Top-𝑘-convolution and the quest for near-linear output-sensitive subset sum

Bringmann, Karl; Nakos, Vasileios

doi:10.1145/3357713.3384308

Cited by 16 publications

(11 citation statements)

References 45 publications

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“…Our -notation hides the same number of logfactors and comparable constants in Theorems 1.1 and 1.2. We also remark that the recent trend of additive-combinatorics-based algorithm design typically leads to improved, but nasty running times [17,13,37], so our clean running time of O(n) is an exception.…”

Section: Algorithmmentioning

confidence: 94%

On Near-Linear-Time Algorithms for Dense Subset Sum

Bringmann¹,

Wellnitz²

2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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In the Subset Sum problem we are given a set of n positive integers X and a target t and are asked whether some subset of X sums to t. Natural parameters for this problem that have been studied in the literature are n and t as well as the maximum input number mxX and the sum of all input numbers ΣX . In this paper we study the dense case of Subset Sum, where all these parameters are polynomial in n. In this regime, standard pseudo-polynomial algorithms solve Subset Sum in polynomial time n O(1) .Our main question is: When can dense Subset Sum be solved in near-linear time O(n)? We provide an essentially complete dichotomy by designing improved algorithms and proving conditional lower bounds, thereby determining essentially all settings of the parameters n, t, mxX , ΣX for which dense Subset Sum is in time O(n). For notational convenience we assume without loss of generality that t ≥ mxX (as larger numbers can be ignored) and t ≤ ΣX /2 (using symmetry). Then our dichotomy reads as follows:By reviving and improving an additive-combinatorics-based approach by Galil and Margalit [SICOMP'91], we show that Subset Sum is in near-linear time O(n) if t mxX ΣX /n 2 . We prove a matching conditional lower bound: If Subset Sum is in near-linear time for any setting with t mxX ΣX /n 2 , then the Strong Exponential Time Hypothesis and the Strong k-Sum Hypothesis fail. We also generalize our algorithm from sets to multi-sets, albeit with non-matching upper and lower bounds.

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Section: Algorithmmentioning

confidence: 94%

On Near-Linear-Time Algorithms for Dense Subset Sum

Bringmann¹,

Wellnitz²

2021

Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA)

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“…Jin and Wu proposed a simpler randomised algorithm [14] achieving the same bounds as [4], which however seems to only solve the decision version of the problem. Recently, Bringmann and Nakos [5] have presented an O(|S t (Z)| 4/3 poly(log t)) algorithm, where S t (Z) is the set of all subset sums of the input set Z that are smaller than t, based on top-k convolution.…”

Section: Related Workmentioning

confidence: 99%

Faster Algorithms for $k$-Subset Sum and Variations

Antonopoulos¹,

Pagourtzis²,

Petsalakis³

et al. 2021

Preprint

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We present new, faster pseudopolynomial time algorithms for the k-Subset Sum problem, defined as follows: given a set Z of n positive integers and k targets t1, . . . , t k , determine whether there exist k disjoint subsets Z1, . . . , Z k ⊆ Z, such that Σ(Zi) = ti, for i = 1, . . . , k. Assuming t = max{t1, . . . , t k } is the maximum among the given targets, a standard dynamic programming approach based on Bellman's algorithm [3] can solve the problem in O(nt k ) time. We build upon recent advances on Subset Sum due to Koiliaris and Xu [17] and Bringmann [4] in order to provide faster algorithms for k-Subset Sum. We devise two algorithms: a deterministic one of time complexity Õ(n k/(k+1) t k ) and a randomised one of Õ(n + t k ) complexity. Additionally, we show how these algorithms can be modified in order to incorporate cardinality constraints enforced on the solution subsets. We further demonstrate how these algorithms can be used in order to cope with variations of k-Subset Sum, namely Subset Sum Ratio, k-Subset Sum Ratio and Multiple Subset Sum.

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“…Output-Sensitive Subset Sum: Given a set X of integers and a threshold τ , compute the set S of all numbers less than τ which can be expressed as a subset sum of X. The bestknown randomized algorithm runs in time O(|S| 4/3 ) [12], and it can be derandomized in same running time. N -fold Boolean Convolution: Given N Boolean vectors A 1 , .…”

Section: Theorem 1 (Deterministicmentioning

confidence: 99%

“…It has been a vital component in fields like signal processing, deep learning (convolutional neural networks) and computer vision. Inside traditional algorithm design it is crucially used as a subroutine in k-SUM [15], Subset Sum [9,30,12,14] and various string problems [21,27,18], to name a few.…”

Section: Introductionmentioning

confidence: 99%

“…The need for such a primitive appears in many situations, e.g. [18,25,15,12,13], as one may often be interested in algorithms that run in time proportional to the actual complexity of the output rather the space the output points live in. These type of problems have been investigated by different communities, including fine-grained complexity [15] and string algorithms [18], computer algebra [38] and compressed sensing [26,22], and they are very closely related to the famous sparse recovery problem, see e.g.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Deterministic and Las Vegas Algorithms for Sparse Nonnegative Convolution

Bringmann¹,

Fischer²,

Nakos³

2021

Preprint

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Computing the convolution A B of two length-n integer vectors A, B is a core problem in several disciplines. It frequently comes up as a subroutine in various problem domains, e.g. in algorithms for Knapsack, k-SUM, All-Pairs Shortest Paths, and string pattern matching problems. For these applications it typically suffices to compute convolutions of nonnegative vectors. This problem can be classically solved in time O(n log n) using the Fast Fourier Transform.However, in many applications the involved vectors are sparse and hence one could hope for output-sensitive algorithms to compute nonnegative convolutions. This question was raised by Muthukrishnan and solved by Cole and Hariharan (STOC '02) by a randomized algorithm running in near-linear time in the (unknown) output-size t. Chan and Lewenstein (STOC '15) presented a deterministic algorithm with a 2 O( √ log t•log log n) overhead in running time and the additional assumption that a small superset of the output is given; this assumption was later removed by Bringmann and Nakos (ICALP '21).In this paper we present the first deterministic near-linear-time algorithm for computing sparse nonnegative convolutions. This immediately gives improved deterministic algorithms for the stateof-the-art of output-sensitive Subset Sum, block-mass pattern matching, N -fold Boolean convolution, and others, matching up to log-factors the fastest known randomized algorithms for these problems. Our algorithm is a blend of algebraic and combinatorial ideas and techniques.Additionally, we provide two fast Las Vegas algorithms for computing sparse nonnegative convolutions. In particular, we present a simple O(t log 2 t) time algorithm, which is an accessible alternative to Cole and Hariharan's algorithm. Subsequently, we further refine this new algorithm to run in Las Vegas time O(t log t • log log t), which matches the running time of the dense case apart from the log log t factor.

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Top-𝑘-convolution and the quest for near-linear output-sensitive subset sum

Cited by 16 publications

References 45 publications

On Near-Linear-Time Algorithms for Dense Subset Sum

On Near-Linear-Time Algorithms for Dense Subset Sum

Faster Algorithms for $k$-Subset Sum and Variations

Deterministic and Las Vegas Algorithms for Sparse Nonnegative Convolution

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