EE 31331
PROGRAMMING METHODOLOGY
AND SOFTWARE ENGINEERING
PROGRAMMING METHDOLOGY AND SOFTWARE ENGINEERING
Copyright©1999 Angus Wu
RECURSION
It is a repetitive process in which an algorithm calls
itself.
Why recursion?
It provides a simple mechanism to perform iterative
process.
It provides much simpler coding.
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RECURSION FUNDAMENTAL
The are two commonly used statements in data structure
analysis:
• Proof by Induction
• Proof by Contradiction/Counter Example
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PROOF BY INDUCTION
Prove that the Fibonacci numbers, F0=1, F1=1, F2=2, F3=3,
F4 = 5, …. Fi = Fi-1 + Fi-2, and satisfy Fi < (5/3)i
Proof of induction starts with the simple trivial case to
establish the base case.
Then, assuming that the theorem is true for the k th case,
based on the given conditions, to prove that it is also true
for the k+1 th case.
If it is true for the case k+1, by the principle of induction,
the theorem will be true for any number if n is finite.
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PROOF BY INDUCTION
Prove that the Fibonacci numbers, F0=1, F1=1, F2=2, F3=3,
F4 = 5, …. Fi = Fi-1 + Fi-2, and satisfy Fi < (5/3)i
It is quite obvious that F1 = 1 < 5/3, F2 = 2 < 25/9.
We need to show that Fk+1 < (5/3)k+1
Since Fk+1 = Fk + Fk-1, then Fk+1 < (5/2)k + (5/2)k-1
(5/2)k + (5/2)k-1 = (3/5)(5/3)k+1 + (3/5)2(5/3)k+1
= (3/5 + 9/25) (5/3)k+1
= (24/25) (5/3)k+1 < (5/3)k+1
Thus, Fk+1 < (5/3)k+1
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Example: Factorial
Factorial (n) = [ 1
n x (n-1) x ….. 2 x 1
Iterative algorithm
Factorial (n) = [ 1
n x (Factorial (n-1))
Recursive algorithm
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if n
=0
if n
>0
if n
=0
if n
>0
Copyright©1999 Angus Wu
i= 1;
factN =1;
loop ( i< n)
factN = factN *i;
i = i +1;
return factN;
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recursiveFactorial( val n <integer>)
if ( n = 0 ) return 1;
else
return ( n* recursiveFactorial (n-1));
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RECURSION
Most mathematical functions are described by a simple formula.
However, some are in more complicated forms.
Define a function, F, valid on positive integers, that satisfies
F(0) = 0, and F(x) = 2F(x-1) + x2
From definition, we have:
F(1) = 1, F(2) = 6, F(3) = 21, and F(4) = 58.
Here, we have a function defines on itself. We call it recursive
function.
The idea is to implement the recursive function by computer
program. Then, how ?
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RECURSION
int F(int x)
{
/* 1 */
if (x = = 0) return 0;
/* 2 */
else return 2*F(x-1) + x*x;
}
In line 1, it is similar to induction case that establishes the base
case. It is the case that solved without recursion. The value for
which the function is directly known without resorting to
recursion.
Simply declare the function, F(x) = 2F(x-1) +x2 without the
base case is ambiguous mathematically.
line 2 makes the recursion call. (function calls itself)
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RECURSION
int F(int x)
{
/* 1 */
if (x = = 0) return 0;
/* 2 */
else return 2*F(x-1) + x*x;
}
What will happen if the function is called to evaluate F(-1)?
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RECURSION
int Bad(unsigned int N)
{
/* 1 */
if (N = = 0) return 0;
/* 2 */
else return Bad (N/3 +1) + N -1;
}
if Bad(1) is called, then line 2 will be executed as it is defined
by line 2. But what is the value Bad(1)? It is not defined in the
base case. Then, the computer will keep on executing line 2
until the system runs out of space. In fact, the program does not
work for any number except Bad(0).
