Question

Prove that for any segment tree T(l, r) the maximum number of standard intervals that can get allocated for an arbitrary interval [b, e] is equal to log2 n + log2(n - 2) where n = r - I, the total number of elementary intervals. Give an example interval for the segment tree T(1, 16) for which this maximum is achieved.

Answer 1

The maximum number of standard intervals that can be allocated for an arbitrary interval in a segment tree T(l, r) is equal to log2 n + log2(n - 2). An example interval for the segment tree T(1, 16) is [1, 16].

Step-by-step explanation:

Proof

To prove that the maximum number of standard intervals that can be allocated for an arbitrary interval [b, e] in a segment tree T(l, r) is equal to log2 n + log2(n - 2), we need to consider the number of levels in the segment tree.

Each level in the segment tree represents a different division of the interval [l, r]. The root level divides the interval into two equal halves, and each subsequent level divides the previous level's intervals into two equal halves.

Therefore, the number of levels in the segment tree is log2 n + 1. Since we are looking for the number of standard intervals that can be allocated for [b, e], we subtract 1 from the number of levels.

The maximum number of standard intervals is achieved when the interval [b, e] spans the entire segment tree T(l, r). In the given example, T(1, 16), the maximum is achieved when [b, e] is equal to [l, r], which means [b, e] = [1, 16].