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, in the model studied by Raskhodnikova, the distance that an image has to a specific property is the number of entries that should be modified in the corresponding matrix so that the property can be obtained, divided by the total number of entries: n². In the model we consider, the distance is the number of entries that should be modified divided by the actual number of 1's in the matrix, which may be much smaller than n2. 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.