Minimum Path Sum in Grid in DSA Typescript

Mental Model

Find the smallest total by moving only right or down from the top-left to the bottom-right of a grid.

Analogy: Imagine walking through a city grid where each block has a cost to cross. You want to find the cheapest path from your home (top-left) to your friend's house (bottom-right) by only moving east or south.

Start:
[1] -> [3] -> [1]
 ↓      ↓      ↓
[1] -> [5] -> [1]
 ↓      ↓      ↓
[4] -> [2] -> [1]

You can only move -> or ↓ to reach the bottom-right.

Dry Run Walkthrough

Input: grid = [[1,3,1],[1,5,1],[4,2,1]]

Goal: Find the minimum sum path from top-left to bottom-right moving only right or down.

Step 1: Start at top-left cell (0,0) with value 1

[1] -> [3] -> [1]
 ↓      ↓      ↓
[1] -> [5] -> [1]
 ↓      ↓      ↓
[4] -> [2] -> [1]

Why: We begin from the starting point with initial cost 1.

Step 2: Calculate minimum path sums for first row by adding left cell sums

[1] -> [4] -> [5]
 ↓      ↓      ↓
[1] -> [5] -> [1]
 ↓      ↓      ↓
[4] -> [2] -> [1]

Why: Only right moves possible in first row, so sums accumulate.

Step 3: Calculate minimum path sums for first column by adding above cell sums

[1] -> [4] -> [5]
 ↓      ↓      ↓
[2] -> [5] -> [1]
 ↓      ↓      ↓
[6] -> [2] -> [1]

Why: Only down moves possible in first column, so sums accumulate.

Step 4: Calculate cell (1,1) as min(top, left) + current value: min(4,2)+5=7

[1] -> [4] -> [5]
 ↓      ↓      ↓
[2] -> [7] -> [1]
 ↓      ↓      ↓
[6] -> [2] -> [1]

Why: Choose cheaper path between coming from top or left.

Step 5: Calculate cell (1,2): min(top, left)+value = min(5,7)+1=6

[1] -> [4] -> [5]
 ↓      ↓      ↓
[2] -> [7] -> [6]
 ↓      ↓      ↓
[6] -> [2] -> [1]

Why: Pick minimum sum path to this cell.

Step 6: Calculate cell (2,1): min(top, left)+value = min(7,6)+2=8

[1] -> [4] -> [5]
 ↓      ↓      ↓
[2] -> [7] -> [6]
 ↓      ↓      ↓
[6] -> [8] -> [1]

Why: Choose minimum path sum to this cell.

Step 7: Calculate cell (2,2): min(top, left)+value = min(6,8)+1=7

[1] -> [4] -> [5]
 ↓      ↓      ↓
[2] -> [7] -> [6]
 ↓      ↓      ↓
[6] -> [8] -> [7]

Why: Final cell sum is minimum path sum to bottom-right.

Result:

[1] -> [4] -> [5]
 ↓      ↓      ↓
[2] -> [7] -> [6]
 ↓      ↓      ↓
[6] -> [8] -> [7]
Minimum path sum = 7

Annotated Code

DSA Typescript

class Solution {
  minPathSum(grid: number[][]): number {
    const m = grid.length;
    const n = grid[0].length;

    // Update first row sums
    for (let j = 1; j < n; j++) {
      grid[0][j] += grid[0][j - 1];
    }

    // Update first column sums
    for (let i = 1; i < m; i++) {
      grid[i][0] += grid[i - 1][0];
    }

    // Update rest of grid with min path sums
    for (let i = 1; i < m; i++) {
      for (let j = 1; j < n; j++) {
        grid[i][j] += Math.min(grid[i - 1][j], grid[i][j - 1]);
      }
    }

    return grid[m - 1][n - 1];
  }
}

// Driver code
const grid = [
  [1, 3, 1],
  [1, 5, 1],
  [4, 2, 1]
];
const sol = new Solution();
console.log(sol.minPathSum(grid));

for (let j = 1; j < n; j++) { grid[0][j] += grid[0][j - 1]; }

Accumulate sums along the first row since only right moves possible

for (let i = 1; i < m; i++) { grid[i][0] += grid[i - 1][0]; }

Accumulate sums along the first column since only down moves possible

for (let i = 1; i < m; i++) { for (let j = 1; j < n; j++) { grid[i][j] += Math.min(grid[i - 1][j], grid[i][j - 1]); } }

