ADVANCED COMPUTATIONAL MODELS AND ALGORITHMS Instructor: Dr. Gautam Das Lecture 23 April 30, 2009 Class notes by Prashanth Kurabalana hundi Hombe gowda[ ]

Outline Chernoff Bounds Proff Routing in parallel computers Oblivious routing Algorithm – Hypercube – Bit fixing strategy

Chernoff Bounds Proff Randomized algorithm is faster than any deterministic algorithm for a problem X= Where X is the sum of Xi and Xi is random variable.

Negative tail (℮ δ /(1+ δ) (1+ δ) ) μ e -μδ 2 /2 ≤E[Y]/EXP(t μ(1+δ)) ΠE[e tx i ] /denominator π [1+p i (e t -1)] 1+ α < e α ≤ πexp([p i (e t -1))/denominator =exp((Σp i )(e t -1))/denominator =exp((μ(et-1)/exp(t μ (1+δ)) But t= ln(1+δ) =(eδ/(1+δ) (1+δ) ) μ

Routing in Parallel computing Permutation routing problem Source ………..n Destination ………. Input to this problem: A graph A permutation Output: Route(i,d(i)) Objective: minimizing the time We use oblivious routing algorithm for this

Oblivious Routing Algorithm Lower bound algorithm: If we have a graph with each node of degree d and N nodes. For any deterministic oblivious routing algorithm A permutation Such that the total time taken=Ω( )

Hypercube: N=2 n x yz n d=n

Bit fixing strategy Consider a transpose permutation Imagine “n” to be even lrrl

2 n =N =2 n = lr rr rl rr Hi= Boolean random variable H= E[H]2n logn0 N