Competitive Paging Algorithms

Report
Competitive Paging
Algorithms
Amos Fiat, Richard Karp, Michael Luby,
Lyle McGeoch, Daniel Sleator, Neal Young
presented by Seth Phillips
Overview
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Introduction
Server Problems
Marking Algorithm
EATR
Lower Bound
Competitive Against Other Algorithms
Introduction – Paging Problem
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System with k pages of fast memory
(cache/RAM)
Has n-k pages of slow memory (RAM/virtual)
Requesting a page not in fast memory
Eject a page to make room (page fault)
Online paging algorithm
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Decision made without knowledge of future
requests
Performance analyzed against off-line algorithm
– complete knowledge
Deterministic algorithms are k-competitive
Randomized Algorithms
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Analyzed as a sum cost of the algorithm
Cost on a sequence of input averaged over all
the random choices that the algorithm makes
while processing the sequence
Overview
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Introduction
Server Problems
Marking Algorithm
EATR
Lower Bound
Competitive Against Other Algorithms
K-server Problem
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Let G be an n-vertex graph with positive edge
lengths obeying the triangle inequality
Let k mobile servers occupy vertices of G
Given a sequence of requests, each of which
specifies a vertex, decide how to move the
servers in response to each request
K-server continued
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Requests must be satisfied in order of their
occurrence in the sequence
Cost of handling a sequence is equal to the total
distance moved by the servers
It is conjectured that there exists a k-competitive
k-server algorithm for any graph
Uniform k-server problem
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Cost of moving a server from any vertex to any
other vertex is 1
Isomorphic to the paging problem: any vertex
with a server is in fast memory and the vertices
of the graph represent the address space
Overview
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Introduction
Server Problems
Marking Algorithm
EATR
Lower Bound
Competitive Against Other Algorithms
Marking Algorithm
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Randomized algorithm for uniform k-server
problem on graph with n vertices
Servers are initially on vertices 1 through k
Algorithm maintains a set of marked vertices
Initially these are the vertices covered by the
servers
Marking
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Each time a vertex is requested, it is marked
If k+1 vertices are marked, all marks except the
most recently requested vertex are erased
Serving
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If vertex is covered, then no servers move
If vertex is not covered, then a server is chosen
uniformly at random from the unmarked
vertices
That server is moved to cover
Competitiveness
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2*H_k competitive (H_k on the order of ln(k))
Algorithm implicitly divides request sequence
into phases.
First phase begins with r(i), where I is the
smallest integer such that r(i) is not in the set {1
through k}.
Comp. continued
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In general the phase starting with r(i) ends with
r(j)
j is the smallest integer such that the set {r(i),
r(i+1), . . ., r(j+1)} has cardinality k+1
Comp. continued
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Implicitly the first request of every phase is
made to an unmarked vertex
A vertex is clean if it was not requested in the
previous phase and has not yet been requested
in this phase – we’ll say there are l of these
A vertex is stale if it was requested in the
previous phase and has not yet been requested
in this phase
Adversary Amortized Cost
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At least l/2 because:
Let d be the number of servers that do not
coincide with marking’s at the beginning of the
phase
Let d’ be this quantity at the end of the phase
Cost(A) >= l – d because l requests have to be
met, mitigated by a possible d different server
locations
Adversary Cost Continued
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C(Adv) >= d’ because:
The vertices of S are those covered by Marking
at the end of the phase, so d’ servers are not in
S
Since Adversary is lazy (does not unnecessarily
move servers) at least d’ of A’s servers were
outside of S for the entire phase
Adversary Cost Continued
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C(Adv) >= max(l – d, d’) >= ½(l – d + d’)
D and d’ naturally telescope so simply C(Adv)
>= l/2
Marking Expected Cost
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l requests to clean vertices – each costs 1
k – l requests to stale vertices – each based on
the probability that there is no server there
Highest cost is when the l requests to clean
vertices come first (allows the most stale vertices
to be reassigned)
That leaves the cost of the k-l stales to be:
l/k + l/(k-1) + l/(k-2) + .. . + l/(l+1) = l*(H_k
– H_l)
Final Cost Competitiveness
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l*H_k – l*H_l + l <= l*H_k
Since the cost of the adversary is l/2:
The Marking algorithm is 2*H_k competitive
For the n-1 server problem it is H_n-1
competitive, but for time constraints I’ll skip this
Overview
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Introduction
Server Problems
Marking Algorithm
EATR
Lower Bound
Competitive Against Other Algorithms
EATR
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Stands for End After Twice Requested
This is an algorithm specific to the uniform 2server problem
Servers initially on vertices 1 + 2
Stale redefined to not clean and not the most
recently requested vertex
When a stale vertex is requested, the servers are
placed on the two most recently requested
vertices
EATR Analysis
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Let l be the number of clean vertices requested
during a phase
The number of stale vertices before the request
that terminates the phase is l+1
The probably of a server on each of these is
1/(l+1)
The expected cost of each phase is l + l/(l+1)
Adversary Analysis
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The best possible cost incurred by any algorith
for each phase is at least l
The competitive factor is therefore: (l +
l/(l+1))/l = 1 + 1/(l+1) <= 3/2
Overview
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Introduction
Server Problems
Marking Algorithm
EATR
Lower Bound
Competitive Against Other Algorithms
Lower Bound
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There is no c-competitive randomized algorithm
for the uniform (n-1) server problem on n
vertices with c < H_n-1
(Time constraints) The article proves this, which
also makes the marking algorithm in a class of
best possible algorithms for this problem
Overview
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Introduction
Server Problems
Marking Algorithm
EATR
Lower Bound
Competitive Against Other Algorithms
Algorithms Competitive Against
Several Others
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Adopt a viewpoint where each algorithm is
tailored for a specific choice of k and n.
The ordered pair (k, n) is called the type of
algorithm
Let A be adeterministic and B be a deterministic
on-line algorithms of the same type and c be a
positive constant
If for every sequence r of requests C_A(r) <=
c*C_B(r) + alpha then A is c-competitive with B
ACASO continued
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Let c* = (c(1), c(2), . .. C(m)) be a sequence of
postive real numbers
c* is realizable if, for every (k,n) and every
sequence B(1), B(2) . . .B(m) there exists an
algorithm A of type (k,n) such then A is c(i)
competitive with B(i)
c* realizable iff:
Sum from 1 to m of (1/c(i)) <= 1

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