Corrected minor typos. Infeasible items are now additionally called "overweight" items (WRS-N-P). Enriched the Introduction (Section 1) with more text and references to related work. Revised the description of sampling with a bounded number of replacements (Section 4.2)
In this work, we present a comprehensive treatment of weighted random sampling (WRS) over data streams. More precisely, we examine two natural interpretations of the item weights, describe an existing algorithm for each case ([2, 4]), discuss sampling with and without replacement and show adaptations of the algorithms for several WRS problems and evolving data streams.
We consider the problem of approximating a maximum weighted matching, when the edges of an underlying weighted graph $G(V,E)$ are revealed in a streaming fashion. We analyze a variant of the previously best-known $(4+ε)$-approximation algorithm due to Crouch and Stubbs (APPROX, 2014), and prove their conjecture that it achieves a tight approximation factor of $3.5+ε$. The algorithm splits the stream into substreams on which it runs a greedy maximum matching algorithm. At the end of the stream, the selected edges are given as input to an optimal maximum weighted matching algorithm. To analyze t…
This paper investigates parallel random sampling from a potentially-unending data stream whose elements are revealed in a series of element sequences (minibatches). While sampling from a stream was extensively studied sequentially, not much has been explored in the parallel context, with prior parallel random-sampling algorithms focusing on the static batch model. We present parallel algorithms for minibatch-stream sampling in two settings: (1) sliding window, which draws samples from a prespecified number of most-recently observed elements, and (2) infinite window, which draws samples from al…
We consider message-efficient continuous random sampling from a distributed stream, where the probability of inclusion of an item in the sample is proportional to a weight associated with the item. The unweighted version, where all weights are equal, is well studied, and admits tight upper and lower bounds on message complexity. For weighted sampling with replacement, there is a simple reduction to unweighted sampling with replacement. However, in many applications the stream has only a few heavy items which may dominate a random sample when chosen with replacement. Weighted sampling \textit{w…
In this paper we study how to perform distinct sampling in the streaming model where data contain near-duplicates. The goal of distinct sampling is to return a distinct element uniformly at random from the universe of elements, given that all the near-duplicates are treated as the same element. We also extend the result to the sliding window cases in which we are only interested in the most recent items. We present algorithms with provable theoretical guarantees for datasets in the Euclidean space, and also verify their effectiveness via an extensive set of experiments.
This is paper introduces a new single-pass reservoir weighted-sampling stream aggregation algorithm, Priority-Based Aggregation (PBA). While order sampling is a powerful and e cient method for weighted sampling from a stream of uniquely keyed items, there is no current algorithm that realizes the benefits of order sampling in the context of stream aggregation over non-unique keys. A naive approach to order sample regardless of key then aggregate the results is hopelessly inefficient. In distinction, our proposed algorithm uses a single persistent random variable across the lifetime of each key…
We study sketching and streaming algorithms for the Longest Common Subsequence problem (LCS) on strings of small alphabet size $|Σ|$. For the problem of deciding whether the LCS of strings $x,y$ has length at least $L$, we obtain a sketch size and streaming space usage of $\mathcal{O}(L^{|Σ| - 1} \log L)$. We also prove matching unconditional lower bounds. As an application, we study a variant of LCS where each alphabet symbol is equipped with a weight that is given as input, and the task is to compute a common subsequence of maximum total weight. Using our sketching algorithm, we obtain an $\…
We consider communication-efficient weighted and unweighted (uniform) random sampling from distributed data streams presented as a sequence of mini-batches of items. This is a natural model for distributed streaming computation, and our goal is to showcase its usefulness. We present and analyze fully distributed, communication-efficient algorithms for both versions of the problem. An experimental evaluation of weighted reservoir sampling on up to 256 nodes (5120 processors) shows good speedups, while theoretical analysis promises further scaling to much larger machines.
Improved approximation guarantees for weighted matching in the semi-streaming model
我们研究了半流模型中的最大权重匹配问题，并通过设计一种性能保证为4.91+ k的确定性方法来改进Zelke（Proc. of STACS 2008，第669-680页）提出的当前最佳单遍算法。此外，我们研究抢占式在线算法，一个子类的一遍算法，我们只允许在内存中保持一个可行的匹配在任何时间点。所有已知的结果之前，泽尔克的属于这个子类。我们提供了一个下界4.967的竞争比任何这样的确定性算法，因此表明，未来的改进将不得不存储在内存中的一组边缘，这不一定是一个可行的匹配。
The manuscript introduces a method to select a random sample from a stream by deciding on each sampling unit immediately after observing it. The process could be applied to unequal as well as equal probability sampling. The implementation is straightforward. Algorithm selects a unit in the sample based on a single condition. It is particularly effective to make direct decisions on stream data, despite the data arriving in groups or the stream being linear.
