Develop recursive algorithm for the following problems.. Find and return the maximum element in an array, where the array and its size are given as parameters.. Develop recursive algorit
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DATA STRUCTURES & ALGORITHMS
Tutorial 3 Questions Recursion and Binary Tree Part 1 Recursion
Required Questions
Question 1
What would be the contents of queue Q1 after the following code is executed and the
following data are entered?
1 Q1 = createQueue
2 S1 = createStack
3 loop (not end of file)
1 read number
2 if (number not 0)
1 pushStack (S1, number)
3 else
1 popStack (S1, x)
2 popStack (S1, x)
3 loop (not empty S1)
1 popStack (S1, x)
2 enqueue (Q1, x)
4 end loop
4 end if
4 end loop
The data are: 9, 5, 10, 4, 0, 5, 4, 6, 8, 67, 32, 25, 51, 0, 54, 23, 20, 6, 10
Question 2
The following algorithm is to reverse a queue
Algorithm reverse (val q <Queue>)
Pre true
Return a reversed queue of q
1 S = createStack
2 Q = createQueue
3 while (not empty(q))
1 dequeue(q,temp)
2 pushStack(S,temp)
4 while (not empty(S))
1 popStack(S,temp)
2 enqueu (Q,temp)
4 return Q
End reverse
Develop a similar algorithm to append a stack to a queue
Algorithm append (val q <Queue>, val s <Stack>)
Pre true
S1=
Q1= 5, 4, 6, 8, 67, 32,9,5
10,6,20,23,54
Algorithm append ((val q <Queue>, val s
<Stack>)) Pre true Return a reversed queue of q
1 S = createStack
2 Q = creatQueue
3 while(not empty(S))
1 popStack(S,temp)
2 enqueue(q,temp) 4
while(not empty(q)) 1
reverse(Q) 5
dequeue(q,temp)
2 enqueue(Q,temp)
6 return Q End append
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Return element of s is appended into q with the same order For
example if q = {1,2,3}, s = {4,5,6} then q = {1,2,3,4,5,6} after
append.
Queue {front rear}
Stack {bottom top}
Question 3
Consider the following algorithm:
Algorithm fun1 (x <integer>)
1 if (x < 5)
1 return (2 * x)
2 else
1 return (2 * fun1 (x – 2) + 5)
3 end if
end fun1
What would be returned if fun1 is called as
a fun1 (4)?
b fun1 (5)?
c fun1 (8)?
d fun1 (20)?
Question 4
Develop recursive algorithm for the following problems
a Compute the sum of all numbers from 1 to n, where n is given as parameter
Algorithm compute (val n <integer>)
Pre n >=0
Return the sum 0 + 1+ 2+ 3+ + n
b Find and return the maximum element in an array, where the array and its size are
given as parameters
Algorithm compute (val a <array>, val n <integer> )
Pre n >=0
Return the maximum element in a[]
Advanced Questions
Question 5
Develop recursive algorithm for the following problems
a Find and return an element in an array, where the array and its size are given as
parameters This element should be in middle position when the array is
re-ordered increasingly
8 17 47 (2*(2*2
Algorithm compute (x <integer>)
1 if (x =0)
1 return (0)
2 else
1 return (n+compute(n-1))
3 end if aánh
Algorithm compute (val a <array>, val n <integer> )
1 if (n=0)
1 return a[0]
2.else return (compute(a,n)>compute(a,n-1))?compute(a,n):compute(a,n-1)
3233
6,5,4 4,5,6
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Algorithm compute (val a <array>, val n <integer> )
Pre n >=0
Return the the element in the middle position when the array
is reordered increasingly in a[]
For example if a = {4,1,5,2,3}, then the value of the last element should
be returned
b Could we design an algorithm for solving question (a) without sorting the array?
Algorithm compute (val a <array>, val n <integer> )
Pre n >=0
Return the the element in the middle position when the array
is reordered increasingly in a[]
Question 6
Develop algorithms for the following problems The algorithms must be fully recursive
in that they contain no loops at all (neither for, while or do-while)
a
Algorithm listnumber (val start <integer>, val end <integer>)
Pre start <=end
Return printout the numbers from start to end to screen
b
Algorithm mul (val a <integer>, val b <integer>)
Pre a,b >=0
Return the product of two integers a and b The only two
problem are addition + and subtraction
-c
Algorithm pow (val a <float>, int b <integer>)
Pre a,b >=0
Return the power a b The only two arithmetic operations that
you are allowed to use in this problem are multiple * and
subtraction
-Question 7
Develop fully recursive algorithms for the functions reverse and append in Question 2
return (start<=end)?end:listnumber(start-1,end)
if(b=1) return a;
else return a+mul(a,b-1) return (b==1)?a:a+mul(a,b-1)
return (b==1)?a:a*pow(a,b-1)
append (Queue*Q,Stack *S) Pre true
Return a reversed queue of q S=new Stack();
Q = new Queue();
while(S->top!=NULL)
popStack(S,temp) enqueue(q,temp) while(Q->front!=NULL) reverse(Q)
dequeue(q,temp) enqueue(Q,temp) End append
return Q
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Part 2 Binary Tree
Required Questions
Question 8
For each of the following key sequences determining the binary search tree obtained
when the keys are inserted one-by-one in the order given into an initially empty tree:
a) 1, 2, 3, 4, 5, 6, 7
b) 4, 2, 1, 3, 6, 5, 7
c) 1, 6, 7, 2, 4, 3, 5
Question 9
For each of the binary search trees obtained in Question 1, determine the tree obtained
when the root is withdrawn
Question 10
Write a global function in pseudocode to generate a BST from an input list by insert
elements in the list into an initial empty BST Refer to Question 1 for an example
algorithm generateBSTfromList (val list <List>)
This algorithm generate a BST from the input list
Pre
Post the BST is built by inserting elements in the list into an initial empty tree
one-by-one from the beginning of the list
Return the BST
end generateBSTfromList
Advanced Questions
Question 11
Devise an algorithm that takes two values, a and b such that a < b, and which visits all
the keys x in a binary search tree such that a x b
1 2 3 4 5 6 7
1
2 3 4
5
6 7
1
2
3 4
6 7 1
2
3 4 6 7 5
2
6 7
4 5
3
1
2 3
5 6 7
1 Allocate subroot
2 subroot->data = DataIn
3 return success
2 else if (DataIn.key < subroot->data.key)
1 return recursive_Insert(subroot->left, DataIn)
3 else if (DataIn.key > subroot->data.key)
1 return recursive_Insert(subroot->right, DataIn)
4 else
1 return duplicate_error
5 End recursive_Insert