arXiv subjects

Chhaya Trehan

Publications and source records attributed to Chhaya Trehan.

8 recordsLinked to original sources

arXiv2025

Constructing Long Paths in Graph Streams

In the graph stream model of computation, an algorithm processes the edges of an input graph in one or more sequential passes while using a memory sublinear in the input size. This model poses significant challenges for constructing long paths. Many known algorithms tasked with extending an existing path as a subroutine require an entire pass to add a single additional edge. This raises a fundamental question: Are multiple passes inherently necessary to construct paths of non-trivial lengths, or can a single pass suffice? To address this question, we study the Longest Path problem in the one-pass streaming model. In this problem, given a desired approximation factor $α$, the objective is to compute a path of length at least $\lp(G) / α$, where $\lp(G)$ is the length of a longest path in the input graph. We give algorithms as well as space lower bounds for both undirected and directed graphs. Our results include: We show that for undirected graphs, in both the insertion-only and the insertion-deletion models, there are semi-streaming algorithms, that compute a path of length at least $d /3$ with high probability, where $d$ is the average degree of the graph. These algorithms can also yield an $α$-approximation to Longest Path using space $\tilde{O}(n^2 / α)$. Next, we show that such a result cannot be achieved for directed graphs, even in the insertion-only model. We show that computing a $(n^{1 - o(1)})$-approximation to Longest Path in directed graphs in the insertion-only model requires space $Ω(n^2)$. We further show two additional lower bounds. First, we show that semi-streaming space is insufficient for small constant factor approximations to Longest Path for undirected graphs in the insertion-only model. Last, in undirected graphs in the insertion-deletion model, we show that computing an $α$-approximation requires space $Ω(n^2 / α^3)$.

cs.DS↗

arXiv2025

Faster Multi-Source Reachability and Approximate Distances via Shortcuts, Hopsets and Matrix Multiplication

Given an $n$-vertex $m$-edge digraph $G = (V,E)$ and a subset $S \subseteq V$ of $|S| = n^{\sigma}$ (for some $0 \le \sigma \le 1$) designated sources, the $S \times V$ reachability problem is to compute the sets $\mathcal V_s$ of vertices reachable from $s$, for every $s \in S$. Naive centralized algorithms run BFS/DFS from each source in $O(m \cdot n^{\sigma})$ time or compute $G$'s transitive closure in $\hat O(n^{\omega})$ time, where $\omega \le 2.371552\ldots$ is the matrix multiplication exponent. Thus, the best known bound is $\hat O(n^{\min \{ 2 + \sigma, \omega\}})$. Leveraging shortcut constructions by Kogan and Parter [SODA 2022, ICALP 2022], we develop a centralized algorithm with running time $\hat O(n^{1 + \frac{2}{3} \omega(\sigma)})$, where $\omega(\sigma)$ is the rectangular matrix multiplication exponent. Using current estimates on $\omega(\sigma)$, our exponent improves upon $\min \{2 + \sigma, \omega \}$ for $\tilde \sigma \leq \sigma \leq 0.53$, where $1/3 < \tilde \sigma < 0.3336$ is a universal constant. In a classical result, Cohen [Journal of Algorithms, 1996] devised parallel algorithms for $S \times V$ reachability on graphs admitting balanced recursive separators of size $n^{\rho}$ for $\rho < 1$, requiring polylogarithmic time and work $n^{\max \{\omega \rho, 2\rho + \sigma \} + o(1)}$. We significantly improve, extend, and generalize Cohen's result. First, our parallel algorithm for graphs with small recursive separators has lower work complexity than Cohen's in boraod paramater ranges. Second, we generalize our algorithm to graphs of treewidth at most $n^{\rho}$ ($\rho < 1$) and provide a centralized algorithm that outperforms existing bounds for $S \times V$ reachability on such graphs. We also do this for some other graph familes with small separators. Finally, we extend these results to $(1 + \epsilon)$-approximate distance computation.