For each cell, add the minimum of top or left cell sums to current cell value

return grid[m - 1][n - 1];

Return the minimum path sum at bottom-right cell

Output

7

Complexity Analysis

Time: O(m*n) because we visit each cell once to compute minimum sums

Space: O(1) because we update the input grid in place without extra space

vs Alternative: Compared to recursive solutions without memoization which are exponential, this dynamic programming approach is efficient and uses constant extra space.

Edge Cases

Empty grid

Function returns undefined or error since no cells to process

DSA Typescript

const m = grid.length; const n = grid[0].length;

Single cell grid [[5]]

Returns the single cell value 5 as minimum path sum

DSA Typescript

return grid[m - 1][n - 1];

Grid with one row [[1,2,3]]

Accumulates sums along the row correctly, returns 6

DSA Typescript

for (let j = 1; j < n; j++) { grid[0][j] += grid[0][j - 1]; }

When to Use This Pattern

When you see a grid and need to find a path with minimum or maximum sum moving only right or down, use dynamic programming to build solutions from smaller subproblems.

Common Mistakes

Mistake: Not updating the first row and first column separately before filling the rest of the grid
Fix: Accumulate sums along first row and first column before processing inner cells

Mistake: Using max instead of min when choosing between top and left cells
Fix: Use Math.min to pick the smaller sum path

Summary

Calculates the minimum sum path from top-left to bottom-right in a grid moving only right or down.

Use when you need the cheapest path sum in a grid with movement constraints.

Build the solution by accumulating minimum sums row-wise and column-wise, then fill the rest using the minimum of top or left neighbors.

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 maximum number in the grid

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

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

D. The path with the largest 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 the grid that has the smallest sum of all numbers along the path.

2. Step 2: Clarify path movement

   Movement is only allowed down or right at each step, so the path is constrained.

3. Final Answer:

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

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 constraints (only right/down)

2. Which TypeScript syntax correctly initializes a 2D array named grid with values [[1,2],[3,4]]?

easy

A. const grid = [1,2,3,4];

B. const grid = new Array(1,2,3,4);

C. const grid = [[1,2],[3,4]];

D. const grid = Array(2,2);

Solution

1. Step 1: Understand 2D array initialization

   To create a 2D array with specific values, use nested arrays like [[1,2],[3,4]].

2. Step 2: Check each option

   const grid = [[1,2],[3,4]]; correctly uses nested arrays. Options B and D create arrays but not nested with those values. const grid = [1,2,3,4]; is a flat array.

3. Final Answer:

   const grid = [[1,2],[3,4]]; -> Option C

4. Quick Check:

   Nested arrays create 2D arrays [OK]

Hint: Use nested brackets for 2D arrays in TypeScript [OK]

Common Mistakes:

• Using flat arrays instead of nested arrays
• Using Array constructor incorrectly
• Confusing array length with array content

3. What is the minimum path sum for the grid below?

const grid = [
  [1, 3, 1],
  [1, 5, 1],
  [4, 2, 1]
];

medium

A. 7

B. 8

C. 9

D. 10

Solution

1. Step 1: Trace possible paths and sums

   Possible minimum path: 1 -> 3 -> 1 -> 1 -> 1 = 7 (wrong path, check carefully). Another path: 1 -> 1 -> 5 -> 1 -> 1 = 9. Correct minimum path is 1 -> 3 -> 1 -> 1 -> 1 = 7 is invalid because movement is only right/down. Let's check carefully:
   Path: (0,0)=1 -> (1,0)=1 -> (2,0)=4 -> (2,1)=2 -> (2,2)=1 sum=1+1+4+2+1=9
   Path: (0,0)=1 -> (0,1)=3 -> (0,2)=1 -> (1,2)=1 -> (2,2)=1 sum=1+3+1+1+1=7
   But movement only right/down, so path is valid.
   Check if any smaller sum exists.

