Question

Problem 2. (25%) This problem is about binomial heap: (a)Suppose a binomial heap H has a total of n nodes. Prove that H has at most [logn] + 1 binomial trees.

(b) Prove that the union operation of two binomial heaps takes O(log n) steps. (c) The binomial heap H₁, which starts from empty, is generated by inserting the odd numbers from 11 to 40 sequentially, and the heap H2, which starts from empty also, is generated by inserting the even numbers from 11 to 40 sequentially. (1)Show the structure of H and H2. (2) Show the structure of the union of Hy and H2-

Answer 1

The question focuses on proving the properties of binomial heaps, showing the maximum number of trees for a heap with 'n' nodes, demonstrating that the union operation takes logarithmic time, and describing the structure of heaps with specific insertions and their union.

Step-by-step explanation:

The student's question pertains to the properties of binomial heaps, a specific data structure in computer science. There are three parts to the problem:

1. Proof that a binomial heap with n nodes has at most [log n] + 1 binomial trees.
2. Proof that the union operation of two binomial heaps takes O(log n) steps.
3. Determination of the structure of binomial heaps H1 and H2 after inserting certain elements and the structure resulting from their union.

For part (a), the maximum number of binomial trees in a binomial heap is determined by the binary representation of the number of nodes, n. Since the height of a binomial tree in a binomial heap corresponds to the powers of 2 that sum up to the number of nodes, the number of trees is limited by the number of bits in the binary representation of n, which is [log n] + 1.

For part (b), the union operation requires merging and possibly linking binomial trees based on degree, which involves comparing and linking at most two trees of the same degree. Considering that the degree of the trees is at most [log n] and each operation is constant time, the union takes O(log n) steps in total.

For part (c), the structure of H1 and H2 would be constructed by successively inserting the given odd and even numbers into two separate binomial heaps. The result of the union operation would be another binomial heap containing all numbers, following the appropriate heap properties.