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Characteristics of Recursion
if this is a simple case
solve it
else
redefine the problem using recursion
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RECURSION
For any valid recursion, the fundamental rules are:
1. Base case. It must include some base cases, which can be
solved without recursion.
2. Making progress. For the cases that are to be solved
recursively, the recursive call must always be to a case that
makes progress toward a base case.
In general, every recursive call must either solve a part of
the
problem or reduce the size of the problem.
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recursiveFactorial( val n <integer>)
if ( n = 0 ) return 1;
else
return ( n* recursiveFactorial (n-1));
Base Case:
if ( n = 0) return 1;
Making Progress
recursiveFactorial(n-1)
*for each call, the argument is towards the base case, n= 0
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Copyright©1999 Angus Wu
Selection Sort
34
34
23
15
15
45
23
15
23
15
34
23
34
23
34
15
45
45
45
45
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Copyright©1999 Angus Wu
Selection Sort
Find largest element in the array, switch it with the
bottom element.
Repeat the same action until the whole array is sorted.
Algorithm
if n is 1
the array is sorted
else
place the largest array element in the last position
Sort the subarray which excludes the last array element
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Algorithm
if n is 1
the array is sorted
else
place the largest array element in the last position
Sort the subarray which excludes the last array element
void select_sort(int array[], int n)
{
if (n ==1) return;
else { place_largest(array, n);
select_sort(array, n-1); }
}
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Copyright©1999 Angus Wu
Recursion Development
1. Base case.
It must include some base cases, which can be solved without
recursion.
* Termination of the recursion.
2. Making progress.
For the cases that are to be solved recursively, the recursive call
must always be to a case that makes progress toward a base
case.
* Dividing the problem into sub-problem with “smaller scale”.
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TREE ADT
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Basic Tree Concepts
A tree consists of a finite set of elements called node, and
a finite set of directed lines, called branches, that connect
the nodes.
A
branch
node
B
E
F
I
C
D
G
H
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Copyright©1999 Angus Wu
root
A
branch 1
B
node
E
F
I
C
D
G
H
for node B, branch 1 is an indegree branch.
indegree branch is a branch directed towards a node
for node A, branch 1 is an outdegree branch.
outdegree branch is a branch directed away from a node
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root
A
B
E
F
I
C
D
G
H
A leaf is any node with an outdegree of zero.
(C, D, E, G, H, I)
Nodes are not the root or leaves, called internal nodes.
(B, F)
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parent
A
B
E
F
parent and
child
I
C
D
child
G
H
A node is a parent if its has child/or successor.
Any node with a predecessor is a child.
Two or more nodes with the same parent are siblings.
{(C,D), (G,H, I)}
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ancestor
A
B
E
descendent
F
I
C
D
G
H
Ancestor- is any node in the path from the root to the node
(A, B, F)
Descendent - is any node in the path below the parent node
(B, E, F, C, D, G, H, I)
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Copyright©1999 Angus Wu
level 0
A
B
E
level 1
F
I
C
D
level 2
G
H
The level of a node is its distance from the root. The height
of the tree is the level of the leaf in the longest path from the
root plus 1. By definition, the height of an empty tree is -1.
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BINARY TREE
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BINARY TREE
It is a tree in which no node can have more than two subtrees.
These subtrees are designated as the left subtree, and right
subtree.
A
B
C
E
D
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F
Copyright©1999 Angus Wu
PROPERTIES OF BINARY TREE
Height of Binary Tree
Given that there are N nodes in a tree. The H max. is N
The H min is [log2 N] + 1.
Given a height of the binary tree, H, the min. and max. no. of
nodes in the tree are:
N min = H, and N max = 2H -1
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PROPERTIES OF BINARY TREE
Balance Factor
The balance factor of a binary tree is the difference in height
between its left and right subtrees.
