Optimal Path To Target : Dynamic Programming

The DP problems belonging to this category, in its simplest form, looks like below or some kind of variations of it:
  • Given a target find minimum (maximum) cost / path / sum to reach the target.

The solution for this kind of problems, in a very generalized form, would often look like below:
  • Choose optimal (minimal or maximal, as the case may be) path among all possible paths that lead to the current state, and then add value for the current state.
  • routes[curr] = min(routes[curr - 1], routes[curr - 2], ... , routes[curr - k]) + cost[i]
    where current target can be reached only from (curr - 1), (curr - 2), ... (curr - k).
  • Overall the solution would look like this :
    for (int curr = 1; curr <= target; curr++) {
       for (int k = 0; k < waysToReachCurrentTarget.size(); k++) {
        dp[i] = min(dp[curr], dp[waysToReachCurrentTarget[k]] + cost / path ) ;
    return dp[target];

The below problem and its solution beautifully demonstrate this approach:

Problem Statement:

Given a m x n grid filled with non-negative numbers, find a path from top left to bottom right which minimizes the sum of all numbers along its path.
Note: You can only move either down or right at any point in time.
Output: 7
Explanation: Because the path 1→3→1→1→1 minimizes the sum.

Naive Solution:

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Optimized Solution:

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Problem Solving:


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