Minimum Path Sum in Grid in DSA Rust - Deep Dive

Overview - Minimum Path Sum in Grid

What is it?

Minimum Path Sum in Grid is a problem where you find the smallest total sum of numbers on a path from the top-left corner to the bottom-right corner of a grid. You can only move either down or right at any point in time. The goal is to choose a path that results in the lowest possible sum of the numbers you pass through. This helps understand how to optimize choices step-by-step in a grid layout.

Why it matters

This problem teaches how to make the best decisions when moving through a grid, which is common in maps, games, and robot navigation. Without this concept, systems might take longer or cost more by choosing poor paths. It also introduces dynamic programming, a powerful way to solve problems efficiently by remembering past results.

Where it fits

Before this, learners should know basic arrays and loops. After this, they can explore more complex dynamic programming problems like unique paths, longest common subsequence, or grid-based pathfinding algorithms.

Mental Model

Core Idea

Find the cheapest path by adding the smallest sums from the top-left to each cell, using only moves down or right.

Think of it like...

Imagine walking through a city grid where each block has a toll cost. You want to get from your home (top-left) to your friend's house (bottom-right) by paying the least toll, moving only east or south.

Grid example:
┌─────┬─────┬─────┐
│ 1   │ 3   │ 1   │
├─────┼─────┼─────┤
│ 1   │ 5   │ 1   │
├─────┼─────┼─────┤
│ 4   │ 2   │ 1   │
└─────┴─────┴─────┘

Path sums:
┌─────┬─────┬─────┐
│ 1   │ 4   │ 5   │
├─────┼─────┼─────┤
│ 2   │ 7   │ 6   │
├─────┼─────┼─────┤
│ 6   │ 8   │ 7   │
└─────┴─────┴─────┘

Build-Up - 7 Steps

1

Foundation

Understanding the Grid and Moves

Concept: Introduce the grid structure and allowed moves (right and down).

A grid is a 2D array of numbers. You start at the top-left cell (0,0). You can only move right (increase column) or down (increase row). The goal is to reach the bottom-right cell (last row, last column). Each cell has a cost, and you want to find a path with the smallest total cost.

Result

You know how to move through the grid and what the problem asks: find the path with the smallest sum of visited cells.

Understanding allowed moves and the grid layout is essential before trying to find any path or sum.

2

Foundation

Brute Force Path Enumeration

Concept: Explore all possible paths to find the minimum sum by checking every route.

You can try every path from top-left to bottom-right by moving right or down. For each path, add the numbers in the cells visited. Compare all sums and pick the smallest. This method is simple but slow because the number of paths grows quickly with grid size.

Result

You see that brute force works but becomes very slow for bigger grids.

Knowing brute force helps appreciate why we need smarter methods like dynamic programming.

3

Intermediate

Dynamic Programming Table Setup

Concept: Use a table to store minimum sums for each cell to avoid repeated calculations.

Create a 2D table of the same size as the grid. Each cell in this table will hold the minimum path sum to reach that cell. Start by copying the top-left cell value. Then fill the first row by adding the previous cell's sum plus the current grid cell. Similarly, fill the first column. This sets the base for the rest of the table.

Result

You have a table with minimum sums for the first row and column, ready to fill the rest.

Filling the first row and column separately sets a foundation for building the solution step-by-step.

4

Intermediate

Filling the DP Table with Minimum Sums

🤔Before reading on: do you think the minimum sum for a cell depends on the cell above, the cell to the left, or both? Commit to your answer.

Concept: Calculate minimum sums for each cell using the minimum of the top or left neighbor plus current cell cost.

For each cell not in the first row or column, calculate the minimum path sum by taking the smaller of the sums from the cell above or the cell to the left, then add the current grid cell's value. This way, you build the solution from smaller subproblems.

Result

The table now contains the minimum path sums for all cells, with the bottom-right cell holding the final answer.

Understanding that each cell's minimum sum depends only on its top and left neighbors is key to efficient computation.