B = HL-HR
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STUCTURE OF BINARY TREE
NODE
left subtree <pointer to Node>
data
<dataType>
rightSubtree <pointer to Node>
End NODE
typedef struct node *NodePtr;
struct node {
int info;
NodePtr left;
NodePtr right;
};
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Node
left
right
Copyright©1999 Angus Wu
BINARY TREE TRAVERSALS
A binary tree traversal requires that each node of the tree
be processed There are three way of traversals for a binary
tree:
preorder, inorder, and postorder
In the preorder traversal, the root node is processed first,
followed by the left subtree and the the right subtree. The
root goes before the subtree.
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Copyright©1999 Angus Wu
BINARY TREE PREORDER TRAVERSALS
A
B
C
E
D
F
ABCDEF
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Copyright©1999 Angus Wu
BINARY TREE TRAVERSALS
A
B
C
E
D
algorithm preorder (val root <nodepointer>)
if (root is not NULL)
process(root);
preorder(root-> LeftSubtree);
preorder(root-> RightSubtree);
return
end preorder
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F
BINARY TREE INORDER TRAVERSALS
A
B
C
E
D
F
Inorder traversal processes the left subtree first, the the
rootm and finally the right subtree.
C B DAE F
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Copyright©1999 Angus Wu
BINARY TREE TRAVERSALS
A
B
C
E
D
algorithm inorder (val root <nodepointer>)
if (root is not NULL)
inorder(root-> LeftSubtree);
process(root);
inorder(root-> RightSubtree);
return
end inorder
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Copyright©1999 Angus Wu
F
BINARY TREE POSTORDER TRAVERSALS
A
B
C
E
D
F
Postorder traversal processes the leftmost leaf then
followed by the right subtrees and finally the root
CDBFEA
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Copyright©1999 Angus Wu
BINARY TREE TRAVERSALS
A
B
C
E
D
algorithm postorder (val root <nodepointer>)
if (root is not NULL)
postorder(root-> LeftSubtree);
postorder(root-> RightSubtree);
process(root);
return
end postorder
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Copyright©1999 Angus Wu
F
EXPRESSION BINARY TREE
An expression tree is a binary tree with the following
properties:
1. Each leaf is an operand
2. The root and the internal nodes are operators
(+-*/)
3. Subtrees are sub-expressions with the root being an
operator.
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Copyright©1999 Angus Wu
EXPRESSION BINARY TREE
+
a*(b+c) + d
An infix tree with parenthesis
*
d
a
((a*(b+c)) + d)
+
b
c
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PRINTING AN INFIX EXPRESSION BINARY TREE
+
*
d
a
+
b
c
((a*(b+c)) + d)
algorithm infix (val tree <tree pointer>)
if (tree not empty)
if (tree->token is an operand)
print (tree->token);
else
{ print (open parenthesis);
infix(tree->left);
print(tree->token);
infix(tree->right);
print (close parenthesis);}
return;
end infix;
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Copyright©1999 Angus Wu
PRINTING AN PREFIX EXPRESSION BINARY TREE
+
*
d
a
+
b
+*a+bcd
c
algorithm prefix (val tree <tree pointer>)
if (tree not empty)
{ print (tree->token);
prefix(tree->LeftPointer);
prefix(tree->RightPointer);
}
return;
end prefix;
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Copyright©1999 Angus Wu
PRINTING AN POSTFIX EXPRESSION BINARY TREE
+
*
d
a
+
b
abc+*d+
c
algorithm postfix (val tree <tree pointer>)
if (tree not empty)
{
postfix(tree->LeftPointer);
postfix(tree->RightPointer);
print (tree->token);
}
return;
end postfix;
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Copyright©1999 Angus Wu
CREATING AN EXPRESSION TREE
Consider the expression :
The corresponding postfix is:
(a+b)*(c*(d+e))
ab+cde+**
*
+
a
*
b
c
+
d
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e
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CREATING AN EXPRESSION TREE
algorithm create_tree
ab+cde+**
{ do until the end of the expression;
{read one value from the expression;
if it is an operand { create a one node tree;
push it to the stack; }
else if it is an operator
{ pop two elements from the stack;
create a tree with the operator as the root;
create a right leaf with the first element;
create a left leaf with the second element;
push the root to the stack;}
}}
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BINARY SEARCH TREE
A binary search tree is a binary tree with the following
properties:
