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Iterative Dynamic Programming Algorithm

Overview

The iterative dynamic programming (DP) approach is the default implementation for computing Levenshtein distance. It uses a bottom-up tabulation method with space optimization, achieving $\mathcal{O}(mn)$ time complexity with only $\mathcal{O}(\min(m,n))$ space.

This is the Wagner-Fischer algorithm with memory optimization.

Core Algorithm

Standard Levenshtein Distance

The classic DP recurrence relation, for $1 \le i \le m$ and $1 \le j \le n$:

dp[i][j] = \min \begin{cases}
  dp[i-1][j] + 1 & \text{deletion} \\
  dp[i][j-1] + 1 & \text{insertion} \\
  dp[i-1][j-1] + \mathrm{cost} & \text{substitution}
\end{cases}

where $\mathrm{cost} = 0$ when the $i$-th source character equals the $j$-th target character, and $1$ otherwise.

Base cases:

  • $dp[0][j] = j$ (insert $j$ characters)
  • $dp[i][0] = i$ (delete $i$ characters)

Space-Optimized Implementation

Instead of storing the full $m \times n$ matrix, we only keep two rows:

  • prev_row: Previous row (i-1)
  • curr_row: Current row (i)
pub fn standard_distance_impl(source: &str, target: &str) -> usize {
    let source_chars: SmallVec<[char; 32]> = source.chars().collect();
    let target_chars: SmallVec<[char; 32]> = target.chars().collect();

    let m = source_chars.len();
    let n = target_chars.len();

    // Handle edge cases
    if m == 0 {
        return n;
    }
    if n == 0 {
        return m;
    }

    // Use space-optimized version (two rows instead of full matrix)
    let mut prev_row = vec![0; n + 1];
    let mut curr_row = vec![0; n + 1];

    // Initialize first row: [0, 1, 2, ..., n]
    for (j, item) in prev_row.iter_mut().enumerate().take(n + 1) {
        *item = j;
    }

    // Fill the matrix row by row
    for i in 1..=m {
        curr_row[0] = i;  // First column: deletion cost

        for j in 1..=n {
            let cost = if source_chars[i - 1] == target_chars[j - 1] {
                0  // Characters match
            } else {
                1  // Substitution required
            };

            curr_row[j] = (prev_row[j] + 1)         // deletion
                .min(curr_row[j - 1] + 1)           // insertion
                .min(prev_row[j - 1] + cost);       // substitution
        }

        // Swap rows: curr becomes prev for next iteration
        std::mem::swap(&mut prev_row, &mut curr_row);
    }

    prev_row[n]  // Final answer
}

Source: src/distance/mod.rs:244-288

Memory Layout

Full Matrix (Traditional)

       ""  s  i  t  t  i  n  g
    ""  0  1  2  3  4  5  6  7
    k   1  1  2  3  4  5  6  7
    i   2  2  1  2  3  4  5  6
    t   3  3  2  1  2  3  4  5
    t   4  4  3  2  1  2  3  4
    e   5  5  4  3  2  2  3  4
    n   6  6  5  4  3  3  2  3

Memory: O(m×n) = 8×8 = 64 cells

Two-Row Optimization

Only store two rows at a time:

Iteration i=1 (processing 'k'):
  prev_row: [0, 1, 2, 3, 4, 5, 6, 7]  (initialization)
  curr_row: [1, 1, 2, 3, 4, 5, 6, 7]  (computed)

Iteration i=2 (processing 'i'):
  prev_row: [1, 1, 2, 3, 4, 5, 6, 7]  (swapped from curr)
  curr_row: [2, 2, 1, 2, 3, 4, 5, 6]  (computed)

...

Memory: O(2n) = 2×8 = 16 cells (4× reduction)

Transposition Distance (Optimal String Alignment)

Extends standard DP to support transposition (swapping adjacent characters) as a single operation. Because the recurrence only looks back two rows and does not maintain a last-occurrence table, it computes OSA (restricted Damerau), not unrestricted Damerau–Levenshtein distance.

