Count the Number of Houses at a Certain Distance II

Problem Description

You are given three positive integers n, x, and y. In a city, there exist houses numbered 1 to n connected by n streets. There is a street connecting the house numbered i with the house numbered i + 1 for all 1 <= i <= n - 1. An additional street connects the house numbered x with the house numbered y. For each k, such that 1 <= k <= n, you need to find the number of pairs of houses (house1, house2) such that the minimum number of streets that need to be traveled to reach house2 from house1 is k. Return a 1-indexed array result of length n where result[k] represents the total number of pairs of houses such that the minimum streets required to reach one house from the other is k.

Key Insights

• The cities form a linear structure with additional connections creating shortcuts.
• The minimum distance between houses can be computed based on their relative positions and the additional connection.
• We can systematically compute the pair counts for each distance from 1 to n using combinatorial logic.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(n)

Solution

To solve the problem, we will utilize a counting approach to determine the number of pairs of houses at each possible distance. The basic idea is to calculate the distance between every pair of houses based on their indices and the additional street. We will maintain a frequency count of how many pairs exist at each distance from 1 to n.

1. Initialize an array result of size n + 1 to hold the count of pairs for each distance k.
2. Loop through each pair of houses (i, j) where i and j range from 1 to n.
3. For each pair, compute the distance based on:
   • The direct distance between i and j.
   • The potential shorter distance using the additional street connecting x and y.
4. Increment the corresponding count in the result array.
5. Return the result array excluding the first element (index 0) since we want a 1-indexed result.

Code Solutions