1. All items in the left subtree are less than the root
2. All items in the right subtree are greater than or equal to
the root
3. Each subtree is itself a binary search tree.
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BINARY SEARCH TREE
17
6
3
17
19
14
valid bst
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6
3
19
22
invalid bst
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BINARY SEARCH TREE
23
18
12
44
20
35
52
preorder : 23 18 12 20 44 35 52
inorder:
12 18 20 23 35 44 52
postorder: 12 20 18 35 52 44 23
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BINARY SEARCH TREE
23
18
12
inorder:
44
20
35
52
12 18 20 23 35 44 52
Note: The inorder traversal of a binary search tree
produces an ordered list.
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OPERATIONS ON BINARY SEARCH TREE
The common operations on BST are:
find min. find max. find the requested data
Find minimum is obvious that the leftmost node is the
least among all the nodes of the tree.
algorithm fmin (val root <pointer>)
{ if (root->left ==NULL) return (root);
return fmin(root->left);
}
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Copyright©1999 Angus Wu
OPERATIONS ON BINARY SEARCH TREE
The common operations on BST are:
find min. find max. find the requested data
Find maximum is obvious that the rightmost node is the
largest among all the nodes of the tree.
algorithm fmax (val root <pointer>)
{ if (root->right ==NULL) return (root);
return fmax(root->right);
}
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Copyright©1999 Angus Wu
BINARY SEARCH TREE AND BINARY SEARCH
Search for the letter L
A
A
C
E
E
E
E
G
H
I
L
N
P
Q
R
H
I
L
N
P
Q
R
H
I
L
L
PROGRAMMING METHDOLOGY AND SOFTWARE ENGINEERING
Copyright©1999 Angus Wu
BINARY SEARCH TREE AND BINARY SEARCH
23
18
12
44
20
35
52
PROGRAMMING METHDOLOGY AND SOFTWARE ENGINEERING
Search for 20.
Starts from the root 23
20 < 23 goes to left tree
18 is the root
20 > 18 goes to right tree
Copyright©1999 Angus Wu
BINARY SEARCH TREE AND BINARY SEARCH
algorithm searchBST (val root <pointer>, val arg <key>)
{ if (root is NULL) return NULL;
if (arg < root->key) return searchBST (root->left, arg);
else if (arg > root->key)
return searchBST (root->right, arg);
else
return root;
}
end searchBST;
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Copyright©1999 Angus Wu
BINARY SEARCH TREE AND BINARY SEARCH
23
18
12
44
20
35
52
19
PROGRAMMING METHDOLOGY AND SOFTWARE ENGINEERING
insert 19 into the tree.
root 23
19 < 23, goes to left
root 18
19 > 18, goes to right
root 20
19 < 20, goes to left
since left is null, then
add at that point
Copyright©1999 Angus Wu
BINARY SEARCH TREE AND BINARY SEARCH
23
18
12
44
20
35
52
38
PROGRAMMING METHDOLOGY AND SOFTWARE ENGINEERING
insert 38 into the tree.
root 23
38 > 23 goes to right
root 44
38 < 44 goes to left
root 35
38 > 35 goes to right
since right is null, thus
insert at that point
Copyright©1999 Angus Wu
BINARY SEARCH TREE AND BINARY SEARCH
algorithm insert (ref root <pointer>, val new <pointer>)
{ if (root==NULL)
{ root=new; root->left = NULL; root->right=NULL;}
else
if (new->key < root->key) insert (root->left, new);
else
insert(root->right, new);
return;
}
end insert;
PROGRAMMING METHDOLOGY AND SOFTWARE ENGINEERING
Copyright©1999 Angus Wu