5

Intermediate

Extracting the Minimum Path Sum Result

Concept: The bottom-right cell in the DP table holds the minimum path sum for the entire grid.

After filling the DP table, look at the value in the bottom-right cell. This value is the smallest sum possible to reach the end of the grid following the allowed moves.

Result

You get the minimum path sum as a single number representing the cheapest path cost.

Knowing where to find the final answer in the DP table completes the problem-solving process.

6

Advanced

Optimizing Space with One-Dimensional Array

🤔Before reading on: do you think we need to store the entire 2D DP table, or can we use less memory? Commit to your answer.

Concept: Reduce memory usage by using a single array to store minimum sums for the current row only.

Instead of a 2D table, use a 1D array with length equal to the number of columns. Update this array row by row. For each cell, update the array value by taking the minimum of the current value (from the previous row) and the value to the left, then add the current grid cell's cost. This saves memory while keeping the logic intact.

Result

You get the same minimum path sum result using less memory.

Knowing how to optimize space without losing correctness is crucial for handling large grids efficiently.

7

Expert

Handling Edge Cases and Negative Costs

🤔Before reading on: do you think the algorithm works if grid cells have negative numbers? Commit to your answer.

Concept: Understand how the algorithm behaves with negative or zero costs and how to adapt it if needed.

The standard algorithm assumes non-negative costs. If negative costs exist, the problem can become more complex, possibly requiring different algorithms like Bellman-Ford for grids with cycles. For zero or positive costs, the DP approach works perfectly. Also, handle empty grids or single-row/column grids carefully to avoid errors.

Result

You know the limits of the algorithm and how to handle special cases safely.

Recognizing assumptions and edge cases prevents bugs and helps choose the right algorithm for different scenarios.

Under the Hood

The algorithm uses dynamic programming to build the solution from the smallest subproblems. It stores the minimum path sum to each cell by relying on previously computed sums from the top and left neighbors. This avoids recalculating paths multiple times, reducing exponential brute force complexity to polynomial time. Memory optimization uses rolling arrays to keep only necessary data.

Why designed this way?

Dynamic programming was chosen because the problem has overlapping subproblems and optimal substructure: the best path to a cell depends only on the best paths to its neighbors. Alternatives like brute force are too slow. The design balances time and space efficiency, making it practical for large grids.

┌───────────────┐
│ Start (0,0)   │
│   │           │
│   ↓           │
│ (i-1,j)       │
│   │           │
│   ↓           │
│ Current (i,j) │
│   ↑           │
│ (i,j-1)       │
└───────────────┘

Calculation:
min_path_sum(i,j) = grid(i,j) + min(min_path_sum(i-1,j), min_path_sum(i,j-1))

Myth Busters - 4 Common Misconceptions

Quick: Do you think you can move diagonally in the grid to get a smaller path sum? Commit yes or no.

Common Belief: You can move diagonally to find a shorter path.

Reality: Only moves to the right or down are allowed; diagonal moves are not permitted.

Why it matters: Allowing diagonal moves changes the problem and can lead to incorrect solutions if not handled properly.

Quick: Do you think the minimum path sum always equals the sum of the smallest numbers in the grid? Commit yes or no.

Common Belief: The minimum path sum is just the sum of the smallest numbers in the grid regardless of path.

Reality: The path must be continuous moving only right or down; you cannot pick arbitrary smallest numbers scattered in the grid.

Why it matters: Ignoring path constraints leads to underestimating the minimum sum and wrong answers.

Quick: Do you think the dynamic programming table must store all paths to find the minimum sum? Commit yes or no.

Common Belief: You must store every possible path to find the minimum sum.

Reality: You only need to store the minimum sum to each cell, not all paths, which saves memory and time.

Why it matters: Trying to store all paths causes huge memory use and inefficiency.

Quick: Do you think the algorithm works unchanged if grid cells have negative numbers? Commit yes or no.

Common Belief: The algorithm works the same with negative numbers in the grid.

