Minimum Path Sum in Grid in DSA Typescript - Time & Space Complexity

Time Complexity: Minimum Path Sum in Grid

O(m x n)

Understanding Time Complexity

We want to know how the time needed to find the minimum path sum in a grid changes as the grid size grows.

How does the number of steps grow when the grid gets bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function minPathSum(grid: number[][]): number {
  const m = grid.length;
  const n = grid[0].length;
  const dp: number[][] = Array.from({ length: m }, () => 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];
}
    

This code finds the smallest sum path from the top-left to the bottom-right of a grid, moving only down or right.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Nested loops that fill the dp table for each cell in the grid.
• How many times: The inner loop runs for each cell except the first row and column, so roughly m x n times.

How Execution Grows With Input

As the grid size grows, the number of operations grows with the number of cells.

Input Size (m x n)	Approx. Operations
10 x 10	About 100 operations
100 x 100	About 10,000 operations
1000 x 1000	About 1,000,000 operations

Pattern observation: Operations grow roughly proportional to the total number of cells, so doubling rows and columns quadruples work.

Final Time Complexity

Time Complexity: O(m x n)

This means the time needed grows proportionally with the total number of cells in the grid.

Common Mistake

[X] Wrong: "Because there are two loops, the time complexity is O(m + n)."

[OK] Correct: The loops are nested, so each cell is processed once, making the total work multiply, not add.

Interview Connect

Understanding this time complexity helps you explain how dynamic programming efficiently solves grid path problems, a common interview topic.

Self-Check

"What if we allowed moving diagonally as well? How would the time complexity change?"

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