- Series
- Combinatorics Seminar
- Time
- Friday, August 31, 2012 - 3:05pm for 1 hour (actually 50 minutes)
- Location
- Skiles 005
- Speaker
- Jeong Han Kim – Professor, Yonsei University, South Korea
- Organizer
- Prasad Tetali
In this talk, we consider a well-known combinatorial search problem.
Suppose that there are n identical looking coins and some of them are
counterfeit.
The weights of all authentic coins are the same and known a priori.
The weights of counterfeit coins vary but different from the weight of
an authentic coin.
Without loss of generality, we may assume the weight of authentic coins is
0.
The problem is to find all counterfeit coins by weighing (queries) sets of
coins
on a spring scale. Finding the optimal number of queries is difficult even
when there are only 2 counterfeit coins.
We introduce a polynomial time randomized algorithm to find all
counterfeit coins when the number of them is known to be at most
m \geq 2 and the weight w(c) of each counterfeit coin c satisfies
\alpha \leq |w(c)| \leq \beta
for fixed constants \alpha, \beta > 0. The query complexity of the
algorithm is O(\frac{m \log n }{\log m}), which is optimal up to a
constant factor. The algorithm uses, in part, random walks.
The algorithm may be generalized to find all edges of a hidden
weighted graph using a similar type of queries. This graph finding
algorithm
has various applications including DNA sequencing.