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Minimum Path Sum

Minimum Path Sum is a classic problem in computer science, particularly in the area of dynamic programming, which involves finding the path through a grid that minimizes the sum of the values in the cells visited. Let's break down the problem and understand it from first principles.

Problem Statement

Given a two-dimensional grid of non-negative integers, where each cell contains a value, the task is to find a path from the top-left corner to the bottom-right corner which minimizes the sum of the values of the cells on the path. You can only move either down or right at any point in time.

First Principles Analysis

  1. Dynamic Programming (DP): This is a method for solving complex problems by breaking them down into simpler subproblems. It is applicable here because the problem has overlapping subproblems and optimal substructure properties. The solution to the whole problem depends on the solutions to its subproblems.

  2. Optimal Substructure: The problem can be broken down into smaller, similar problems. The optimal solution to the Minimum Path Sum problem involves finding the optimal solutions to the smaller instances of the problem, i.e., the minimum path to reach a cell in the grid depends on the minimum path to reach its adjacent cells to the top and left.

  3. Overlapping Subproblems: In the process of solving the problem using a brute-force approach, you would notice that the solution to a particular cell would be calculated multiple times if approached naively. Dynamic programming solves this by storing the solution to each subproblem in a table (or grid, in this case) so that each subproblem is solved only once.

Solution Strategy

The strategy to solve the Minimum Path Sum problem involves the following steps:

  1. Initialization: First, create a 2D array (let's call it dp) with the same dimensions as the input grid. dp[i][j] will represent the minimum path sum to reach cell (i, j) from the top-left corner.

  2. Base Case: The value at dp[0][0] is the same as the input grid's top-left corner, as there's only one path to that cell (which is the cell itself).

  3. Filling the DP Table: For each cell (i, j) in dp, calculate the minimum path sum to reach that cell. This can be done by taking the minimum of the sum of the cell above (i-1, j) and the cell to the left (i, j-1), then adding the value of the current cell from the input grid. This captures the decision to move either right or down.

    • For the first row and first column, since there's only one path (either right or down), the dp values are cumulative sums of the grid values along the row or column.

  4. Answer: The value at dp[m-1][n-1] (bottom-right corner of the dp table) will give the minimum path sum from the top-left to the bottom-right corner, where m and n are the dimensions of the grid.

Example

Consider a grid:

[1, 3, 1]
[1, 5, 1]
[4, 2, 1]
  • The minimum path sum to reach the bottom-right corner is 7, which corresponds to the path 1 -> 3 -> 1 -> 1 -> 1.

Implementation

This problem is typically solved using a 2D dynamic programming array, but space optimization techniques can reduce the space complexity to linear. The essence of solving it efficiently lies in recognizing the repetitive nature of the subproblems and systematically solving and recording their solutions in a structured manner, thus avoiding recalculating solutions for the same subproblems.

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