Correction to: Rooted Spanning Superpixels

Rooted Uniform Monotone Minimum Spanning Trees

We study the construction of the minimum cost spanning geometric graph of a given rooted point set P where each point of P is connected to the root by a path that satisfies a given property. We focus on two properties, namely the monotonicity w.r.t. a single direction (y-monotonicity) and the monotonicity w.r.t. a single pair of orthogonal directions (xy-monotonicity). We propose algorithms tha...

Counting rooted spanning forests in complete multipartite graphs

Jin and Liu discovered an elegant formula for the number of rooted spanning forests in the complete bipartite graph a1;a2 , with b1 roots in the rst vertex class and b2 roots in the second vertex class. We give a simple proof to their formula, and a generalization for complete m-partite graphs, using the multivariate Lagrange inverse. Y. Jin and C. Liu [3] give a formula for f(m; l;n; k), the n...

Reduction of Ultrametric Minimum Cost Spanning Tree Games to Cost Allocation Games on Rooted Trees

A minimum cost spanning tree game is called ultrametric if the cost function on the edges of the underlying network is an ultrametric. We show that every ultrametric minimum cost spanning tree game is reduced to a cost allocation game on a rooted tree. It follows that there exist O(n) time algorithms for computing the Shapley value, the nucleolus and the egalitarian allocation of the ultrametri...

Using a Solution Archive to Enhance Metaheuristics for the Rooted Delay-Constrained Minimum Spanning Tree Problem

SLIC Superpixels

Superpixels are becoming increasingly popular for use in computer vision applications. However, there are few algorithms that output a desired number of regular, compact superpixels with a low computational overhead. We introduce a novel algorithm that clusters pixels in the combined five-dimensional color and image plane space to efficiently generate compact, nearly uniform superpixels. The si...