Complexity of Nondeterministic Functions
Alexander E. Andreev February 1994 |
Abstract:The complexity of a nondeterministic function is the minimum possible complexity of its determinisation. The entropy of a nondeterministic function, F, is minus the logarithm of the ratio between the number of determinisations of F and the number of all deterministic functions. We obtain an upper bound on the complexity of a nondeterministic function with restricted entropy for the worst case. These bounds have strong applications in the problem of algorithm derandomization. A lot of randomized algorithms can be converted to deterministic ones if we have an effective hitting set with certain parameters (a set is hitting for a set system if it has a nonempty intersection with any set from the system). Linial,
Luby, Saks and Zuckerman (1993) constructed the best effective hitting set
for the system of k-value, n-dimensional rectangles. The set size is
polynomial in Our bounds of nondeterministic
functions complexity offer a possibility to construct an effective hitting
set for this system with almost linear size in Available as PostScript, PDF, DVI. |