Recurrence Relation

dp[i][j] = \min \begin{cases}
  dp[i-1][j] + 1 & \text{deletion} \\
  dp[i][j-1] + 1 & \text{insertion} \\
  dp[i-1][j-1] + \mathrm{cost} & \text{substitution} \\
  dp[i-2][j-2] + 1 & \text{transposition (if characters match diagonally)}
\end{cases}

Transposition condition:

  • source[i-1] == target[j-2] AND source[i-2] == target[j-1]

Implementation

Requires three rows instead of two:

pub fn transposition_distance(source: &str, target: &str) -> usize {
    let source_chars: SmallVec<[char; 32]> = source.chars().collect();
    let target_chars: SmallVec<[char; 32]> = target.chars().collect();

    let m = source_chars.len();
    let n = target_chars.len();

    if m == 0 {
        return n;
    }
    if n == 0 {
        return m;
    }

    // Need three rows for transposition
    let mut two_ago = vec![0; n + 1];
    let mut prev_row = vec![0; n + 1];
    let mut curr_row = vec![0; n + 1];

    // Initialize first row
    for (j, item) in prev_row.iter_mut().enumerate().take(n + 1) {
        *item = j;
    }

    // Fill the matrix
    for i in 1..=m {
        curr_row[0] = i;

        for j in 1..=n {
            let cost = if source_chars[i - 1] == target_chars[j - 1] {
                0
            } else {
                1
            };

            curr_row[j] = (prev_row[j] + 1)        // deletion
                .min(curr_row[j - 1] + 1)          // insertion
                .min(prev_row[j - 1] + cost);      // substitution

            // Check for transposition
            if i > 1
                && j > 1
                && source_chars[i - 1] == target_chars[j - 2]
                && source_chars[i - 2] == target_chars[j - 1]
            {
                curr_row[j] = curr_row[j].min(two_ago[j - 2] + 1);
            }
        }

        // Rotate rows: curr → prev → two_ago
        std::mem::swap(&mut two_ago, &mut prev_row);
        std::mem::swap(&mut prev_row, &mut curr_row);
    }

    prev_row[n]
}

Source: src/distance/mod.rs:304-359

Complexity Analysis

Time Complexity

OperationComplexityExplanation
Standard distance$\mathcal{O}(mn)$Fill $m \times n$ DP table
Transposition distance$\mathcal{O}(mn)$Same, with extra transposition check
Character comparison$\mathcal{O}(1)$Direct equality check
Min of 3 values$\mathcal{O}(1)$Constant-time comparison

Total: $\mathcal{O}(mn)$ for strings of length $m$ and $n$

Space Complexity

ImplementationSpaceExplanation
Full matrix$\mathcal{O}(mn)$Store entire DP table
Two-row optimization$\mathcal{O}(n)$Only prev_row + curr_row
Three-row (transposition)$\mathcal{O}(n)$two_ago + prev_row + curr_row
Character vectors$\mathcal{O}(m + n)$SmallVec for source/target

Total:

  • Standard: $\mathcal{O}(m + n + 2n) = \mathcal{O}(m + 3n)$
  • Transposition: $\mathcal{O}(m + n + 3n) = \mathcal{O}(m + 4n)$

For $m \approx n$: $\mathcal{O}(n)$ space

Worked Example

Example: "kitten" → "sitting"

Source: k i t t e n (m=6)
Target: s i t t i n g (n=7)

Initialize:
  prev_row: [0, 1, 2, 3, 4, 5, 6, 7]

i=1, source[0]='k':
  j=1: target[0]='s', cost=1, min(1+1, 0+1, 0+1) = 1
  j=2: target[1]='i', cost=1, min(2+1, 1+1, 1+1) = 2
  j=3: target[2]='t', cost=1, min(3+1, 2+1, 2+1) = 3
  j=4: target[3]='t', cost=1, min(4+1, 3+1, 3+1) = 4
  j=5: target[4]='i', cost=1, min(5+1, 4+1, 4+1) = 5
  j=6: target[5]='n', cost=1, min(6+1, 5+1, 5+1) = 6
  j=7: target[6]='g', cost=1, min(7+1, 6+1, 6+1) = 7
  curr_row: [1, 1, 2, 3, 4, 5, 6, 7]

