GRAPHLET COUNTING FOR TOPOLOGICAL DATA ANALYSIS In the present work we focus on computing the shortest possible basis of the first homology group H_1 of a weighted simplicial complex. The task is proved to be an NP-hard problem for H_k with k > 1, but the k=1 case is subject of recent research. Taking the lead from the recent work of Dey, which borrows ideas from previous studies to improve computational complexity, we have implemented a polynomial-time algorithm for the shortest homology basis of a simplicial complex over Z_2 coefficients and extended it to compute persistence over a family of its refinements.