Finding the Largest Land for Noddy's House in Toyland

In the city of Toyland, Noddy is on a quest to find the largest piece of land to build his house. The city is lined with houses, each having a unique number and position. To help Noddy, we need to determine which two houses are positioned in such a way that the distance between them is the largest. The result should return the house numbers between which Noddy can build his largest house, with some important constraints to consider.

Problem Overview

The houses in Toyland are arranged in a straight line, and each house has a specific number and a position. Noddy is looking for the largest possible land between two houses. The task is to determine the two house numbers between which Noddy can build his house.

Input Parameters

We are provided with two inputs:

numOfHouse (integer): The total number of houses in the city.
houseList (list): A list of lists where each element contains two integers:
- The first integer represents the house number.
- The second integer represents the position of the house.

Constraints

The number of houses will be greater than 2 and less than $10^6$ .
No two houses will have the same position.
The house numbers are between 1 and $n u m O f Ho u se$ .
The position of each house is greater than 0 and less than $10^6$ .

Example

Input: Output: Explanation:The largest distance between houses is 15, and it is between house numbers 4 and 5, which are positioned at 30 and 15, respectively. The output should return these two house numbers in ascending order.

Approach to Solve the Problem

We need to determine the pair of houses that have the largest distance between them. Here’s the step-by-step approach to solve the problem:

Step 1: Sort the House List by Position

First, we need to sort the houses based on their positions. This is important because the largest distance between two houses will always occur between two houses that are closest in position after sorting.

Step 2: Calculate the Distances Between Consecutive Houses

Once the house list is sorted by position, we calculate the distance between each consecutive pair of houses. The largest distance will give us the house numbers between which Noddy can build the largest house.

Step 3: Return the Pair of House Numbers

Finally, we return the house numbers of the pair that has the largest distance. If there are multiple pairs with the same distance, we return the one that is closest to the reference point (position 0).

Python Code

python

def findLargestLand(numOfHouse, houseList):
# Sort the house list based on position
houseList.sort(key=lambda x: x[1])

max_distance = 0 result = []# Find the pair with the largest distance for i in range(1, len(houseList)): distance = houseList[i][1] – houseList[i-1][1] if distance > max_distance: max_distance = distance result = [houseList[i-1][0], houseList[i][0]]# Return the result in ascending order return sorted(result)# Example usage numOfHouse = 5 houseList = [[3, 7], [1, 9], [2, 0], [5, 15], [4, 30]] print(findLargestLand(numOfHouse, houseList))

Explanation of the Code

Sorting: We first sort the houseList by the position of the houses using Python’s sort() function. This ensures that we can easily calculate distances between consecutive houses.
Distance Calculation: We loop through the sorted list, calculating the distance between each consecutive pair of houses.
Result: Whenever we find a larger distance, we update the max_distance and store the house numbers of the pair with the largest distance.
Returning the Result: Finally, we return the two house numbers sorted in ascending order.

Time Complexity

Sorting the houseList takes $\log n)$ , where $n$ is the number of houses.
Calculating the distances and finding the largest one takes $O (n)$ . Thus, the overall time complexity is $\log n)$ , which is efficient enough for large inputs.

Conclusion

This approach allows Noddy to find the largest piece of land available in Toyland, between two houses that are positioned in the best possible way. The solution is efficient and works well within the given constraints.