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Deterministic streaming algorithms for non-monotone submodular maximization

FRONTIERS OF COMPUTER SCIENCE 2025年第6期19卷 196404-196404页

作者： Sun, Xiaoming Zhang, Jialin Zhang, Shuo Chinese Acad Sci Inst Comp Technol State Key Lab Processors Beijing 100190 Peoples R China Univ Chinese Acad Sci Sch Comp Sci & Technol Beijing 100049 Peoples R China

Submodular maximization is a significant area of interest in combinatorial optimization. It has various real-world applications. In recent years, streaming algorithms for submodular maximization have gained attention, allowing realtime processing of large data sets by examining each piece of data only once. However, most of the current state-of-the-art algorithms are only applicable to monotone submodular maximization. There are still significant gaps in the approximation ratios between monotone and non-monotone objective functions. In this paper, we propose a streaming algorithm framework for non-monotone submodular maximization and use this framework to design deterministic streaming algorithms for the d-knapsack constraint and the knapsack constraint. Our 1-pass streaming algorithm for the d-knapsack constraint has a 1/4(d+1)-& varepsilon;approximation ratio, using O((B) over tilde log (B) over tilde/& varepsilon;) memory, and O(log (B) over tilde & varepsilon;) query time per element, where (B) over tilde = min(n,b) is the maximum number of elements that the knapsack can store. As a special case of the d-knapsack constraint, we have the 1-pass streaming algorithm with a 1/8 - & varepsilon;approximation ratio to the knapsack constraint. To our knowledge, there is currently no streaming algorithm for this constraint when the objective function is non-monotone, even when d = 1. In addition, we propose a multi-pass streaming algorithm with 1/6 - & varepsilon;approximation, which stores O((B) over tilde )elements.

关键词： submodular maximization streaming algorithms cardinality constraint knapsack constraint

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streaming algorithms for Estimating High Set Similarities in LogLog Space

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IEEE TRANSACTIONS ON KNOWLEDGE AND DATA ENGINEERING 2021年第10期33卷 3438-3452页

作者： Qi, Yiyan Wang, Pinghui Zhang, Yuanming Zhai, Qiaozhu Wang, Chenxu Tian, Guangjian Lui, John C. S. Guan, Xiaohong Xi An Jiao Tong Univ MOE Key Lab Intelligent Networks & Network Secur POB 108828 Xianning West Rd Xian 710049 Shaanxi Peoples R China Xi An Jiao Tong Univ Shenzhen Res Inst Shenzhen Peoples R China Tsinghua Univ Ctr Intelligent & Networked Syst Tsinghua Natl Lab Informat Sci & Technol Beijing 100084 Peoples R China Xi An Jiao Tong Univ Sch Software Engn Xian 710049 Shaanxi Peoples R China Huawei Noahs Ark Lab Hong Kong Peoples R China Chinese Univ Hong Kong Hong Kong Peoples R China

Estimating set similarity and detecting highly similar sets are fundamental problems in areas such as databases and machine learning. MinHash is a well-known technique for approximating Jaccard similarity of sets and has been successfully used for many applications. Its two compressed versions, b-bit MinHash and Odd Sketch, can significantly reduce the memory usage of the MinHash, especially for estimating high similarities (i.e., similarities around 1). Although MinHash can be applied to static sets as well as streaming sets, of which elements are given in a streaming fashion, unfortunately, b-bit MinHash and Odd Sketch fail to deal with streaming data. To solve this problem, we previously designed a memory-efficient sketch method, MaxLogHash, to accurately estimate Jaccard similarities in streaming sets. Compared with MinHash, our method uses smaller sized registers (each register consists of less than 7 bits) to build a compact sketch for each set. In this paper, we further develop a faster method, MaxLogOPH++. Compared with MaxLogHash, MaxLogOPH++ reduces the time complexity for updating each coming element from O(k) to O(1) with a small additional memory. We conduct experiments on a variety of datasets, and experimental results demonstrate the efficiency and effectiveness of our methods.

