About

Virginia Vassilevska Williams is a Cecil H. Green Professor in Electrical Engineering and Computer Science at MIT, with affiliations in the departments of Computer Science and Artificial Intelligence + Decision-making. Her research areas include the theory of computation, with a focus on developing groundbreaking algorithms and systems that address complex computational challenges. She is recognized for her contributions to the analysis and synthesis of systems that interact with the external world through perception, communication, and action, as well as systems that learn, make decisions, and adapt to changing environments. Her work leverages computational, theoretical, and experimental tools to advance understanding in these fields and to develop innovative solutions for shared human challenges.

Research topics

• Mathematics
• Computer Science
• Physics
• Combinatorics
• Algorithm
• Arithmetic
• Pure mathematics
• Chemistry
• Discrete mathematics
• Materials science
• Mathematical analysis
• Quantum mechanics

Selected publications

• New Diameter Approximations via Distance Oracle Techniques

  arXiv (Cornell University) · 2026-04-29

  preprintOpen accessSenior author

  Computing the diameter of a graph is a problem of great interest both in general algorithms research and specifically within fine-grained complexity, where it is a cornerstone hard problem. Recent work has achieved a full conditional lower bound tradeoff curve for both directed and undirected graphs. However, the best known upper bounds do not match the lower bounds. In particular, the best known approximation scheme for undirected graph diameter has not been improved. Moreover, this scheme is randomized and no similar deterministic scheme is known. Another fundamental field of research in shortest paths computation is the construction of approximate distance oracles. Thorup and Zwick [JACM'05] provided the first such distance oracle with constant query time and (conditionally) optimal space, and in the years since many advances have led to a vast toolbox of techniques and data structures. These two areas of research seem natural to combine since they both concern approximating shortest paths. However, the known diameter approximation algorithms only use a small subset of the techniques used in distance oracles research. In this work we show that in fact approximate diameter and distance oracles are intricately connected. We first demonstrate a strong connection between the current best known diameter approximation scheme of Cairo, Grossi and Rizzi ("CGR") and the $(2k-1)$-approximate distance oracle of Thorup and Zwick. This allows us to derandomize the CGR algorithm and obtain the first deterministic diameter approximation tradeoff. We further derandomize other central techniques in the field of distance oracles and use them to achieve new deterministic diameter approximation algorithms. Finally, we show how these new techniques can be used to derandomize many current best known results in various fields of shortest paths approximations.

  PublisherDOI

• New Diameter Approximations via Distance Oracle Techniques

  ArXiv.org · 2026-04-29

  articleOpen accessSenior author

  Computing the diameter of a graph is a problem of great interest both in general algorithms research and specifically within fine-grained complexity, where it is a cornerstone hard problem. Recent work has achieved a full conditional lower bound tradeoff curve for both directed and undirected graphs. However, the best known upper bounds do not match the lower bounds. In particular, the best known approximation scheme for undirected graph diameter has not been improved. Moreover, this scheme is randomized and no similar deterministic scheme is known. Another fundamental field of research in shortest paths computation is the construction of approximate distance oracles. Thorup and Zwick [JACM'05] provided the first such distance oracle with constant query time and (conditionally) optimal space, and in the years since many advances have led to a vast toolbox of techniques and data structures. These two areas of research seem natural to combine since they both concern approximating shortest paths. However, the known diameter approximation algorithms only use a small subset of the techniques used in distance oracles research. In this work we show that in fact approximate diameter and distance oracles are intricately connected. We first demonstrate a strong connection between the current best known diameter approximation scheme of Cairo, Grossi and Rizzi ("CGR") and the $(2k-1)$-approximate distance oracle of Thorup and Zwick. This allows us to derandomize the CGR algorithm and obtain the first deterministic diameter approximation tradeoff. We further derandomize other central techniques in the field of distance oracles and use them to achieve new deterministic diameter approximation algorithms. Finally, we show how these new techniques can be used to derandomize many current best known results in various fields of shortest paths approximations.

