Competitive Analysis
2
Beyond Worst Case Analysis
Worst-case analysis.
■Analyze running time as function of worst input of a given size.
Average case analysis.
■Analyze average running time over some distribution of inputs.
■Ex: quicksort.
Amortized analysis.
■Worst-case bound on sequence of operations.
■Ex: splay trees, union-find.
Competitive analysis.
■Make quantitative statements about online algorithms.
■Ex: paging, load balancing.
3
Online Algorithm and Competitive Analysis
Paging problem: Given two-level store consisting of fast memory
(cache) that can hold k pages, and slow memory that can store
infinitely many pages.
■Sequence of page requests p:
–if page p already in cache, no cost incurred
–otherwise, eject some other page q from cache and replace with
p, and pay unit cost for page fault.
■If p not in cache, which page q should you evict?
■Most fundamental and practically important online problem in CS.
4
Online Algorithm and Competitive Analysis
Competitive analysis. (Sleator-Tarjan)
■Algorithm A is ρ-competitive if there exists some constant b such
that for every sequence of inputs σ:
where OPT is optimal offline algorithm.
■OPT = MIN: evict page whose next access is furthest away.
■A = LRU: evict page whose most recent access was earliest
!Traditional analysis completely uninformative.
!We show LRU is k-competitive.
■A = LIFO: evict page brought in most recently.
!LIFO can have arbitrarily bad competitive ratio.
■Fact: no online paging algorithm is better than k-competitive.
.)(cost)(cost b
OPTA +≤
σ
ρ
σ
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