关键词： streaming algorithms sketch jaccard similarity

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streaming algorithms for Independent Sets in Sparse Hypergraphs

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ALGORITHMICA 2016年第2期76卷 490-501页

作者： Halldorsson, Bjarni V. Halldorsson, Magnus M. Losievskaja, Elena Szegedy, Mario Reykjavik Univ Sch Sci & Engn ICE TCS IS-101 Reykjavik Iceland deCODE Genet IS-101 Reykjavik Iceland Reykjavik Univ Sch Comp Sci ICE TCS IS-101 Reykjavik Iceland Rutgers State Univ Dept Comp Sci 110 Frelinghuysen Rd Piscataway NJ USA

We give the first treatment of the classic independent set problem in graphs and hypergraphs in the streaming setting. The objective is to find space-efficient algorithms that output independent sets that are "combinatorially optimal", that is, with size guarantee in terms of the degree sequence alone. Our main result is a randomized algorithm that achieves this using space in bits that is linear in the number of vertices. We use this to examine assumptions about the streaming model, and advocate the study of output-efficient algorithms that measure space usage relative to the size of the output solution. In that sense, our main algorithm uses space linear in the output size. We also examine algorithms that use little or no space in addition to the bits storing the output. Our algorithms fall also into an online streaming model, where output-changes can go only in one direction. In particular a feasible solution must be maintained at all times, and items that are removed from the solution can never reenter. We obtain tight bounds on deterministic algorithms for independent sets in graphs in that model.

关键词： streaming algorithms Independent sets Turan bound Online streaming

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streaming algorithms for Bin Packing and Vector Scheduling

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THEORY OF COMPUTING SYSTEMS 2021年第6期65卷 916-942页

作者： Cormode, Graham Vesely, Pavel Univ Warwick Dept Comp Sci Coventry W Midlands England

Problems involving the efficient arrangement of simple objects, as captured by bin packing and makespan scheduling, are fundamental tasks in combinatorial optimization. These are well understood in the traditional online and offline cases, but have been less well-studied when the volume of the input is truly massive, and cannot even be read into memory. This is captured by the streaming model of computation, where the aim is to approximate the cost of the solution in one pass over the data, using small space. As a result, streaming algorithms produce concise input summaries that approximately preserve the optimum value. We design the first efficient streaming algorithms for these fundamental problems in combinatorial optimization. For Bin Packing, we provide a streaming asymptotic (1 + epsilon)-approximation with (O) over tilde (1/epsilon), where (O) over tilde hides logarithmic factors. Moreover, such a space bound is essentially optimal. Our algorithm implies a streaming (d + epsilon)-approximation for VECTOR BIN PACKING in d dimensions, running in space (O) over tilde (d/epsilon). For the related VECTOR SCHEDULING problem, we show how to construct an input summary in space (O) over tilde (d(2) . m/epsilon(2)) that preserves the optimum value up to a factor of 2 - 1/m + epsilon, where m is the number of identical machines.

关键词： streaming algorithms Bin packing Scheduling

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streaming algorithms for Subspace Analysis: Comparative Review and Implementation on IoT Devices

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IEEE INTERNET OF THINGS JOURNAL 2023年第14期10卷 12798-12810页

作者： Marchioni, Alex Prono, Luciano Mangia, Mauro Pareschi, Fabio Rovatti, Riccardo Setti, Gianluca Univ Bologna Dept Elect Elect & Informat Engn I-40136 Bologna Italy Univ Bologna Adv Res Ctr Elect Syst I-40125 Bologna Italy Politecn Torino Dept Elect & Telecommun I-10129 Turin Italy Univ Bologna Adv Res Ctr Elect Syst Bologna Italy King Abdullah Univ Sci & Technol CEMSE Thuwal 23955 Saudi Arabia

Subspace analysis (SA) is a widely used technique for coping with high-dimensional data and is becoming a fundamental step in the early treatment of many signal-processing tasks. However, traditional SA often requires a large amount of memory and computational resources, as it is equivalent to eigenspace determination. To address this issue, specialized streaming algorithms have been developed, allowing SA to be run on low-power devices, such as sensors or edge devices. Here, we present a classification and a comparison of these methods by providing a consistent description and highlighting their features and similarities. We also evaluate their performance in the task of subspace identification with a focus on computational complexity and memory footprint for different signal dimensions. Additionally, we test the implementation of these algorithms on common hardware platforms typically employed for sensors and edge devices.