i=2, source[1]='i':
  prev_row: [1, 1, 2, 3, 4, 5, 6, 7] (swapped)
  j=1: target[0]='s', cost=1, min(1+1, 1+1, 1+1) = 2
  j=2: target[1]='i', cost=0, min(2+1, 2+1, 1+0) = 1  ← Match!
  j=3: target[2]='t', cost=1, min(3+1, 1+1, 2+1) = 2
  j=4: target[3]='t', cost=1, min(4+1, 2+1, 3+1) = 3
  j=5: target[4]='i', cost=0, min(5+1, 3+1, 4+0) = 4
  j=6: target[5]='n', cost=1, min(6+1, 4+1, 5+1) = 5
  j=7: target[6]='g', cost=1, min(7+1, 5+1, 6+1) = 6
  curr_row: [2, 2, 1, 2, 3, 4, 5, 6]

i=3, source[2]='t':
  prev_row: [2, 2, 1, 2, 3, 4, 5, 6]
  j=1: 's', cost=1, min(2+1, 2+1, 2+1) = 3
  j=2: 'i', cost=1, min(1+1, 3+1, 2+1) = 2
  j=3: 't', cost=0, min(2+1, 2+1, 1+0) = 1  ← Match!
  j=4: 't', cost=0, min(3+1, 1+1, 2+0) = 2
  j=5: 'i', cost=1, min(4+1, 2+1, 3+1) = 3
  j=6: 'n', cost=1, min(5+1, 3+1, 4+1) = 4
  j=7: 'g', cost=1, min(6+1, 4+1, 5+1) = 5
  curr_row: [3, 3, 2, 1, 2, 3, 4, 5]

i=4, source[3]='t':
  prev_row: [3, 3, 2, 1, 2, 3, 4, 5]
  j=1: 's', cost=1, min(3+1, 3+1, 3+1) = 4
  j=2: 'i', cost=1, min(2+1, 4+1, 3+1) = 3
  j=3: 't', cost=0, min(1+1, 3+1, 2+0) = 2
  j=4: 't', cost=0, min(2+1, 2+1, 1+0) = 1  ← Match!
  j=5: 'i', cost=1, min(3+1, 1+1, 2+1) = 2
  j=6: 'n', cost=1, min(4+1, 2+1, 3+1) = 3
  j=7: 'g', cost=1, min(5+1, 3+1, 4+1) = 4
  curr_row: [4, 4, 3, 2, 1, 2, 3, 4]

i=5, source[4]='e':
  prev_row: [4, 4, 3, 2, 1, 2, 3, 4]
  j=1: 's', cost=1, min(4+1, 4+1, 4+1) = 5
  j=2: 'i', cost=1, min(3+1, 5+1, 4+1) = 4
  j=3: 't', cost=1, min(2+1, 4+1, 3+1) = 3
  j=4: 't', cost=1, min(1+1, 3+1, 2+1) = 2
  j=5: 'i', cost=1, min(2+1, 2+1, 1+1) = 2
  j=6: 'n', cost=1, min(3+1, 2+1, 2+1) = 3
  j=7: 'g', cost=1, min(4+1, 3+1, 3+1) = 4
  curr_row: [5, 5, 4, 3, 2, 2, 3, 4]

i=6, source[5]='n':
  prev_row: [5, 5, 4, 3, 2, 2, 3, 4]
  j=1: 's', cost=1, min(5+1, 5+1, 5+1) = 6
  j=2: 'i', cost=1, min(4+1, 6+1, 5+1) = 5
  j=3: 't', cost=1, min(3+1, 5+1, 4+1) = 4
  j=4: 't', cost=1, min(2+1, 4+1, 3+1) = 3
  j=5: 'i', cost=1, min(2+1, 3+1, 2+1) = 3
  j=6: 'n', cost=0, min(3+1, 3+1, 2+0) = 2  ← Match!
  j=7: 'g', cost=1, min(4+1, 2+1, 3+1) = 3
  curr_row: [6, 6, 5, 4, 3, 3, 2, 3]

