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 ( m log n /log m ) additive queries. This solves the open problem in [S. Choi, J. H. Kim. Optimal Query Complexity Bounds for Finding Graphs. Proc. of the 40th annual ACM Symposium on Theory of Computing , 749-758, 2008]. Choi and Kim's proof holds for m ≥ (log n ) α for a sufficiently large constant α and uses graph theory. We use the algebraic approach for the problem. Our proof is simple and holds for any m .