The recursive memoization approach implements Levenshtein distance computation using top-down recursion with caching (memoization). This is the C++-style implementation that trades memory for speed through aggressive optimizations.
Key Features:
Writing $s_0$ for the first character of $s$ and $s_{1..}$ for its remaining suffix, the edit distance satisfies the recurrence
\mathrm{distance}(s, t) = \begin{cases}
\lvert t\rvert & \text{if } s \text{ is empty} \\
\lvert s\rvert & \text{if } t \text{ is empty} \\
\mathrm{distance}(s_{1..},\ t_{1..}) & \text{if } s_0 = t_0 \\
1 + \min\bigl(\mathrm{distance}(s_{1..},\ t),\ \mathrm{distance}(s,\ t_{1..}),\ \mathrm{distance}(s_{1..},\ t_{1..})\bigr) & \text{otherwise}
\end{cases}
The three arguments to $\min$ are the costs of deletion (advance $s$), insertion (advance $t$), and substitution (advance both), respectively.
pub fn standard_distance_recursive(source: &str, target: &str, cache: &MemoCache) -> usize {
// Check cache first
let cache_key = SymmetricPair::new(source, target);
if let Some(distance) = cache.get(&cache_key) {
return distance; // Cache hit!
}
// Handle base cases
if source.is_empty() {
return target.chars().count();
}
if target.is_empty() {
return source.chars().count();
}
// Strip common prefix and suffix (major optimization)
let (prefix_len, adjusted_source_len, adjusted_target_len) =
strip_common_affixes(source, target);
// If strings are identical after stripping, distance is 0
if adjusted_source_len == 0 && adjusted_target_len == 0 {
cache.insert(cache_key, 0);
return 0;
}
// If one string is fully consumed, distance is remaining chars in other
if adjusted_source_len == 0 {
let result = adjusted_target_len;
cache.insert(cache_key, result);
return result;
}
if adjusted_target_len == 0 {
let result = adjusted_source_len;
cache.insert(cache_key, result);
return result;
}
// Extract the core substrings (after prefix, before suffix)
let source_chars: SmallVec<[char; 32]> = source.chars().collect();
let target_chars: SmallVec<[char; 32]> = target.chars().collect();
let s_remaining: String = source_chars[prefix_len..prefix_len + adjusted_source_len]
.iter()
.collect();
let t_remaining: String = target_chars[prefix_len..prefix_len + adjusted_target_len]
.iter()
.collect();
let a = source_chars[prefix_len];
let b = target_chars[prefix_len];
// Compute substrings for recursion
let s = substring_from(&s_remaining, 1); // source without first char
let t = substring_from(&t_remaining, 1); // target without first char
let mut distance;
if a == b {
// Characters match - no cost
distance = standard_distance_recursive(s, t, cache);
// Early exit optimization
if distance == 0 {
cache.insert(cache_key, distance);
return distance;
}
} else {
// Characters differ - try all three operations
// Deletion: advance source
distance = standard_distance_recursive(s, &t_remaining, cache);
// Early exit
if distance == 0 {
cache.insert(cache_key, 1);
return 1;
}
// Insertion: advance target
let ins_dist = standard_distance_recursive(&s_remaining, t, cache);
distance = distance.min(ins_dist);
// Early exit
if distance == 0 {
cache.insert(cache_key, 1);
return 1;
}
// Substitution: advance both
let sub_dist = standard_distance_recursive(s, t, cache);
distance = distance.min(sub_dist);
distance += 1; // Cost of operation
}
cache.insert(cache_key, distance);
distance
}
Source: src/distance/mod.rs:383-480
Exploits the symmetric property: $d(a,b) = d(b,a)$
struct SymmetricPair {
first: Arc<str>, // Lexicographically smaller
second: Arc<str>, // Lexicographically larger
}
impl SymmetricPair {
fn new(a: &str, b: &str) -> Self {
match a.cmp(b) {
Ordering::Less | Ordering::Equal => Self {
first: Arc::from(a),
second: Arc::from(b),
},
Ordering::Greater => Self {
first: Arc::from(b),
second: Arc::from(a),
},
}
}
}
Benefits:
("test", "best") and ("best", "test") map to same cache keyArc<str> enables efficient cloning without full string copyTwo implementations based on feature flags:
#[cfg(feature = "eviction-dashmap")]
pub struct MemoCache {
cache: DashMap<SymmetricPair, usize>,
}
Pros: Lock-free concurrent access, high throughput Cons: ~24 bytes overhead per entry
#[cfg(not(feature = "eviction-dashmap"))]
pub struct MemoCache {
cache: RwLock<FxHashMap<SymmetricPair, usize>>,
}
Pros: Faster hash function (FxHash), lower memory overhead Cons: Read/write locks (still very fast for mostly-read workloads)
Strip identical characters from both ends before recursion.
