Edit Distance Problem Levenshtein in DSA Javascript - Time & Space Complexity

Time Complexity: Edit Distance Problem Levenshtein

O(m * n)

Understanding Time Complexity

We want to understand how the time needed to find the edit distance between two words grows as the words get longer.

The question is: how does the number of steps change when the input words become bigger?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

function levenshteinDistance(str1, str2) {
  const dp = Array(str1.length + 1).fill(null).map(() => Array(str2.length + 1).fill(0));

  for (let i = 0; i <= str1.length; i++) {
    for (let j = 0; j <= str2.length; j++) {
      if (i === 0) dp[i][j] = j;
      else if (j === 0) dp[i][j] = i;
      else if (str1[i - 1] === str2[j - 1]) dp[i][j] = dp[i - 1][j - 1];
      else dp[i][j] = 1 + Math.min(dp[i - 1][j], dp[i][j - 1], dp[i - 1][j - 1]);
    }
  }
  return dp[str1.length][str2.length];
}
    

This code calculates the minimum number of edits needed to change one word into another using a table to store results.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Nested loops filling a 2D table of size (length of str1 + 1) by (length of str2 + 1).
• How many times: The inner loop runs for each character of str2 for every character of str1, so total steps are roughly (str1.length + 1) * (str2.length + 1).

How Execution Grows With Input

As the words get longer, the number of steps grows by multiplying their lengths.

Input Size (n)	Approx. Operations
10	About 100 steps
100	About 10,000 steps
1000	About 1,000,000 steps

Pattern observation: Doubling the length of both words roughly quadruples the work needed.

Final Time Complexity

Time Complexity: O(m * n)

This means the time needed grows proportionally to the product of the lengths of the two words.

Common Mistake

[X] Wrong: "The algorithm runs in linear time because it just compares characters one by one."

[OK] Correct: The algorithm compares all pairs of characters in two words, so it must check a grid of size m by n, not just a single line.

Interview Connect

Understanding this time complexity helps you explain how dynamic programming solves problems efficiently by avoiding repeated work.

Self-Check

"What if we used recursion without storing results? How would the time complexity change?"

Practice

1. What does the Levenshtein distance between two strings measure?

easy

A. The length difference between the two strings

B. The number of matching characters in the same positions

C. The minimum number of single-character edits needed to change one string into the other

D. The number of vowels in both strings combined

Solution

1. Step 1: Understand the definition of Levenshtein distance

   It measures how many single-character edits (insertions, deletions, or substitutions) are needed to convert one string into another.

2. Step 2: Compare with other options

   Options B, C, and D do not describe edit distance but other unrelated string properties.

3. Final Answer:

   The minimum number of single-character edits needed to change one string into the other -> Option C

4. Quick Check:

   Levenshtein distance = minimum edits [OK]

Hint: Think of edits needed to transform one word into another [OK]

Common Mistakes:

• Confusing edit distance with matching characters count
• Thinking it measures length difference only
• Assuming it counts only substitutions

2. Which of the following JavaScript function signatures correctly represents a function to compute Levenshtein distance between two strings s1 and s2?

easy

A. function levenshtein(s1, s2) { /* code */ }

B. function levenshtein(s1, s2, s3) { /* code */ }

C. function levenshtein() { /* code */ }

D. function levenshtein(s1) { /* code */ }

Solution

1. Step 1: Identify required parameters for Levenshtein function

   The function needs exactly two strings to compare, so it should accept two parameters.

2. Step 2: Check options for correct parameter count

   function levenshtein(s1, s2) { /* code */ } has two parameters (s1, s2), which is correct. Options A, B, and C have incorrect parameter counts.

