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John Kallaugher

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8 papers
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8

FOCS Conference 2022 Conference Paper

Factorial Lower Bounds for (Almost) Random Order Streams

  • Ashish Chiplunkar
  • John Kallaugher
  • Michael Kapralov
  • Eric Price 0001

In this paper we introduce and study the STREAMINGCYCLES problem, a random order streaming version of the Boolean Hidden Hypermatching problem that has been instrumental in streaming lower bounds over the past decade. In this problem the edges of a graph G, comprising n/$\ell$ disjoint length-$\ell$ cycles on n vertices, are partitioned randomly among n players. Every edge is annotated with an independent uniformly random bit, and the players’ task is to output, for some cycle in G, the sum (modulo 2) of the bits on its edges, after one round of sequential communication. Our main result is an $\ell^{\Omega(\ell)}$ lower bound on the communication complexity of STREAMINGCYCLES, which is tight up to constant factors in the exponent. Applications of our lower bound for STREAMINGCYCLES include an essentially tight lower bound for component collection in (almost) random order graph streams, making progress towards a conjecture of Peng and Sohler [SODA’18] and the first exponential space lower bounds for random walk generation.

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FOCS Conference 2022 Conference Paper

The Quantum and Classical Streaming Complexity of Quantum and Classical Max-Cut

  • John Kallaugher
  • Ojas Parekh

We investigate the space complexity of two graph streaming problems: MAX-CUT and its quantum analogue, QUANTUM MAX-CUT. Previous work by Kapralov and Krachun [STOC 19] resolved the classical complexity of the classical problem, showing that any (2 – ε)-approximation requires Ω(n) space (a 2-approximation is trivial with O(log n) space). We generalize both of these qualifiers, demonstrating Ω(n) space lower bounds for (2 – ε)-approximating MAX-CUT and QUANTUM MAX-CUT, even if the algorithm is allowed to maintain a quantum state. As the trivial approximation algorithm for QUANTUM MAX-CUT only gives a 4-approximation, we show tightness with an algorithm that returns a (2 + ε)-approximation to the QUANTUM MAX-CUT value of a graph in O(log n) space. Our work resolves the quantum and classical approximability of quantum and classical Max-Cut using o(n) space. We prove our lower bounds through the techniques of Boolean Fourier analysis. We give the first application of these methods to sequential one-way quantum communication, in which each player receives a quantum message from the previous player, and can then perform arbitrary quantum operations on it before sending it to the next. To this end, we show how Fourier-analytic techniques may be used to understand the application of a quantum channel.

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FOCS Conference 2021 Conference Paper

A Quantum Advantage for a Natural Streaming Problem

  • John Kallaugher

Data streaming, in which a large dataset is received as a “stream” of updates, is an important model in the study of space-bounded computation. Starting with the work of Le Gall [SPAA '06], it has been known that quantum streaming algorithms can use asymptotically less space than their classical counterparts for certain problems. However, so far, all known examples of quantum advantages in streaming are for problems that are either specially constructed for that purpose, or require many streaming passes over the input. We give a one-pass quantum streaming algorithm for one of the best-studied problems in classical graph streaming-the triangle counting problem. Almost-tight parametrized upper and lower bounds are known for this problem in the classical setting; our algorithm uses polynomially less space in certain regions of the parameter space, resolving a question posed by Jain and Nayak in 2014 on achieving quantum advantages for natural streaming problems.

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STOC Conference 2020 Conference Paper

Separations and equivalences between turnstile streaming and linear sketching

  • John Kallaugher
  • Eric Price 0001

A longstanding observation, which was partially proven by Li, Nguyen, and Woodruff in 2014, and extended by Ai, Hu, Li, and Woodruff in 2016, is that any turnstile streaming algorithm can be implemented as a linear sketch (the reverse is trivially true). We study the relationship between turnstile streaming and linear sketching algorithms in more detail, giving both new separations and new equivalences between the two models. It was shown by Li, Nguyen, and Woodruff in 2014 that, if a turnstile algorithm works for arbitrarily long streams with arbitrarily large coordinates at intermediate stages of the stream, then the turnstile algorithm is equivalent to a linear sketch. We show separations of the opposite form: if either the stream length or the maximum value of the stream are substantially restricted, there exist problems where linear sketching is exponentially harder than turnstile streaming. A further limitation of the Li, Nguyen, and Woodruff equivalence is that the turnstile sketching algorithm is neither explicit nor uniform, but requires an exponentially long advice string. We show how to remove this limitation for deterministic streaming algorithms: we give an explicit small-space algorithm that takes the streaming algorithm and computes an equivalent module.

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FOCS Conference 2018 Conference Paper

The Sketching Complexity of Graph and Hypergraph Counting

  • John Kallaugher
  • Michael Kapralov
  • Eric Price 0001

Subgraph counting is a fundamental primitive in graph processing, with applications in social network analysis (e. g. , estimating the clustering coefficient of a graph), database processing and other areas. The space complexity of subgraph counting has been studied extensively in the literature, but many natural settings are still not well understood. In this paper we revisit the subgraph (and hypergraph) counting problem in the sketching model, where the algorithm's state as it processes a stream of updates to the graph is a linear function of the stream. This model has recently received a lot of attention in the literature, and has become a standard model for solving dynamic graph streaming problems. In this paper we give a tight bound on the sketching complexity of counting the number of occurrences of a small subgraph H in a bounded degree graph G presented as a stream of edge updates. Specifically, we show that the space complexity of the problem is governed by the fractional vertex cover number of the graph H. Our subgraph counting algorithm implements a natural vertex sampling approach, with sampling probabilities governed by the vertex cover of H. Our main technical contribution lies in a new set of Fourier analytic tools that we develop to analyze multiplayer communication protocols in the simultaneous communication model, allowing us to prove a tight lower bound. We believe that our techniques are likely to find applications in other settings. Besides giving tight bounds for all graphs H, both our algorithm and lower bounds extend to the hypergraph setting, albeit with some loss in space complexity.

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