The k-server Problem

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The k-server Problem

• Study Group Randomized Algorithm
• Presented by Ray Lam
• August 16, 2003

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Outline

• Background and problem definition
• The Harmonic k-server Algorithm
• Proving the claimed performance of the algorithm

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Background

• And Problem Definition

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The Metric Space

• Definition A metric space M (V, d) consists of
  a set of points V with a distance function dV
  R satisfying the following properties
• d(u,v) 0 for all u, v V.
• d(u,v) 0 iff u v.
• d(u,v) d(v,u) for all u, v V.
• d(u,v) d(v,w) d(u,w) for all u, v, w V.

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The Metric Space

• Think of it as a complete weighted graph
• Weight corresponds to distance between points

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1
2
4
1
3
2
1
2
2
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The k-server Problem

• k servers in the metric space
• Located at particular points
• Request of service
• Happens at the points
• To serve the request move a server to the point
  of request
• A request sequence , where
  is a point in M, is a finite sequence of requests

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The k-server Problem

• Two competing algorithms
• An adversary offline algorithm
• An online algorithm to be designed
• The adversary algorithm
• Knows all of right from the beginning and
  serves them optimally with his own k servers
• Thus it is offline

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The k-server Problem

• Algorithm to be designed
• Online
• Only knows the next request and has to serve it
  immediately
• Cost measure
• Total distance moved by all the servers to serve
• total cost incurred by the
  optimal offline algorithm

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The k-server Problem

• Let denote the cost of algorithm A on
  request sequence .
• Definition A randomized algorithm A is
  c-competitive (compared to the optimal offline
  algorithm), if for all starting configurations
  there is a real a, independent of , such that

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Lower Bound of Performance

• Theorem For any metric space, the competitive
  ratio of the k-server problem is at least k (i.e.
  k-competitive).
• Note This lower bound holds for any randomized
  algorithm against an optimal online adversary
• The proof is skipped

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The Harmonic k-server Algorithm
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The Harmonic Algorithm

• Suppose node r makes a request
• The algorithm works as follows
• Let di be the distance from server i to the
  request node r
• If any di 0, do nothing (server i will serve
  the request no server moves)
• Else, use server i with probability inversely
  proportional to di......

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The Harmonic Algorithm

• i.e. letand choose server i with probability
  .
• We denote the Harmonic k-server algorithm by
  Harmonic or H in the following slides
• Eddie Grove proved that H is
  -competitive for all .

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Eddie Groves Proof

• Showing H is -competitive

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Process of Serving Requests

• Let be a request sequence of length m
• Let be the ith request
• Think of the process of serving requests as
  follows
• For each request , first the adversary moves a
  server, if necessary, to serve the request
• Then H flips a coin (takes a decision at random
  according to the pdf mentioned) to choose a
  server to serve

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Process of Serving Requests

• In this way, we have 2m phases
• Odd phase (phase ) adversary serves
• Even phase (phase 2i) H serves
• Let Dj be the distance moved by the server during
  phase j
• Odd j Distance moved by adversarys server
• Even j Distance moved by Hs server

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Introducing the Potential Function

• To analyze, a function is used
• Define to be the value of at the end
  of phase t. is chosen in such a fashion that
  the following three conditions hold
• ,
  where ck is the constant to be determined later
• Referred as Condition (1), (2) and (3) in the
  following slides

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Introducing the Potential Function

• What means?
• From Vijay Guptas lecture represents the
  amount of work that H can be forced to do if the
  offline servers do not move
• My intuitionPotential energy, reserved by
  adversary moves, consumed by Hs moves
• Why introduce ?
• Lemma If Condition (1), (2) and (3) hold, then H
  is ck-competitive.

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Lemma from 3 Conditions

• Proof

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Lemma from 3 Conditions

• Now,

(1)
(2)
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Lemma from 3 Conditions

• Using Equation (1) and (2), we havePutAlso,
  by the linearity of expectation, we haveBut,
  from Condition (1),Hence,

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More Notations

• k offline and k online servers
• Lower-case letter online serverCapital letter
  offline server
• Perfect matchings M between online and offline
  servers
• Denote by M(x) the mate of x
• Initial condition every online server coincides
  with one offline server
• i.e. In the 0th phase, d(x, M(x)) 0 for each
  online server x

