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- Faculty of Computer Science and Engineering
Department of Computer Science
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
Q1= 5, 4, 6, 8, 67, 32,9,5
1 popStack (S1, x)
2 popStack (S1, x)
3 loop (not empty S1)
S1= 10,6,20,23,54
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 append ((val q , val s
Algorithm reverse (val q )
))
Pre true
Pre true
Return a reversed queue of q
Return a reversed queue of q
1 S = createStack
2 Q = createQueue 1 S = createStack
3 while (not empty(q)) 2 Q = creatQueue
1 dequeue(q,temp) 3 while(not empty(S))
2 pushStack(S,temp)
1 popStack(S,temp)
4 while (not empty(S))
2 enqueue(q,temp)
1 popStack(S,temp)
reverse(Q)
4
2 enqueu (Q,temp)
5 while(not empty(q))
4 return Q
1 dequeue(q,temp)
End reverse
2 enqueue(Q,temp)
6 return Q
Develop a similar algorithm to append a stack to a queue
Algorithm append (val q , val s ) End append
Pre true
1/ 4
- Faculty of Computer Science and Engineering
Department of Computer Science
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}
6,5,4
Question 3. 4,5,6
Consider the following algorithm:
Algorithm fun1 (x )
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)? 8
b. fun1 (5)? 17
c. fun1 (8)? 47
aánh 3233
(2*(2*2
d. fun1 (20)?
Question 4.
Algorithm compute (x )
Develop recursive algorithm for the following problems. 1 if (x =0)
1 return (0)
a. Compute the sum of all numbers from 1 to n, where n is given as parameter.
2 else
Algorithm compute (val n )
1 return (n+compute(n-1))
Pre n >=0
Return the sum 0 + 1+ 2+ 3+ ...+ n 3 end if
b. Find and return the maximum element in an array, where the array and its size are
given as parameters.
Algorithm compute (val a , val n )
Algorithm compute (val a , val n )
Pre n >=0
1. if (n=0)
Return the maximum element in a[]
1 return a[0]
2.else
Advanced Questions
return (compute(a,n)>compute(a,n-1))?compute(a,n):compute(a,n-1)
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.
2/ 4
- Faculty of Computer Science and Engineering
Department of Computer Science
Algorithm compute (val a , val n )
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 , val n )
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 , val end )
Pre start =0
Return the power ab. The only two arithmetic operations that
you are allowed to use in this problem are multiple * and
subtraction -
return (b==1)?a:a*pow(a,b-1)
append (Queue*Q,Stack *S)
Question 7. Pre true
Develop fully recursive algorithms for the functions reverse and aReturn ia reversed queue of q
ppend n Question 2
S=new Stack();
Q = new Queue();
while(S->top!=NULL)
popStack(S,temp)
enqueue(q,temp)
reverse(Q)
while(Q->front!=NULL)
dequeue(q,temp)
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enqueue(Q,temp)
return Q
End append
- Faculty of Computer Science and Engineering
Department of Computer Science
1. if (subroot is NULL)
1. Allocate subroot
Part 2. Binary Tree 2. subroot->data = DataIn
Required Questions 3. return success
2. else if (DataIn.key < subroot->data.key)
Question 8. 1. return recursive_Insert(subroot->left, DataIn)
3. else if (DataIn.key >tsubroot->data.key)
For each of the following key sequences determining he binary search tree obtained
when the keys are inserted one-by-one in 1. return recursive_Insert(subroot->right, DataIn) 4
the order given into an initially empty tree:
1
4. else
6
2
1. return duplicate_error2
1
a) 1, 2, 3, 4, 5, 6, 7. 3 3
5. End recursive_Insert
6
b) 4, 2, 1, 3, 6, 5, 7. 7
1 5
4
2
c) 1, 6, 7, 2, 4, 3, 5.
5
7
4 6
Question 9.
2 1 7
For each of the binary search trees obtained3in Question 1, d2 etermine the tree obtained
6 5
3
when the root is withdrawn. 4 6
2
4
7
5 3
3
Question 10. 5 7
1
Write a global function in pseudocode to generate a BST from an input list 6 insert by
elements in the list into an initial empty BST. Refer to Question 1 for an example. 7
algorithm generateBSTfromList (val 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.
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nguon tai.lieu . vn
