Spectral Analysis of Internet Topologies

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Lecture 4, Wed. Aug 27 03.
Today
Quicksort Design Paradigm Recursion, Divide
Conquer, Randomization !!! Performance O(n
log n) comparisons, In Place ! Reading CLRS,
Chapter 7, and these
lecture notes.
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Quicksort is a Recursive Divide Conquer
Algorithm
sort elements smaller than s
sort elements larger than s
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By picking random splitters we create, on the
average, O( log n) recursion depth,
for a total number of comparisons, on the
average, O(n log n) .
Why do we expect recursion depth log n
? Because, even if we are unlucky and pick bad
splitters for a few rounds, we expect that, it
is unlikely to continue picking bad splitters for
too long, and pretty soon we are bound to pick a
good splitter.
Say that a splitter is good if it creates two
partitions whose sizes are within constant
Factors of each other, eg 1/2 ,1/2 or 1/4,3/4,
otherwise say that the splitter is bad. If all
splitters are good then the recursion depth is
definetely O( log n).
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Complexity Analysis
T(n) Expected Number of Comparisons
Notice the partition is balanced, on the average
!!!
Formula is too messy to start substitutions. Will
bring it to slightly easier form.
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Solve by Substitution