2014-15 Colloquia

Tuesdays at 4:10 in
Carver 202 or as announced

 

For more information, contact
Songting Luo
Derrick Stolee
Steve Butler

 
 

Fall 2014

 

September 2 - Derrick Stolee: Automated discharging arguments for density problems in grids
Poster

 

Abstracts

Derrick Stolee: Automated discharging arguments for density problems in grids

Motivated by problems in wireless sensor networks, we consider minimizing the density of an identifying code in the hexagonal grid. An identifying code is a set of vertices where every vertex in the grid is uniquely identified by its adjacent code elements. While the minimum density of an identifying code is known for the square and triangular grids, there is still a gap between the upper and lower bounds for the hexagonal grid. Most lower bounds are found using discharging, which is a method to demonstrate the interaction between local structure and global averages.

Verifying a discharging proof is straightforward but usually very tedious. However, the creation of a discharging argument can be very mysterious. We will present a new computer-automated approach to not only verify discharging arguments, but also to generate them from scratch. A critical component to this method is solving a linear program that will assign value to the specified discharging rules, resulting in the best possible proof using those rules. Using this method, we find a new lower bound of 23/55 (approximately 0.4181818) on the density of an identifying code in the hexagonal grid, improving on the current-best human-created proofs.

Mridul Nandi (Indian Statistical Institute, Kolkata) Minimum number of multiplication to compute a Delta-universal function

Delta universal function is a family of (multivariate polynomial) function with small differential probability. It is an important combinatorial object which has application in many areas of computer science. Any multivariate polynomial can be computed by a sequence of multiplication and addition. As multiplications are usually costlier operations than addition, we study the lower bounds on the number of multiplication to compute a delta universal hash function. In this talk, we obtain a concrete form of the lower bound. We also discuss memory requirement of computing such functions and how to tweak an optimum construction to reduce the memory constraint.


Archived

August 26 - Mridul Nandi (Indian Statistical Institute, Kolkata) Minimum number of multiplication to compute a Delta-universal function
Poster