pub fn strip_common_affixes(a: &str, b: &str) -> (usize, usize, usize) {
let a_chars: SmallVec<[char; 32]> = a.chars().collect();
let b_chars: SmallVec<[char; 32]> = b.chars().collect();
let len_a = a_chars.len();
let len_b = b_chars.len();
if len_a == 0 || len_b == 0 {
return (0, len_a, len_b);
}
// Find common prefix
let mut prefix_len = 0;
let min_len = len_a.min(len_b);
while prefix_len < min_len && a_chars[prefix_len] == b_chars[prefix_len] {
prefix_len += 1;
}
if prefix_len == min_len {
// One string is a prefix of the other
return (prefix_len, len_a - prefix_len, len_b - prefix_len);
}
// Find common suffix (but don't overlap with prefix)
let mut suffix_len = 0;
while suffix_len < (min_len - prefix_len)
&& a_chars[len_a - 1 - suffix_len] == b_chars[len_b - 1 - suffix_len]
{
suffix_len += 1;
}
(
prefix_len,
len_a - prefix_len - suffix_len,
len_b - prefix_len - suffix_len,
)
}
Example:
Input: "http://example.com/page1" vs "http://example.com/page2"
Prefix: "http://example.com/page" (23 chars)
Suffix: "" (no common suffix)
Core: "1" vs "2" (1 char each)
Recursion depth: 1 instead of 25!
Impact: 5-10× speedup for strings with 50%+ overlap
Return immediately when distance=0 is found.
if a == b {
distance = standard_distance_recursive(s, t, cache);
// Early exit: if remaining distance is 0, we're done
if distance == 0 {
cache.insert(cache_key, distance);
return distance;
}
}
Example:
Input: "test" vs "test"
After stripping prefix "test": both strings empty
→ Return 0 immediately (no recursion!)
Impact: 2-3× speedup for near-identical strings
Subsequent queries on similar strings benefit from cached subproblems.
// First query: computes distance + caches subproblems
let d1 = standard_distance_recursive("testing", "test", &cache);
// Second query: uses cached subproblems from first query
let d2 = standard_distance_recursive("testing", "tested", &cache);
Impact: 100-500× speedup for cache hits (150ns vs 75µs)
| Scenario | Complexity | Explanation |
|---|---|---|
| Worst case (cold cache) | $\mathcal{O}(mn)$ | Explore all subproblems |
| Best case (cache hit) | $\mathcal{O}(1)$ | Instant lookup |
| Common prefix (80% overlap) | $\mathcal{O}(kl)$ | $k, l$ = remaining lengths after stripping |
| Early exit (identical strings) | $\mathcal{O}(\text{prefix length})$ | Linear scan for prefix |
| Component | Space | Explanation |
|---|---|---|
| Recursion stack | $\mathcal{O}(\max(m,n))$ | Worst case depth |
| Cache entries | $\mathcal{O}(\lvert \text{unique pairs}\rvert)$ | ~24 bytes per pair |
| Character vectors | $\mathcal{O}(m + n)$ | Temporary for each call |
Total: $\mathcal{O}(\max(m,n) + \text{cache size})$
With prefix stripping: $\mathcal{O}(\max(k,l) + \text{cache size})$ where $k, l \ll m, n$
Call 1: distance_recursive("kitten", "sitting", cache)
Cache miss: cache_key = ("kitten", "sitting")
strip_common_affixes("kitten", "sitting"):
No common prefix
No common suffix
→ (0, 6, 7)
First chars: 'k' vs 's' (different)
Try deletion: distance_recursive("itten", "sitting", cache)
Cache miss
strip_common_affixes("itten", "sitting"):
Prefix: "" (no match)
Suffix: "tting" (5 chars)
→ (0, 1, 2) // "i" vs "si"
First chars: 'i' vs 's' (different)
Try deletion: distance_recursive("", "si", cache)
→ return 2 (base case)
Try insertion: distance_recursive("i", "i", cache)
Cache miss
strip_common_affixes("i", "i"):
Prefix: "i" (1 char)
→ (1, 0, 0) // Identical after stripping!
