Problem statement - Circular Intervals
Problem Statement
Chef is the head of a commercial logging industry that recently bought a farm containing N trees. You are given the initial height of the i-th tree, denoted by H_i, and the rate of growth of height as R_i meters per month. All the trees can be considered perfect cylinders of equal radius, which allows us to focus solely on their height when discussing the amount of wood.
In Chef’s country, laws require that a tree must be cut completely to gather wood, and trees with heights strictly lower than L meters cannot be cut. Chef has received an order for W meters of wood and wants to determine the minimum number of months he must wait to fulfill this order. You can assume that the sawing machines used by Chef’s company are very efficient and take negligible time to cut the trees.
Approach
The code idea is to use a binary search technique to find the minimum number of months required to gather at least W meters of wood. Start with two variables representing the lower and upper bounds for the months. For each midpoint value of months, calculate the total height of all trees after that many months, only considering those that meet the minimum height requirement. Adjust the search bounds based on whether the collected wood meets the order requirement. Continue until the optimal number of months is found, then return that value.
Time Complexity
The time complexity of this approach is O(N log(max(H_i + R_i * m))), where N is the number of trees and m is the number of months.
Space Complexity
The space complexity is O(N), primarily for storing the array of trees’ height and growth information.