Data Structures and Algorithms
CS245-2009S-03
Recursive Function Analysis
David Galles
Department of Computer Science
University of San Francisco
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Recursive Functions and Algorithm Analysis - Prof. David J. Galles, Study notes of Data Structures and Algorithms
An analysis of recursive functions and recurrence relations, with a focus on determining the running time of recursive programs. It includes examples of recursive functions, the use of recurrence relations, and methods for solving them. The document also covers the importance of base cases and induction in proving bounds.
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Data Structures and Algorithms
CS245-2009S-
Recursive Function Analysis
David Galles
Department of Computer Science
University of San Francisco
03-0:
Algorithm Analysis
for (i=1; i<=n*n; i++)
for (j=0; j<i; j++)
sum++;
03-2:
Algorithm Analysis
for (i=1; i<=n*n; i++)
for (j=0; j<i; j++)
sum++;
Exact # of times
sum++
is executed:
(^2) n ∑ i=
i^
n
n
2
n
4
n
2 2
n
03-3:
Recursive Functions
long power(long x, long n) {
if (n == 0)
return 1; else
return x * power(x, n-1);
03-5:
Recurrence Relations
long power(long x, long n) {
if (n == 0)
return 1; else
return x * power(x, n-1);
}^ T
c
1
for some constant
c
1
T
n
c
2
T
n
for some constant
c
2
03-6:
Solving Recurrence Relations
T
c
1
T
n
T
n
c
2
If we knew
T
n
, we could solve
T
n
T
n
)^
=
T
n
c
2
03-8:
Solving Recurrence Relations
T
c
1
T
n
T
n
c
2
If we knew
T
n
, we could solve
T
n
T
n
)^
=
T
n
c
2
T
n
T
n
c
2
=
T
n
c
2
c
2
=
T
n
c^2
T
n
T
n
c
2
=
T
n
c
2
c^2
=
T
n
c^2
03-9:
Solving Recurrence Relations
T
c
1
T
n
T
n
c
2
If we knew
T
n
, we could solve
T
n
T
n
)^
=
T
n
c
2
T
n
T
n
c
2
=
T
n
c
2
c
2
=
T
n
c^2
T
n
T
n
c
2
=
T
n
c
2
c^2
=
T
n
c^2
T
n
T
n
c
2
=
T
n
c^2
03-11:
Solving Recurrence Relations
T
c
1
T
n
T
n
k
k
c
2
for all
k
If we set
k
n
, we have:
T
n
)^
=
T
n
n
nc
2
=
T
nc
2
=
c
1
nc
2
n
03-12:
Building a Better
Power
long power(long x, long n) {
if (n==0) return 1;if (n==1) return x;if ((n % 2) == 0)
return power(x*x, n/2); else
return power(x*x, n/2) * x;
03-14:
Solving Recurrence Relations
T
n
)^
=
T
n/
c
3
03-15:
Solving Recurrence Relations
T
n
)^
=
T
n/
c
3
T
n/
T
n/
c
3
=
T
n/
c
3
c
3
=
T
n/
c^3
03-17:
Solving Recurrence Relations
T
n
)^
=
T
n/
c
3
T
n/
T
n/
c
3
=
T
n/
c
3
c
3
=
T
n/
c^3
T
n/
T
n/
c
3
=
T
n/
c
3
c^3
=
T
n/
c^3
T
n/
T
n/
c
3
=
T
n/
c
3
c^3
=
T
n/
c^3
03-18:
Solving Recurrence Relations
T
n
)^
=
T
n/
c
3
T
n/
T
n/
c
3
=
T
n/
c
3
c
3
=
T
n/
c^3
T
n/
T
n/
c
3
=
T
n/
c
3
c^3
=
T
n/
c^3
T
n/
T
n/
c
3
=
T
n/
c
3
c^3
=
T
n/
c^3
T
n/
T
n/
c
3
=
T
n/
c
3
c^3
=
T
n/
c^3