9,589 research outputs found
Deterministic Approximation of Random Walks in Small Space
We give a deterministic, nearly logarithmic-space algorithm that given an undirected graph G, a positive integer r, and a set S of vertices, approximates the conductance of S in the r-step random walk on G to within a factor of 1+epsilon, where epsilon>0 is an arbitrarily small constant. More generally, our algorithm computes an epsilon-spectral approximation to the normalized Laplacian of the r-step walk.
Our algorithm combines the derandomized square graph operation [Eyal Rozenman and Salil Vadhan, 2005], which we recently used for solving Laplacian systems in nearly logarithmic space [Murtagh et al., 2017], with ideas from [Cheng et al., 2015], which gave an algorithm that is time-efficient (while ours is space-efficient) and randomized (while ours is deterministic) for the case of even r (while ours works for all r). Along the way, we provide some new results that generalize technical machinery and yield improvements over previous work. First, we obtain a nearly linear-time randomized algorithm for computing a spectral approximation to the normalized Laplacian for odd r. Second, we define and analyze a generalization of the derandomized square for irregular graphs and for sparsifying the product of two distinct graphs. As part of this generalization, we also give a strongly explicit construction of expander graphs of every size
Efficient Subgraph Matching on Billion Node Graphs
The ability to handle large scale graph data is crucial to an increasing
number of applications. Much work has been dedicated to supporting basic graph
operations such as subgraph matching, reachability, regular expression
matching, etc. In many cases, graph indices are employed to speed up query
processing. Typically, most indices require either super-linear indexing time
or super-linear indexing space. Unfortunately, for very large graphs,
super-linear approaches are almost always infeasible. In this paper, we study
the problem of subgraph matching on billion-node graphs. We present a novel
algorithm that supports efficient subgraph matching for graphs deployed on a
distributed memory store. Instead of relying on super-linear indices, we use
efficient graph exploration and massive parallel computing for query
processing. Our experimental results demonstrate the feasibility of performing
subgraph matching on web-scale graph data.Comment: VLDB201
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