Graph reordering is a powerful technique to increase the locality of the representations of graphs, which can be helpful in several applications. We study how the technique can be used to improve compression of graphs and inverted indexes. We extend the recent theoretical model of (KDD 2009) for graph compression, and show how it can be employed for compression-friendly reordering of social networks and web graphs and for assigning document identifiers in inverted indexes. We design and implement a novel theoretically sound reordering algorithm that is based on recursive graph bisection. Our experiments show a significant improvement of the compression rate of graph and indexes over existing heuristics. The new method is relatively simple and allows efficient parallel and distributed implementations, which is demonstrated on graphs with billions of vertices and hundreds of billions of edges.
@cite_25 consider the theoretical aspect of the reordering problem motivated by compressing social networks. They develop a simple but practical heuristic for the problem, called shingle ordering . The heuristic is based on obtaining a fingerprint of the neighbors of a vertex and positioning vertices with identical fingerprints close to each other. If the fingerprint can capture locality and similarity of the vertices, then it can be effective for compression. This approach is also called minwise hashing and was originally applied by Broder @cite_12 for finding duplicate web pages.
@cite_25 @cite_12
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We study a fingerprinting game in which the number of colluders and the collusion channel are unknown. The encoder embeds fingerprints into a host sequence and provides the decoder with the capability to trace back pirated copies to the colluders. Fingerprinting capacity has recently been derived as the limit value of a sequence of maximin games with mutual information as their payoff functions. However, these games generally do not admit saddle-point solutions and are very hard to solve numerically. Here under the so-called Boneh-Shaw marking assumption, we reformulate the capacity as the value of a single two-person zero-sum game, and show that it is achieved by a saddle-point solution. If the maximal coalition size is k and the fingerprinting alphabet is binary, we show that capacity decays quadratically with k. Furthermore, we prove rigorously that the asymptotic capacity is 1 (k221n2) and we confirm our earlier conjecture that Tardos' choice of the arcsine distribution asymptotically maximizes the mutual information payoff function while the interleaving attack minimizes it. Along with the asymptotics, numerical solutions to the game for small k are also presented. This paper is motivated by two applications, namely i) generalizations of cuckoo hashing, a computationally simple approach to assigning keys to objects, and ii) load balancing in content distribution networks, where one is interested in determining the impact of content replication on performance. These two problems admit a common abstraction: in both scenarios, performance is characterized by the maximum weight of a generalization of a matching in a bipartite graph, featuring node and edge capacities. Our main result is a law of large numbers characterizing the asymptotic maximum weight matching in the limit of large bipartite random graphs, when the graphs admit a local weak limit that is a tree. This result specializes to the two application scenarios, yielding new results in both contexts. In contrast with previous results, the key novelty is the ability to handle edge capacities with arbitrary integer values. An analysis of belief propagation algorithms (BP) with multivariate belief vectors underlies the proof. In particular, we show convergence of the corresponding BP by exploiting monotonicity of the belief vectors with respect to the so-called upshifted likelihood ratio stochastic order. This auxiliary result can be of independent interest, providing a new set of structural conditions which ensure convergence of BP. We consider the problem of finding a matching between two sets of features, given complex relations among them, going beyond pairwise. Each feature set is modeled by a hypergraph where the complex relations are represented by hyper-edges. A match between the feature sets is then modeled as a hypergraph matching problem. We derive the hyper-graph matching problem in a probabilistic setting represented by a convex optimization. First, we formalize a soft matching criterion that emerges from a probabilistic interpretation of the problem input and output, as opposed to previous methods that treat soft matching as a mere relaxation of the hard matching problem. Second, the model induces an algebraic relation between the hyper-edge weight matrix and the desired vertex-to-vertex probabilistic matching. Third, the model explains some of the graph matching normalization proposed in the past on a heuristic basis such as doubly stochastic normalizations of the edge weights. A key benefit of the model is that the global optimum of the matching criteria can be found via an iterative successive projection algorithm. The algorithm reduces to the well known Sinkhorn [15] row column matrix normalization procedure in the special case when the two graphs have the same number of vertices and a complete matching is desired. Another benefit of our model is the straight-forward scalability from graphs to hyper-graphs. In (2015), we proposed a graph-based formulation that links and clusters person hypotheses over time by solving a minimum cost subgraph multicut problem. In this paper, we modify and extend (2015) in three ways: (1) We introduce a novel local pairwise feature based on local appearance matching that is robust to partial occlusion and camera motion. (2) We perform extensive experiments to compare different pairwise potentials and to analyze the robustness of the tracking formulation. (3) We consider a plain multicut problem and remove outlying clusters from its solution. This allows us to employ an efficient primal feasible optimization algorithm that is not applicable to the subgraph multicut problem of (2015). Unlike the branch-and-cut algorithm used there, this efficient algorithm used here is applicable to long videos and many detections. Together with the novel pairwise feature, it eliminates the need for the intermediate tracklet representation of (2015). We demonstrate the effectiveness of our overall approach on the MOT16 benchmark ( 2016), achieving state-of-art performance.