→ return 0 (early exit)
min(2, 0) = 0
Early exit: return 1 (0 + 1)
Cache insert: ("itten", "sitting") → 1
distance = 1
Try insertion: distance_recursive("kitten", "itting", cache)
Cache miss
[Similar recursion, computes distance = 2]
Cache insert: ("kitten", "itting") → 2
distance = min(1, 2) = 1
Try substitution: distance_recursive("itten", "itting", cache)
Cache miss
strip_common_affixes("itten", "itting"):
Prefix: "i" (1 char)
Suffix: "tting" (5 chars)
→ (1, 0, 0) // Identical after stripping!
→ return 0 (early exit)
Cache insert: ("itten", "itting") → 0
distance = min(1, 0) = 0
distance = 0 + 1 = 1
[Continue recursion...]
Final: distance = 3
Cache insert: ("kitten", "sitting") → 3
Cache state after this query:
Subsequent query: distance_recursive("sitting", "kitten", cache)
→ Cache hit: return 3 (symmetric pair)
Extends to support transposition (swapping adjacent chars):
pub fn transposition_distance_recursive(source: &str, target: &str, cache: &MemoCache) -> usize {
// ... (same cache check, base cases, prefix stripping)
if a == b {
distance = transposition_distance_recursive(s, t, cache);
// ... early exit
} else {
// Standard operations
distance = transposition_distance_recursive(s, &t_remaining, cache); // deletion
let ins_dist = transposition_distance_recursive(&s_remaining, t, cache); // insertion
distance = distance.min(ins_dist);
let sub_dist = transposition_distance_recursive(s, t, cache); // substitution
distance = distance.min(sub_dist);
// Check for transposition
if !s.is_empty() && !t.is_empty() {
let s_chars: SmallVec<[char; 32]> = s.chars().collect();
let t_chars: SmallVec<[char; 32]> = t.chars().collect();
let a1 = s_chars[0];
let b1 = t_chars[0];
// Transposition: source[0] == target[1] && source[1] == target[0]
if a == b1 && a1 == b {
let ss = substring_from(s, 1);
let tt = substring_from(t, 1);
let trans_dist = transposition_distance_recursive(ss, tt, cache);
distance = distance.min(trans_dist);
}
}
distance += 1;
}
cache.insert(cache_key, distance);
distance
}
Source: src/distance/mod.rs:496-602
| Scenario | Time | Speedup vs Cold |
|---|---|---|
| Cold cache (first query) | 72µs | 1× |
| Warm cache (exact match) | 150ns | 480× |
| Symmetric query | 150ns | 480× |
| Substring query (shared prefix) | 8µs | 9× |
| Prefix Overlap | Time (Recursive) | Time (Iterative) | Speedup |
|---|---|---|---|
| 0% | 75µs | 72µs | 0.96× |
| 25% | 52µs | 72µs | 1.38× |
| 50% | 28µs | 72µs | 2.57× |
| 75% | 12µs | 72µs | 6.00× |
| 90% | 4µs | 72µs | 18.00× |
Key Insight: Recursive wins for strings with significant overlap.
| Cached Pairs | Memory (DashMap) | Memory (RwLock+FxHash) |
|---|---|---|
| 100 | ~2.4 KB | ~1.6 KB |
| 1,000 | ~24 KB | ~16 KB |
| 10,000 | ~240 KB | ~160 KB |
| 100,000 | ~2.4 MB | ~1.6 MB |
Formula: ~24 bytes/pair (DashMap), ~16 bytes/pair (FxHash)
\mathcal{O}(\text{depth})$ stack frames (mitigated by prefix stripping)src/distance/mod.rs:365-730Can you improve this documentation?Edit on GitHub
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