More Like this

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). 

   more » « less
2. 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). 

   more » « less
3. On the fine-grained complexity of small-size geometric set cover and discrete k-center for small k

   https://doi.org/10.4230/LIPIcs.ICALP.2023.34  

   Chan, Timothy M. ; He, Qizheng ; Yu, Yuancheng ( July 2023 , Proc. 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023))

   Etessami, Kousha ; Feige, Uriel ; Puppis, Gabriele (Ed.)

   We study the time complexity of the discrete k-center problem and related (exact) geometric set cover problems when k or the size of the cover is small. We obtain a plethora of new results: - We give the first subquadratic algorithm for rectilinear discrete 3-center in 2D, running in Õ(n^{3/2}) time. - We prove a lower bound of Ω(n^{4/3-δ}) for rectilinear discrete 3-center in 4D, for any constant δ > 0, under a standard hypothesis about triangle detection in sparse graphs. - Given n points and n weighted axis-aligned unit squares in 2D, we give the first subquadratic algorithm for finding a minimum-weight cover of the points by 3 unit squares, running in Õ(n^{8/5}) time. We also prove a lower bound of Ω(n^{3/2-δ}) for the same problem in 2D, under the well-known APSP Hypothesis. For arbitrary axis-aligned rectangles in 2D, our upper bound is Õ(n^{7/4}). - We prove a lower bound of Ω(n^{2-δ}) for Euclidean discrete 2-center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of Õ(n^ω), if the matrix multiplication exponent ω is equal to 2. - We similarly prove an Ω(n^{k-δ}) lower bound for Euclidean discrete k-center in O(k) dimensions for any constant k ≥ 3, under the Hyperclique Hypothesis. This lower bound again nearly matches known upper bounds if ω = 2. - We also prove an Ω(n^{2-δ}) lower bound for the problem of finding 2 boxes to cover the largest number of points, given n points and n boxes in 12D . This matches the straightforward near-quadratic upper bound. 

   more » « less
4. Adversarially Robust Coloring for Graph Streams

   Chakrabarti, Amit ; Ghosh, Prantar ; Stoeckl, Manuel ( January 2022 , Leibniz international proceedings in informatics)

   A streaming algorithm is considered to be adversarially robust if it provides correct outputs with high probability even when the stream updates are chosen by an adversary who may observe and react to the past outputs of the algorithm. We grow the burgeoning body of work on such algorithms in a new direction by studying robust algorithms for the problem of maintaining a valid vertex coloring of an n-vertex graph given as a stream of edges. Following standard practice, we focus on graphs with maximum degree at most Δ and aim for colorings using a small number f(Δ) of colors. A recent breakthrough (Assadi, Chen, and Khanna; SODA 2019) shows that in the standard, non-robust, streaming setting, (Δ+1)-colorings can be obtained while using only Õ(n) space. Here, we prove that an adversarially robust algorithm running under a similar space bound must spend almost Ω(Δ²) colors and that robust O(Δ)-coloring requires a linear amount of space, namely Ω(nΔ). We in fact obtain a more general lower bound, trading off the space usage against the number of colors used. From a complexity-theoretic standpoint, these lower bounds provide (i) the first significant separation between adversarially robust algorithms and ordinary randomized algorithms for a natural problem on insertion-only streams and (ii) the first significant separation between randomized and deterministic coloring algorithms for graph streams, since deterministic streaming algorithms are automatically robust. We complement our lower bounds with a suite of positive results, giving adversarially robust coloring algorithms using sublinear space. In particular, we can maintain an O(Δ²)-coloring using Õ(n √Δ) space and an O(Δ³)-coloring using Õ(n) space. 

   more » « less
5. Online Spanners in Metric Spaces

   Bhore, Sujoy ; Filtser, Arnold ; Khodabandeh, Hadi ; Toth, Csaba D. ( September 2022 , Proc. 30th Annual European Symposium on Algorithms (ESA))

   Given a metric space ℳ = (X,δ), a weighted graph G over X is a metric t-spanner of ℳ if for every u,v ∈ X, δ(u,v) ≤ δ_G(u,v) ≤ t⋅ δ(u,v), where δ_G is the shortest path metric in G. In this paper, we construct spanners for finite sets in metric spaces in the online setting. Here, we are given a sequence of points (s₁, …, s_n), where the points are presented one at a time (i.e., after i steps, we have seen S_i = {s₁, … , s_i}). The algorithm is allowed to add edges to the spanner when a new point arrives, however, it is not allowed to remove any edge from the spanner. The goal is to maintain a t-spanner G_i for S_i for all i, while minimizing the number of edges, and their total weight. Under the L₂-norm in ℝ^d for arbitrary constant d ∈ ℕ, we present an online (1+ε)-spanner algorithm with competitive ratio O_d(ε^{-d} log n), improving the previous bound of O_d(ε^{-(d+1)}log n). Moreover, the spanner maintained by the algorithm has O_d(ε^{1-d}log ε^{-1})⋅ n edges, almost matching the (offline) optimal bound of O_d(ε^{1-d})⋅ n. In the plane, a tighter analysis of the same algorithm provides an almost quadratic improvement of the competitive ratio to O(ε^{-3/2}logε^{-1}log n), by comparing the online spanner with an instance-optimal spanner directly, bypassing the comparison to an MST (i.e., lightness). As a counterpart, we design a sequence of points that yields a Ω_d(ε^{-d}) lower bound for the competitive ratio for online (1+ε)-spanner algorithms in ℝ^d under the L₁-norm. Then we turn our attention to online spanners in general metrics. Note that, it is not possible to obtain a spanner with stretch less than 3 with a subquadratic number of edges, even in the offline setting, for general metrics. We analyze an online version of the celebrated greedy spanner algorithm, dubbed ordered greedy. With stretch factor t = (2k-1)(1+ε) for k ≥ 2 and ε ∈ (0,1), we show that it maintains a spanner with O(ε^{-1}logε^{-1})⋅ n^{1+1/k} edges and O(ε^{-1}n^{1/k}log² n) lightness for a sequence of n points in a metric space. We show that these bounds cannot be significantly improved, by introducing an instance that achieves an Ω(1/k⋅ n^{1/k}) competitive ratio on both sparsity and lightness. Furthermore, we establish the trade-off among stretch, number of edges and lightness for points in ultrametrics, showing that one can maintain a (2+ε)-spanner for ultrametrics with O(ε^{-1}logε^{-1})⋅ n edges and O(ε^{-2}) lightness. 

   more » « less