关键词： Index Terms-Edge computing minor component analysis minor subspace (MSA) principal components analysis principal subspace (PSA) streaming algorithms

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streaming algorithms for monotone non-submodular function maximization under a knapsack constraint on the integer lattice

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THEORETICAL COMPUTER SCIENCE 2022年 937卷 39-49页

作者： Tan, Jingjing Wang, Fengmin Ye, Weina Zhang, Xiaoqing Zhou, Yang Weifang Univ Sch Math & Informat Sci Weifang 261061 Peoples R China Beijing Jinghang Res Inst Comp & Commun Beijing 100074 Peoples R China Beijing Univ Technol Beijing Inst Sci & Engn Comp Beijing 100124 Peoples R China Shandong Normal Univ Sch Math & Stat Jinan 250014 Peoples R China

The study of non-submodular maximization on the integer lattice is an important extension of submodular optimization. In this paper, streaming algorithms for maximizing non -negative monotone non-submodular functions with knapsack constraint on integer lattice are considered. We first design a two-pass streamingKnapsack algorithm combining with BinarySearch as a subroutine for this problem. By introducing the DR ratio gamma and the weak DR ratio gamma w of the non-submodular objective function, we obtain that the approximation ratio is min{gamma 2(1 - epsilon)/2 gamma +1, 1 - 1/gamma w2 gamma - epsilon}, the total memory complexity is O (K log K /epsilon), and the total query complexity for each element is O (log K log(K/epsilon 2)/epsilon). Then, we design a one-pass streaming algorithm by dynamically updating the maximal function value among unit vectors along with the currently arriving element. Finally, in order to decrease the memory complexity, we design an improved streamingKnapsack algorithm and reduce the memory complexity to O (K /epsilon 2).(c) 2022 Elsevier B.V. All rights reserved.

关键词： Integer lattice Non-submodular Knapsack constraint streaming algorithms

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streaming algorithms for Extent Problems in High Dimensions

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ALGORITHMICA 2015年第1期72卷 83-98页

作者： Agarwal, Pankaj K. Sharathkumar, R. Duke Univ Dept Comp Sci Durham NC 27708 USA Stanford Univ Dept Comp Sci Stanford CA 94305 USA

We present (single-pass) streaming algorithms for maintaining extent measures of a stream S of n points in . We focus on designing streaming algorithms whose working space is polynomial in d (poly(d)) and sub-linear in n. For the problems of computing diameter, width and minimum enclosing ball of S, we obtain lower bounds on the worst-case approximation ratio of any streaming algorithm that uses poly(d) space. On the positive side, we introduce the notion of blurred ball cover and use it for answering approximate farthest-point queries and maintaining approximate minimum enclosing ball and diameter of S. We describe a streaming algorithm for maintaining a blurred ball cover whose working space is linear in d and independent of n.

关键词： streaming algorithms Computational geometry Extent problems Minimum enclosing ball Diameter

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streaming algorithms for Geometric Steiner Forest

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ACM TRANSACTIONS ON algorithms 2024年第4期20卷 1-38页

作者： Czumaj, Artur Jiang, Shaofeng H. -C Krauthgamer, Robert Vesely, Pavel Univ Warwick Coventry England Peking Univ Beijing Peoples R China Weizmann Inst Sci Rehovot Israel Charles Univ Prague Prague Czech Republic

We consider a generalization of the Steiner tree problem, the Steiner forest problem, in the Euclidean plane: the input is a multiset X subset of R2, partitioned into k color classes C1,. .. , C k subset of X . The goal is to find a minimum-cost Euclidean graph G such that every color class C is connected in G . We study this Steiner forest problem in the streaming setting, where the stream consists of insertions and deletions of points to X . Each input point x is an element of X arrives with its color color(x) is an element of [k], and as usual for dynamic geometric streams, the input is restricted to the discrete grid {1, ... , Delta }2. We design a single-pass streaming algorithm that uses poly(k log Delta) space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio a 2 (currently 1.1547 <= a 2 <= 1.214). This approximation guarantee matches the state-of-the-art bound for streaming Steiner tree, i.e., when k = 1, and it is a major open question to improve the ratio to 1 + & even for this special case. Our approach relies on a novel combination of streaming techniques, like sampling and linear sketching, with the classical Arora-style dynamic-programming framework for geometric optimization problems, which usually requires large memory and so far has not been applied in the streaming setting. We complement our streaming algorithm for the Steiner forest problem with simple arguments showing that any finite multiplicative approximation requires Omega (k) bits of space.

