Lecture 4 — September 16, 2014

Claim: E(time to insert x) ≤ O(1). Proof: A cuckoo graph has m vertices (one per cell of A) and n edges (since for each x, we connect g(x) to h(x)). Consider the path we get from an insertion of x. We could get a simple path, a single cycle, or a double cycle. Let us define the following random variables: T , the runtime; Pk, the indicator random variable of a path being at least length k; Ck, the indicator random variable for single-cycle config of length ≥ k; and D the indicator for a random variable for having a 2-cycle config. Note also that the probability of the insertion process taking more than N = C log n steps implies that one of either D, PN , or CN occurred. Therefore We know that: ET = E ∑