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arXiv2024

A Distributed Conductance Tester Without Global Information Collection

We propose a simple and time-optimal algorithm for property testing a graph for its conductance in the CONGEST model. Our algorithm takes only $O(\log n)$ rounds of communication (which is known to be optimal), and consists of simply running multiple random walks of $O(\log n)$ length from a certain number of random sources, at the end of which nodes can decide if the underlying network is a good conductor or far from it. Unlike previous algorithms, no aggregation is required even with a smaller number of walks. Our main technical contribution involves a tight analysis of this process for which we use spectral graph theory. We introduce and leverage the concept of sticky vertices which are vertices in a graph with low conductance such that short random walks originating from these vertices end in a region around them. The present state-of-the-art distributed CONGEST algorithm for the problem by Fichtenberger and Vasudev [MFCS 2018], runs in $O(\log n)$ rounds using three distinct phases : building a rooted spanning tree (\emph{preprocessing}), running $O(n^{100})$ random walks to generate statistics (\emph{Phase~1}), and then convergecasting to the root to make the decision (\emph{Phase~2}). The whole of our algorithm is, however, similar to their Phase~1 running only $O(m^2) = O(n^4)$ walks. Note that aggregation (using spanning trees) is a popular technique but spanning tree(s) are sensitive to node/edge/root failures, hence, we hope our work points to other more distributed, efficient and robust solutions for suitable problems.

cs.DC↗

arXiv2024

Faster Multi-Source Directed Reachability via Shortcuts and Matrix Multiplication

Given an $n$-vertex $m$-edge digraph $G = (V,E)$ and a set $S \subseteq V$, $|S| = n^σ$ (for some $0 < σ\le 1$) of designated sources, the $S \times V$-direachability problem is to compute for every $s \in S$, the set of all the vertices reachable from $s$ in $G$. Known naive algorithms for this problem either run a BFS/DFS separately from every source, and as a result require $O(m \cdot n^σ)$ time, or compute the transitive closure of $G$ in $\tilde O(n^ω)$ time, where $ω< 2.371552\ldots$ is the matrix multiplication exponent. Hence, the current state-of-the-art bound for the problem on graphs with $m = Θ(n^μ)$ edges in $\tilde O(n^{\min \{μ+ σ, ω\}})$. Our first contribution is an algorithm with running time $\tilde O(n^{1 + \tiny{\frac{2}{3}} ω(σ)})$ for this problem, where $ω(σ)$ is the rectangular matrix multiplication exponent. Using current state-of-the-art estimates on $ω(σ)$, our exponent is better than $\min \{2 + σ, ω\}$ for $\tilde σ\le σ\le 0.53$, where $1/3 < \tilde σ< 0.3336$ is a universal constant. Our second contribution is a sequence of algorithms $\mathcal A_0, \mathcal A_1, \mathcal A_2, \ldots$ for the $S \times V$-direachability problem. We argue that under a certain assumption that we introduce, for every $\tilde σ\le σ< 1$, there exists a sufficiently large index $k = k(σ)$ so that $\mathcal A_k$ improves upon the current state-of-the-art bounds for $S \times V$-direachability with $|S| = n^σ$, in the densest regime $μ=2$. We show that to prove this assumption, it is sufficient to devise an algorithm that computes a rectangular max-min matrix product roughly as efficiently as ordinary $(+, \cdot)$ matrix product. Our algorithms heavily exploit recent constructions of directed shortcuts by Kogan and Parter.