关键词： Steiner forest Steiner tree streaming algorithms

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streaming algorithms for Line Simplification

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DISCRETE & COMPUTATIONAL GEOMETRY 2010年第3期43卷 497-515页

作者： Abam, M. A. de Berg, M. Hachenberger, P. Zarei, A. Aarhus Univ Dept Comp Sci MADALGO DK-8000 Aarhus Denmark TU Eindhoven Dept Comp Sci NL-5600 MB Eindhoven Netherlands Sharif Univ Technol Dept Comp Engn Tehran Iran IPM Sch Comp Sci Tehran Iran

We study the following variant of the well-known line-simplification problem: we are getting a (possibly infinite) sequence of points p (0),p (1),p (2),aEuro broken vertical bar in the plane defining a polygonal path, and as we receive the points, we wish to maintain a simplification of the path seen so far. We study this problem in a streaming setting, where we only have a limited amount of storage, so that we cannot store all the points. We analyze the competitive ratio of our algorithms, allowing resource augmentation: we let our algorithm maintain a simplification with 2k (internal) points and compare the error of our simplification to the error of the optimal simplification with k points. We obtain the algorithms with O(1) competitive ratio for three cases: convex paths, where the error is measured using the Hausdorff distance (or Fr,chet distance), xy-monotone paths, where the error is measured using the Hausdorff distance (or Fr,chet distance), and general paths, where the error is measured using the Fr,chet distance. In the first case the algorithm needs O(k) additional storage, and in the latter two cases the algorithm needs O(k (2)) additional storage.

关键词： Line simplification streaming algorithms

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streaming algorithms for language recognition problems

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THEORETICAL COMPUTER SCIENCE 2013年 494卷 13-23页

作者： Babu, Ajesh Limaye, Nutan Radhakrishnan, Jaikumar Varma, Girish Yahoo Labs Bangalore Karnataka India Indian Inst Technol Dept Comp Sci & Engn Mumbai 400076 Maharashtra India Tata Inst Fundamental Res Sch Technol & Comp Sci Bombay 400005 Maharashtra India

We study the complexity of the following problems in the streaming model. Membership testing for DLIN. We show that every language in DLIN can be recognized by a randomized one-pass O(log n) space algorithm with an inverse polynomial one-sided error and by a deterministic p-pass O(n/p) space algorithm. We show that these algorithms are optimal. Membership testing for LL (k). For languages generated by LL (k) grammars with a bound of r on the number of nonterminals at any stage in the left-most derivation, we show that membership can be tested by a randomized one-pass O(r log n) space algorithm with an inverse polynomial (in n) one-sided error. Membership testing for DCFL. We show that randomized algorithms as efficient as the ones described above for DLIN and LL(k) (which are subclasses of DCFL) cannot exist for all of DCFL: there is a language in VPL (a subclass of DCFL) for which any randomized p-pass algorithm with an error bounded by epsilon < 1/2 must use Omega(n/p) space. Degree sequence problem. We study the problem of determining, given a sequence d(1), d(2), . . . , d(n), and a graph G, whether the degree sequence of G is precisely d(1), d(2), . . . , d(n). We give a randomized one-pass O(log n) space algorithm with an inverse polynomial one-sided error probability. We show that our algorithms are optimal. Our randomized algorithms are based on the recent work of Magniez et al. [1];our lower bounds are obtained by considering related communication complexity problems. (c) 2013 Elsevier B.V. All rights reserved.

关键词： streaming algorithms Randomized algorithms Communication complexity Context free language

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