  PublisherOA PDF

• Preprocessed 3SUM for Unknown Universes with Subquadratic Space

  Open MIND · 2026-02-11

  preprintSenior author

  We consider the classic 3SUM problem: given sets of integers $A, B, C $, determine whether there is a tuple $(a, b, c) \in A \times B \times C$ satisfying $a + b + c = 0$. The 3SUM Hypothesis, central in fine-grained complexity, states that there does not exist a truly subquadratic time 3SUM algorithm. Given this long-standing barrier, recent work over the past decade has explored 3SUM from a data structural perspective. Specifically, in the 3SUM in preprocessed universes regime, we are tasked with preprocessing sets $A, B$ of size $n$, to create a space-efficient data structure that can quickly answer queries, each of which is a 3SUM problem of the form $A', B', C'$, where $A' \subseteq A$ and $B' \subseteq B$. A series of results have achieved $\tilde{O}(n^2)$ preprocessing time, $\tilde{O}(n^2)$ space, and query time improving progressively from $\tilde{O}(n^{1.9})$ [CL15] to $\tilde{O}(n^{11/6})$ [CVX23] to $\tilde{O}(n^{1.5})$ [KPS25]. Given these series of works improving query time, a natural open question has emerged: can one achieve both truly subquadratic space and truly subquadratic query time for 3SUM in preprocessed universes? We resolve this question affirmatively, presenting a tradeoff curve between query and space complexity. Specifically, we present a simple randomized algorithm achieving $\tilde{O}(n^{1.5 + \varepsilon})$ query time and $\tilde{O}(n^{2 - 2\varepsilon/3})$ space complexity. Furthermore, our algorithm has $\tilde{O}(n^2)$ preprocessing time, matching past work. Notably, quadratic preprocessing is likely necessary for our tradeoff as either the preprocessing or the query time must be at least $n^{2-o(1)}$ under the 3SUM Hypothesis.

  DOI

• Undirected Replacement Paths: Dual Fault Reduces to Single Source

  arXiv (Cornell University) · 2026-05-04

  preprintOpen accessSenior author

  Given a graph and two fixed vertices $s$ and $t$, the Replacement Path Problem (RP) is to compute for every edge $e$, the distance between $s$ and $t$ when $e$ is removed. There are two natural extensions to RP: (1) Single Source Replacement Paths (SSRP): Given a graph $G$ and a source node $s$, compute for every vertex $v$ and every edge $e$ the $s$-$v$ distance in $G \setminus \{e\}$. That is, we do not fix the target anymore. (2) $2$-Fault Replacement Paths (2-FRP): Given a graph $G$ and two nodes $s$ and $t$, compute for every pair of edges $e, e'$ the $s$-$t$ distance in $G \setminus \{e, e'\}$. That is, we consider two failures instead of one. Previously, there was no known reduction between SSRP and 2-FRP. It seemed plausible that 2-FRP would be computationally harder because there are no settings where 2-FRP admits a faster algorithm than SSRP. In directed unweighted graphs there is a provable gap in complexity, and in undirected graphs many of the known 2-FRP algorithms in a variety of settings are much slower than those for SSRP in the same setting. The main contribution of this paper is a tight reduction from undirected $2$-FRP to undirected SSRP, showing that contrary to prior intuition, 2-FRP is not harder than SSRP. As our reduction is weight-preserving, we obtain the first algorithms for $2$-FRP that match the best-known runtimes for SSRP: (1) $\tilde{O}(M n^ω)$ for weights in $[1, M]$ [GVW19], improving upon $O(Mn^{2.87})$ [CZ24]; (2) $n^3/2^{Ω(\sqrt{\log n})}$ for weights in $[1, \text{poly}(n)]$ [GVW19], improving over the previous $n^3\text{polylog}(n)$ running time [VWWX22]; (3) $\tilde{O}(mn^{1/2}+n^{2})$ combinatorial time for unweighted graphs [CC19], and more generally for rational weights in $[1, 2]$ [CM20], improving upon $\tilde{O}(n^{3-1/18})$ [CZ24]. We complement these upper bounds with tight lower bounds under fine-grained hypotheses.

  PublisherDOI

• Preprocessed 3SUM for Unknown Universes with Subquadratic Space

  arXiv (Cornell University) · 2026-02-11

  articleOpen accessSenior author

  We consider the classic 3SUM problem: given sets of integers $A, B, C $, determine whether there is a tuple $(a, b, c) \in A \times B \times C$ satisfying $a + b + c = 0$. The 3SUM Hypothesis, central in fine-grained complexity, states that there does not exist a truly subquadratic time 3SUM algorithm. Given this long-standing barrier, recent work over the past decade has explored 3SUM from a data structural perspective. Specifically, in the 3SUM in preprocessed universes regime, we are tasked with preprocessing sets $A, B$ of size $n$, to create a space-efficient data structure that can quickly answer queries, each of which is a 3SUM problem of the form $A', B', C'$, where $A' \subseteq A$ and $B' \subseteq B$. A series of results have achieved $\tilde{O}(n^2)$ preprocessing time, $\tilde{O}(n^2)$ space, and query time improving progressively from $\tilde{O}(n^{1.9})$ [CL15] to $\tilde{O}(n^{11/6})$ [CVX23] to $\tilde{O}(n^{1.5})$ [KPS25]. Given these series of works improving query time, a natural open question has emerged: can one achieve both truly subquadratic space and truly subquadratic query time for 3SUM in preprocessed universes? We resolve this question affirmatively, presenting a tradeoff curve between query and space complexity. Specifically, we present a simple randomized algorithm achieving $\tilde{O}(n^{1.5 + \varepsilon})$ query time and $\tilde{O}(n^{2 - 2\varepsilon/3})$ space complexity. Furthermore, our algorithm has $\tilde{O}(n^2)$ preprocessing time, matching past work. Notably, quadratic preprocessing is likely necessary for our tradeoff as either the preprocessing or the query time must be at least $n^{2-o(1)}$ under the 3SUM Hypothesis.

