4 research outputs found
Optimal Query Complexity for Reconstructing Hypergraphs
In this paper we consider the problem of reconstructing a hidden weighted
hypergraph of constant rank using additive queries. We prove the following: Let
be a weighted hidden hypergraph of constant rank with n vertices and
hyperedges. For any there exists a non-adaptive algorithm that finds the
edges of the graph and their weights using
additive queries. This solves the open problem in [S. Choi, J. H. Kim. Optimal
Query Complexity Bounds for Finding Graphs. {\em STOC}, 749--758,~2008].
When the weights of the hypergraph are integers that are less than
where is the rank of the hypergraph (and therefore for
unweighted hypergraphs) there exists a non-adaptive algorithm that finds the
edges of the graph and their weights using additive queries.
Using the information theoretic bound the above query complexities are tight
Finding Weighted Graphs by Combinatorial Search
We consider the problem of finding edges of a hidden weighted graph using a
certain type of queries. Let be a weighted graph with vertices. In the
most general setting, the vertices are known and no other information about
is given. The problem is finding all edges of and their weights using
additive queries, where, for an additive query, one chooses a set of vertices
and asks the sum of the weights of edges with both ends in the set. This model
has been extensively used in bioinformatics including genom sequencing.
Extending recent results of Bshouty and Mazzawi, and Choi and Kim, we present a
polynomial time randomized algorithm to find the hidden weighted graph when
the number of edges in is known to be at most and the weight
of each edge satisfies \ga \leq |w(e)|\leq \gb for fixed constants
\ga, \gb>0. The query complexity of the algorithm is , which is optimal up to a constant factor