Reality: Negative numbers can break the assumptions of the algorithm, possibly causing incorrect results or infinite loops.

Why it matters: Using the algorithm blindly on negative costs can cause wrong answers or crashes.

Expert Zone

1

The order of filling the DP table matters; row-wise or column-wise traversal ensures dependencies are met.

2

Space optimization using a single array works because each cell depends only on the current and previous row, not the entire table.

3

In some cases, reconstructing the actual path requires backtracking from the bottom-right cell using stored decisions.

When NOT to use

Avoid this approach if the grid allows moves other than right and down, or if negative cycles exist. For such cases, graph algorithms like Dijkstra or Bellman-Ford are better suited.

Production Patterns

Used in robotics for path planning on grids, in games for AI movement cost calculation, and in logistics for optimizing routes on grid-like maps.

Connections

Dynamic Programming

Minimum Path Sum is a classic example of dynamic programming applied to grid traversal.

Understanding this problem deepens comprehension of dynamic programming principles like overlapping subproblems and optimal substructure.

Graph Shortest Path Algorithms

The grid can be seen as a graph where each cell is a node connected to neighbors; minimum path sum relates to shortest path problems.

Recognizing grids as graphs helps apply graph algorithms when movement rules become more complex.

Operations Research - Cost Minimization

Minimum path sum mirrors cost minimization problems in operations research where the goal is to find the cheapest route through a network.

Knowing this connection shows how computer science algorithms solve real-world optimization problems in logistics and planning.

Common Pitfalls

#1 Trying to move diagonally or in other directions not allowed.

Wrong approach:

for i in 0..rows {
for j in 0..cols {
// Trying to use diagonal move
dp[i][j] = grid[i][j] + min(dp[i-1][j], dp[i][j-1], dp[i-1][j-1]);
}
}

Correct approach:

for i in 0..rows {
for j in 0..cols {
dp[i][j] = grid[i][j] + min(dp[i-1][j], dp[i][j-1]);
}
}

Root cause:

Misunderstanding allowed moves and adding invalid directions.

#2 Not initializing the first row and first column correctly.

Wrong approach:

dp[0][0] = grid[0][0];
for j in 1..cols {
dp[0][j] = grid[0][j]; // Missing cumulative sum
}
for i in 1..rows {
dp[i][0] = grid[i][0]; // Missing cumulative sum
}

Correct approach:

dp[0][0] = grid[0][0];
for j in 1..cols {
dp[0][j] = dp[0][j-1] + grid[0][j];
}
for i in 1..rows {
dp[i][0] = dp[i-1][0] + grid[i][0];
}

Root cause:

Forgetting that the first row and column sums accumulate from previous cells.

#3 Using the algorithm on grids with negative numbers without adjustments.

Wrong approach:

Applying the same DP logic directly on grids with negative values without checks or alternative algorithms.

Correct approach:

Use graph algorithms like Bellman-Ford or detect negative cycles before applying DP.

Root cause:

Assuming all grid values are non-negative without validating input constraints.

Key Takeaways

Minimum Path Sum in Grid finds the cheapest path from top-left to bottom-right moving only right or down.

Dynamic programming efficiently solves this by building solutions from smaller subproblems and storing intermediate results.

The minimum sum for each cell depends only on the minimum sums of its top and left neighbors plus its own cost.

Space can be optimized by using a single array since only the previous row's data is needed at each step.

Understanding problem constraints like allowed moves and input values is crucial to applying the correct algorithm.

Practice

1. What does the "Minimum Path Sum in Grid" problem ask you to find?
Given a grid of numbers, find the path from top-left to bottom-right with the smallest sum.

easy

A. The path with the largest sum of numbers from top-left to bottom-right

B. The maximum number in the grid

C. The number of paths from top-left to bottom-right

D. The path with the smallest sum of numbers from top-left to bottom-right

Solution

1. Step 1: Understand the problem goal

   The problem asks to find a path from the top-left corner to the bottom-right corner of a grid that has the smallest sum of all numbers along the path.

