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Self-Organizing Lists Competitive Analysis. Lee Wee Sun SoC1 5-26 [email protected] Modified by Orgad Keller Modified again by Ariel Rosenfeld. Self-Organizing List. List L of n elements Operations Access(x) cost rank(x)=position of x in the list

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self organizing lists competitive analysis

Self-Organizing ListsCompetitive Analysis

Lee Wee Sun

SoC1 5-26

[email protected]

Modified by Orgad Keller

Modified again by Ariel Rosenfeld

Self-Organizing Lists

self organizing list
Self-Organizing List
  • List L of n elements
    • Operations Access(x) cost

rank(x)=position of x in the list

    • L can be reordered by transposing adjacent elements – cost = 1.

Self-Organizing Lists

on line problem
On-line Problem
  • Problem: Sequence S of operations to which the algorithm must respond to without knowing the future inputs.
  • Goal: Minimize the total cost of responding to the operations

Self-Organizing Lists

analysis
Analysis
  • Worst case:
    • Adversary can always access the tail element of L
    • Cost: in worst case
  • Average case:
    • Suppose element x is accessed with a probability of p(x)
      • Minimized when L sorted in decreasing order WRT p.
    • Keep count of #times the element is accessed and maintain a list in order to decrease count.

Self-Organizing Lists

heuristic
Heuristic
  • Move to Front (MTF):
    • After accessing x, move x to head of L (by transposes)
      • Cost is
    • Responds well to locality in access pattern.

Self-Organizing Lists

competitive analysis
Competitive analysis
  • An online algorithm A is -competitive if there exists a constant k such that for any sequence S of operations
  • is the algorithm that performs optimally for that particular sequence
    • Knows the whole sequence in advance and has infinite computation power to find the method for achieving the optimal cost.

Self-Organizing Lists

mtf is competitive
MTF is competitive
  • Theorem: MTF is 4-competitive for self-organizing lists.
  • Proof:
    • Let Li be MTF’s list after i-th access.
    • Let Li* be OPT’s list after i-th access
    • Let ci= MTF’s cost for i-th operation
    • Let ci* = OPT’s cost for the i-th operation = rank*(x) + ti if OPT performs titransposes

Self-Organizing Lists

slide8
Define potential function by
    • x preceded y in Li and y precedes x in Li* = #inversions
    • Example:
      • Li = <E,C,A,D,B>
      • Li* = <C,A,B,D,E>
      • |{(E,C),(E,A),(E,D),(E,B),(D,B)| = 10
    • Note:
      • for every i
      • if MTF and OPT have the same list

Self-Organizing Lists

slide9
How much does change in 1 transpose?
    • Transpose either creates or destroys 1 inversion
  • Let = amortized cost WRT of MTF for the i-th operation
  • Note (#inversions created - #inversions destroyed).

Self-Organizing Lists

slide11

why ?

r = rank(x) = |A| + |B| + 1

A  B

C  D

Li-1:

x

  • A = elements preceding x in Li-1 and preceding x in Li-1*
  • B = elements preceding x in Li-1 and following x in Li-1*
  • C = elements following x in Li-1 and preceding x in Li-1*
  • D = elements following x in Li-1 and following x in Li-1*

A  C

B  D

Li-1*:

x

r* = rank*(x)  |A| + 1

Self-Organizing Lists

slide12
When x is moved to front
    • # inversions created = |A|
    • # inversions destroyed = |B|
  • Each transpose by OPT creates  1 inversions

Transposes by OPT



Actual cost by OPT

Self-Organizing Lists

slide13
If we count transposes that move x towards the front as “free”, then MTF is 2-competitive (see Sleator and Tarjan)
  • What if L0 L0*?
    • might be (n2)
    • COSTMTF(S)  4COSTOPT(S) + (n2)
      • Still 4-competitive

Self-Organizing Lists

summary
Summary
  • Definition of competitive analysis
  • MTF is competitive for self organizing lists
    • Potential function method
      • Number of inversions with the OPT’s list is the potential.

Self-Organizing Lists

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