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In the {\em distributed all-pairs shortest paths} problem (APSP), every node in the weighted undirected distributed network (the CONGEST model) needs to know the distance from every other node using least number of communication rounds…
In this paper we provide new randomized algorithms with improved runtimes for solving linear programs with two-sided constraints. In the special case of the minimum cost flow problem on $n$-vertex $m$-edge graphs with integer…
We give a deterministic $m^{1+o(1)}$ time algorithm that computes exact maximum flows and minimum-cost flows on directed graphs with $m$ edges and polynomially bounded integral demands, costs, and capacities. As a consequence, we obtain the…
The minimum degree algorithm is one of the most widely-used heuristics for reducing the cost of solving large sparse systems of linear equations. It has been studied for nearly half a century and has a rich history of bridging techniques…
Given a directed graph and a source vertex, the fully dynamic single-source reachability problem is to maintain the set of vertices that are reachable from the given vertex, subject to edge deletions and insertions. It is one of the most…
Expander graphs are known to be robust to edge deletions in the following sense: for any online sequence of edge deletions $e_1, e_2, \ldots, e_k$ to an $m$-edge graph $G$ that is initially a $\phi$-expander, the algorithm can grow a set $P…
Finding dense subgraphs is a fundamental problem with applications to community detection, clustering, and data mining. Our work focuses on finding approximate densest subgraphs in directed graphs in computational models for processing…
We initiate the study of dynamic algorithms for graph sparsification problems and obtain fully dynamic algorithms, allowing both edge insertions and edge deletions, that take polylogarithmic time after each update in the graph. Our three…
In a graph $G$ with a source $s$, we design a distance oracle that can answer the following query: Query$(s,t,e)$ -- find the length of shortest path from a fixed source $s$ to any destination vertex $t$ while avoiding any edge $e$. We…
We continue the study of distance sensitivity oracles (DSOs). Given a directed graph $G$ with $n$ vertices and edge weights in $\{1, 2, \dots, M\}$, we want to build a data structure such that given any source vertex $u$, any target vertex…
For $n$-vertex $m$-edge graphs with integer polynomially-bounded costs and capacities, we provide a randomized parallel algorithm for the minimum cost flow problem with $\tilde O(m+n^ {1.5})$ work and $\tilde O(\sqrt{n})$ depth. On…
In this paper we introduce a general framework for casting fully dynamic transitive closure into the problem of reevaluating polynomials over matrices. With this technique, we improve the best known bounds for fully dynamic transitive…
Fully dynamic graph is a data structure that (1) supports edge insertions and deletions and (2) answers problem specific queries. The time complexity of (1) and (2) are referred to as the update time and the query time respectively. There…
An $\alpha$-spanner of a graph $ G $ is a subgraph $ H $ such that $ H $ preserves all distances of $ G $ within a factor of $ \alpha $. In this paper, we give fully dynamic algorithms for maintaining a spanner $ H $ of a graph $ G $…
In this work we derandomize two central results in graph algorithms, replacement paths and distance sensitivity oracles (DSOs) matching in both cases the running time of the randomized algorithms. For the replacement paths problem, let G =…
In this paper we study the dynamic versions of two basic graph problems: Minimum Dominating Set and its variant Minimum Connected Dominating Set. For those two problems, we present algorithms that maintain a solution under edge insertions…
In the semi-streaming model, an algorithm must process any $n$-vertex graph by making one or few passes over a stream of its edges, use $O(n \cdot \text{polylog }n)$ words of space, and at the end of the last pass, output a solution to the…
Finding dense subgraphs is a fundamental algorithmic tool in data mining, community detection, and clustering. In this problem, one aims to find an induced subgraph whose edge-to-vertex ratio is maximized. We study the directed case of this…
One of the most basic graph problems, All-Pairs Shortest Paths (APSP) is known to be solvable in $n^{3-o(1)}$ time, and it is widely open whether it has an $O(n^{3-\epsilon})$ time algorithm for $\epsilon > 0$. To better understand APSP,…
In the fully dynamic edge connectivity problem, the input is a simple graph $G$ undergoing edge insertions and deletions, and the goal is to maintain its edge connectivity, denoted $\lambda_G$. We present two simple randomized algorithms…