1/21/04 CS 5633 Analysis of Algorithms 1
CS 5633 -- Spring 2004
Recurrences and Divide & Conquer
Carola Wenk
Slides courtesy of Charles Leiserson with small
changes by Carola Wenk
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Merge Sort Algorithm Analysis - Prof. Carola Wenk, Study notes of Algorithms and Programming
An analysis of the merge sort algorithm, including its recurrence relation, divide-and-conquer design paradigm, and recursion tree. It also covers the master method for solving recurrences and provides examples of its application.
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Pre 2010
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1/21/
CS 5633 Analysis of Algorithms
CS 5633 -- Spring 2004
Recurrences and Divide & Conquer
Carola Wenk
Slides courtesy of Charles Leiserson with small
changes by Carola Wenk
1/21/
CS 5633 Analysis of Algorithms
Merge sortM
ERGE
-S
ORT
A
[..
n
]
1. If
n
= 1, done.
2. Recursively sort
A
[ 1..
n
^
]
and
A
[^
n
n
].
Merge
” the 2 sorted lists.
Key subroutine:
M
ERGE
1/21/
CS 5633 Analysis of Algorithms
Merging two sorted arrays
1/21/
CS 5633 Analysis of Algorithms
Merging two sorted arrays
1/21/
CS 5633 Analysis of Algorithms
Merging two sorted arrays
1/21/
CS 5633 Analysis of Algorithms
Merging two sorted arrays
1/21/
CS 5633 Analysis of Algorithms
Merging two sorted arrays
1/21/
CS 5633 Analysis of Algorithms
Merging two sorted arrays
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CS 5633 Analysis of Algorithms
13
Merging two sorted arrays
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CS 5633 Analysis of Algorithms
14
Merging two sorted arrays
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CS 5633 Analysis of Algorithms
16
Analyzing merge sort
M
ERGE
-S
ORT
A
[..
n
]
1. If
n
= 1, done.
2. Recursively sort
A
[ 1..
n
^
]
and
A
[^
n
n
].
3. “Merge”
the 2 sorted lists
T
( n
d
0
T
( n
dn
Sloppiness:
Should be
T
(^
n
^
T
(^
n
^
but it turns out not to matter asymptotically.
1/21/
CS 5633 Analysis of Algorithms
Recurrence for merge sort
T
( n
d
0
if
n
T
( n
dn
if
n
• We shall often omit stating the base case
when
T
( n
(1) for sufficiently small
n
but only when it has no effect on theasymptotic solution to the recurrence.
• But what does
T
( n
) solve to? I.e., is it
O(n) or O(n
2 ) or O(n
3 ) or …?
1/21/
CS 5633 Analysis of Algorithms
Example: merge sort
1. Divide:
Trivial.
2. Conquer:
Recursively sort 2 subarrays.
3. Combine:
Linear-time merge.
T
( n
T
( n
( n
# subproblems
subproblem size
work dividingand combining
1/21/
CS 5633 Analysis of Algorithms