705132. The Light Problem
Here is the Markdown blog post for the "The Light Problem" problem:
The Light Problem
Summary
The Light Problem involves finding the minimum number of lights needed to light up all points in a given grid. The problem requires a combination of logic and algorithmic thinking, making it an interesting challenge for beginners.
Detailed Explanation
To solve this problem, we will iterate through each point in the grid and count the number of lights that can be seen from that point. We then update our total count based on this count. This process is repeated until all points have been processed.
Here's a step-by-step breakdown of the solution:
- Initialize a variable
lightsto 0, which will store the minimum number of lights needed. - Iterate through each row in the grid from top to bottom and for each column from left to right.
- For each point in the grid, initialize a variable
visibleLightsto 0, which will count the number of lights that can be seen from this point. - Check if there are any lights to the left or above the current point. If yes, increment
visibleLights. - Check if there are any lights to the right or below the current point. If yes, increment
visibleLights. - Update the total count of lights (
lights) by adding 1 and the count of visible lights. - Repeat steps 2-6 until all points in the grid have been processed.
The time complexity of this solution is O(m*n), where m is the number of rows and n is the number of columns in the grid. The space complexity is O(1) as we only use a constant amount of space to store our variables.
Optimized Solutions
Java
public class LightProblem {
public static int minLights(int[][] grid) {
int lights = 0;
for (int i = 0; i < grid.length; i++) {
for (int j = 0; j < grid[0].length; j++) {
int visibleLights = 0;
if (i > 0 || j > 0) {
visibleLights++;
}
if (i < grid.length - 1 || j < grid[0].length - 1) {
visibleLights++;
}
lights += 1 + visibleLights;
}
}
return lights;
}
}