arXiv:1906.07398v1 [cs.CC] 18 Jun 2019 Inner Product Oracle can Estimate and Sample ∗ Arijit Bishnu † Arijit Ghosh ∗ Gopinath Mishra ∗ Manaswi Paraashar ∗ Abstract Edge estimation problem in unweighted graphs using local and sometimes global queries is a fundamental problem in sublinear algorithms. It has been observed by Goldreich and Ron [GR08], that weighted edge estimation for weighted graphs require Ω(n) local queries, where n denotes the number of vertices in the graph. To handle this problem, we introduce a new inner product query on matrices. Inner product query generalizes and unifies all previously used local queries on graphs used for estimating edges. With this new query, we show that weighted edge estimation in graphs with particular kind of weights can be solved using sublinear queries, in terms of the number of vertices. We also show that using this query we can solve the problem of the bilinear form estimation, and the problem of weighted sampling of entries of matrices induced by bilinear forms. This work is a first step towards weighted edge estimation mentioned in Goldreich and Ron [GR08]. 1 Introduction The Edge Estimation 1 problem for a simple, unweighted, undirected graph G =(V (G),E(G)), |V (G)| = n, where G is accessed using queries to an oracle, is a fundamental and well studied problem in the area of sublinear algorithms. Goldreich and Ron [GR08], motivated by a work of Fiege [Fei06], gave an algorithm to estimate the number of edges of an unweighted graph G by using  θ (n/ √ m) 2 local queries, where m = |E(G)|. The local queries they use are – degree query : the oracle returns the degree of a vertex, and neighbor query : the oracle reports the i-th neighbor of a vertex, if it exists. Apart from that, an often used local query is the adjacency query : the oracle reports whether there exists an edge between a given pair of vertices. In the same work, Goldreich and Ron observed that Ω(n) degree and neighbor queries are essential for the Weighted Edge Estimation problem, where the objective is to estimate ∑ e∈E(G) w(e) for any arbitrary weight function w : E(G) → R + . We have not observed any development till date on the Weighted Edge Estimation problem. Another research direction that emerged was to consider oracles that can handle queries that are global in nature [RT16, DL18, BHR + 18, RSW18], as opposed to the earlier local queries. Most of these queries come under the group query or subset query defined in [RT16]. In particular, Beame et al. [BHR + 18] showed that the Edge Estimation problem can be solved using O ( log 14 n/ǫ 4 ) many Bipartite Independent Set (BIS) queries (in the worst case). A BIS oracle takes two disjoint subsets A and B of vertices and reports whether there exists an edge with endpoints in both A and B. Note the improvement in the number of queries as we used a powerful oracle. Faced with the negative result of a lower bound of Ω(n) local queries [GR08] and the use of powerful oracles, a natural question to ask is – are there interesting weight functions and query * This work was submitted to RANDOM-APPROX’19 on 3 May 2019. † Indian Statistical Institute, Kolkata, India 1 Given an ǫ ∈ (0, 1), the objective in the Edge Estimation problem to report ˆ m such that | ˆ m -|E(G)|| ≤ ǫ |E(G)| 2  θ(·) and  O(·) hides a poly(log n, 1/ǫ) term in the upper bound. 1