A generic framework for median graph computation based on a recursive embedding approach

Abstract

The median graph has been shown to be a good choice to obtain a representative of a set of graphs. However, its computation is a complex problem. Recently, graph embedding into vector spaces has been proposed to obtain approximations of the median graph. The problem with such an approach is how to go from a point in the vector space back to a graph in the graph space. The main contribution of this paper is the generalization of this previous method, proposing a generic recursive procedure that permits to recover the graph corresponding to a point in the vector space, introducing only the amount of approximation inherent to the use of graph matching algorithms. In order to evaluate the proposed method, we compare it with the set median and with the other state-of-the-art embedding-based methods for the median graph computation. The experiments are carried out using four different databases (one semi-artificial and three containing real-world data). Results show that with the proposed approach we can obtain better medians, in terms of the sum of distances to the training graphs, than with the previous existing methods.

Exact solution to median surface problem using 3D graph search and application to parameter space exploration

“…Both frameworks are approximative only and therefore suitable for those median problems with inherently high computational complexity. Indeed, they have been applied to computing generalized median of strings [11,23], graphs [14], and clusterings [24], all being of NP-hard class.…”

“…One such framework described in [14] is based on an embedding into the vector space. The median vector is computed by means of the Weiszfeld algorithm [8] and inversely transformed to the original space.…”

“…The median concept has been concretized to many domains including vectors [8], tensors [9], rotations [10], strings [11], graphs [12][13][14], clusterings [15,16], tree-like shapes [17], time series [18], ranking (permutation) [19], images (for atlas construction) [20], and image segmentations [21,22]. In [4] the 2D median contour problem is investigated.…”

Efficient Answering of Why-Not Questions in Similar Graph Matching

“…The median graph is used as a representative of a set of graphs [26], [27], [28]. The computation of the generalized median graph is NP-hard and therefore, approximate solutions are preferred [26], [29]. An alternative to the generalized median graph, which is computationally less challenging, is the set median graph [30] (if g is a member of X, then g is called the set median graph).…”

Gauging Association Patterns of Chromosome Territories via Chromatic Median

“…The most popular one is median graph [16], which has received considerable attentions in recent years [11,12,14,15]. Given a set of input graphs with possible labels associated with vertices, the objective of a median graph problem is to compute a new graph, called median graph, that has the minimum distance to the input graphs.…”