  PublisherOA PDF

• Undirected Replacement Paths: Dual Fault Reduces to Single Source

  ArXiv.org · 2026-05-04

  articleOpen accessSenior author

  Given a graph and two fixed vertices $s$ and $t$, the Replacement Path Problem (RP) is to compute for every edge $e$, the distance between $s$ and $t$ when $e$ is removed. There are two natural extensions to RP: (1) Single Source Replacement Paths (SSRP): Given a graph $G$ and a source node $s$, compute for every vertex $v$ and every edge $e$ the $s$-$v$ distance in $G \setminus \{e\}$. That is, we do not fix the target anymore. (2) $2$-Fault Replacement Paths (2-FRP): Given a graph $G$ and two nodes $s$ and $t$, compute for every pair of edges $e, e'$ the $s$-$t$ distance in $G \setminus \{e, e'\}$. That is, we consider two failures instead of one. Previously, there was no known reduction between SSRP and 2-FRP. It seemed plausible that 2-FRP would be computationally harder because there are no settings where 2-FRP admits a faster algorithm than SSRP. In directed unweighted graphs there is a provable gap in complexity, and in undirected graphs many of the known 2-FRP algorithms in a variety of settings are much slower than those for SSRP in the same setting. The main contribution of this paper is a tight reduction from undirected $2$-FRP to undirected SSRP, showing that contrary to prior intuition, 2-FRP is not harder than SSRP. As our reduction is weight-preserving, we obtain the first algorithms for $2$-FRP that match the best-known runtimes for SSRP: (1) $\tilde{O}(M n^ω)$ for weights in $[1, M]$ [GVW19], improving upon $O(Mn^{2.87})$ [CZ24]; (2) $n^3/2^{Ω(\sqrt{\log n})}$ for weights in $[1, \text{poly}(n)]$ [GVW19], improving over the previous $n^3\text{polylog}(n)$ running time [VWWX22]; (3) $\tilde{O}(mn^{1/2}+n^{2})$ combinatorial time for unweighted graphs [CC19], and more generally for rational weights in $[1, 2]$ [CM20], improving upon $\tilde{O}(n^{3-1/18})$ [CZ24]. We complement these upper bounds with tight lower bounds under fine-grained hypotheses.

  PublisherOA PDF

• Improved Additive Approximation Algorithms for APSP

  Society for Industrial and Applied Mathematics eBooks · 2026-01-01

  book-chapterSenior author

  The All-Pairs Shortest Paths (APSP) is a foundational problem in theoretical computer science. Approximating APSP in undirected unweighted graphs has been studied for many years, beginning with the work of Dor, Halperin and Zwick [SICOMP’01]. Many recent works have attempted to improve these original algorithms using the algebraic tools of fast matrix multiplication. We improve on these results for the following problems.

  PublisherDOI

• Output-sensitive approximate counting via a measure-bounded hyperedge oracle, or: How asymmetry helps estimate $k$-clique counts faster

  ArXiv.org · 2025-03-27

  preprintOpen accessSenior author

  Dell, Lapinskas and Meeks [DLM SICOMP 2022] presented a general reduction from approximate counting to decision for a class of fine-grained problems that can be viewed as hyperedge counting or detection problems in an implicit hypergraph, thus obtaining tight equivalences between approximate counting and decision for many key problems such as $k$-clique, $k$-sum and more. Their result is a reduction from approximately counting the number of hyperedges in an implicit $k$-partite hypergraph to a polylogarithmic number of calls to a hyperedge oracle that returns whether a given subhypergraph contains an edge. The main result of this paper is a generalization of the DLM result for {\em output-sensitive} approximate counting, where the running time of the desired counting algorithm is inversely proportional to the number of witnesses. Our theorem is a reduction from approximately counting the (unknown) number of hyperedges in an implicit $k$-partite hypergraph to a polylogarithmic number of calls to a hyperedge oracle called only on subhypergraphs with a small ``measure''. If a subhypergraph has $u_i$ nodes in the $i$th node partition of the $k$-partite hypergraph, then its measure is $\prod_i u_i$. Using the new general reduction and by efficiently implementing measure-bounded colorful independence oracles, we obtain new improved output-sensitive approximate counting algorithms for $k$-clique, $k$-dominating set and $k$-sum. In graphs with $n^t$ $k$-cliques, for instance, our algorithm $(1\pm ε)$-approximates the $k$-clique count in time $$\tilde{O}_ε(n^{ω(\frac{k-t-1}{3},\frac{k-t}{3},\frac{k-t+2}{3}) }+n^2),$$ where $ω(a,b,c)$ is the exponent of $n^a\times n^b$ by $n^b\times n^c$ matrix multiplication. For large $k$ and $t>2$, this is a substantial improvement over prior work, even if $ω=2$.

