Sublinear Time and Space Algorithms 2016B – Lecture 7 Sublinear-Time Algorithms for Sparse Graphs∗

Problem definition: Input: A graph represented (say) as the adjacency list for each vertex (or even just the degree of each vertex) Goal: Compute the average degree (equiv. number of edges) Concern: Seems to be impossible e.g. if all degrees ≤ 1, except possibly for a few vertices whose degree is about n. Theorem 1 [Feige, 2004]: There is an algorithm that estimates the average degree d of a connected graph within factor 2 + ε in time O((1ε ) O(1) √ n/d0), given a lower bound d0 ≤ d and ε ∈ (0, 1). We will prove the case of d0 = 1 (i.e., suffices to know G is connected). Algorithm: 1. Choose a set S by choosing at random s = c √ n/εO(1) vertices, and compute the average degree dS of these vertices. 2. Repeat the above 8/ε times, and report the smallest seen dS . Analysis: We will need 2 claims. Claim 1a: In each iteration, Pr[dS < ( 1 2 − ε)d] ≤ ε/64. Claim 1b: In each iteration, Pr[dS > (1 + ε)d] ≤ 1− ε/2. Proof of theorem: Follows easily from the two claims, as seen in class. Proof of Claim 1b: Follows from Markov’s inequality, as seen in class. Proof of Claim 1a: Was seen in class. Here we really used the fact the degrees form a graph. ∗These notes summarize the material covered in class, usually skipping proofs, details, examples and so forth, and possibly adding some remarks, or pointers. The exercises are for self-practice and need not be handed in. In the interest of brevity, most references and credits were omitted.