3. Final Answer:

   function levenshtein(s1, s2) { /* code */ } -> Option A

4. Quick Check:

   Two strings needed as input [OK]

Hint: Levenshtein compares two strings, so function needs two parameters [OK]

Common Mistakes:

• Using wrong number of parameters
• Omitting parameters entirely
• Adding unnecessary extra parameters

3. What is the output of the following JavaScript code snippet?

function min(a, b, c) {
  return Math.min(a, b, c);
}

function levenshtein(s1, s2) {
  const len1 = s1.length;
  const len2 = s2.length;
  const dp = Array(len1 + 1).fill(null).map(() => Array(len2 + 1).fill(0));

  for (let i = 0; i <= len1; i++) dp[i][0] = i;
  for (let j = 0; j <= len2; j++) dp[0][j] = j;

  for (let i = 1; i <= len1; i++) {
    for (let j = 1; j <= len2; j++) {
      if (s1[i - 1] === s2[j - 1]) {
        dp[i][j] = dp[i - 1][j - 1];
      } else {
        dp[i][j] = min(dp[i - 1][j] + 1, dp[i][j - 1] + 1, dp[i - 1][j - 1] + 1);
      }
    }
  }
  return dp[len1][len2];
}

console.log(levenshtein('kitten', 'sitting'));

medium

A. 3

B. 4

C. 5

D. 6

Solution

1. Step 1: Understand the Levenshtein distance calculation for 'kitten' and 'sitting'

   Transform 'kitten' to 'sitting' requires 3 substitutions and 1 insertion: k->s, i->i (same), t->t (same), t->t (same), e->i, n->n (same), plus insert 'g'. Total edits = 3.

2. Step 2: Trace the code logic

   The code correctly implements the dynamic programming approach to calculate the minimum edits, returning 3.

3. Final Answer:

   3 -> Option A

4. Quick Check:

   Levenshtein('kitten', 'sitting') = 3 [OK]

Hint: Count edits: substitutions + insertions + deletions [OK]

Common Mistakes:

• Counting only substitutions
• Ignoring insertions or deletions
• Misindexing string characters

4. Identify the bug in the following Levenshtein distance function and select the fix:

function levenshtein(s1, s2) {
  const len1 = s1.length;
  const len2 = s2.length;
  const dp = Array(len1).fill(null).map(() => Array(len2).fill(0));

  for (let i = 0; i <= len1; i++) dp[i][0] = i;
  for (let j = 0; j <= len2; j++) dp[0][j] = j;

  for (let i = 1; i <= len1; i++) {
    for (let j = 1; j <= len2; j++) {
      if (s1[i - 1] === s2[j - 1]) {
        dp[i][j] = dp[i - 1][j - 1];
      } else {
        dp[i][j] = Math.min(dp[i - 1][j] + 1, dp[i][j - 1] + 1, dp[i - 1][j - 1] + 1);
      }
    }
  }
  return dp[len1][len2];
}

medium

A. Change loops to run from 0 to len1-1 and len2-1

B. Replace Math.min with Math.max in dp calculation

C. Remove the initialization loops for dp first row and column

D. Change dp array initialization to size (len1+1) x (len2+1)

Solution

1. Step 1: Analyze dp array initialization

   The dp array is initialized with size len1 x len2, but the loops access indices up to len1 and len2, causing out-of-bounds errors.

2. Step 2: Fix dp size to (len1+1) x (len2+1)

   Allocating dp with one extra row and column prevents index errors and allows proper initialization of base cases.

3. Final Answer:

   Change dp array initialization to size (len1+1) x (len2+1) -> Option D

4. Quick Check:

   dp size must match loop bounds [OK]

Hint: dp size must be string length + 1 for indexing [OK]

Common Mistakes:

• Not allocating dp with extra row/column
• Using wrong loop ranges causing errors
• Replacing min with max breaks logic

5. Given two strings s1 = 'intention' and s2 = 'execution', what is the Levenshtein distance between them? Use the concept of minimum edits (insertions, deletions, substitutions) to find the answer.

hard

A. 4

B. 6

C. 5

D. 7

Solution

1. Step 1: Analyze differences between 'intention' and 'execution'

   Transform 'intention' to 'execution' involves substitutions and insertions: i->e, n->x, t->c, e->u, n->t, plus insert 'o'. Total edits count to 6.

2. Step 2: Count minimum edits carefully

   Counting substitutions and insertions needed gives 6 as the minimum number of edits.

3. Final Answer:

   6 -> Option B

4. Quick Check:

   Minimum edits between 'intention' and 'execution' = 6 [OK]

Hint: Count each letter change and insertion carefully [OK]

Common Mistakes:

• Underestimating substitutions needed
• Ignoring insertions or deletions
• Confusing with string length difference