  PublisherOA PDFDOI

• Average-Case Hardness of Parity Problems: Orthogonal Vectors, k-SUM and More

  ArXiv.org · 2025-03-27

  preprintOpen accessSenior author

  This work establishes conditional lower bounds for average-case {\em parity}-counting versions of the problems $k$-XOR, $k$-SUM, and $k$-OV. The main contribution is a set of self-reductions for the problems, providing the first specific distributions, for which: $\mathsf{parity}\text{-}k\text{-}OV$ is $n^{Ω(\sqrt{k})}$ average-case hard, under the $k$-OV hypothesis (and hence under SETH), $\mathsf{parity}\text{-}k\text{-}SUM$ is $n^{Ω(\sqrt{k})}$ average-case hard, under the $k$-SUM hypothesis, and $\mathsf{parity}\text{-}k\text{-}XOR$ is $n^{Ω(\sqrt{k})}$ average-case hard, under the $k$-XOR hypothesis. Under the very believable hypothesis that at least one of the $k$-OV, $k$-SUM, $k$-XOR or $k$-Clique hypotheses is true, we show that parity-$k$-XOR, parity-$k$-SUM, and parity-$k$-OV all require at least $n^{Ω(k^{1/3})}$ (and sometimes even more) time on average (for specific distributions). To achieve these results, we present a novel and improved framework for worst-case to average-case fine-grained reductions, building on the work of Dalirooyfard, Lincoln, and Vassilevska Williams, FOCS 2020.

  PublisherOA PDFDOI

• Improved girth approximation in weighted undirected graphs

  ArXiv.org · 2025-07-18

  preprintOpen access

  Let $G = (V,E,\ell)$ be a $n$-node $m$-edge weighted undirected graph, where $\ell: E \rightarrow (0,\infty)$ is a real \emph{length} function defined on its edges, and let $g$ denote the girth of $G$, i.e., the length of its shortest cycle. We present an algorithm that, for any input, integer $k \geq 1$, in $O(kn^{1+1/k}\log{n} + m(k+\log{n}))$ expected time finds a cycle of length at most $\frac{4k}{3}g$. This algorithm nearly matches a $O(n^{1+1/k}\log{n})$-time algorithm of \cite{KadriaRSWZ22} which applied to unweighted graphs of girth $3$. For weighted graphs, this result also improves upon the previous state-of-the-art algorithm that in $O((n^{1+1/k}\log n+m)\log (nM))$ time, where $\ell: E \rightarrow [1, M]$ is an integral length function, finds a cycle of length at most $2kg$~\cite{KadriaRSWZ22}. For $k=1$ this result improves upon the result of Roditty and Tov~\cite{RodittyT13}.

  PublisherOA PDFDOI

Recent grants

• AF: Medium: Collaborative Research: Hardness in Polynomial Time

  NSF · $292k · 2015–2017

• BSF:2012338:Shortest Paths: Upper and lower bounds

  NSF · $45k · 2013–2017

• AF: Small: Graphs and structures for distance estimation

  NSF · $300k · 2015–2017

• AF: Small: Average-Case Fine-Grained Complexity

  NSF · $500k · 2019–2022

• AF: Small: Graphs and structures for distance estimation

  NSF · $219k · 2017–2019

Frequent coauthors

• Amir Abboud

  39 shared

• Yinzhan Xu

  Massachusetts Institute of Technology

  35 shared

• Mina Dalirrooyfard

  26 shared

• Nicole Wein

  University of Michigan–Ann Arbor

  26 shared

• Liam Roditty

  Bar-Ilan University

  23 shared

• Andrea Lincoln

  21 shared

• Ryan Williams

  17 shared

• Artūrs Bačkurs

  16 shared

Education

• Ph.D., Computer Science

  Carnegie Mellon University

Awards & honors

• 2023-24 EECS Faculty Award Roundup
• Five MIT faculty members named 2023 Simons Investigators
• Two EECS professors awarded 2022 Faculty Research Innovation…