17 research outputs found
Optimal Reconstruction of Graphs under the Additive Model
Article dans revue scientifique avec comité de lecture.We study the problem of reconstructing unknown graphs under the additive combinatorial search model. The main result concerns the reconstruction of {\em bounded degree} \/graphs, i.e.\ graphs with the degree of all vertices bounded by a constant . We show that such graphs can be reconstructed in non--adaptive queries, which matches the information--theoretic lower bound. The proof is based on the technique of se\-pa\-rating matrices. A central result here is a new upper bound for a general class of separating matrices. As a particular case, we obtain a tight upper bound for the class of --separating matrices, which settles an open question stated by Lindström in~\cite{Lindstrom75}. Finally, we consider several particular classes of graphs. We show how an optimal non--adaptive solution of queries for general graphs can be obtained. We also prove that trees with unbounded vertex degree can be reconstructed in a linear number of queries by a non--adaptive algorithm
Optimal Nested Test Plan for Combinatorial Quantitative Group Testing
We consider the quantitative group testing problem where the objective is to
identify defective items in a given population based on results of tests
performed on subsets of the population. Under the quantitative group testing
model, the result of each test reveals the number of defective items in the
tested group. The minimum number of tests achievable by nested test plans was
established by Aigner and Schughart in 1985 within a minimax framework. The
optimal nested test plan offering this performance, however, was not obtained.
In this work, we establish the optimal nested test plan in closed form. This
optimal nested test plan is also order optimal among all test plans as the
population size approaches infinity. Using heavy-hitter detection as a case
study, we show via simulation examples orders of magnitude improvement of the
group testing approach over two prevailing sampling-based approaches in
detection accuracy and counter consumption. Other applications include anomaly
detection and wideband spectrum sensing in cognitive radio systems
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
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
Parallel Feature Selection Using Only Counts
Count queries belong to a class of summary statistics routinely used in basket analysis, inventory tracking, and study cohort finding. In this article, we demonstrate how it is possible to use simple count queries for parallelizing sequential data mining algorithms. Specifically, we parallelize a published algorithm for finding minimum sets of discriminating features and demonstrate that the parallel speedup is close to the expected optimum. 
Deterministic Symmetry Breaking in Ring Networks
We study a distributed coordination mechanism for uniform agents located on a
circle. The agents perform their actions in synchronised rounds. At the
beginning of each round an agent chooses the direction of its movement from
clockwise, anticlockwise, or idle, and moves at unit speed during this round.
Agents are not allowed to overpass, i.e., when an agent collides with another
it instantly starts moving with the same speed in the opposite direction
(without exchanging any information with the other agent). However, at the end
of each round each agent has access to limited information regarding its
trajectory of movement during this round.
We assume that mobile agents are initially located on a circle unit
circumference at arbitrary but distinct positions unknown to other agents. The
agents are equipped with unique identifiers from a fixed range. The {\em
location discovery} task to be performed by each agent is to determine the
initial position of every other agent.
Our main result states that, if the only available information about movement
in a round is limited to %information about distance between the initial and
the final position, then there is a superlinear lower bound on time needed to
solve the location discovery problem. Interestingly, this result corresponds to
a combinatorial symmetry breaking problem, which might be of independent
interest. If, on the other hand, an agent has access to the distance to its
first collision with another agent in a round, we design an asymptotically
efficient and close to optimal solution for the location discovery problem.Comment: Conference version accepted to ICDCS 201