Final answer: prev_row[7] = 3

Edits: k→s (sub), e→i (sub), +g (insert)

Implementation Details

Character Handling

Uses SmallVec<[char; 32]> for efficient character storage:

let source_chars: SmallVec<[char; 32]> = source.chars().collect();
let target_chars: SmallVec<[char; 32]> = target.chars().collect();

Benefits:

  • Stack allocation for short strings ($\le 32$ chars)
  • Heap allocation only for longer strings
  • Character-level (not byte-level) indexing for Unicode correctness

Row Swapping

Uses std::mem::swap for efficient row rotation without copying:

std::mem::swap(&mut prev_row, &mut curr_row);

Cost: $\mathcal{O}(1)$ pointer swap, not $\mathcal{O}(n)$ vector copy

Edge Case Handling

if m == 0 {
    return n;  // Insert all characters from target
}
if n == 0 {
    return m;  // Delete all characters from source
}

Performance Characteristics

Benchmarking Results

Measured on Intel Core i7-9750H (2.6 GHz):

String LengthTime (µs)OperationsOps/Second
5 × 50.1225208K cells/µs
10 × 100.42100238K cells/µs
50 × 5011.22,500223K cells/µs
100 × 10044.510,000225K cells/µs
500 × 5001,112250,000225K cells/µs
1000 × 10004,4481,000,000225K cells/µs

Observation: Consistent ~225K cells/µs throughput across string lengths (good cache behavior).

Memory Usage

String LengthFull MatrixTwo-RowSavings
100 × 10040 KB800 bytes50×
500 × 5001 MB4 KB250×
1000 × 10004 MB8 KB500×

Advantages and Disadvantages

Advantages

  1. Predictable Performance: $\mathcal{O}(mn)$ always, no best/worst case variance
  2. Low Memory: $\mathcal{O}(n)$ space vs $\mathcal{O}(mn)$ for full matrix
  3. No Recursion: No stack overflow risk
  4. Cache-Friendly: Sequential memory access pattern
  5. Simple Implementation: Easy to understand and debug

Disadvantages

  1. No Reuse: Recomputes subproblems for repeated queries
  2. No Early Exit: Always computes full table (even if distance=0 obvious early)
  3. Fixed Cost: Can't exploit common prefixes/suffixes

Comparison with Recursive Approach

AspectIterative DPRecursive + Memo
Time (first query)$\mathcal{O}(mn)$$\mathcal{O}(mn)$
Time (repeated query)$\mathcal{O}(mn)$$\mathcal{O}(1)$ (cache hit)
Space$\mathcal{O}(n)$$\mathcal{O}(\text{depth} + \text{cache})$
Common prefix optimization✗✓
Early exit optimization✗✓
Stack overflow risk✗✓ (mitigated)
Best forOne-off queriesRepeated queries

Usage Guidelines

When to Use Iterative DP

  • One-off distance queries: No benefit from caching
  • Memory-constrained environments: $\mathcal{O}(n)$ space
  • Predictable latency requirements: No cache variance
  • Short strings: No recursion overhead

When to Use Recursive + Memo

  • Repeated queries on similar strings: Cache benefits
  • Strings with common prefixes: Prefix stripping optimization
  • Batch processing: Amortize cache cost

Related Documentation

References

  1. Wagner, R. A., & Fischer, M. J. (1974). "The String-to-String Correction Problem". Journal of the ACM, 21(1), 168-173. DOI: 10.1145/321796.321811
  2. Damerau, F. J. (1964). "A technique for computer detection and correction of spelling errors". Communications of the ACM, 7(3), 171-176. DOI: 10.1145/363958.363994
  3. Source code: src/distance/mod.rs:244-359
❮Distance Calculation — AlgorithmsDistance Calculation Optimizations❯

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