Analyzing Graph Structure via Linear Measurements

Abstract

We initiate the study of graph sketching, i.e., algorithms that use a limited number of linear measurements of a graph to determine the properties of the graph. While a graph on n nodes is essentially O(n 2 )-dimensional, we show the existence of a distribution over random projections into d-dimensional "sketch" space (d n 2 ) such that the relevant properties of the original graph can be inferred from the sketch with high probability. Specifically, we show that:including connectivity, k-connectivity, bipartiteness, and to return any constant approximation of the weight of the minimum spanning tree.2. d = O(n 1+γ ) suffices to compute graph sparsifiers, the exact MST, and approximate the maximum weighted matchings if we permit O(1/γ)-round adaptive sketches, i.e., a sequence of projections where each projection may be chosen dependent on the outcome of earlier sketches.Our results have two main applications, both of which have the potential to give rise to fruitful lines of further research. First, our results can be thought of as giving the first compressed-sensing style algorithms for graph data. Secondly, our work initiates the study of dynamic graph streams. There is already extensive literature on processing massive graphs in the data-stream model. However, the existing work focuses on graphs defined by a sequence of inserted edges and does not consider edge deletions. We think this is a curious omission given the existing work on both dynamic graphs in the non-streaming setting and dynamic geometric streaming. Our results include the first dynamic graph semi-streaming algorithms for connectivity, spanning trees, sparsification, and matching problems.

When Distributed Computation Is Communication Expensive

“…1. Statistical problems: computing the number of distinct elements, known as F 0 in the database literature; and finding the element with the maximum frequency, known as the ∞ or iceberg query problem.…”

“…For graph problems, Ahn, Guha and McGregor [1,2] developed an elegant technique for sketching graphs, and showed its applicability to many graph problems including connectivity, bipartiteness, and minimum spanning tree. Each sketching step in these algorithms can be implemented in the message-passing model as follows: each site computes a sketch of its local graph and sends its sketch to P 1 .…”

“…In [7], a (1 + ε)-approximation algorithm (protocol) with O(k(log n + 1/ε 2 log 1/ε)) bits of communication was given in the distributed streaming model, which is also a protocol in the message-passing model. In a typical setting, we could have ε = 0.01, n = 10 9 and k = 1000, in which case the communication cost is about 6.6 × 10 7 bits 1 . On the other hand, our result shows that if exact computation is required, then the communication cost among the k sites needs to be at least be Ω(kF 0 / log k) (See Corollary 1), which is already 10 9 bits even when F 0 = n/100.…”

Spectral Sparsification in Dynamic Graph Streams

“…This use of linear projections is well-studied in the context of processing numerical data, e.g., signal reconstruction in compressed sensing [5,8], Johnson-Lindenstrauss style dimensionality reduction [1,12], and estimating properties of the frequency vectors that arise in data stream applications [7,13]. However, it is only recently that it has been shown that this technique can be applied to more structured data such as graphs [2,3].…”

“…It was recently shown that there exist O( −2 n polylog n)-dimensional sketches from which a combinatorial sparsifier can be constructed [2,3]. This naturally gave rise to the first data stream algorithm for cut estimation when the stream is fully-dynamic, i.e., contains both edge insertions and deletions.…”