18 1 21 master method proof tủ tài liệu training pdf

Master Method

Proof (Part 1)
Design and Analysis
of Algorithms I


The Master Method


Assume : recurrence is
I.

( For some
constant c
ti

II.
And n is a power of b.
(general case is similar, but more tedious

ti
Idea : generalize MergeSort analysis.
(i.e., use a recursion tree ti

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Shangari



THE RECURSION TREE

Level 0
Level 1
.
.
.
.
Level logbn

a braches

Base cases
(size 1ti


Work at a Single Level
Total work at level j [ignoring work in recursive calls]


Total Work
Summing over all levels j = 0,1,2,…, logbn :

Total
work


Design and Analysis
of Algorithms I

Master Method
Intui3on for

the 3 Cases
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HOW TO THINK ABOUT (*)

Interpreta3on
a = rate of subproblem prolifera3on (RSP)
bd = rate of work shrinkage (RWS)
(per subproblem)
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Which of the following statements are true?
(Check all that apply.ti


INTUITION FOR THE 3 CASES
1. RSP = RWS => Same amount of work each level (like
[expect
Merge
dlog(nti]
2. RSP
< RWS => less work eachO(n
level
=> most work at the
Sortti
[might expect
root
dti] => most work at

O(n
3. RSP > RWS => more work each
level
the leaves
[might expect O(# leavesti]
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Master Method

Proof (Part II)
Design and Analysis
of Algorithms I


THE STORY SO FAR/CASE 1
= 1 for
all j
=1
= (logbn + 1)

[ end Case 1 ]
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Basic Sums Fact
For

, we have

Proof : by induction (you check)
Upshot:
1. If r<1 is constant, RHS is <=

= a constant

2. If r>1 is constant, RHS is <=
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Shangari


CASE 2

:= r

•
<= a constant
( independent of nti
[ by basic sums fact ]

[ total work dominated by top level ]
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Shangari


:= r > 1

•
<= constant *
largest term

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Level 0
A CHILDREN

Level 1

Level logbn

The number of levels of the recursion tree.
The number of nodes of the recursion tree.
The number of edges of the recursion tree.

# of leaves =


•

More intuiKve
Simpler to apply

[End Case 3]


The Master Method
•