Transcript SCE Assessment

CSCI 3333 Data Structures
Introduction to Recursion
Acknowledgement
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Dr. Bun Yue
Mr. Charles Moen
Dr. Wei Ding
Ms. Krishani Abeysekera
Dr. Michael Goodrich
Dr. Behrouz A. Forouzan
Dr. Richard F. Gilberg
Recursion
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Recursion is defining a function that
calls itself as a subroutine.
Recursion is a way to combat
complexity and solve problems.
Solving a problem by:
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Solving the problem for ‘trivial’ cases,
and
Solving the same problem, but with
smaller sizes.
Recursive Definition
A recursive definition: a definition in
terms of itself.
Example:
 0 is a natural number.
 If x is a natural number, then x+1
is a natural number. (or x is a
natural number if x-1 is a natural
number).
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Recursive Methods
A recursive method may calls itself.
Example: factorial.
Iterative definition:
n! = n . (n-1) . (n-2) … 3 . 2 . 1
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Recursive definition:
1
if n  0

f ( n)  
else
n  f (n  1)
Recursive Factorial Algorithm
//By Forouzan
Factorial Implementation: Fac-1
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Found here in C++:
int factorial(int number)
{ int temp;
if (number <= 1) return 1;
temp = number * factorial(number - 1);
return temp;
}
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What do you think?
Another Implementation: Fac-2
int recursiveFactorial(int n)
{
if (n == 0)
return 1; // base case
else
return n * recursiveFactorial(n- 1);
// recursive case /general case
}
Content of a Recursive Method
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Base case(s): exit conditions
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Values of the input variables (exit conditions) for
which we perform no recursive calls are called base
cases (there should be at least one base case).
Every possible chain of recursive calls must
eventually reach a base case.
Recursive calls: recursive conditions
 Calls to the current method.
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Each call must reduce the size of the problem and
move it toward the base case; reduces the size of the
problem. .
Visualizing Recursion
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Example recursion trace:
Recursion trace
return 4*6 = 24
final answer
A box for each
recursiveFactorial (4)
recursive call
return 3*2 = 6
An arrow from each
recursiveFactorial (3)
caller to callee
return 2*1 = 2
An arrow from each
recursiveFactorial (2)
callee to caller
return 1*1 = 1
showing return value
recursiveFactorial (1)
call
call
call
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call
call
recursiveFactorial (0)
return 1
Tips on Recursive Analysis
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Identify exit and recursive
conditions.
Ensure that the conditions cover all
input ranges.
Input Range Analysis of Fac
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Fac-1 and Fac-2: n (or number) has
the type of int.
Fac-2 may be caught in circularity
forever for negative numbers.
Reversing an Array
Algorithm ReverseArray(A, i, j):
Input: An array A and nonnegative
integer indices i and j
Output: The reversal of the elements in
A starting at index i and ending at j
if i < j then
Swap A[i] and A[ j]
ReverseArray(A, i + 1, j - 1)
return
Tips on Constructing Recursion
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In creating recursive methods, it is
important to define the methods in
ways that facilitate recursion.
This sometimes requires we may
define additional parameters that
are passed to the method.
For example, we defined the array
reversal method as ReverseArray(A,
i, j), not ReverseArray(A).
Computing Powers
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The power function, p(x,n)=xn, can be
defined recursively:
1
if n  0

