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Bipartite (BP) has been seen to be a fast and accurate suboptimal algorithm to solve the Error-Tolerant Graph Matching problem. Recently, Fast Bipartite (FBP) has been presented that obtains the same distance value and node labelings but in a reduced time. Both algorithms approximate the quadratic problem in a linear problem and they do it through a speci¯c cost matrix. FBP imposes the Edit costs to be de¯ned such as the Edit distance is a distance function. Originally, the Hungarian method was used but it has been seen the Jonker-Volgenant linear solver obtains similar results than the Hungarian method but with an important run time reduction. Nevertheless, this second solver has some convergence problems on some speci¯c cost matrices. The aim of this paper is to de¯ne a new cost matrix such that the Jonker-Volgenant solver converges and the matching algorithm obtains the same distance value than the BP algorithm.

Extended reduced graphs provide summary representations of chemical structures using pharmacophore-type node descriptions to encode the relevant molecular properties. Commonly used similarity measures using reduced graphs convert these graphs into 2D vectors like fingerprints, before chemical comparisons are made. This study investigates the effectiveness of a graph-only driven molecular comparison by using extended reduced graphs along with graph edit distance methods for molecular similarity calculation as a tool for ligand-based virtual screening applications, which estimate the bioactivity of a chemical on the basis of the bioactivity of similar compounds. The results proved to be very stable and the graph editing distance method performed better than other methods previously used on reduced graphs. This is exemplified with six publicly available data sets: DUD-E, MUV, GLL&GDD, CAPST, NRLiSt BDB, and ULS-UDS. The screening and statistical tools available on the ligand-based virtual screening benchmarking platform and the RDKit were also used. In the experiments, our method performed better than other molecular similarity methods which use array representations in most cases. Overall, it is shown that extended reduced graphs along with graph edit distance is a combination of methods that has numerous applications and can identify bioactivity similarities in a structurally diverse group of molecules.

Abstract. The aim of this paper is to present a new method to compare histograms. The main advantage is that there is an important time-complexity reduction respect the methods presented before. This reduction is statistically and analytically demonstrated in the paper. The distances between histograms that we present are defined on a structure called signature, which is a lossless representation of histograms. Moreover, the type of the elements of the sets that the histograms represent are ordinal, nominal and modulo. We show that the computational cost of these distances is O(z') for the ordinal and nominal types and O(z'2 ) for the modulo one, being z' the number of non-empty bins of the histograms. The computational cost of the algorithms presented in the literature depends on the number of bins of the histograms. In most of the applications, the obtained histograms are sparse, then considering only the non-empty bins makes the time consuming of the comparison drastically decrease. The distances and algorithms presented in this paper are experimentally validated on the comparison of images obtained from public databases and positioning of mobile robots through the recognition of indoor scenes (captured in a learning stage).

Abstract. The aim of this article is to present a random graph representation, that is based on 2 nd order relations between graph elements, for modeling sets of attributed graphs (AGs). We refer to these models as second-order random graphs (SORGs). The basic feature of SORGs is that they include both marginal probability functions of graph elements and 2 ndorder joint probability functions. This allows a more precise description of both the structural and semantic information contents in a set of AGs and, consequently, an expected improvement in graph matching and object recognition. The article presents a probabilistic formulation of SORGs that includes as particular cases the two previously proposed approaches based on random graphs, namely the first-order random graphs (FORGs) and the function-described graphs (FDGs). We then propose a distance measure derived from the probability of instantiating a SORG into an AG and an incremental procedure to synthesize SORGs from sequences of AGs.Finally, SORGs are shown to improve the performance of FORGs, FDGs and direct AG-to-AG matching in three experimental recognition tasks: one -2 -in which AGs are randomly generated and the other two in which AGs represent multiple views of 3D objects (either synthetic or real) that have been extracted from color images. In the last case, object learning is achieved through the synthesis of SORG models.