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Algorithms for Generalized Caching

Nikhil Bansal

IBM Research

Niv Buchbinder

Open Univ. Israel

Seffi Naor

Technion

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The Paging/Caching Problem

• Set of n pages, cache of size k<n.• Request sequence of pages 1, 6, 4, 1, 4, 7, 6, 1, …

If requested page already in cache, no penalty. Else, cache miss: Fetch page in cache (possibly) evicting

some other page.

Goal: Minimize the number of cache misses.

Main Question: Which page to evict?

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Previous Results: Paging

Paging (Deterministic) [Sleator Tarjan 85]:

• Any det. algorithm ¸ k-competitive.

• LRU is k-competitive (also other algorithms)

Paging (Randomized):

• Rand. Marking O(log k) [Fiat, Karp, Luby, McGeoch, Sleator, Young 91].

• Lower bound Hk [Fiat et al. 91], tight results known.

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Generalized Caching

First Extension:

• Pages have different fetching costs.

• Models scenarios in which the fetching cost is not uniform:

Main memory, disk, internet …

Second (Orthogonal) Extension:

• Pages have different sizes (in range [1,k])

• Models web-caching problems (Proxy Servers, local cache in browser)

web

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Generalized Paging

Deterministic: (k+1)-competitive [Cao, Irani 97], [ Young 98]

Randomized:

• General Problem: Nothing better

• Special cases [Irani 97]

Bit Model (wt = size): O(log2 k)

Fault Model (wt = 1): O(log2 k)

Offline: NP-Hard, 4-approximation

[Bar-noy, Bar-yehuda, Freund, Naor, Schieber 01]

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Our Results

Thm: O(log2 k) competitive alg. for Generalized Caching

Thm: O(log k) for bit and fault model.

Main Technique: Primal Dual Method for Online Algorithms(formalized by Buchbinder and Naor 05)

[based on multiplicative updates, several previous works…]

Recent Update: Improved to O(log k) [Adamaszek, Czumaj, Englert, Räcke]

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High level approach

First step:• O(log k)-competitive for fractional generalized caching. Maintains fractions on pages.

Second Step: Transform fractional solution into Randomized Alg.:

• Maintain distribution on cache states that is “consistent” with the fractional solution.

• Keep costs comparable We lose O(1) for Bit/Fault model, O(log k) for general model.

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Rest of the Talk

• Online Primal Dual Method• Formulation of Caching Problem• Knapsack cover inequalities• Final Algorithm

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An Abstract Online Problem

min 3 x1 + 5 x2 + x3 + 4 x4 + …

2 x1 + x3 + x6 + … ¸ 3x3 + x14 + x19 + … ¸ 8x2 + 7 x4 + x12 + … ¸ 2

Goal: Find feasible solution x* with min cost.

Requirements: 1) Upon arrival constraint must be satisfied2) Cannot decrease a variable.

Covering LP (non-negative entries)

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Example

min x1 + x2 + … + xn

x1 + x2 + x3 + … + xn ¸ 1

x2 + x3 + … + xn ¸ 1

x3 + … + xn ¸ 1

…

xn ¸ 1

Online ¸ ln n (1+1/2+ 1/3+ … + 1/n)

Opt = 1 ( xn=1 suffices)

Set all xi to 1/n

Increase x2 ,x3,…,xn to 1/n-1

…

Increase xn to 1

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General Covering Results

Thm [Buchbinder-Naor 05]: O(log n) competitive fractional alg.

(n – number of variables).

– Can improve if more structure

(e.g. O(log k) for caching, O(ln d) for d-sparse matrix)

– Can add “box” constraints to covering LP (e.g. x · 1)

– Tradeoff feasibility vs. competitive ratio

– Allow variables to increase or decrease

– More general forms than covering/packing

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Online Primal Dual Approach

Primal updated based on dual.

Recall, any dual solution is a lower bound on optimum.

At each step: Primal · c ( Dual)(if primal and dual feasible) ! c competitive fractional algorithm

Algorithm is always similar.

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Key Idea for Online Primal Dual

Primal: Min i ci xi Dual

Step t, new constraint: New variable yt

a1x1 + a2x2 + … + ajxj ¸ bt + bt yt in dual objective

How much: xi ? yt ! yt + 1 (additive update)

primal cost =

dxi/dy = ai xi/ci so, x varies as exp(y)

= Dual Cost

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How to initialize

A problem: dx/dy is proportional to x, but x=0 initially.

So, x will remain equal to 0 ?

Answer: Initialize to 1/n.

Complementary slackness : x > 0 only if dual constraint for x is tight.

Set x=1/n when its dual constraint becomes tight.

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Rest of the Talk

• Online Primal Dual Method• Formulation of Caching Problem• Knapsack cover inequalities• Final Algorithm

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Fractional covering LP

At each time we have a profile of how much“fraction” of each page has been evicted.

i wi xi ¸ W - k (W = total size of all pages)

Real variables: x(i,j) j-th request for page iInitially 0 when page requested, then increases as page

evicted over time.

If x(i,j) increases by , pay c(i)

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Large Integrality Gap

Input: Two pages of size k/2+1 requested alternately.

1) Any solution must fault on every step.

2) But LP can hold 1-O(1/k) fraction of each page,incurring only O(1/k) cost at each step.

Cannot hope to show o(k) guarantees.

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Rest of the Talk

• Online Primal Dual Method• Formulation of Caching Problem• Knapsack cover inequalities• Final Algorithm

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Knapsack Cover Inequalities

Knapsack Cover Problem: Knapsack of size D,

n items with sizes w1,w2,…,wn and cost c1,..,cn

Goal: Choose min cost subset, that covers knapsack (i.e. total size ¸ D)

LP: min ci xi s.t. i wi xi ¸ D, xi = {0,1} 2 [0,1]

Bad example: Only one item of size 1000 D.LP can just choose 1/1000 of this item.

Simple fix: min ci xi s.t. i min(wi, D) xi ¸ D

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Still Bad

Bad example: 2 items, s1 = s2 = D-1 c1=0, c2= L

Any solution must choose item 2. (Integer cost = L)

LP: sets x1=1 and x2 = 1/(D-1) (LP cost = L/(D-1))

Consider a subset of jobs S. Even if choose all items in S, still need to cover at least D-W(S) size with remaining items.

Thm: Factor 2 in general (Carr, Fleischer, Leung, Phillips’ 97)

Exponential constraints, but poly time.

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• Consider the LP strengthened with Knapsack inequalites (exponentially many per time step)

• While there exist an unsatisfied primal constraint of set of pages S and time t:

• Increase the dual variable y(t,S).

And update primal according to general recipe.

Thm: Gives O(log k)-competitive fractional algorithm.

Sketch of Primal-Dual algorithm

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Fractional -> Randomized

At each time t, LP gives a fractional state p = (p1,…,pn) Say p ! p’ at fractional cost =

Randomized alg: Distribution on cache statesD -> D’ cost should be close to

Offline rounding (K.C. inequalities) + Online maintainability

Easy: If lose O(log W) and O(log P) (reduce to uniform)

We avoid for bit and fault model, but lose O(log k) for general model.

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Concluding Remarks

• Primal-dual approach gives simple unifying framework for caching and many online problems.

• Strong relation to multiplicative updates technique

• Recently, used to obtain the first polylog(n,k) competitive algorithm for k-server problem on general metrics [B., Buchbinder, Naor, Madry 11]

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Thank You!