Single-source Queries Research Articles

Efficient Resistance Distance Computation: The Power of Landmark-based Approaches

Resistance distance is a fundamental metric to measure the similarity between two nodes in graphs which has been widely used in many real-world applications. In this paper, we study two problems on approximately computing resistance distance: (i) single-pair query which aims at calculating the resistance distance r(s, t) for a given pair of nodes (s, t); and (ii) single-source query which is to compute all the resistance distances r(s, u) for all nodes u in the graph with a given source node s. Existing algorithms for these two resistance distance query problems are often costly on large graphs. To efficiently solve these problems, we first establish several interesting connections among resistance distance, a new concept called v-absorbed random walk, random spanning forests, and a newly-developed v-absorbed push procedure. Based on such new connections, we propose three novel and efficient sampling-based algorithms as well as a deterministic algorithm for single-pair query; and we develop an online and two index-based approximation algorithms for single-source query. We show that the two index-based algorithms for single-source query take almost the same running time as the algorithms for single-pair query with the aid of a linear-size index. The striking feature of all our algorithms is that they are allowed to select an easy-to-hit node by random walks on the graph. Such an easy-to-hit landmark node v can make the v-absorbed random walk sampling, spanning tree sampling, as well as the v-absorbed push more efficient, thus significantly improving the performance of our algorithms. Extensive experiments on 5 real-life datasets show that our algorithms substantially outperform the state-of-the-art algorithms for two resistance distance query problems in terms of both running time and estimation errors.

A matrix sampling approach for efficient SimRank computation

Evaluating similarities between node pairs in a graph is an important task for data analytics and mining. Among various similarity measures proposed in recent years, SimRank is regarded as one of the most influential measures. However, the computation of SimRank is very expensive especially for large graphs. Although pruning technique and random walk based methods were proposed to accelerate the computation, the accuracy of SimRank score is still very low. In this paper, we propose a novel matrix random sampling approach to accelerate computation speed and reduce memory cost. The matrix random sampling technique not only guarantees the sparsity of the involved matrices, but also enhances the precision of estimated SimRank scores. Moreover, we design a fast sparse matrix–matrix multiplication technique which makes the time complexity of single-source query free of the graph size. We further exploit the Steepest Decent technique to accelerate the speed of convergence. The experimental results show our proposed algorithms outperform the state-of-the-art SimRank algorithms.

SimTab

SimRank is a classic measure of vertex-pair similarity according to the structure of graphs. Top- k and thresholding SimRank queries are two important types of similarity search with numerous applications in web mining, social network analysis, spam detection, etc. However, extensive studies for SimRank most focus on single-pair and single-source queries and fail to provide any feasible solution for the top- k and thresholding queries, e.g., with theoretical accuracy guarantee or acceptable empirical performance. In this paper, we propose SimTab (<u>Sim</u>Rank queries with <u>T</u>ighter confidence bounds and multi-<u>a</u>rmed <u>b</u>andits) to answer top- k and thresholding queries in a unified manner. First, we integrate several techniques with random walk sampling to tighten the confidence bound of SimRank estimation, which enhances the query efficiency. Second, we answer top- k and thresholding queries from the perspective of the Multi-Armed Bandits (MAB) problems. The proposed algorithms significantly improve the theoretical efficiency over state of the art, whereas the algorithmic complexity closely matches the hardness of the problem. We further propose a novel sampling strategy specially tailored for node similarity queries, which improves both the theoretical and practical query efficiency of the MAB-based algorithms. Our method is the first with query accuracy guarantee for these two queries, and the sole algorithm to achieve high-quality query results on large graphs. Moreover, all proposed algorithms are index-free, and thus can be naturally applied to dynamic graphs. Extensive experiments on several large graph datasets demonstrate that our algorithms achieve much superior effectiveness with comparable or less query time cost than all index-free and index-based state of the art. Besides, our work proposes the first thorough empirical evaluation of the existing SimRank algorithms over top- k and thresholding queries.

Walking in the cloud

Despite its popularity, SimRank is computationally costly, in both time and space. In particular, its recursive nature poses a great challenge in using modern distributed computing power, and also prevents querying similarities individually. Existing solutions suffer greatly from these practical issues. In this paper, we break such dependency for maximum efficiency possible. Our method consists of offline and online phases. In offline phase, a length- n indexing vector is derived by solving a linear system in parallel. At online query time, the similarities are computed instantly from the index vector. Throughout, the Monte Carlo method is used to maximally reduce time and space. Our algorithm, called CloudWalker, is highly parallelizable, with only linear time and space. Remarkably, it responses to both single-pair and single-source queries in constant time. CloudWalker is orders of magnitude more efficient and scalable than existing solutions for large-scale problems. Implemented on Spark with 10 machines and tested on the web-scale clue-web graph with 1 billion nodes and 43 billion edges, it takes 110 hours for offline indexing, 64 seconds for a single-pair query, and 188 seconds for a single-source query. To the best of our knowledge, our work is the first to report results on clue-web, which is 10x larger than the largest graph ever reported for SimRank computation.

Algorithms for Approximate Shortest Path Queries on Weighted Polyhedral Surfaces

We consider the well-known geometric problem of determining shortest paths between pairs of points on a polyhedral surface P, where P consists of triangular faces with positive weights assigned to them. The cost of a path in P is defined to be the weighted sum of Euclidean lengths of the sub-paths within each face of P. We present query algorithms that compute approximate distances and/or approximate shortest paths on P. Our all-pairs query algorithms take as input an approximation parameter e∈(0,1) and a query time parameter $\mathfrak{q}$, in a certain range, and build a data structure $\mathrm{APQ}(P,\varepsilon;\mathfrak{q})$, which is then used for answering e-approximate distance queries in $O(\mathfrak{q})$time. As a building block of the $\mathrm{APQ}(P,\varepsilon;\mathfrak{q})$data structure, we develop a single-source query data structure SSQ(a;P,e) that can answer e-approximate distance queries from a fixed point a to any query point on P in logarithmic time. Our algorithms answer shortest path queries in weighted surfaces, which is an important extension, both theoretically and practically, to the extensively studied Euclidean distance case. In addition, our algorithms improve upon previously known query algorithms for shortest paths on surfaces. The algorithms are based on a novel graph separator algorithm introduced and analyzed here, which extends and generalizes previously known separator algorithms.