2. Step 2: Clarify path rules

   Movement is only allowed down or right at any point, so the path must be continuous and only move in these directions.

3. Final Answer:

   The path with the smallest sum of numbers from top-left to bottom-right -> Option D

4. Quick Check:

   Minimum path sum means smallest total sum [OK]

Hint: Look for smallest sum path from top-left to bottom-right [OK]

Common Mistakes:

• Confusing minimum with maximum sum
• Counting number of paths instead of sum
• Ignoring movement restrictions (only right/down)

2. Which Rust code snippet correctly initializes a 2D vector grid with values for the Minimum Path Sum problem?

easy

A. let grid = vec![vec![1, 3, 1], vec![1, 5, 1], vec![4, 2, 1]];

B. let grid = [1, 3, 1, 1, 5, 1, 4, 2, 1];

C. let grid = vec![1, 3, 1; 1, 5, 1; 4, 2, 1];

D. let grid = vec![[1, 3, 1], [1, 5, 1], [4, 2, 1]];

Solution

1. Step 1: Identify correct 2D vector syntax in Rust

   Rust uses vec![vec![...], vec![...], ...] to create a 2D vector. let grid = vec![vec![1, 3, 1], vec![1, 5, 1], vec![4, 2, 1]]; uses this correct syntax.

2. Step 2: Check other options for errors

   let grid = [1, 3, 1, 1, 5, 1, 4, 2, 1]; is a 1D array, not 2D. let grid = vec![1, 3, 1; 1, 5, 1; 4, 2, 1]; uses invalid syntax with semicolons inside vec!. let grid = vec![[1, 3, 1], [1, 5, 1], [4, 2, 1]]; uses nested arrays inside vec! which is invalid.

3. Final Answer:

   let grid = vec![vec![1, 3, 1], vec![1, 5, 1], vec![4, 2, 1]]; -> Option A

4. Quick Check:

   vec![vec![...], vec![...]] creates 2D vector [OK]

Hint: Use vec![vec![...], vec![...]] for 2D vectors in Rust [OK]

Common Mistakes:

• Using semicolons inside vec! macro
• Confusing 1D and 2D vector syntax
• Using arrays inside vec! without vec! wrapper

3. What is the output of this Rust code that calculates the minimum path sum in a 3x3 grid?

fn min_path_sum(grid: &mut Vec) -> i32 {
    let m = grid.len();
    let n = grid[0].len();
    for i in 1..m {
        grid[i][0] += grid[i - 1][0];
    }
    for j in 1..n {
        grid[0][j] += grid[0][j - 1];
    }
    for i in 1..m {
        for j in 1..n {
            grid[i][j] += grid[i - 1][j].min(grid[i][j - 1]);
        }
    }
    grid[m - 1][n - 1]
}

fn main() {
    let mut grid = vec![
        vec![1, 3, 1],
        vec![1, 5, 1],
        vec![4, 2, 1],
    ];
    println!("{}", min_path_sum(&mut grid));
}

medium

A. 9

B. 7

C. 8

D. 10

Solution

1. Step 1: Calculate first column cumulative sums

   grid[1][0] = 1 + 1 = 2, grid[2][0] = 4 + 2 = 6

2. Step 2: Calculate first row cumulative sums

   grid[0][1] = 3 + 1 = 4, grid[0][2] = 1 + 4 = 5

3. Step 3: Calculate remaining cells

   grid[1][1] = 5 + min(4,2) = 5 + 2 = 7, grid[1][2] = 1 + min(5,7) = 1 + 5 = 6, grid[2][1] = 2 + min(7,6) = 2 + 6 = 8, grid[2][2] = 1 + min(6,8) = 1 + 6 = 7

4. Final Answer:

   7 -> Option B

5. Quick Check:

   Minimum path sum = 7 [OK]

Hint: Add min of top or left cell to current cell [OK]

Common Mistakes:

• Not updating first row and column sums correctly
• Adding max instead of min of neighbors
• Mixing indices in loops

