Java Program to Count the Number of Ways to Color the Boundary of Each Block in an M×N Table

The role of colouring of the boundary of each block in an M×N grid can be described on the basis of a particular pattern that is used for determining the number of possible ways of coloring the perimeters of cells that incorporate the block.

This kind of problem needs high consideration of the grid structure and the coloring rules occurred on each block. But if we analyze this step by step it would clearly show us how to compute the number of ways to color the boundary of each block in an M×N table.

Problem Explanation

What if instead of one-dimensional or two-dimensional blocks you had M×N of them? Every cell of the grid has four edges, top, bottom, left and right, and each of those edges can be painted in a number of colors equal to C defined beforehand.

The problem is to determine the number of different Paint jobs for all boundaries in the grid of blocks where each side of the block can be painted otherwise. However, it can be seen that the blocks adjacent to each other share edges so the colors on these edges should be same.

Therefore, the issue is shifted to determination of how it is possible to color the boundary of every block and ensure that the coloring is uniform across the entire grid.

Step-by-Step Breakdown

1. Grid Layout:
   • We are given an M×N grid, which means there are M rows and N columns of blocks. Each block has four sides that can be colored.
   • For each block (i, j), the sides are: top, bottom, left, and right.
2. Shared Edges:
   • Each block shares its edges with neighboring blocks. The top side of block (i, j) must be the same as the bottom side of block (i-1, j), and similarly for other shared edges:
     • Right side of (i, j) = Left side of (i, j+1)
     • Bottom side of (i, j) = Top side of (i+1, j)
     • Left side of (i, j) = Right side of (i, j-1)
3. Color Choices:
   • Each edge of a block can be colored using one of C colors. But since neighboring blocks share edges, the actual number of ways to color the grid depends on how many valid colorings there are for the shared edges.
4. Boundary Conditions:
   • Blocks on the boundary of the grid (i.e., the outermost blocks) have some sides that are not shared with any other block. These can be colored independently.

Formula for Coloring the Grid

The total number of ways to color the grid can be determined by:

• First, considering how many ways there are to color each individual block, ignoring edge constraints.
• Then, applying constraints for shared edges between blocks.

Let's assume that each side of the block can be painted in C different colors.

• For a single block: Each block has 4 sides, and each side can independently be colored in C ways. Thus, the number of ways to color the boundary of a single block is C4C^4C4.
• For the entire grid: To compute the number of ways to color the entire grid, taking shared edges into account, we must carefully handle the color constraints imposed by shared boundaries between adjacent blocks.

File Name: GridColoring.java

Output:

Total number of ways to color the boundary of the grid: 68719476736

Complexity Analysis

Time Complexity: O(log (m * n))

Auxiliary Space: O(1)