Recognition of d-dimensional Monge arrays

AbstractIt is known that the d-dimensional axial transportation (assignment) problem can easily be solved by a greedy algorithm if and only if the underlying cost array fulfills the d-dimensional Monge property. In this paper the following question is solved: Is it possible to find d permutations in such a way that the permuted array becomes a Monge array? Furthermore we give an algorithm which constructs such permutations in the affirmative case. If the cost array has the dimensions n1×n2×⋯×nd with n1⩽n2⩽⋯⩽nd, then the algorithm has time complexity O(d2n2⋯nd(n1+lognd)). By using this algorithm a wider class of d-dimensional axial transportation problems and in particular of the d-dimensional axial assignment problems can be solved efficiently

On Monge sequences in d-dimensional arrays

AbstractLet C be an n × m matrix. Then the sequence j:= ((i1, j1), (i2, j2), …, (inm, jnm)) of pairs of indices is called a Monge sequence with respect to the given matrix C if and only if, whenever (i, j) precedes both (i, s) and (r, j) in j, then c[i, j] + c[r, s] ≤ c[i, s] + c[r, j]. Monge sequences play an important role in greedily solvable transportation problems. Hoffman showed that the greedy algorithm which maximizes all variables along a sequence j in turn solves the classical Hitchcock transportation problem for all supply and demand vectors if and only if j is a Monge sequence with respect to the cost matrix C. In this paper we generalize Hoffman's approach to higher dimensions. We first introduce the concept of a d-dimensional Monge sequence. Then we show that the d-dimensional axial transportation problem is solved to optimality for arbitrary right-hand sides if and only if the sequence j applied in the greedy algorithm is a d-dimensional Monge sequence. Finally we present an algorithm for obtaining a d-dimensional Monge sequence which runs in polynomial time for fixed d

Exploiting monge structures in optimum subwindow search

Optimum subwindow search for object detection aims to find a subwindow so that the contained subimage is most similar to the query object. This problem can be formulated as a four dimensional (4D) maximum entry search problem wherein each entry corresponds to the quality score of the subimage contained in a subwindow. For n x n images, a naive exhaustive search requires O(n4) sequential computations of the quality scores for all subwindows. To reduce the time complexity, we prove that, for some typical similarity functions like Euclidian metric, χ2 metric on image histograms, the associated 4D array carries some Monge structures and we utilise these properties to speed up the optimum subwindow search and the time complexity is reduced to O(n3). Furthermore, we propose a locally optimal alternating column and row search method with typical quadratic time complexity O(n2). Experiments on PASCAL VOC 2006 demonstrate that the alternating method is significantly faster than the well known efficient subwindow search (ESS) method whilst the performance loss due to local maxima problem is negligible

Graph based study of allergen cross-reactivity of plant lipid transfer proteins (LTPs) using microarray in a multicenter study.

The study of cross-reactivity in allergy is key to both understanding. the allergic response of many patients and providing them with a rational treatment In the present study, protein microarrays and a co-sensitization graph approach were used in conjunction with an allergen microarray immunoassay. This enabled us to include a wide number of proteins and a large number of patients, and to study sensitization profiles among members of the LTP family. Fourteen LTPs from the most frequent plant food-induced allergies in the geographical area studied were printed into a microarray specifically designed for this research. 212 patients with fruit allergy and 117 food-tolerant pollen allergic subjects were recruited from seven regions of Spain with different pollen profiles, and their sera were tested with allergen microarray. This approach has proven itself to be a good tool to study cross-reactivity between members of LTP family, and could become a useful strategy to analyze other families of allergens