Radio Networks (RN) is one of the fundamental models for network communication where nodes can broadcast messages locally but their simultaneous transmissions can interfere with each other at their shared neighbors. This work focuses on performing the very fundamental primitive of Local Broadcast, in spite of the interferences. We investigate to what extent local knowledge, called advice, relating to the 2-local domination number gamma 2 may speed up Local Broadcast. Specifically for each node and some dominating set, knowledge about some neighboring dominating node and the local number among the neighbors of that dominating node. We show that such advice is sufficient to build an efficient oblivious transmission schedule. Along those lines, we present three algorithms trading the level of adaptiveness (from oblivious to adaptive) for bits of advice per node (from O(log(Delta gamma(2))) to 1). All our algorithms complete Local Broadcast in (O) over tilde(Delta gamma(2)(2)) rounds, where Delta is the maximum degree of the network. On the side of lower bounds, we show that, for each quasi-adaptive deterministic Local Broadcast algorithm, there is some RN that requires Omega(min{(min{Delta,gamma(2)}/ log n)(2), n}) communication rounds, where n is the number of network nodes. In quasi-adaptive protocols nodes may stop executing once its computational task is completed. To the best of our knowledge, this is the first (nearly) quadratic Local Broadcast (same message for all neighbors) lower bound in the RN model. Our lower bound is stronger than previous works in multiple ways: i) it is nearly quadratically better than the best known general lower bound for this class of algorithms, ii) it applies to a wider class of algorithms than previous work for fully oblivious, iii) it achieves similar time lower bound than previous work proved for a much more demanding Local Broadcast where each node sends a possibly different message to each neighbor, and iv) it takes into account the local domination parameter gamma(2). 2012 ACM Subject Classification Theory of computation -> Distributed algorithms
In the Pinwheel Packing problem, we are given a set of recurring tasks, each associated with a positive integer a(i) for task i. The objective is to select one task to perform each day such that every task i is performed at least once within every a(i) consecutive days. The exact computational complexity of this problem, where Sigma 1/ a(i) = 1, has remained an open question for more than 30 years; in particular, it is still unknown whether the problem is NP-hard. The first contribution of this paper is to show that Pinwheel Packing cannot be solved in polynomial time under a standard complexity assumption, improving upon the hardness result shown by Jacobs and Longo. Additionally, we present fixed-parameter algorithms for variants of Pinwheel Packing, parameterized by the number of tasks. 2012 ACM Subject Classification Theory of computation -> Design and analysis of algorithms; Theory of computation -> Computational complexity and cryptography
We study the online service with deadlines (or delays) problem, in which a server must serve requests for points in a metric space while balancing travel distance and promptness of service. While the problem has been extensively studied (STOC 2017), (FOCS 2019), (FOCS 2023), the main open question whether a constant competitive ratio can be achieved remains wide open. We prove a logarithmic lower bound for a natural class of algorithms already on uniform line metrics. Our lower bound applies to, and is tight for, the best known algorithms for general metrics and uniform line metrics. 2012 ACM Subject Classification Theory of computation -> Online algorithms; Theory of computation -> K-server algorithms
Let P be a polygonal curve with n vertices in the plane. We construct a data structure of size O( n log n) suited for simplification queries of the following kind. Given a query line l and an integer k >= 1, find a curve Q on l with at most k vertices that minimizes the discrete Frechet distance to P, among all such curves. Using our data structure, a query can be handled in O(k(2) log(3) n + k log(4) n) time. More generally, a geometric tree T on n vertices in the plane can be preprocessed into a nearlinear-size structure so that, given a pair u, v of its vertices, a line l, and an integer k >= 1, one can find a curve Q on l with at most k vertices that minimizes the discrete Frechet distance to the path from u to v in T, in time O(k(2) polylog n). For the general dimension-reduction problem, where P is a curve in R-d ( d >= 3), 0 < epsilon(0) < 1 is a real parameter, and a query specifies a g-flat h (1 = g = d - 1) and an integer k >= 1, we construct a data structure of size O( n log n + f(epsilon(0))n), where f(epsilon(0)) = (1 + 1/epsilon(0))(( d-1)/2), that allows us to find a curve Q on h with at most k vertices, whose discrete Frechet distance to P is at most 1 + epsilon(0) times the distance of Q(*) to P, where Q(*) is such a curve that minimizes the distance to P. The query handling time is O(f(epsilon(0))k(2) log(2) n). 2012 ACM Subject Classification Theory of computation ->. Computational geometry
Let B be a collection of n arbitrary balls in R-3, and let G(0)(B) be their intersection graph. We provide an algorithm for performing BFS on G(0)(B), which runs in O*(n(4/3)) time, where the O *( center dot) notation hides subpolynomial factors. For r >= 0, let G(gamma)(B) be the intersection graph of the set B-gamma = {B + gamma | B is an element of B}, where B + gamma is the ball concentric with B whose radius is larger by gamma than the radius of B. We provide an efficient algorithm for the reverse shortest path (RSP) problem, where we are given two designated balls B-s, B-t of B and a parameter 0 < lambda < n, and seek the smallest value gamma* for which G(gamma)* (B) contains a path from B-s to B-t of at most lambda edges. For the special case of congruent balls (equivalently, for points in R-3), the algorithm runs in O*(n(29/21)) approximate to O*(n(1.381)) time. For the general case, the algorithm runs in O*(n(56/39)) approximate to O*(n(1.436)) time. We also extend the technique to handle other measures of expansion and higher dimensions. 2012 ACM Subject Classification Theory of computation -> Computational geometry
