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CS-2852
Data Structures
LECTURE 12B
Andrew J. Wozniewicz
Image copyright © 2010 andyjphoto.com
Agenda
• Recursion
– Definition
– Examples
• Math Functions
• Algorithms
• Data
– Computability
CS-2852 Data Structures, Andrew J. Wozniewicz
What Is Recursion?
DEFINITION
Recursion in computer science is a method where
the solution to a problem depends on solutions to
smaller instances of the same problem.
• Recursion, n. see Recursion
• Some functional programming
languages don’t have loops (equivalent
to imperative looping constructs)
• A function “calls itself” – directly or not
CS-2852 Data Structures, Andrew J. Wozniewicz
Visual Recursion
Source: Wikipedia
CS-2852 Data Structures, Andrew J. Wozniewicz
Recursion in Mathematics
SET THEORY DEFINITION FOR NATURAL NUMBERS
1 is a natural number, and each natural
number has a successor, which is also a
natural number.
• By this base case and recursive rule,
one can generate the set of all natural
numbers
CS-2852 Data Structures, Andrew J. Wozniewicz
Recursive Function
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…
Fibonacci Series
• fib(0) = 0
• fib(1) = 1
• fib(n) = fib(n-1) + fib(n-2)
(the growth of a population of rabbits
under idealized assumptions)
Leonardo Pisano Bigollo
c. 1170 – c. 1250
Liber Abaci
(The Book of Calculation)
Leonardo of Pisa, Leonardo Pisano, Leonardo
Bonacci, Leonardo Fibonacci, Fibonacci
1.618 – Golden Ratio
CS-2852 Data Structures, Andrew J. Wozniewicz
Fibonacci Numbers
BINARY REPRESENTATION
http://mathworld.wolfram.com
The first 511 terms of the Fibonacci sequence represented in binary.
CS-2852 Data Structures, Andrew J. Wozniewicz
Fibonacci Function
1: int fib(int n) {
2:
if (n==0)
3:
return 0;
4:
if (n==1)
5:
return 1;
6:
return fib(n-1) + fib(n-2);
7: }
DEMO!
CS-2852 Data Structures, Andrew J. Wozniewicz
Fibonacci Numbers - Plotted
9E+09
8E+09
7E+09
50000
45000
40000
6E+09
5E+09
35000
30000
25000
4E+09
20000
15000
3E+09
10000
2E+09
5000
0
1E+09
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
CS-2852 Data Structures, Andrew J. Wozniewicz
Factorial Function !
0! = 1
n! = n * (n-1)
1: int fact(int n) {
2:
if (n==0)
3:
return 0;
6:
return n*fact(n-1);
7: }
CS-2852 Data Structures, Andrew J. Wozniewicz
Greatest Common Divisor
EUCLID’S ALGORITHM
long gcd(long a, long b) {
// a > b, b != 0
if (b==0)
return a;
return gcd(b, a % b);
}
CS-2852 Data Structures, Andrew J. Wozniewicz
Recursively Count From n Downto 0
9876543210
void countDown(int n) {
for
System.out.print(i);
System.out.print(n);
(int i = n; i >= 0; i--)
countDown(n-1);
if
(n
System.out.print(i);
== 0)
}
return;
countDown(n-1);
}
CS-2852 Data Structures, Andrew J. Wozniewicz
Other Examples of Recursive
Algorithms
• Linear search through an array
• Binary search in an array
CS-2852 Data Structures, Andrew J. Wozniewicz
Recursive Algorithm Design
• Identify the base case
– the small problem that can be solved
directly
• Split the big problem into smaller
subproblems
– Divide-and-Conquer
• Combine the solutions to the smaller
subproblems
CS-2852 Data Structures, Andrew J. Wozniewicz
Tail Recursion
• Occurs when the recursive call is at the “end”
of the recursive function.
• The algorithm can usually be rewritten to use
(pure) iteration instead.
• Computational Equivalence:
– f1() { while(C) { S; }; return Q; }
– f2() { if (C) then { S; return f(); } else { return Q; } }
CS-2852 Data Structures, Andrew J. Wozniewicz
Recursion versus Iteration
• Both allow to repeat an operation
• You can always write a recursive
solution to a problem solvable by
iteration.
• The reverse is not necessarily true:
– Some recursion problems (non-tail
recursion) cannot be solved by iteration
alone.
CS-2852 Data Structures, Andrew J. Wozniewicz
“Proving” Correctness
• Verify that the base case is recognized
and solved correctly.
• Verify that each recursive case makes
progress towards the base case.
• Verify that if all smaller problems are
solved correctly, then the original
problem is also solved correctly.
CS-2852 Data Structures, Andrew J. Wozniewicz
Recursive Data Structures
• Linked-List:
– Node
– Node, Linked-List
• Tree
– Node
– Left-(Sub) Tree
– Right-(Sub)Tree
CS-2852 Data Structures, Andrew J. Wozniewicz
Summary
• Recursion
– Recursive Functions
• Fibonacci
• Factorial
• Greatest Common Divisor
– Recursive Algorithms
•
•
•
•
Designing Recursive Algorithms
Recursion versus Iteration
Proving Correctness – by Induction
Tail Recursion
– Recursive Data Structures
CS-2852 Data Structures, Andrew J. Wozniewicz
Questions?
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