4. Identify the bug in this Rust function for minimum path sum and select the fix:

fn min_path_sum(grid: &mut Vec) -> i32 {
    let m = grid.len();
    let n = grid[0].len();
    for i in 1..m {
        grid[i][0] = grid[i - 1][0]; // Bug here
    }
    for j in 1..n {
        grid[0][j] += grid[0][j - 1];
    }
    for i in 1..m {
        for j in 1..n {
            grid[i][j] += grid[i - 1][j].min(grid[i][j - 1]);
        }
    }
    grid[m - 1][n - 1]
}

medium

A. Change grid[i][j] += grid[i - 1][j].min(grid[i][j - 1]); to grid[i][j] = grid[i - 1][j].max(grid[i][j - 1]);

B. Change grid[0][j] += grid[0][j - 1]; to grid[0][j] = grid[0][j - 1];

C. Change grid[i][0] = grid[i - 1][0]; to grid[i][0] += grid[i - 1][0];

D. No bug, code is correct

Solution

1. Step 1: Check first column update

   The line grid[i][0] = grid[i - 1][0]; overwrites the current cell instead of adding the previous sum, so it loses the current cell's value.

2. Step 2: Correct the update to accumulate sums

   It should be grid[i][0] += grid[i - 1][0]; to add the previous cell's sum to the current cell's value.

3. Final Answer:

   Change grid[i][0] = grid[i - 1][0]; to grid[i][0] += grid[i - 1][0]; -> Option C

4. Quick Check:

   First column sums must accumulate [OK]

Hint: Add previous cell value, don't overwrite it [OK]

Common Mistakes:

• Overwriting instead of adding in first column
• Changing correct addition in first row
• Using max instead of min for path sum

5. Given a 4x4 grid:

[[1, 3, 1, 2],
 [1, 5, 1, 1],
 [4, 2, 1, 3],
 [2, 1, 2, 1]]

What is the minimum path sum from top-left to bottom-right moving only right or down?

hard

A. 10

B. 11

C. 12

D. 13

Solution

1. Step 1: Calculate first row and column cumulative sums

   First row: [1, 4, 5, 7]
   First column: [1, 2, 6, 8]

2. Step 2: Calculate inner cells using min of top or left

   Row 1: grid[1][1] = 5 + min(4,2) = 7
   grid[1][2] = 1 + min(5,7) = 6
   grid[1][3] = 1 + min(7,6) = 7
   Row 2: grid[2][1] = 2 + min(7,6) = 8
   grid[2][2] = 1 + min(6,8) = 7
   grid[2][3] = 3 + min(7,7) = 10
   Row 3: grid[3][1] = 1 + min(8,8) = 9
   grid[3][2] = 2 + min(7,9) = 9
   grid[3][3] = 1 + min(10,9) = 10

3. Step 3: Final minimum path sum

   The bottom-right cell value is 10, but we must check carefully:
   Recalculate carefully:
   First row: 1,4,5,7
   First col: 1,2,6,8
   grid[1][1] = 5 + min(4,2) = 5 + 2 = 7
   grid[1][2] = 1 + min(5,7) = 1 + 5 = 6
   grid[1][3] = 1 + min(7,6) = 1 + 6 = 7
   grid[2][1] = 2 + min(7,6) = 2 + 6 = 8
   grid[2][2] = 1 + min(6,8) = 1 + 6 = 7
   grid[2][3] = 3 + min(7,7) = 3 + 7 = 10
   grid[3][1] = 1 + min(8,8) = 1 + 8 = 9
   grid[3][2] = 2 + min(7,9) = 2 + 7 = 9
   grid[3][3] = 1 + min(10,9) = 1 + 9 = 10

4. Final Answer:

   10 -> Option A

5. Quick Check:

   Bottom-right cell after DP = 10 [OK]

Hint: Build sums row-wise and column-wise, then fill inner cells [OK]

Common Mistakes:

• Miscomputing min between top and left cells
• Forgetting to update first row or column sums
• Adding max instead of min