18 1 21 master method proof tủ tài liệu training pdf

Master MethodProof Part 1 Design and Analysis of Algorithms I... Level logbn a braches Base cases size 1ti... Work at a Single LevelTotal work at level j [ignoring work in recursive call

Master Method

Proof (Part 1)

Design and Analysis

of Algorithms I

The Master Method

Nextcore AI -Gopal Shangari

Assume : recurrence is

I

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

( For some constant c ti

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

Nextcore AI -Gopal Shangari

Master Method Intui3on for

the 3 Cases

Design and Analysis

of Algorithms I

Nextcore AI -Gopal Shangari

HOW TO THINK ABOUT (*)

Interpreta3on

a = rate of subproblem prolifera3on (RSP)

bd = rate of work shrinkage (RWS)

(per subproblem)

Which of the following statements are true? (Check all that apply.ti

[might expect O(# leavesti]

Nextcore AI - Gopal Shangari

INTUITION FOR THE 3 CASES

1 RSP = RWS => Same amount of work each level (like

Merge

Sortti

[expect O(ndlog(nti]

2 RSP < RWS => less work each level => most work at the

O(ndti]

3 RSP > RWS => more work each level => most work at

the leaves

Master Method

Proof (Part II)

Design and Analysis

of Algorithms I

THE STORY SO FAR/CASE 1

Nextcore AI -Gopal Shangari

[ end Case 1 ]

=1

= (logbn + 1)

= 1 for all j

Basic Sums Fact

, we have

Proof : by induction (you check)

Upshot:

1 If r<1 is constant, RHS is <=

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

= a constant

Nextcore AI -Gopal Shangari

<= a constant ( independent of nti

[ by basic sums fact ]

[ total work dominated by top level ]

CASE 2

Nextcore AI -Gopal Shangari

:= r

Nextcore AI -Gopal Shangari

•

<= constant * largest term

:= r > 1

The number of levels of the recursion tree.

The number of nodes of the recursion tree.

The number of edges of the recursion tree.

Level 0

Level 1

Level logbn

# of leaves =

[End Case 3]

More intuiKve Simpler to apply

The Master Method

•