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1. Estimation of the Size of Union of Delphic Sets: Achieving Independence from Stream Size

   https://doi.org/10.1145/3517804.3526222  

   Meel, Kuldeep S.; Chakraborty, Sourav; Vinodchandran, N. V. (June 2022, PODS '22: Proceedings of the 41st ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems)

   Given a family of sets (S1, S2,... SM) over a universe Ω, estimating the size of their union in the data streaming model is a fundamental computational problem with a wide variety of applications. The holy grail in the field of streaming is to seek design of algorithms that achieve (ε, δ)-approximation with poly(log |Ω|, ε-1, log δ-1) space and update time complexity. Earlier investigations achieve algorithms with desired space and update time complexity for restricted cases such as singletons (Distinct Elements problem), one-dimensional ranges, arithmetic progressions, and sub-cubes. However, techniques used in these works fail for many other simple structured sets. A prominent example is that of Klee's Measure Problem (KMP), wherein every set Si is represented by an axis-parallel rectangle in d-dimensional spaces. Despite extensive prior work, the best-known streaming algorithms for many of these cases depend on the size of the stream, and therefore the problem of whether there exists a streaming algorithm for estimations of size of the union of sets with poly(log |Ω|, ε-1, log δ-1) space and update time complexity has remained open. In this work, we focus on certain general families of sets called Delphic families (which allows efficient membership, sampling, and cardinality queries). Such families of sets capture several well-known problems, including KMP, test coverage, and hypervolume estimation. The primary contribution of our work is to resolve the above-mentioned open problem for streams over Delphic families. In particular, we design the first streaming algorithm for estimating |⋃i=1M Si| with poly(log |Ω|, ε-1, log δ-1) space and update time complexity (independent of M, the length of the stream) when each Si is a member from a Delphic family of sets. We further generalize our results to larger families of sets, called approximate-Delphic families, for which the size of a set can be known approximately but not exactly. Our results resolve two of the open problems listed in Meel, Vinodchandran, Chakraborty (PODS-21). 

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2. Space Complexity of Minimum Cut Problems in Single-Pass Streams

   https://doi.org/10.4230/LIPIcs.ITCS.2025.43  

   Ding, Matthew; Garces, Alexandro; Li, Jason; Lin, Honghao; Nelson, Jelani; Shah, Vihan; Woodruff, David P (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   We consider the problem of finding a minimum cut of a weighted graph presented as a single-pass stream. While graph sparsification in streams has been intensively studied, the specific application of finding minimum cuts in streams is less well-studied. To this end, we show upper and lower bounds on minimum cut problems in insertion-only streams for a variety of settings, including for both randomized and deterministic algorithms, for both arbitrary and random order streams, and for both approximate and exact algorithms. One of our main results is an Õ(n/ε) space algorithm with fast update time for approximating a spectral cut query with high probability on a stream given in an arbitrary order. Our result breaks the Ω(n/ε²) space lower bound required of a sparsifier that approximates all cuts simultaneously. Using this result, we provide streaming algorithms with near optimal space of Õ(n/ε) for minimum cut and approximate all-pairs effective resistances, with matching space lower-bounds. The amortized update time of our algorithms is Õ(1), provided that the number of edges in the input graph is at least (n/ε²)^{1+o(1)}. We also give a generic way of incorporating sketching into a recursive contraction algorithm to improve the post-processing time of our algorithms. In addition to these results, we give a random-order streaming algorithm that computes the exact minimum cut on a simple, unweighted graph using Õ(n) space. Finally, we give an Ω(n/ε²) space lower bound for deterministic minimum cut algorithms which matches the best-known upper bound up to polylogarithmic factors. 

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3. Testing Intersecting and Union-Closed Families

   https://doi.org/10.4230/LIPICS.ITCS.2024.33  

   Chen, Xi; De, Anindya; Li, Yuhao; Nadimpalli, Shivam; Servedio, Rocco A (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Guruswami, Venkatesan (Ed.)

