Lecture 17: Space-bounded Derandomization
نویسندگان
چکیده
The randomized result was obtained by viewing random bit sequences as vertices of an expander graph and performing a random walk upon choosing a start vertex uniformly at random, and casting a majority vote. The error (probability of majority vote resulting in error) exponentially decreases with the length of the random walk. We also saw a stronger statement based on Chernoff bounds for random walks expander graphs. In this lecture, we will discuss space-bounded derandomization. We construct a Pseudorandom Generator (PRG) for space-bounded computations based on expanders. The idea is to decrease the required number of seed (random) bits and simulate the algorithm on all possibilities of seed values. The following section (Section 1) defines Pseudorandom Generators (PRGs) and its various parameters. Section 2 outlines the use of pseudorandom generators in complexity theory. Section 3 concludes with the construction of an efficient PRG for BPL that has seed length of O((log n)2).
منابع مشابه
On Probabilistic Space-Bounded Machines with Multiple Access to Random Tape
We investigate probabilistic space-bounded machines that are allowed to make multiple passes over the random tape. As our main contribution, we establish a connection between derandomization of such probabilistic space-bounded classes to the derandomization of probabilistic time-bounded classes. Our main result is the following. For some integer k > 0, if all the languages accepted by boundeder...
متن کاملDerandomization and Circuit Lower Bounds
1 Introduction Primality testing is the following problem: Given a number n in binary, decide whether n is prime. In 1977, Solovay and Strassen [SS77] proposed a new type of algorithm for testing whether a given number is a prime, the celebrated randomized Solovay-Strassen primality test. This test and similar ones proved to be very useful. This fact changed the common notion of " feasible comp...
متن کاملRandomization and Derandomization in Space_Bounded Computation
This is a survey of space-bounded probabilistic computation, summarizing the present state of knowledge about the relationships between the various complexity classes associated with such computation. The survey especially emphasizes recent progress in the construction of pseudorandom generators that fool probabilistic space-bounded computations, and the application of such generators to obtain...
متن کاملLecture Space-Bounded Derandomization
We now prove Theorem 1. Let M be a probabilistic machine running in space S (and time 2S), using R random bits, and deciding a language L with two-sided error. (Note that S, R are functions of the input length n, and the theorem requires S = Ω(log n).) We will assume without loss of generality that M always uses exactly R random bits on all inputs. Fixing an input x and letting ` be some parame...
متن کامل