Testing Properties of Sparse Images

We initiate the study of testing properties of images that correspond to sparse 0/1-valued matrices of size n × n. Our study is related to but different from the study initiated by Raskhodnikova (Proceedings of RANDOM, 2003), where the images correspond to dense 0/1-valued matrices. Specifically, while distance between images in the model studied by Raskhodnikova is the fraction of entries on which the images differ taken with respect to all n² entries, the distance measure in our model is defined by the fraction of such entries taken with respect to the actual number of 1´s in the matrix. We study several natural properties: connectivity, convexity, monotonicity, and being a line. In all cases we give testing algorithms with sublinear complexity, and in some of the cases we also provide corresponding lower bounds.