An efficient algorithm for the detection of Eden

In this paper, a polynomial time algorithm is presented for solving the Eden problem for graph cellular automata. The algorithm is based on our neighborhood elimination operation which removes local neighborhood configurations which cannot be used in a pre-image of a given configuration. This paper presents a detailed derivation of our algorithm from first principles, and a detailed complexity and accuracy analysis is also given. In the case of time complexity, it is shown that the average case time complexity of the algorithm is \Theta(n^2), and the best and worst cases are \Omega(n) and O(n^3) respectively. This represents a vast improvement in the upper bound over current methods, without compromising average case performance.

Decomposition construction for secret sharing schemes with graph access structures in polynomial time

The purpose of this paper is to describe a new decomposition construction for perfect secret sharing schemes with graph access structures. The previous decomposition construction proposed by Stinson is a recursive method that uses small secret sharing schemes as building blocks in the construction of larger schemes. When the Stinson method is applied to the graph access structures, the number of such “small” schemes is typically exponential in the number of the participants, resulting in an exponential algorithm. Our method has the same flavor as the Stinson decomposition construction; however, the linear programming problem involved in the construction is formulated in such a way that the number of “small” schemes is polynomial in the size of the participants, which in turn gives rise to a polynomial time construction. We also show that if we apply the Stinson construction to the “small” schemes arising from our new construction, both have the same information rate.