Reconstructing weighted graphs with minimal query complexity

In this paper, we consider the problem of reconstructing a hidden weighted graph using additive queries. We prove the following. Let G be a weighted hidden graph with n vertices and m edges such that the weights on the edges are bounded between n^-a and n^b for any positive constants a and b. For any m, there exists a non-adaptive algorithm that finds the edges of the graph using O(mlogn/logm) additive queries. This solves the open problem in [S. Choi, J.H. Kim, Optimal query complexity bounds for finding graphs, in: STOC, 2008, pp. 749758]. Choi and Kim's proof holds for m<(logn) ^α for a sufficiently large constant α and uses the graph theory. We use the algebraic approach for the problem. Our proof is simple and holds for any m.