2. Step 2: Confirm minimum sum

   The minimum path sum is 8 by moving right, down, down, right.
   Path: (0,0)=1 -> (0,1)=3 -> (1,1)=5 -> (2,1)=2 -> (2,2)=1 sum=1+3+5+2+1=12 (too large)
   Try path: (0,0)=1 -> (1,0)=1 -> (1,1)=5 -> (1,2)=1 -> (2,2)=1 sum=1+1+5+1+1=9
   Try path: (0,0)=1 -> (1,0)=1 -> (2,0)=4 -> (2,1)=2 -> (2,2)=1 sum=1+1+4+2+1=9
   Try path: (0,0)=1 -> (0,1)=3 -> (0,2)=1 -> (1,2)=1 -> (2,2)=1 sum=1+3+1+1+1=7 (valid)
   So minimum sum is 7.

3. Final Answer:

   7 -> Option A

4. Quick Check:

   Sum of path 1->3->1->1->1 = 7 [OK]

Hint: Add numbers along right/down paths, pick smallest sum [OK]

Common Mistakes:

• Adding numbers from invalid paths (not only right/down)
• Forgetting to include starting cell value
• Mixing row and column indices

4. Identify the error in this TypeScript function to find minimum path sum:

function minPathSum(grid: number[][]): number {
  const m = grid.length;
  const n = grid[0].length;
  const dp = Array(m).fill(Array(n).fill(0));
  dp[0][0] = grid[0][0];
  for (let i = 1; i < m; i++) dp[i][0] = dp[i-1][0] + grid[i][0];
  for (let j = 1; j < n; j++) dp[0][j] = dp[0][j-1] + grid[0][j];
  for (let i = 1; i < m; i++) {
    for (let j = 1; j < n; j++) {
      dp[i][j] = Math.min(dp[i-1][j], dp[i][j-1]) + grid[i][j];
    }
  }
  return dp[m-1][n-1];
}

medium

A. grid is not checked for empty input

B. Loop ranges are incorrect, should start from 0

C. Return statement returns wrong cell

D. dp array initialization shares references causing incorrect updates

Solution

1. Step 1: Analyze dp array initialization

   Using Array(m).fill(Array(n).fill(0)) creates m references to the same inner array, so updating one row updates all rows.

2. Step 2: Understand impact on dp updates

   Because all rows share the same array, dp updates overwrite each other, causing wrong results.

3. Final Answer:

   dp array initialization shares references causing incorrect updates -> Option D

4. Quick Check:

   Array.fill with objects shares references [OK]

Hint: Use loops or map to create independent inner arrays [OK]

Common Mistakes:

• Using fill for 2D array causes shared references
• Ignoring empty grid input checks
• Incorrect loop start indices

5. Given a grid:

const grid = [
  [2, 1, 3],
  [6, 5, 4],
  [7, 8, 9]
];

Write a TypeScript function to find the minimum path sum from top-left to bottom-right, moving only right or down. What is the minimum path sum?

hard

A. 19

B. 20

C. 21

D. 18

Solution

1. Step 1: Calculate dp values row-wise and column-wise

   Start dp[0][0] = 2.
   First row: dp[0][1] = 2+1=3, dp[0][2] = 3+3=6.
   First column: dp[1][0] = 2+6=8, dp[2][0] = 8+7=15.

2. Step 2: Fill remaining dp cells using min of top or left + current grid value

   dp[1][1] = min(dp[0][1], dp[1][0]) + 5 = min(3,8)+5=8
   dp[1][2] = min(dp[0][2], dp[1][1]) + 4 = min(6,8)+4=10
   dp[2][1] = min(dp[1][1], dp[2][0]) + 8 = min(8,15)+8=16
   dp[2][2] = min(dp[1][2], dp[2][1]) + 9 = min(10,16)+9=19

3. Final Answer:

   19 -> Option A

4. Quick Check:

   Minimum path sum dp[m-1][n-1] = 19 [OK]

Hint: Use dp table, fill from top-left to bottom-right [OK]

Common Mistakes:

• Adding values incorrectly in dp table
• Mixing up indices for rows and columns
• Not considering only right/down moves