   Inspired by the classic problem of Boolean function monotonicity testing, we investigate the testability of other well-studied properties of combinatorial finite set systems, specifically intersecting families and union-closed families. A function f: {0,1}ⁿ → {0,1} is intersecting (respectively, union-closed) if its set of satisfying assignments corresponds to an intersecting family (respectively, a union-closed family) of subsets of [n]. Our main results are that - in sharp contrast with the property of being a monotone set system - the property of being an intersecting set system, and the property of being a union-closed set system, both turn out to be information-theoretically difficult to test. We show that: - For ε ≥ Ω(1/√n), any non-adaptive two-sided ε-tester for intersectingness must make 2^{Ω(n^{1/4}/√{ε})} queries. We also give a 2^{Ω(√{n log(1/ε)})}-query lower bound for non-adaptive one-sided ε-testers for intersectingness. - For ε ≥ 1/2^{Ω(n^{0.49})}, any non-adaptive two-sided ε-tester for union-closedness must make n^{Ω(log(1/ε))} queries. Thus, neither intersectingness nor union-closedness shares the poly(n,1/ε)-query non-adaptive testability that is enjoyed by monotonicity. To complement our lower bounds, we also give a simple poly(n^{√{nlog(1/ε)}},1/ε)-query, one-sided, non-adaptive algorithm for ε-testing each of these properties (intersectingness and union-closedness). We thus achieve nearly tight upper and lower bounds for two-sided testing of intersectingness when ε = Θ(1/√n), and for one-sided testing of intersectingness when ε = Θ(1). 

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4. Streaming Algorithms with Few State Changes

   https://doi.org/10.1145/3651145  

   Jayaram, Rajesh; Woodruff, David P; Zhou, Samson (May 2024, Proceedings of the ACM on Management of Data)

   In this paper, we study streaming algorithms that minimize the number of changes made to their internal state (i.e., memory contents). While the design of streaming algorithms typically focuses on minimizing space and update time, these metrics fail to capture the asymmetric costs, inherent in modern hardware and database systems, of reading versus writing to memory. In fact, most streaming algorithms write to their memory on every update, which is undesirable when writing is significantly more expensive than reading. This raises the question of whether streaming algorithms with small space and number of memory writes are possible. We first demonstrate that, for the fundamental F_pmoment estimation problem with p ≥ 1, any streaming algorithm that achieves a constant factor approximation must make Ω(n^1-1/p) internal state changes, regardless of how much space it uses. Perhaps surprisingly, we show that this lower bound can be matched by an algorithm which also has near-optimal space complexity. Specifically, we give a (1+ε)-approximation algorithm for F_pmoment estimation that use a near-optimal ~O_ε(n^1-1/p) number of state changes, while simultaneously achieving near-optimal space, i.e., for p∈[1,2), our algorithm uses poly(log n,1/ε) bits of space for, while for p>2, the algorithm uses ~O_ε(n^1-1/p) space. We similarly design streaming algorithms that are simultaneously near-optimal in both space complexity and the number of state changes for the heavy-hitters problem, sparse support recovery, and entropy estimation. Our results demonstrate that an optimal number of state changes can be achieved without sacrificing space complexity. 

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5. Matchings in Low-Arboricity Graphs in the Dynamic Graph Stream Model

   https://doi.org/10.4230/LIPIcs.FSTTCS.2024.29  

   Konrad, Christian; McGregor, Andrew; Sengupta, Rik; Than, Cuong (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Barman, Siddharth; Lasota, Sławomir (Ed.)

   We consider the problem of estimating the size of a maximum matching in low-arboricity graphs in the dynamic graph stream model. In this setting, an algorithm with limited memory makes multiple passes over a stream of edge insertions and deletions, resulting in a low-arboricity graph. Let n be the number of vertices of the input graph, and α be its arboricity. We give the following results. 1) As our main result, we give a three-pass streaming algorithm that produces an (α + 2)(1 + ε)-approximation and uses space O(ε^{-2}⋅α²⋅n^{1/2}⋅log n). This result should be contrasted with the Ω(α^{-5/2}⋅n^{1/2}) space lower bound established by [Assadi et al., SODA'17] for one-pass algorithms, showing that, for graphs of constant arboricity, the one-pass space lower bound can be achieved in three passes (up to poly-logarithmic factors). Furthermore, we obtain a two-pass algorithm that uses space O(ε^{-2}⋅α²⋅n^{3/5}⋅log n). 2) We also give a (1+ε)-approximation multi-pass algorithm, where the space used is parameterized by an upper bound on the size of a largest matching. For example, using O(log log n) passes, the space required is O(ε^{-1}⋅α²⋅k⋅log n), where k denotes an upper bound on the size of a largest matching. Finally, we define a notion of arboricity in the context of matrices. This is a natural measure of the sparsity of a matrix that is more nuanced than simply bounding the total number of nonzero entries, but less restrictive than bounding the number of nonzero entries in each row and column. For such matrices, we exploit our results on estimating matching size to present upper bounds for the problem of rank estimation in the dynamic data stream model. 

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