Involved Coding Problems

Dynamic Programming

In many occasions the most difficult programming problems arise when an efficient solution requires a dynamic programming class of algorithms. These are both challenging because the solution requires a lot of attention to detail but also the analysis to identify the steps needed are not always evident. Key to this is the observation that every problem is made by subproblems that relate recursively and the subproblem solution may be needed in many other superproblems during computation.

Problem: `"Optimal Path in a Grid"

Description: Given a 2D grid of size m x n, filled with different integer values. Each integer represents the cost of moving to that grid cell. You can only move either down or right at any point in time from the top left corner to the bottom right corner. Your task is to find and implement a dynamic programming solution that determines the minimum cost path.

Constraints:

1. You can only move either down or right from a cell in the grid.
2. The grid size does not exceed 500 x 500.
3. The cost of moving to any cell is represented by a non-negative integer.

The goal is to minimize the total cost incurred to move from the top-left cell of the grid to the bottom-right cell.

Solution

Here is a Python solution using dynamic programming:

def minPathSum(grid):
    m, n = len(grid), len(grid[0])
    dp = [[0]*n for _ in range(m)]
    dp[0][0] = grid[0][0]

    # Initialize the first row
    for i in range(1, n):
        dp[0][i] = dp[0][i-1] + grid[0][i]

    # Initialize the first column
    for i in range(1, m):
        dp[i][0] = dp[i-1][0] + grid[i][0]

    # Fill up the dp table
    for i in range(1, m):
        for j in range(1, n):
            dp[i][j] = min(dp[i-1][j], dp[i][j-1]) + grid[i][j]

    return dp[-1][-1]

In this solution, we first initialize a 2D dp table with the same size as the input grid. The dp[i] [j] represents the minimum cost to reach the cell (i, j). We first fill up the first row and the first column of the dp table, because they only have one way to go (right or down). Then we fill up the rest of the dp table. For each cell, we can either come from the cell above it or the cell to its left, and we choose the one with the smaller cost; there are not other subproblems for each problem at [i] [j]. Finally, we return the value in the bottom-right cell of the dp table, which represents the minimum cost to reach the destination.

Dynamic Programming Revisit

Dynamic Programming (DP) is mainly used for optimization problems, where we seek the best solution among a set of possible solutions. The main idea behind DP is to solve each subproblem only once, store the result of each subproblem to avoid redundant work, and use these stored results to construct solutions for bigger problems.

HOWTO:

1. Identify the subproblems: In the grid problem above, the subproblems are finding the minimum cost path to reach each cell in the grid from the top-left cell.

2. Recursion: Define the value of an optimal solution recursively in terms of smaller subproblems. In the grid problem, the minimum cost to reach a cell is the cost of the cell itself plus the minimum cost to reach either the cell above it or the cell to its left.

3. Compute the value of an optimal solution in a bottom-up fashion: Start with smaller subproblems and use their solutions to solve larger subproblems. In the grid problem, we start from the top-left cell and move right and down, computing the minimum cost path for each cell.

4. Construct an optimal solution to the problem from the computed information: The minimum cost path from the top-left cell to the bottom-right cell is the value in the bottom-right cell of the DP table.

The basics of dynamic programming involve understanding the concept of 'state' and 'transition'. A 'state' represents a subproblem, and a 'transition' is the process of making a decision or a move to go from one state to another. In the grid problem, a 'state' is a cell in the grid, and a 'transition' is a move from one cell to another cell either to its right or down.

IDENTIFICATION

To identify if a problem can be solved using dynamic programming, look for overlapping subproblems and optimal substructure. Overlapping subproblems means that the problem can be broken down into subproblems which are reused several times. Optimal substructure means that the optimal solution of the main problem can be obtained from the optimal solutions of its subproblems.