The algorithm leverages dynamic programming and modular arithmetic to efficiently calculate the number of ways to build a LEGO wall of height ( n ) and width ( m ). Let's break down the mechanism behind this algorithm step-by-step.

Problem Breakdown

1. Single Row Configurations (f Array):

   • The f array represents the number of ways to fill a row of various widths using LEGO blocks of widths 1, 2, 3, and 4.
   • Initial configurations for widths 0 to 4 are known:
     • f[0] = 0 (no way to build a width 0 row)
     • f[1] = 1 (one way to build a width 1 row: [1])
     • f[2] = 2 (two ways to build a width 2 row: [1,1] or [2])
     • f[3] = 4 (four ways to build a width 3 row: [1,1,1], [1,2], [2,1], [3])
     • f[4] = 8 (eight ways to build a width 4 row: [1,1,1,1], [1,1,2], [1,2,1], [2,1,1], [2,2], [1,3], [3,1], [4])

2. Extending f for Larger Widths:

   • For widths greater than 4, the number of ways to build the row can be derived from the previous four widths:
     • f[k] = f[k-1] + f[k-2] + f[k-3] + f[k-4]
   • This recurrence relation holds because a row of width k can end with a block of width 1, 2, 3, or 4, leaving a subproblem of width k-1, k-2, k-3, or k-4, respectively.

3. Building Multi-Row Walls (pn Array):

   • pn[k] represents the number of ways to build a wall of width k and height n without any constraints on solidness (i.e., it might have vertical gaps).
   • This is computed using pn[k] = (f[k]^n) % MOD because each row is independent, and the number of ways to stack n rows is simply f[k] raised to the power of n, taken modulo ( 10^9 + 7 ).

4. Solid Walls (gn Array):

   • The gn array represents the number of ways to build a solid wall (without vertical gaps) of width k and height n.
   • To compute gn[k], we start by considering pn[k], which includes both solid and non-solid walls.
   • We then subtract the non-solid configurations:
     • For each possible partition of the width k into two parts, we compute the number of ways to build a solid wall of the first part and any wall (solid or non-solid) of the remaining part.
     • gn[k] = pn[k] - sum(gn[i] * pn[k-i] for i in range(1, k)) % MOD
     • This effectively subtracts all configurations where the wall is not solid, ensuring gn[k] only counts solid walls.

Why the Algorithm Works

1. Dynamic Programming:

   • The use of the f, pn, and gn arrays allows the algorithm to build up solutions to larger problems from solutions to smaller subproblems.
   • This avoids redundant calculations and ensures that each required value is computed only once, making the algorithm efficient.

2. Modular Arithmetic:

   • By taking all operations modulo ( 10^9 + 7 ), the algorithm keeps numbers manageable and avoids overflow.
   • This is essential for handling the large numbers that result from exponentiation and large sums.

3. Separation of Concerns:

   • The algorithm separates the problem into manageable parts: computing row configurations, building multi-row configurations, and ensuring solidity.
   • This clear separation makes the logic easier to follow and verify.

Conclusion

The algorithm leverages dynamic programming and modular arithmetic to efficiently compute the number of ways to build a solid LEGO wall of given dimensions. By breaking down the problem into smaller parts and building up solutions, it ensures both correctness and efficiency.