Efficient Dynamic Approximate Distance Oracles for Vertex-Labeled Planar Graphs

Let $G$ be a graph where each vertex is associated with a label. A Vertex-Labeled Approximate Distance Oracle is a data structure that, given a vertex $v$ and a label $\lambda$, returns a $(1+\varepsilon)$-approximation of the distance from $v$ to the closest vertex with label $\lambda$ in $G$. Such an oracle is dynamic if it also supports label changes. In this paper we present three different dynamic approximate vertex-labeled distance oracles for planar graphs, all with polylogarithmic query and update times, and nearly linear space requirements

Speeding up shortest path algorithms

Given an arbitrary, non-negatively weighted, directed graph $G=(V,E)$ we present an algorithm that computes all pairs shortest paths in time $\mathcal{O}(m^* n + m \lg n + nT_\psi(m^*, n))$, where $m^*$ is the number of different edges contained in shortest paths and $T_\psi(m^*, n)$ is a running time of an algorithm to solve a single-source shortest path problem (SSSP). This is a substantial improvement over a trivial $n$ times application of $\psi$ that runs in $\mathcal{O}(nT_\psi(m,n))$. In our algorithm we use $\psi$ as a black box and hence any improvement on $\psi$ results also in improvement of our algorithm. Furthermore, a combination of our method, Johnson's reweighting technique and topological sorting results in an $\mathcal{O}(m^*n + m \lg n)$ all-pairs shortest path algorithm for arbitrarily-weighted directed acyclic graphs. In addition, we also point out a connection between the complexity of a certain sorting problem defined on shortest paths and SSSP.Comment: 10 page

A simpler and more efficient algorithm for the next-to-shortest path problem

Given an undirected graph $G=(V,E)$ with positive edge lengths and two vertices $s$ and $t$, the next-to-shortest path problem is to find an $st$-path which length is minimum amongst all $st$-paths strictly longer than the shortest path length. In this paper we show that the problem can be solved in linear time if the distances from $s$ and $t$ to all other vertices are given. Particularly our new algorithm runs in $O(|V|\log |V|+|E|)$ time for general graphs, which improves the previous result of $O(|V|^2)$ time for sparse graphs, and takes only linear time for unweighted graphs, planar graphs, and graphs with positive integer edge lengths.Comment: Partial result appeared in COCOA201

Dynamic Range Majority Data Structures

Given a set $P$ of coloured points on the real line, we study the problem of answering range $\alpha$-majority (or "heavy hitter") queries on $P$. More specifically, for a query range $Q$, we want to return each colour that is assigned to more than an $\alpha$-fraction of the points contained in $Q$. We present a new data structure for answering range $\alpha$-majority queries on a dynamic set of points, where $\alpha \in (0,1)$. Our data structure uses O(n) space, supports queries in $O((\lg n) / \alpha)$ time, and updates in $O((\lg n) / \alpha)$ amortized time. If the coordinates of the points are integers, then the query time can be improved to $O(\lg n / (\alpha \lg \lg n) + (\lg(1/\alpha))/\alpha))$. For constant values of $\alpha$, this improved query time matches an existing lower bound, for any data structure with polylogarithmic update time. We also generalize our data structure to handle sets of points in d-dimensions, for $d \ge 2$, as well as dynamic arrays, in which each entry is a colour.Comment: 16 pages, Preliminary version appeared in ISAAC 201

Fast Locality-Sensitive Hashing Frameworks for Approximate Near Neighbor Search

The Indyk-Motwani Locality-Sensitive Hashing (LSH) framework (STOC 1998) is a general technique for constructing a data structure to answer approximate near neighbor queries by using a distribution $\mathcal{H}$ over locality-sensitive hash functions that partition space. For a collection of $n$ points, after preprocessing, the query time is dominated by $O(n^{\rho} \log n)$ evaluations of hash functions from $\mathcal{H}$ and $O(n^{\rho})$ hash table lookups and distance computations where $\rho \in (0,1)$ is determined by the locality-sensitivity properties of $\mathcal{H}$. It follows from a recent result by Dahlgaard et al. (FOCS 2017) that the number of locality-sensitive hash functions can be reduced to $O(\log^2 n)$, leaving the query time to be dominated by $O(n^{\rho})$ distance computations and $O(n^{\rho} \log n)$ additional word-RAM operations. We state this result as a general framework and provide a simpler analysis showing that the number of lookups and distance computations closely match the Indyk-Motwani framework, making it a viable replacement in practice. Using ideas from another locality-sensitive hashing framework by Andoni and Indyk (SODA 2006) we are able to reduce the number of additional word-RAM operations to $O(n^\rho)$.Comment: 15 pages, 3 figure

Cross-Document Pattern Matching

We study a new variant of the string matching problem called cross-document string matching, which is the problem of indexing a collection of documents to support an efficient search for a pattern in a selected document, where the pattern itself is a substring of another document. Several variants of this problem are considered, and efficient linear-space solutions are proposed with query time bounds that either do not depend at all on the pattern size or depend on it in a very limited way (doubly logarithmic). As a side result, we propose an improved solution to the weighted level ancestor problem

Compressed Subsequence Matching and Packed Tree Coloring

We present a new algorithm for subsequence matching in grammar compressed strings. Given a grammar of size $n$ compressing a string of size $N$ and a pattern string of size $m$ over an alphabet of size $\sigma$, our algorithm uses $O(n+\frac{n\sigma}{w})$ space and $O(n+\frac{n\sigma}{w}+m\log N\log w\cdot occ)$ or $O(n+\frac{n\sigma}{w}\log w+m\log N\cdot occ)$ time. Here $w$ is the word size and $occ$ is the number of occurrences of the pattern. Our algorithm uses less space than previous algorithms and is also faster for $occ=o(\frac{n}{\log N})$ occurrences. The algorithm uses a new data structure that allows us to efficiently find the next occurrence of a given character after a given position in a compressed string. This data structure in turn is based on a new data structure for the tree color problem, where the node colors are packed in bit strings.Comment: To appear at CPM '1

Widest Paths and Global Propagation in Bounded Value Iteration for Stochastic Games

Solving stochastic games with the reachability objective is a fundamental problem, especially in quantitative verification and synthesis. For this purpose, bounded value iteration (BVI) attracts attention as an efficient iterative method. However, BVI's performance is often impeded by costly end component (EC) computation that is needed to ensure convergence. Our contribution is a novel BVI algorithm that conducts, in addition to local propagation by the Bellman update that is typical of BVI, global propagation of upper bounds that is not hindered by ECs. To conduct global propagation in a computationally tractable manner, we construct a weighted graph and solve the widest path problem in it. Our experiments show the algorithm's performance advantage over the previous BVI algorithms that rely on EC computation.Comment: v2: an URL to the implementation is adde

‘Equally unequal or unequally equal’: Adopting a substantive equality approach to gender discrimination in Nigeria

The purpose of this article is to critically assess the approach of Nigerian courts to interpreting section 42 of the Constitution. This article argues that Nigerian courts are yet to develop a substantive equality approach to interpreting section 42 of the Constitution. Rather, the courts have tended to adopt the formal equality approach to interpreting the section. Analysing some decisions of the Court of Appeal and the Supreme Court, the article argues that in order to safeguard women’s rights and address gender inequality in the country, Nigerian courts should lean towards substantive equality approach to the interpretation of section 42 of the Constitution. This is not only consistent with Nigeria’s obligations under international law but also crucial to addressing historical imbalances between men and women in the country

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