A meld operation on two instances of a data structure combines them into one single instance of the same data structure. Consider the following data structures:
P: Unsorted doubly linked list with pointers to the head node and tail node of the list.
Q: Min-heap implemented using an array.
R: Binary Search Tree.
Which ONE of the following options gives the worst-case time complexities for meld operation on instances of size \( n \) of these data structures?
A meld operation on two instances of a data structure combines them into one single instance of the same data structure. Consider the following data structures:
P: Unsorted doubly linked list with pointers to the head node and tail node of the list.
Q: Min-heap implemented using an array.
R: Binary Search Tree.
Which ONE of the following options gives the worst-case time complexities for meld operation on instances of size \( n \) of these data structures?
Show Hint
- \( P: \Theta(1), \quad Q: \Theta(n), \quad R: \Theta(n) \)
- \( P: \Theta(1), \quad Q: \Theta(n \log n), \quad R: \Theta(n) \)
- \( P: \Theta(n), \quad Q: \Theta(n \log n), \quad R: \Theta(n^2) \)
- \( P: \Theta(1), \quad Q: \Theta(n), \quad R: \Theta(n \log n) \)
The Correct Option is A
Solution and Explanation
- Min-Heap Implemented Using an Array (Q): Merging two heaps requires inserting all elements of one heap into the other, which takes \( O(n) \) time in the worst case. Inserting an element into a heap typically takes \( O(\log n) \) time, and since all elements from the second heap need to be inserted, the total time complexity is \( O(n) \). Hence, the worst-case time complexity is \( \mathbf{\Theta(n)} \).
- Binary Search Tree (R): In the worst case (e.g., skewed BSTs), merging two BSTs requires an inorder traversal to collect all elements of both trees and then reconstructing a new BST. The inorder traversal takes \( O(n) \) time, and reconstructing the tree from the sorted list of elements also takes \( O(n) \) time. Thus, the worst-case time complexity is \( \mathbf{\Theta(n)} \).
Thus, the worst-case complexities are \( \mathbf{\Theta(1)} \) for P, \( \mathbf{\Theta(n)} \) for Q, and \( \mathbf{\Theta(n)} \) for R, which matches option (A).
Top Questions on Trees
Consider the following algorithm someAlgo that takes an undirected graph \( G \) as input.
someAlgo(G) Let \( v \) be any vertex in \( G \).
1. Run BFS on \( G \) starting at \( v \). Let \( u \) be a vertex in \( G \) at maximum distance from \( v \) as given by the BFS.
2. Run BFS on \( G \) again with \( u \) as the starting vertex. Let \( z \) be the vertex at maximum distance from \( u \) as given by the BFS. 3. Output the distance between \( u \) and \( z \) in \( G \).
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\( A = (49, 77, 83, -) \)
\( B = (7, 19, 33, 44) \)
\( C = (20^*, 22^*, 25^*, 26^*) \)
The *-marked keys signify that these are data entries in a leaf. Assume that a pointer between keys \( k_1 \) and \( k_2 \) points to a subtree containing keys in \([ k_1, k_2 )\), and that when a leaf is created, the smallest key in it is copied up into its parent. A record with key value 23 is inserted into the B+- tree. The smallest key value in the parent of the leaf that contains 25* is __________ . (Answer in integer)Suppose the values 10, −4, 15, 30, 20, 5, 60, 19 are inserted in that order into an initially empty binary search tree. Let \( T \) be the resulting binary search tree. The number of edges in the path from the node containing 19 to the root node of \( T \) is __________. (Answer in integer)
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What is the output of the following C code?
void foo(int *p, int x) { *p = x; } void main() { int *z; int a = 20, b = 25; z = a; // Incorrect: Should be z = a; foo(z, b); printf("%d", a); }Issue: The statement
z = a;is invalid becauseais an integer, andzis a pointer.
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Consider the following C program:
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