Finding maximum degrees in hidden bipartite graphs

Abstract

An (edge) hidden graph is a graph whose edges are not explicitly given. Detecting the presence of an edge requires expensive edgeprobing queries. We consider the k most connected vertex problem on hidden bipartite graphs. Specifically, given a bipartite graph G with independent vertex sets B and W , the goal is to find the k vertices in B with the largest degrees using the minimum number of queries. This problem can be regarded as a top-k extension of a semi-join, and is encountered in many applications in practice (e.g., top-k spatial join with arbitrarily complex join predicates).If B and W have n and m vertices respectively, the number of queries needed to solve the problem is nm in the worst case. This, however, is a pessimistic estimate on how many queries are necessary on practical data. In fact, on some easy inputs, the problem can be efficiently settled with only km+n edges, which is significantly lower than nm for k n. The huge difference between km + n and nm makes it interesting to design an adaptive algorithm that is guaranteed to achieve the best possible performance on every input G. We give such an algorithm, and prove that it is instance optimal among a broad class of solutions. This means that, for any G, our algorithm can perform more queries than the optimal solution (which is currently unknown) by only a constant factor, which can be shown to be at most 2. Extensive experiments demonstrate that, in practice, the number of queries required by our technique is far less than nm, and agrees with our theoretical findings very well.

Identifying the Most Connected Vertices in Hidden Bipartite Graphs Using Group Testing

“…Another branch of hidden graph research is graph learning: Given a hidden graph G, the objective is to reconstruct the whole graph using a minimal number of edge probe tests [18], [4], [3], [7], [15]. As argued by Tao et al [29], [28], the kMCV problem is different from those work because it neither tests the possession of any property of the hidden graph, nor reconstructs the whole graph.…”

“…Comparing with SOE [29], [28], the use of group testing raises at least two new technical aspects: 1) In terms of algorithm design, a kMCV algorithm that exploits the group test model has to determine the group size carefully, in which algorithms that based on the 2-vertex model do not. 2) In terms of solution analysis, the analysis has to base on the external testing cost, which depends on a) the number of executed group tests, b) the group size, and c) the cost function of various group testing implementations.…”

“…connects with a white vertex w 2 W: The 2-vertex testing method is used by SOE [29], [28]. As mentioned earlier, in many applications, for example, distance join, protein-protein interaction, we can test a group of vertices together.…”

“…Graph pattern matching. Applications like drug discovery often need to identify the graph patterns that match the most number of data graphs [29], [28]. The discovery process usually involves testing whether a graph pattern b is a sub/supergraph of a data graph w. An edge is present if such a containment relationship exists between b and w. Such information, however, remains hidden unless an explicit sub/supergraph containment test is carried out.…”

“…As the pioneering work, Tao et al [29], [28] developed an algorithm, SOE, 1 to solve the kMCV problem. SOE is based on 2-vertex edge probe testing, or simply 2-vertex testing [7], i.e., each edge probe test Qðb; wÞ takes as inputs one black vertex b 2 B and one white vertex w 2 W, and returns 1 if b and w possess an edge in the hidden bipartite graph G and 0 otherwise.…”