Mergesort

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Mergesort
Department of Computer and Information
Science,School of Science, IUPUI
Dale Roberts, Lecturer Computer Science,
IUPUI E-mail droberts_at_cs.iupui.edu
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Computational Complexity

• Framework to study efficiency of algorithms.
  Example sorting.
• MACHINE MODEL count fundamental operations.
• count number of comparisons
• UPPER BOUND algorithm to solve the problem
  (worst-case).
• N log2 N from mergesort
• LOWER BOUND proof that no algorithm can do
  better.
• N log2 N - N log2 e
• OPTIMAL ALGORITHM lower bound upper bound.
• mergesort

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Decision Tree
printa1, a2, a3
printa2, a1, a3
printa1, a3, a2
printa3, a1, a2
printa2, a3, a1
printa3, a2, a1
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Comparison Based Sorting Lower Bound

• Theorem. Any comparison based sorting algorithm
  must use ?(N log2N) comparisons.
• Proof. Worst case dictated by tree height h.
• N! different orderings.
• One (or more) leaves corresponding to each
  ordering.
• Binary tree with N! leaves must have height
• Food for thought. What if we don't use
  comparisons?
• Stay tuned for radix sort.

Stirling's formula
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Mergesort Analysis

• How long does mergesort take?
• Bottleneck merging (and copying).
• merging two files of size N/2 requires N
  comparisons
• T(N) comparisons to mergesort N elements.
• to make analysis cleaner, assume N is a power of
  2
• Claim. T(N) N log2 N.
• Note same number of comparisons for ANY file.
• even already sorted
• We'll prove several different ways to illustrate
  standard techniques.

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Profiling Mergesort Empirically
Striking featureAll numbers SMALL!
comparisonsTheory N log2 N 9,966Actual
9,976
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Sorting Analysis Summary

• Running time estimates
• Home pc executes 108 comparisons/second.
• Supercomputer executes 1012 comparisons/second.
• Lesson 1 good algorithms are better than
  supercomputers.
• Lesson 2 great algorithms are better than good
  ones.

Insertion Sort (N2)
Mergesort (N log N)
computer
thousand
million
billion
thousand
million
billion
home
instant
2.8 hours
317 years
instant
1 sec
18 min
super
instant
1 second
1.6 weeks
instant
instant
instant
Quicksort (N log N)
thousand
million
billion
instant
0.3 sec
6 min
instant
instant
instant
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Acknowledgements

• Sorting methods are discussed in our Sedgewick
  text. Slides and demos are from our texts
  website at princeton.edu.
• Special thanks to Kevin Wayne in helping to
  prepare this material.