arXiv subjects

Vignesh Manoharan

Publications and source records attributed to Vignesh Manoharan.

5 recordsLinked to original sources

arXiv2025

Algorithms for Distance Sensitivity Oracles and other Graph Problems on the PRAM

The distance sensitivity oracle (DSO) problem asks us to preprocess a given graph $G=(V,E)$ in order to answer queries of the form $d(x,y,e)$, which denotes the shortest path distance in $G$ from vertex $x$ to vertex $y$ when edge $e$ is removed. This is an important problem for network communication, and it has been extensively studied in the sequential settingand recently in the distributed CONGEST model. However, no prior DSO results tailored to the parallel setting were known. We present the first PRAM algorithms to construct DSOs in directed weighted graphs, that can answer a query in $O(1)$ time with a single processor after preprocessing. We also present the first work-optimal PRAM algorithms for other graph problems that belong to the sequential $\tilde{O}(mn)$ fine-grained complexity class: Replacement Paths, Second Simple Shortest Path, All Pairs Second Simple Shortest Paths and Minimum Weight Cycle.

cs.DS↗

arXiv2024

Distributed Distance Sensitivity Oracles

We present results for the distance sensitivity oracle (DSO) problem, where one needs to preprocess a given directed weighted graph $G=(V,E)$ in order to answer queries about the shortest path distance in $G$ from vertex $s$ to vertex $t$ avoiding edge $e$, for any $s,t \in V, e \in E$. DSO enables optimal re-routing under a link failure, and can serve as a key component for fault tolerance in a distributed setting. However, no non-trivial results for DSO are known in the distributed CONGEST model. We present DSO algorithms with different tradeoffs between preprocessing and query cost: one that optimizes query response rounds, and another that prioritizes preprocessing rounds. We complement these algorithms with unconditional CONGEST lower bounds for DSO. Our DSO lower bounds build on a lower bound we present for the $k$-source shortest paths problem ($k$-SSP), which may be of independent interest. Additionally, we present almost-optimal upper and lower bounds for the related all pairs second simple shortest path (2-APSiSP) problem.

cs.DS↗

arXiv2023

Improved Approximation Bounds for Minimum Weight Cycle in the CONGEST Model

Minimum Weight Cycle (MWC) is the problem of finding a simple cycle of minimum weight in a graph $G=(V,E)$. This is a fundamental graph problem with classical sequential algorithms that run in $\tilde{O}(n^3)$ and $\tilde{O}(mn)$ time where $n=|V|$ and $m=|E|$. In recent years this problem has received significant attention in the context of fine-grained sequential complexity as well as in the design of faster sequential approximation algorithms, though not much is known in the distributed CONGEST model. We present sublinear-round approximation algorithms for computing MWC in directed graphs, and weighted graphs. Our algorithms use a variety of techniques in non-trivial ways, such as in our approximate directed unweighted MWC algorithm that efficiently computes BFS from all vertices restricted to certain implicitly computed neighborhoods in sublinear rounds, and in our weighted approximation algorithms that use unweighted MWC algorithms on scaled graphs combined with a fast and streamlined method for computing multiple source approximate SSSP. We present $\tilde{\Omega}(\sqrt{n})$ lower bounds for arbitrary constant factor approximation of MWC in directed graphs and undirected weighted graphs.

cs.DS↗

arXiv2022

Near Optimal Bounds for Replacement Paths and Related Problems in the CONGEST Model

We present several results in the CONGEST model on round complexity for Replacement Paths (RPaths), Minimum Weight Cycle (MWC), and All Nodes Shortest Cycles (ANSC). We study these fundamental problems in both directed and undirected graphs, both weighted and unweighted. Many of our results are optimal to within a polylog factor: For an $n$-node graph $G$ we establish near linear lower and upper bounds for computing RPaths if $G$ is directed and weighted, and for computing MWC and ANSC if $G$ is weighted, directed or undirected; near $\sqrt{n}$ lower and upper bounds for undirected weighted RPaths; and $\Theta(D)$ bound for undirected unweighted RPaths. We also present lower and upper bounds for approximation versions of these problems, notably a $(2-(1/g))$-approximation algorithm for undirected unweighted MWC that runs in $\tilde{O}(\sqrt{n}+D)$ rounds, improving on the previous best bound of $\tilde{O}(\sqrt{ng}+D)$ rounds, where $g$ is the MWC length. We present a $(1+\epsilon)$-approximation algorithm for directed weighted RPaths, which beats the linear lower bound for exact RPaths.

cs.DS↗

arXiv2017

AUPCR Maximizing Matchings : Towards a Pragmatic Notion of Optimality for One-Sided Preference Matchings

We consider the problem of computing a matching in a bipartite graph in the presence of one-sided preferences. There are several well studied notions of optimality which include pareto optimality, rank maximality, fairness and popularity. In this paper, we conduct an in-depth experimental study comparing different notions of optimality based on a variety of metrics like cardinality, number of rank-1 edges, popularity, to name a few. Observing certain shortcomings in the standard notions of optimality, we propose an algorithm which maximizes an alternative metric called the Area under Profile Curve ratio (AUPCR). To the best of our knowledge, the AUPCR metric was used earlier but there is no known algorithm to compute an AUPCR maximizing matching. Finally, we illustrate the superiority of the AUPCR-maximizing matching by comparing its performance against other optimal matchings on synthetic instances modeling real-world data.

cs.MA↗