A Combinatorial Algorithm for All-Pairs Shortest Paths in Directed Vertex-Weighted Graphs with Applications to Disc Graphs

We consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. For an $n\times n$ 0-1 matrix $C,$ let $K_{C}$ be the complete weighted graph on the rows of $C$ where the weight of an edge between two rows is equal to their Hamming distance. Let $MWT(C)$ be the weight of a minimum weight spanning tree of $K_{C}.$ We show that the all-pairs shortest path problem for a directed graph $G$ on $n$ vertices with nonnegative real weights and adjacency matrix $A_G$ can be solved by a combinatorial randomized algorithm in time $$\widetilde{O}(n^{2}\sqrt {n + \min\{MWT(A_G), MWT(A_G^t)\}})$$ As a corollary, we conclude that the transitive closure of a directed graph $G$ can be computed by a combinatorial randomized algorithm in the aforementioned time. $\widetilde{O}(n^{2}\sqrt {n + \min\{MWT(A_G), MWT(A_G^t)\}})$ We also conclude that the all-pairs shortest path problem for uniform disk graphs, with nonnegative real vertex weights, induced by point sets of bounded density within a unit square can be solved in time $\widetilde{O}(n^{2.75})$

Theta-3 is connected

In this paper, we show that the $\theta$-graph with three cones is connected. We also provide an alternative proof of the connectivity of the Yao graph with three cones.Comment: 11 pages, to appear in CGT

Exploiting non-constant safe memory in resilient algorithms and data structures

We extend the Faulty RAM model by Finocchi and Italiano (2008) by adding a safe memory of arbitrary size $S$, and we then derive tradeoffs between the performance of resilient algorithmic techniques and the size of the safe memory. Let $\delta$ and $\alpha$ denote, respectively, the maximum amount of faults which can happen during the execution of an algorithm and the actual number of occurred faults, with $\alpha \leq \delta$. We propose a resilient algorithm for sorting $n$ entries which requires $O\left(n\log n+\alpha (\delta/S + \log S)\right)$ time and uses $\Theta(S)$ safe memory words. Our algorithm outperforms previous resilient sorting algorithms which do not exploit the available safe memory and require $O\left(n\log n+ \alpha\delta\right)$ time. Finally, we exploit our sorting algorithm for deriving a resilient priority queue. Our implementation uses $\Theta(S)$ safe memory words and $\Theta(n)$ faulty memory words for storing $n$ keys, and requires $O\left(\log n + \delta/S\right)$ amortized time for each insert and deletemin operation. Our resilient priority queue improves the $O\left(\log n + \delta\right)$ amortized time required by the state of the art.Comment: To appear in Theoretical Computer Science, 201

Dynamic Set Intersection

Consider the problem of maintaining a family $F$ of dynamic sets subject to insertions, deletions, and set-intersection reporting queries: given $S,S'\in F$, report every member of $S\cap S'$ in any order. We show that in the word RAM model, where $w$ is the word size, given a cap $d$ on the maximum size of any set, we can support set intersection queries in $O(\frac{d}{w/\log^2 w})$ expected time, and updates in $O(\log w)$ expected time. Using this algorithm we can list all $t$ triangles of a graph $G=(V,E)$ in $O(m+\frac{m\alpha}{w/\log^2 w} +t)$ expected time, where $m=|E|$ and $\alpha$ is the arboricity of $G$. This improves a 30-year old triangle enumeration algorithm of Chiba and Nishizeki running in $O(m \alpha)$ time. We provide an incremental data structure on $F$ that supports intersection {\em witness} queries, where we only need to find {\em one} $e\in S\cap S'$. Both queries and insertions take O\paren{\sqrt \frac{N}{w/\log^2 w}} expected time, where $N=\sum_{S\in F} |S|$. Finally, we provide time/space tradeoffs for the fully dynamic set intersection reporting problem. Using $M$ words of space, each update costs $O(\sqrt {M \log N})$ expected time, each reporting query costs $O(\frac{N\sqrt{\log N}}{\sqrt M}\sqrt{op+1})$ expected time where $op$ is the size of the output, and each witness query costs $O(\frac{N\sqrt{\log N}}{\sqrt M} + \log N)$ expected time.Comment: Accepted to WADS 201

In pursuit of the dynamic optimality conjecture

In 1985, Sleator and Tarjan introduced the splay tree, a self-adjusting binary search tree algorithm. Splay trees were conjectured to perform within a constant factor as any offline rotation-based search tree algorithm on every sufficiently long sequence---any binary search tree algorithm that has this property is said to be dynamically optimal. However, currently neither splay trees nor any other tree algorithm is known to be dynamically optimal. Here we survey the progress that has been made in the almost thirty years since the conjecture was first formulated, and present a binary search tree algorithm that is dynamically optimal if any binary search tree algorithm is dynamically optimal.Comment: Preliminary version of paper to appear in the Conference on Space Efficient Data Structures, Streams and Algorithms to be held in August 2013 in honor of Ian Munro's 66th birthda