This paper presents a novel algorithm solving the classic problem of generating a random sample of size s from population of size n with non-uniform probabilities. The sampling is done with replacement. The algorithm requires constant additional memory, and works in O(n) time (even when s >> n, in which case the algorithm produces a list containing, for every population member, the number of times it has been selected for sample). The algorithm works online, and as such is well-suited to processing streams. In addition, a novel method of mass-sampling from any discrete distribution using the a…
While the basic greedy algorithm gives a semi-streaming algorithm with an approximation guarantee of $2$ for the \emph{unweighted} matching problem, it was only recently that Paz and Schwartzman obtained an analogous result for weighted instances. Their approach is based on the versatile local ratio technique and also applies to generalizations such as weighted hypergraph matchings. However, the framework for the analysis fails for the related problem of weighted matroid intersection and as a result the approximation guarantee for weighted instances did not match the factor $2$ achieved by the…
An alias table is a data structure that allows for efficiently drawing weighted random samples in constant time and can be constructed in linear time. The PSA algorithm by Hübschle-Schneider and Sanders is able to construct alias tables in parallel on the CPU. In this report, we transfer the PSA algorithm to the GPU. Our construction algorithm achieves a speedup of 17 on a consumer GPU in comparison to the PSA method on a 16-core high-end desktop CPU. For sampling, we achieve an up to 24 times higher throughput. Both operations also require several times less energy than on the CPU. Adaptation…
We recently introduced a formalism for the modeling of temporal networks, that we call stream graphs. It emphasizes the streaming nature of data and allows rigorous definitions of many important concepts generalizing classical graphs. This includes in particular size, density, clique, neighborhood, degree, clustering coefficient, and transitivity. In this contribution, we show that, like graphs, stream graphs may be extended to cope with bipartite structures, with node and link weights, or with link directions. We review the main bipartite, weighted or directed graph concepts proposed in the l…
The Expiration Streaming Model: Diameter, $k$-Center, Counting, Sampling, and Friends
University of Bonn（波恩大学）；University of Ljubljana（卢布尔雅那大学）；Institute of Mathematics, Physics and Mechanics（数学、物理和力学研究所）；University of Maryland, College Park（马里兰大学帕克分校）；Weizmann Institute of Science（魏茨曼科学研究所）；Microsoft Research（微软研究院）；Université Côte d’Azur（蔚蓝海岸大学）
Min-Hash is a popular technique for efficiently estimating the Jaccard similarity of binary sets. Consistent Weighted Sampling (CWS) generalizes the Min-Hash scheme to sketch weighted sets and has drawn increasing interest from the community. Due to its constant-time complexity independent of the values of the weights, Improved CWS (ICWS) is considered as the state-of-the-art CWS algorithm. In this paper, we revisit ICWS and analyze its underlying mechanism to show that there actually exists dependence between the two components of the hash-code produced by ICWS, which violates the condition o…
We design a generic method for reducing the task of finding weighted matchings to that of finding short augmenting paths in unweighted graphs. This method enables us to provide efficient implementations for approximating weighted matchings in the streaming model and in the massively parallel computation (MPC) model. In the context of streaming with random edge arrivals, our techniques yield a $(1/2+c)$-approximation algorithm thus breaking the natural barrier of $1/2$. For multi-pass streaming and the MPC model, we show that any algorithm computing a $(1-δ)$-approximate unweighted matching in …
Hong Kong University of Science and Technology（香港科技大学）
Gradient boosting decision tree (GBDT) is a powerful and widely-used machine learning model, which has achieved state-of-the-art performance in many academic areas and production environment. However, communication overhead is the main bottleneck in distributed training which can handle the massive data nowadays. In this paper, we propose two novel communication-efficient methods over distributed dataset to mitigate this problem, a weighted sampling approach by which we can estimate the information gain over a small subset efficiently, and distributed protocols for weighted quantile problem us…
Data structures for efficient sampling from a set of weighted items are an important building block of many applications. However, few parallel solutions are known. We close many of these gaps both for shared-memory and distributed-memory machines. We give efficient, fast, and practicable parallel algorithms for building data structures that support sampling single items (alias tables, compressed data structures). This also yields a simplified and more space-efficient sequential algorithm for alias table construction. Our approaches to sampling $k$ out of $n$ items with/without replacement and…
If we cannot store all edges in a graph stream, which edges should we store to estimate the triangle count accurately? Counting triangles (i.e., cycles of length three) is a fundamental graph problem with many applications in social network analysis, web mining, anomaly detection, etc. Recently, much effort has been made to accurately estimate global and local triangle counts in streaming settings with limited space. Although existing methods use sampling techniques without considering temporal dependencies in edges, we observe temporal locality in real dynamic graphs. That is, future edges ar…
In this paper, we introduce adversarially robust streaming algorithms for central machine learning and algorithmic tasks, such as regression and clustering, as well as their more general counterparts, subspace embedding, low-rank approximation, and coreset construction. For regression and other numerical linear algebra related tasks, we consider the row arrival streaming model. Our results are based on a simple, but powerful, observation that many importance sampling-based algorithms give rise to adversarial robustness which is in contrast to sketching based algorithms, which are very prevalen…
