Recurrence Examples

1

22

1

)(nc

nT

nc

nT

1

1

)(

ncnb

naT

nc

nT

2

Recursion Tree for T(n)=3T(n/4)+(n2)

T(n)

(a)

cn2

T(n/4) T(n/4) T(n/4)

(b)

cn2

c(n/4)2 c(n/4)2 c(n/4)2

T(n/16) T(n/16) T(n/16)T(n/16)T(n/16)T(n/16)

T(n/16) T(n/16) T(n/16)

(c)cn2

c(n/4)2 c(n/4)2 c(n/4)2

c(n/16)2 c(n/16)2 c(n/16)2c(n/16)2c(n/16)2c(n/16)2 c(n/16)2 c(n/16)2 c(n/16)2

cn2

(3/16)cn2

(3/16)2cn2

T(1)T(1) T(1)T(1) T(1)T(1) (nlog 43)

3log4n= nlog 43 Total O(n2)

log 4n

(d)

3

Solution to T(n)=3T(n/4)+(n2)

The height is log 4n,

#leaf nodes = 3log 4n= nlog 43. Leaf node cost: T(1). Total cost T(n)=cn2+(3/16) cn2+(3/16)2 cn2+ +(3/16)log 4

n-1 cn2+ (nlog 43)

=(1+3/16+(3/16)2+ +(3/16)log 4n-1) cn2 + (nlog 4

3) <(1+3/16+(3/16)2+ +(3/16)m+ ) cn2 + (nlog 4

3) =(1/(1-3/16)) cn2 + (nlog 4

3) =16/13cn2 + (nlog 4

3) =O(n2).

4

Recursion Tree of T(n)=T(n/3)+ T(2n/3)+O(n)

5

Recursion tree for T(n)=aT(n/b)+f(n)

6

General Divide-and-Conquer Recurrence

T(n) = aT(n/b) + f (n) where f(n) (nd), d 0

Master Theorem: If a < bd, T(n) (nd) If a = bd, T(n) (nd log n) If a > bd, T(n) (nlog b a ) Examples: T(n) = T(n/2) + n T(n) (n )

Here a = 1, b = 2, d = 1, a < bd

T(n) = 2T(n/2) + 1 T(n) (n log 2

2 ) = (n )Here a = 2, b = 2, d = 0, a > bd

7

Examples

T(n) = T(n/2) + 1 T(n) (log(n) )Here a = 1, b = 2, d = 0, a = bd

T(n) = 4T(n/2) + n T(n) (n log 2

4 ) = (n 2 )Here a = 4, b = 2, d = 1, a > bd

T(n) = 4T(n/2) + n2 T(n) (n2 log n)Here a = 4, b = 2, d = 2, a = bd

T(n) = 4T(n/2) + n3 T(n) (n3) Here a = 4, b = 2, d = 3, a < bd