A Novel Algorithm for the All-Best-Swap-Edge Problem on Tree Spanners

Given a 2-edge connected, unweighted, and undirected graph $G$ with $n$ vertices and $m$ edges, a $\sigma$-tree spanner is a spanning tree $T$ of $G$ in which the ratio between the distance in $T$ of any pair of vertices and the corresponding distance in $G$ is upper bounded by $\sigma$. The minimum value of $\sigma$ for which $T$ is a $\sigma$-tree spanner of $G$ is also called the {\em stretch factor} of $T$. We address the fault-tolerant scenario in which each edge $e$ of a given tree spanner may temporarily fail and has to be replaced by a {\em best swap edge}, i.e. an edge that reconnects $T-e$ at a minimum stretch factor. More precisely, we design an $O(n^2)$ time and space algorithm that computes a best swap edge of every tree edge. Previously, an $O(n^2 \log^4 n)$ time and $O(n^2+m\log^2n)$ space algorithm was known for edge-weighted graphs [Bil\`o et al., ISAAC 2017]. Even if our improvements on both the time and space complexities are of a polylogarithmic factor, we stress the fact that the design of a $o(n^2)$ time and space algorithm would be considered a breakthrough.Comment: The paper has been accepted for publication at the 29th International Symposium on Algorithms and Computation (ISAAC 2018). 12 pages, 3 figure

Fully Dynamic MIS in Uniformly Sparse Graphs

We consider the problem of maintaining a maximal independent set (MIS) in a dynamic graph subject to edge insertions and deletions. Recently, Assadi, Onak, Schieber and Solomon (STOC 2018) showed that an MIS can be maintained in sublinear (in the dynamically changing number of edges) amortized update time. In this paper we significantly improve the update time for uniformly sparse graphs. Specifically, for graphs with arboricity alpha, the amortized update time of our algorithm is O(alpha^2 * log^2 n), where n is the number of vertices. For low arboricity graphs, which include, for example, minor-free graphs as well as some classes of "real world" graphs, our update time is polylogarithmic. Our update time improves the result of Assadi et al. for all graphs with arboricity bounded by m^{3/8 - epsilon}, for any constant epsilon > 0. This covers much of the range of possible values for arboricity, as the arboricity of a general graph cannot exceed m^{1/2}

Unbounded Regions of High-Order Voronoi Diagrams of Lines and Segments in Higher Dimensions

We study the behavior at infinity of the farthest and the higher-order Voronoi diagram of n line segments or lines in a d-dimensional Euclidean space. The unbounded parts of these diagrams can be encoded by a Gaussian map on the sphere of directions S^(d-1). We show that the combinatorial complexity of the Gaussian map for the order-k Voronoi diagram of n line segments or lines is O(min{k,n-k} n^(d-1)), which is tight for n-k = O(1). All the d-dimensional cells of the farthest Voronoi diagram are unbounded, its (d-1)-skeleton is connected, and it does not have tunnels. A d-cell of the Voronoi diagram is called a tunnel if the set of its unbounded directions, represented as points on its Gaussian map, is not connected. In a three-dimensional space, the farthest Voronoi diagram of lines has exactly n^2-n three-dimensional cells, when n >= 2. The Gaussian map of the farthest Voronoi diagram of line segments or lines can be constructed in O(n^(d-1) alpha(n)) time, while if d=3, the time drops to worst-case optimal O(n^2)

Succinct Dynamic One-Dimensional Point Reporting

In this paper we present a succinct data structure for the dynamic one-dimensional range reporting problem. Given an interval [a,b] for some a,b in [m], the range reporting query on an integer set S subseteq [m] asks for all points in S cap [a,b]. We describe a data structure that answers reporting queries in optimal O(k+1) time, where k is the number of points in the answer, and supports updates in O(lg^epsilon m) expected time. Our data structure uses B(n,m) + o(B(n,m)) bits where B(n,m) is the minimum number of bits required to represent a set of size n from a universe of m elements. This is the first dynamic data structure for this problem that uses succinct space and achieves optimal query time

Longest Increasing Subsequence under Persistent Comparison Errors

We study the problem of computing a longest increasing subsequence in a sequence $S$ of $n$ distinct elements in the presence of persistent comparison errors. In this model, every comparison between two elements can return the wrong result with some fixed (small) probability $p$, and comparisons cannot be repeated. Computing the longest increasing subsequence exactly is impossible in this model, therefore, the objective is to identify a subsequence that (i) is indeed increasing and (ii) has a length that approximates the length of the longest increasing subsequence. We present asymptotically tight upper and lower bounds on both the approximation factor and the running time. In particular, we present an algorithm that computes an $O(\log n)$-approximation in time $O(n\log n)$, with high probability. This approximation relies on the fact that that we can approximately sort $n$ elements in $O(n\log n)$ time such that the maximum dislocation of an element is at most $O(\log n)$. For the lower bounds, we prove that (i) there is a set of sequences, such that on a sequence picked randomly from this set every algorithm must return an $\Omega(\log n)$-approximation with high probability, and (ii) any $O(\log n)$-approximation algorithm for longest increasing subsequence requires $\Omega(n \log n)$ comparisons, even in the absence of errors

Space Efficient Two-Dimensional Orthogonal Colored Range Counting

In the two-dimensional orthogonal colored range counting problem, we preprocess a set, $P$, of $n$ colored points on the plane, such that given an orthogonal query rectangle, the number of distinct colors of the points contained in this rectangle can be computed efficiently. For this problem, we design three new solutions, and the bounds of each can be expressed in some form of time-space tradeoff. By setting appropriate parameter values for these solutions, we can achieve new specific results with (the space are in words and $\epsilon$ is an arbitrary constant in $(0,1)$): ** $O(n\lg^3 n)$ space and $O(\sqrt{n}\lg^{5/2} n \lg \lg n)$ query time; ** $O(n\lg^2 n)$ space and $O(\sqrt{n}\lg^{4+\epsilon} n)$ query time; ** $O(n\frac{\lg^2 n}{\lg \lg n})$ space and $O(\sqrt{n}\lg^{5+\epsilon} n)$ query time; ** $O(n\lg n)$ space and $O(n^{1/2+\epsilon})$ query time. A known conditional lower bound to this problem based on Boolean matrix multiplication gives some evidence on the difficulty of achieving near-linear space solutions with query time better than $\sqrt{n}$ by more than a polylogarithmic factor using purely combinatorial approaches. Thus the time and space bounds in all these results are efficient. Previously, among solutions with similar query times, the most space-efficient solution uses $O(n\lg^4 n)$ space to answer queries in $O(\sqrt{n}\lg^8 n)$ time (SIAM. J. Comp.~2008). Thus the new results listed above all achieve improvements in space efficiency, while all but the last result achieve speed-up in query time as well.Comment: full version of an ESA 2021 pape