p ( x, n )  
else
 x  p( x, n  1)
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This leads to an power function that runs
in linear time (for we make n recursive
calls).
We can do better than this, however.
Recursive Squaring
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We can derive a more efficient linearly
recursive algorithm by using repeated
squaring:
1
if x  0
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p( x, n)   x  p( x, (n  1) / 2) 2 if x  0 is odd
 p( x, n / 2) 2
if x  0 is even
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For example,
24 = 2(4/2)2 = (24/2)2 = (22)2 = 42 = 16
25 = 21+(4/2)2 = 2(24/2)2 = 2(22)2 = 2(42) = 32
26 = 2(6/ 2)2 = (26/2)2 = (23)2 = 82 = 64
27 = 21+(6/2)2 = 2(26/2)2 = 2(23)2 = 2(82) = 128.
Note: This definition leads to linear recursive algorithm
that uses O(n)
A Recursive Squaring Method
Algorithm Power(x, n):
Input: A number x and integer n
Output: The value xn
if n = 0 then
return 1
if n is odd then
y = Power(x, (n - 1)/ 2)
return x · y · y
else
y = Power(x, n/ 2)
return y · y
Analyzing the Recursive Squaring
Method
Algorithm Power(x, n):
Input: A number x and integer
Output: The value xn
if n = 0 then
return 1
if n is odd then
y = Power(x, (n - 1)/ 2)
return x · y · y
else
y = Power(x, n/ 2)
return y · y
n
Each time we make a
recursive call we halve the
value of n; hence, we make
log n recursive calls. That
is, this method runs in
O(log n) time.
It is important that we
used a variable twice here
rather than calling the
method twice.
Further Analysis
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Need to limit the range of n, or.
Add the recursive condition:
if n < 0 return 1 / Power(x, -n);
Tail Recursion 1
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Tail recursion occurs when a linearly
recursive method makes its recursive
call as its last step.
The array reversal method is an
example.
Such methods can be easily converted
to non-recursive methods (which
saves on some resources), often by
the compiler and runtime system.
Tail Recursion Example
Algorithm IterativeReverseArray(A, i, j ):
Input: An array A and nonnegative integer
indices i and j
Output: The reversal of the elements in A
starting at index i and ending at j
while i < j do
Swap A[i ] and A[ j ]
i =i+1
j =j-1
return
Tail Recursion 2
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Tips: for linear recursion, write your
method using tail recursion.
Example: Fac-1 does not use tail
recursion. Fac-2 does.
Binary Recursion
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Binary recursion occurs whenever there are
two recursive calls for each non-base case.
Also known as Higher-Order Recursion.
Example: Fibonacci number
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0, 1, 1, 2, 3, 5, 8, 13, 21, 34
Fibonacci Numbers
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Fibonacci numbers are defined
recursively:
F0 = 0
F1 = 1
Fi = Fi-1 + Fi-2
for i > 1.
Binary Recursion
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Binary Recursions are expensive.
If possible, convert it to linear
recursion.
Recursive Fibonacci Algorithm
Algorithm BinaryFib(k):
Input: Nonnegative integer k
Output: The kth Fibonacci number Fk
if k = 1 then
return k
else
return BinaryFib(k - 1) + BinaryFib(k - 2)
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Binary recursion from the text book.
Any problem here?
A C++ Implementation
long fib(unsigned long n) {
if (n <= 1) {
return n;
} else {
return fib(n-1)+fib(n-2);
}
}
Analyzing the Binary Recursion
Fibonacci Algorithm
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Let nk denote number of recursive calls made by
BinaryFib(k). Then
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n0
n1
n2
n3
n4
n5
n6
n7
n8
=
=
=
=
=
=
=
=
=
1
1
n1
n2
n3
n4
n5
n6
n7
+
+
+
+
+
+
+
n0
n1
n2
n3
n4
n5
n6
+
+
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+
1
1
1
1
1
1
1
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1+1+1=3
3+1+1=5
5+3+1=9
9 + 5 + 1 = 15
15 + 9 + 1 = 25
25 + 15 + 1 = 41
41 + 25 + 1 = 67.
Note that the value at least doubles for every other
value of nk. That is, nk > 2k/2. It is exponential!
Analyzing the Binary Recursion
Fibonacci Algorithm
//By Forouzan
A Better Fibonacci Algorithm
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Use linear recursion instead:
Algorithm LinearFibonacci(k):
Input: A nonnegative integer k
Output: Pair of Fibonacci numbers (Fk, Fk-1)
if k = 1 then
return (k, 0)
else
(i, j) = LinearFibonacci(k - 1)
return (i +j, i)
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Runs in O(k) time.
Questions and Comments?