Derandomizing via the Method of Conditional Expectation

Suppose we want to derandomize A—that is, give a deterministic variant of A which succeeds with probabilty 1 on every input. Sometimes we can do this using the method of conditional expectation. We can think of A as a binary tree which, given x, branches on the sampled value of each random bit Ri in turn. Paths in this tree correspond to different possible random strings R1, . . . , Rm that could be sampled by A, and leaf nodes at level m + 1 are labeled by outputs. The fact that A succeeds with probability at least 2/3 means that at least 2/3s of the leaf nodes are good outputs for the input x.