there is no thread pooling, the safety verification problem becomes 3EXPSPACE-complete. Given the high complexity lower bounds of related problems involving binary parameters, the relatively low complexity of safety verification with thread-pooling comes as a surprise.

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Lower bounds on the estimation performance in low complexity quantize-and-forward cooperative systems

EURASIP Journal on Wireless Communications and Networking ◽

10.1186/s13638-015-0461-8 ◽

2015 ◽

Vol 2015 (1) ◽

Author(s):

Iancu Avram ◽

Nico Aerts ◽

Marc Moeneclaey

Keyword(s):

Lower Bounds ◽

Low Complexity ◽

Cooperative Systems ◽

Estimation Performance

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Chromatic Numbers of Stable Kneser Hypergraphs via Topological Tverberg-Type Theorems

International Mathematics Research Notices ◽

10.1093/imrn/rny135 ◽

2018 ◽

Vol 2020 (13) ◽

pp. 4037-4061 ◽

Cited By ~ 2

Author(s):

Florian Frick

Keyword(s):

Euclidean Space ◽

Chromatic Number ◽

Lower Bounds ◽

Pairwise Disjoint ◽

Convex Hulls ◽

Chromatic Numbers ◽

Set Systems ◽

Equivariant Topology ◽

Disjoint Sets ◽

Combination Of Methods

Abstract Kneser’s 1955 conjecture—proven by Lovász in 1978—asserts that in any partition of the $k$-subsets of $\{1, 2, \dots , n\}$ into $n-2k+1$ parts, one part contains two disjoint sets. Schrijver showed that one can restrict to significantly fewer $k$-sets and still observe the same intersection pattern. Alon, Frankl, and Lovász proved a different generalization of Kneser’s conjecture for $r$ pairwise disjoint sets. Dolnikov generalized Lovász’ result to arbitrary set systems, while Kříž did the same for the $r$-fold extension of Kneser’s conjecture. Here we prove a common generalization of all of these results. Moreover, we prove additional strengthenings by determining the chromatic number of certain sparse stable Kneser hypergraphs, and further develop a general approach to establishing lower bounds for chromatic numbers of hypergraphs using a combination of methods from equivariant topology and intersection results for convex hulls of points in Euclidean space.

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