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arXiv2022

When you come at the kings you best not miss

A tournament is an orientation of a complete graph. We say that a vertex $x$ in a tournament $\vec T$ controls another vertex $y$ if there exists a directed path of length at most two from $x$ to $y$. A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu (SIAM J. Comput., 2003) in investigating the query complexity of finding a king, that is, the number of arcs in $\vec T$ one has to know in order to surely identify at least one vertex as a king. The aforementioned authors showed that one always has to query at least $Ω(n^{4/3})$ arcs and provided a strategy that queries at most $O(n^{3/2})$. While this upper bound has not yet been improved for the original problem, Biswas et al. (Frontiers in Algorithmics, 2017) proved that with $O(n^{4/3})$ queries one can identify a semi-king, meaning a vertex which controls at least half of all vertices. Our contribution is a novel strategy which improves upon the number of controlled vertices: using $O(n^{4/3} \operatorname{polylog} n)$ queries, we can identify a $(\frac{1}{2}+\frac{2}{17})$-king. To achieve this goal we use a novel structural result for tournaments.

math.CO↗

arXiv2022

$(1+ε)$-Approximate Shortest Paths in Dynamic Streams

Computing approximate shortest paths in the dynamic streaming setting is a fundamental challenge that has been intensively studied during the last decade. Currently existing solutions for this problem either build a sparse multiplicative spanner of the input graph and compute shortest paths in the spanner offline, or compute an exact single source BFS tree. Solutions of the first type are doomed to incur a stretch-space tradeoff of $2κ-1$ versus $n^{1+1/κ}$, for an integer parameter $κ$. (In fact, existing solutions also incur an extra factor of $1+ε$ in the stretch for weighted graphs, and an additional factor of $\log^{O(1)}n$ in the space.) The only existing solution of the second type uses $n^{1/2 - O(1/κ)}$ passes over the stream (for space $O(n^{1+1/κ})$), and applies only to unweighted graphs. In this paper we show that $(1+ε)$-approximate single-source shortest paths can be computed in this setting with $\tilde{O}(n^{1+1/κ})$ space using just \emph{constantly} many passes in unweighted graphs, and polylogarithmically many passes in weighted graphs (assuming $ε$ and $κ$ are constant). Moreover, in fact, the same result applies for multi-source shortest paths, as long as the number of sources is $O(n^{1/κ})$. We achieve these results by devising efficient dynamic streaming constructions of $(1 + ε, β)$-spanners and hopsets. We believe that these constructions are of independent interest.

cs.DS↗

arXiv2016

Energy Optimization of Memory Intensive Parallel workloads

Energy consumption is an important concern in modern multicore processors. The energy consumed during the execution of an application can be minimized by tuning the hardware state utilizing knobs such as frequency, voltage etc. The existing theoretical work on energy mini- mization using Global DVFS (Dynamic Voltage and Frequency Scaling), despite being thorough, ignores the energy consumed by the CPU on memory accesses and the dynamic energy consumed by the idle cores. This article presents an analytical model for the performance and the overall energy consumed by the CPU chip on CPU instructions as well as the memory accesses without ignoring the dynamic energy consumed by the idle cores. We present an analytical framework around our energy-performance model to predict the operating frequencies for global DVFS that minimize the overall CPU energy consumption within a performance budget. Finally, we suggest a scheduling criteria for energy aware scheduling of memory intensive parallel applications.

cs.DC↗

arXiv2014

Fast and Compact Distributed Verification and Self-Stabilization of a DFS Tree

We present algorithms for distributed verification and silent-stabilization of a DFS(Depth First Search) spanning tree of a connected network. Computing and maintaining such a DFS tree is an important task, e.g., for constructing efficient routing schemes. Our algorithm improves upon previous work in various ways. Comparable previous work has space and time complexities of $O(n\log Δ)$ bits per node and $O(nD)$ respectively, where $Δ$ is the highest degree of a node, $n$ is the number of nodes and $D$ is the diameter of the network. In contrast, our algorithm has a space complexity of $O(\log n)$ bits per node, which is optimal for silent-stabilizing spanning trees and runs in $O(n)$ time. In addition, our solution is modular since it utilizes the distributed verification algorithm as an independent subtask of the overall solution. It is possible to use the verification algorithm as a stand alone task or as a subtask in another algorithm. To demonstrate the simplicity of constructing efficient DFS algorithms using the modular approach, We also present a (non-sielnt) self-stabilizing DFS token circulation algorithm for general networks based on our silent-stabilizing DFS tree. The complexities of this token circulation algorithm are comparable to the known ones.

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