Edit Distance Problem Levenshtein in DSA Typescript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is the Edit Distance (Levenshtein distance) between two strings?

It is the minimum number of single-character edits (insertions, deletions, or substitutions) required to change one string into the other.

beginner

Which operations are allowed in calculating Levenshtein distance?

Insertion, deletion, and substitution of a single character.

intermediate

What is the base case when using dynamic programming to compute Levenshtein distance?

When one string is empty, the distance is the length of the other string (all insertions or deletions).

intermediate

How does the dynamic programming table for Levenshtein distance get filled?

Each cell represents the minimum edit distance between prefixes of the two strings, calculated using values from neighboring cells representing smaller prefixes.

beginner

Why is Levenshtein distance useful in real life?

It helps in spell checking, DNA sequence analysis, and natural language processing by measuring how similar two strings are.

Which of the following is NOT an allowed operation in Levenshtein distance?

A Transposition

B Deletion

C Substitution

D Insertion

Levenshtein distance allows insertion, deletion, and substitution, but not transposition (swapping two adjacent characters).

What is the Levenshtein distance between "cat" and "cut"?

A 2

B 1

C 3

D 0

Only one substitution (a -> u) is needed, so the distance is 1.

If one string is empty and the other has length 5, what is the Levenshtein distance?

A 0

B 1

C 5

D Cannot be determined

All 5 characters must be inserted or deleted, so the distance is 5.

In dynamic programming for Levenshtein distance, what does each cell in the table represent?

A Minimum edit distance between prefixes of the two strings

B Number of substitutions only

C Length of the longer string

D Number of insertions only

Each cell stores the minimum edit distance between prefixes of the two strings up to that point.

Which real-life application uses Levenshtein distance?

A Rendering graphics

B Sorting numbers

C Calculating factorial

D Spell checking

Spell checking uses Levenshtein distance to find words close to a misspelled word.

Explain how dynamic programming helps compute the Levenshtein distance between two strings.

Think about how you compare smaller parts of the strings step by step.

Describe the three operations allowed in the Edit Distance Problem and how each affects the strings.

Imagine changing one word into another by editing letters.

Practice

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

easy

A. The maximum length of common substring between two strings

B. The number of characters that are the same in both strings

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

D. The total length of both strings combined

Solution

1. Step 1: Understand the definition of Levenshtein distance

   It counts the minimum edits (insertions, deletions, substitutions) needed to convert one string into another.

2. Step 2: Compare with other options

   Options A, C, and D describe different string metrics, not edit distance.

3. Final Answer:

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

4. Quick Check:

   Edit distance = minimum edits [OK]

Hint: Edit distance counts minimal edits to match strings [OK]

Common Mistakes:

• Confusing edit distance with longest common substring
• Thinking it counts only substitutions, not insertions/deletions
• Assuming it measures similarity score instead of edit count

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

easy

A. function levenshtein(s1: string, s2: string): number

B. function levenshtein(s1: number, s2: number): string

C. function levenshtein(s1: string[], s2: string[]): boolean

D. function levenshtein(s1: any, s2: any): void

Solution

1. Step 1: Identify correct parameter and return types

   Levenshtein distance takes two strings and returns a number representing the edit distance.

2. Step 2: Check each option's signature

   function levenshtein(s1: string, s2: string): number matches the expected input and output types. Others mismatch types or return void.

3. Final Answer:

   function levenshtein(s1: string, s2: string): number -> Option A

4. Quick Check:

   Input strings, output number [OK]

Hint: Input strings, output number for edit distance [OK]

Common Mistakes:

• Using wrong parameter types like arrays or numbers
• Returning wrong types like boolean or void
• Using any type without specifying string inputs

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

function levenshtein(s1: string, s2: string): number {
  const dp: number[][] = [];
  for (let i = 0; i <= s1.length; i++) {
    dp[i] = [i];
  }
  for (let j = 0; j <= s2.length; j++) {
    dp[0][j] = j;
  }
  for (let i = 1; i <= s1.length; i++) {
    for (let j = 1; j <= s2.length; 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], dp[i][j - 1], dp[i - 1][j - 1]) + 1;
      }
    }
  }
  return dp[s1.length][s2.length];
}

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

medium

A. 3

B. 4

C. 5

D. 6

Solution

1. Step 1: Initialize DP table for 'kitten' and 'sitting'

   DP table rows = 7, columns = 8. Base cases set for empty prefixes.

2. Step 2: Fill DP table comparing characters and calculating minimum edits

   Calculations show minimum edits needed to convert 'kitten' to 'sitting' is 3 (substitute 'k'->'s', 'e'->'i', insert 'g').

3. Final Answer:

   3 -> Option A

4. Quick Check:

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

Hint: Trace DP table for small strings to find edit count [OK]

Common Mistakes:

• Counting only substitutions, ignoring insertions/deletions
• Off-by-one errors in indexing strings
• Misreading the DP initialization steps

4. Identify the bug in this TypeScript implementation of Levenshtein distance:

function levenshtein(s1: string, s2: string): number {
  const dp: number[][] = [];
  for (let i = 0; i < s1.length; i++) {
    dp[i] = [i];
  }
  for (let j = 0; j < s2.length; j++) {
    dp[0][j] = j;
  }
  for (let i = 1; i <= s1.length; i++) {
    for (let j = 1; j <= s2.length; 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], dp[i][j - 1], dp[i - 1][j - 1]) + 1;
      }
    }
  }
  return dp[s1.length][s2.length];
}

medium

A. The function should return dp[0][0] instead of dp[s1.length][s2.length]

B. The dp array should be initialized with zeros instead of indices

C. The comparison should be s1[i] === s2[j], not s1[i - 1] === s2[j - 1]

D. The loops initializing dp should run up to <= length, not < length

Solution

1. Step 1: Check dp initialization loops

   Loops run from 0 to less than length, but dp needs size length+1 to include empty prefix cases.

2. Step 2: Correct loop ranges to include dp[s1.length] and dp[s2.length]

   Change loops to run up to <= length to properly initialize dp edges.

3. Final Answer:

   The loops initializing dp should run up to <= length, not < length -> Option D

4. Quick Check:

   DP size must cover full string lengths [OK]

Hint: Initialize DP arrays with length+1 for empty string cases [OK]

Common Mistakes:

• Using < length instead of <= length in loops
• Incorrect string index comparisons causing off-by-one errors
• Returning wrong dp cell ignoring full string lengths

5. Given two strings s1 = 'intention' and s2 = 'execution', what is the minimum edit distance using Levenshtein algorithm? Also, explain why this distance is correct.

hard

A. 6

B. 5

C. 4

D. 7

Solution

1. Step 1: Analyze transformations from 'intention' to 'execution'

   Operations include substituting 'i'->'e', 'n'->'x', deleting 't', substituting 'n'->'c', substituting 't'->'u'. Counting these gives 5 edits.

2. Step 2: Confirm minimal edits with DP approach

   DP table confirms minimum edits needed is 5, as fewer edits cannot convert all differing characters.

3. Final Answer:

   5 -> Option B

4. Quick Check:

   Minimum edits = 5 for 'intention' to 'execution' [OK]

Hint: Count substitutions, insertions, deletions carefully [OK]

Common Mistakes:

• Underestimating edits by ignoring deletions
• Counting only substitutions
• Misaligning characters causing wrong counts