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Matching M

• Each time an online server moves, update matching
  M
• Example
• Request placed at offline server A with M(a) A
• Online server b, with M(b) B, moves to the
  request at A
• Change matching to M(b) A, M(a) B
• Matching unchanged for all other servers

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Active Set

• Idea of active set is central to the proof
• Call OFF the set of all k offline servers
• For and any online server x, the
  radius of about x is
• AS(x), the active set of x, is the with
  largest minimizing

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Active Set

• Example
• k 4
• All offline servers shown only online server a
  shown
• M(a) A
• Let
• Two possible minimizing
• AS(x) A,B,D

B
C
5
1
A
1
a
2
D
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Active Set

• Any minimizing set must contain all offline
  servers within distance of x
• Intuitively, the active set includes offline
  servers close to x in comparison to d(x,M(x))
• For convenience
• Definition
• Definition

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The Potential Function

• All the 3 conditions satisfied?

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The Potential Function

• Definition The potential function is computed
  as
• Condition (1) is satisfied
• , hence , is always non-negative
• At t0, every online server and its matched
  offline server at identical point,

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Notes before Analysis

• Condition (2) corresponds to an adversary move
• Condition (3) corresponds to a Harmonic move
• Analyzing an (generic) adversary move and a
  (generic) Harmonic move completes the proof

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Notes before Analysis

• In the following analysis, a request is placed at
  some point
• Let A be the offline server moved in response to
  the request, with M(a)A
• Let b be the online server moved in response to
  the request, with M(b)B
• Unless otherwise specified, all expressions
  describe configuration BEFORE the movement
• Abuse notation same variable for a server and
  the point it occupies

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Analysis of Adversary Moves

• Let Z be the place of request
• A moves a distance D2i1 to Z in phase 2i1
• Consider the set of servers,
• Physical meaning online server with A inside its
  active set, and now A moves out of its active set
  boundary
• For wont increase

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Analysis of Adversary Moves

• Indexing all yh as follows
• If a in , y0a else no y0
• For hgt0, index yh such that
• When an offline server moves a distance D2i1
• increases by at most
  for all
• Other terms do not increase

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Analysis of Adversary Moves

• To estimate the increase in potential, we need to
  estimate S(yh)
• Let Yh be the offline server matched to yh
• Lemma For hgt1,

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Analysis of Adversary Moves

• ProofLet .
  HenceDistance from yh to any Yj in Th is
  bounded byHence,

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Analysis of Adversary Moves

• By the minimality in the definition of ,
  we haveHence

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Analysis of Adversary Moves

• The increase in potential due to a move by an
  offline server of distance D2i1 is at most
• Condition (2) is satisfied with competitive ratio

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Analysis of Harmonic Moves

• Three cases
• Case 1 a serves the request at A (i.e. b is
  identical to a)
• Case 2 B is close to a,
• Case 3 B is at distance greater than R(a) from
  a,
• We will describe sets NS(x) for which AFTER
  update matching M

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Harmonic Moves Case 1

• Case 1 a serves the request at A
• AFTER the move, goes to zero
• Nothing else is changed
• Chance is
• Expected change in potential

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Harmonic Moves Case 2

• Case 2 B is close to a,
• For , let NS(x)AS(x). NS(b)A
• Terms for unaffected
• Potential decreases by at least
• This term is dropped in an inequality in later
  proof

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Harmonic Moves Case 3

• Case 3 B is at distance greater than R(a) from
  a,
• Call Bi the offline server that is ith closest to
  a among offline servers at a distance more than
  R(a) from a
• Break any ties arbitrarily
• Let Bl B
• Call bi the online server matched to Bi
• bl b
• Let dld(A,bl)

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Harmonic Moves Case 3

• For
• R(a,NS(a)) will be at most
• Now
• Since , we have

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Harmonic Moves Case 3

• Only and changes
• Expected increase in potential at most
• The increase happens for each l between 1
  andk-S(a)

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Analysis of Harmonic Moves

• It remains to show that satisfies Condition
  (3)
• From previous results, we see that

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Analysis of Harmonic Moves

• The identity,proves that
• This completes the proof that the Harmonic
  algorithm is -competitive
  for all

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Reference

• V. Gupta, CS497 SHT Spring 1999 Prof. Shang-Hua
  Teng Lecture 12 2nd March, 1999, Mar. 1999
• E.F. Grove, The Harmonic online k-server
  algorithm is competitive, Proceedings of the
  23rd Annual ACM Symposium on Theory of Computing,
  1991