A great variety of complex systems, from user interactions in communication networks to transactions in financial markets, can be modeled as temporal graphs consisting of a set of vertices and a series of timestamped and directed edges. Temporal motifs are generalized from subgraph patterns in static graphs which consider edge orderings and durations in addition to topologies. Counting the number of occurrences of temporal motifs is a fundamental problem for temporal network analysis. However, existing methods either cannot support temporal motifs or suffer from performance issues. Moreover, t…
We propose Graph Priority Sampling (GPS), a new paradigm for order-based reservoir sampling from massive streams of graph edges. GPS provides a general way to weight edge sampling according to auxiliary and/or size variables so as to accomplish various estimation goals of graph properties. In the context of subgraph counting, we show how edge sampling weights can be chosen so as to minimize the estimation variance of counts of specified sets of subgraphs. In distinction with many prior graph sampling schemes, GPS separates the functions of edge sampling and subgraph estimation. We propose two …
Joining records with all other records that meet a linkage condition can result in an astronomically large number of combinations due to many-to-many relationships. For such challenging (acyclic) joins, a random sample over the join result is a practical alternative to working with the oversized join result. Whereas prior works are limited to uniform join sampling where each join row is assigned the same probability, the scope is extended in this work to weighted sampling to support emerging applications such as scientific discovery in observational data and privacy-preserving query answering.…
Graph Sampling provides an efficient yet inexpensive solution for analyzing large graphs. While extracting small representative subgraphs from large graphs, the challenge is to capture the properties of the original graph. Several sampling algorithms have been proposed in previous studies, but they lack in extracting good samples. In this paper, we propose a new sampling method called Weighted Edge Sampling. In this method, we give equal weight to all the edges in the beginning. During the sampling process, we sample an edge with the probability proportional to its weight. When an edge is samp…
Toyota Technological Institute at Chicago（芝加哥丰田技术研究所）；Microsoft Research（微软研究院）；Carnegie Mellon University（卡内基梅隆大学）
Adaptive sampling is a useful algorithmic tool for data summarization problems in the classical centralized setting, where the entire dataset is available to the single processor performing the computation. Adaptive sampling repeatedly selects rows of an underlying matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$, where $n\gg d$, with probabilities proportional to their distances to the subspace of the previously selected rows. Intuitively, adaptive sampling seems to be limited to trivial multi-pass algorithms in the streaming model of computation due to its inherently sequential nature of assigni…
We study the Maximum Independent Set problem for geometric objects given in the data stream model. A set of geometric objects is said to be independent if the objects are pairwise disjoint. We consider geometric objects in one and two dimensions, i.e., intervals and disks. Let $α$ be the cardinality of the largest independent set. Our goal is to estimate $α$ in a small amount of space, given that the input is received as a one-pass stream. We also consider a generalization of this problem by assigning weights to each object and estimating $β$, the largest value of a weighted independent set. W…
Random sampling is a fundamental primitive in modern algorithms, statistics, and machine learning, used as a generic method to obtain a small yet "representative" subset of the data. In this work, we investigate the robustness of sampling against adaptive adversarial attacks in a streaming setting: An adversary sends a stream of elements from a universe $U$ to a sampling algorithm (e.g., Bernoulli sampling or reservoir sampling), with the goal of making the sample "very unrepresentative" of the underlying data stream. The adversary is fully adaptive in the sense that it knows the exact content…
The Barabási-Albert model is a popular scheme for creating scale-free graphs but has been previously shown to have ambiguities in its definition. In this paper we discuss a new ambiguity in the definition of the BA model by identifying the tight relation between the preferential attachment process and unequal probability random sampling. While the probability that each individual vertex is selected is set to be proportional to their degree, the model does not specify the joint probabilities that any tuple of $m$ vertices is selected together for $m>1$. We demonstrate the consequences using ana…
Bipartite networks manifest as a stream of edges that represent transactions, e.g., purchases by retail customers. Many machine learning applications employ neighborhood-based measures to characterize the similarity among the nodes, such as the pairwise number of common neighbors (CN) and related metrics. While the number of node pairs that share neighbors is potentially enormous, only a relatively small proportion of them have many common neighbors. This motivates finding a weighted sampling approach to preferentially sample these node pairs. This paper presents a new sampling algorithm that …
Big data streams are possibly one of the most essential underlying notions. However, data streams are often challenging to handle owing to their rapid pace and limited information lifetime. It is difficult to collect and communicate stream samples while storing, transmitting and computing a function across the whole stream or even a large segment of it. In answer to this research issue, many streaming-specific solutions were developed. Stream techniques imply a limited capacity of one or more resources such as computing power and memory, as well as time or accuracy limits. Reservoir sampling a…
Google Research（谷歌研究院）；Tel Aviv University（特拉维夫大学）；Stanford University（斯坦福大学）；Ben-Gurion University（本-古里安大学）
Common datasets have the form of elements with keys (e.g., transactions and products) and the goal is to perform analytics on the aggregated form of key and frequency pairs. A weighted sample of keys by (a function of) frequency is a highly versatile summary that provides a sparse set of representative keys and supports approximate evaluations of query statistics. We propose private weighted sampling (PWS): A method that ensures element-level differential privacy while retaining, to the extent possible, the utility of a respective non-private weighted sample. PWS maximizes the reporting probab…
