11 research outputs found
Non-adaptive Group Testing on Graphs
Grebinski and Kucherov (1998) and Alon et al. (2004-2005) study the problem
of learning a hidden graph for some especial cases, such as hamiltonian cycle,
cliques, stars, and matchings. This problem is motivated by problems in
chemical reactions, molecular biology and genome sequencing.
In this paper, we present a generalization of this problem. Precisely, we
consider a graph G and a subgraph H of G and we assume that G contains exactly
one defective subgraph isomorphic to H. The goal is to find the defective
subgraph by testing whether an induced subgraph contains an edge of the
defective subgraph, with the minimum number of tests. We present an upper bound
for the number of tests to find the defective subgraph by using the symmetric
and high probability variation of Lov\'asz Local Lemma
Topology Discovery of Sparse Random Graphs With Few Participants
We consider the task of topology discovery of sparse random graphs using
end-to-end random measurements (e.g., delay) between a subset of nodes,
referred to as the participants. The rest of the nodes are hidden, and do not
provide any information for topology discovery. We consider topology discovery
under two routing models: (a) the participants exchange messages along the
shortest paths and obtain end-to-end measurements, and (b) additionally, the
participants exchange messages along the second shortest path. For scenario
(a), our proposed algorithm results in a sub-linear edit-distance guarantee
using a sub-linear number of uniformly selected participants. For scenario (b),
we obtain a much stronger result, and show that we can achieve consistent
reconstruction when a sub-linear number of uniformly selected nodes
participate. This implies that accurate discovery of sparse random graphs is
tractable using an extremely small number of participants. We finally obtain a
lower bound on the number of participants required by any algorithm to
reconstruct the original random graph up to a given edit distance. We also
demonstrate that while consistent discovery is tractable for sparse random
graphs using a small number of participants, in general, there are graphs which
cannot be discovered by any algorithm even with a significant number of
participants, and with the availability of end-to-end information along all the
paths between the participants.Comment: A shorter version appears in ACM SIGMETRICS 2011. This version is
scheduled to appear in J. on Random Structures and Algorithm
Graph Reconstruction via Distance Oracles
We study the problem of reconstructing a hidden graph given access to a
distance oracle. We design randomized algorithms for the following problems:
reconstruction of a degree bounded graph with query complexity
; reconstruction of a degree bounded outerplanar graph with
query complexity ; and near-optimal approximate reconstruction of
a general graph
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