How similar are two strings? It sounds like a simple question, but the answer runs far deeper than you might think. In this article, we'll explore the world of string similarity algorithms, uncover the mathematics behind each one, and see how they solve real-world problems from catching typos to matching financial transactions.
1. Why String Similarity Matters
In the real world, data is rarely perfect. Users make typos. Databases accumulate duplicates. The same customer might appear as "John Smith", "JOHN SMITH", and "Jon Smyth" across three different records. An e-commerce platform might list "iPhone 15 Pro Max" in one catalog and "Apple iPhone15 ProMax" in another. If you rely on exact string matching, these records will never be linked and that's a problem.
This is where string similarity and fuzzy matching come in. These techniques allow us to quantify how "close" two strings are to each other, even when they're not identical. The applications are everywhere: spell checkers suggest corrections when you type "algortihm" instead of "algorithm." Search engines understand that "pythn" probably means "Python." CRM systems merge duplicate customer records despite minor variations in spelling. In bioinformatics, researchers compare DNA sequences to find evolutionary relationships. Fraud detection systems flag suspiciously similar addresses. OCR post-processing corrects misread characters. And record linkage systems connect data across databases that were never designed to talk to each other.
All of these problems boil down to one fundamental question, How similar are these two strings? There are dozens of algorithms that answer this question, and each one has strengths suited to different scenarios. In this article, we'll examine the most important ones in depth.
2. Edit Distance: Measuring Similarity Through Transformation
The edit distance family of algorithms asks a deceptively simple question: what is the minimum number of operations needed to transform one string into another? The definition of "operation" varies by algorithm - it might include insertion, deletion, substitution, or transposition - but the core idea remains the same. The fewer operations required, the more similar the strings are.
2.1 Levenshtein Distance
The Levenshtein Distance, named after Soviet mathematician Vladimir Levenshtein who introduced it in 1965, is the most fundamental and widely used string similarity metric. It measures the minimum number of single-character insertions, deletions, and substitutions required to change one string into the other.
The algorithm works using dynamic programming. It constructs a matrix where each cell represents the edit distance between prefixes of the two input strings. The matrix is filled row by row, with each cell calculated as the minimum of three possible operations: inserting a character (cell above + 1), deleting a character (cell to the left + 1), or substituting a character (diagonal cell + 0 if characters match, +1 if they don't). The final answer sits in the bottom-right corner of the matrix.
Let's walk through a classic example. To transform "kitten" into "sitting", we need exactly 3 operations: substitute 'k' with 's', substitute 'e' with 'i', and insert 'g' at the end. The dynamic programming matrix makes this clear:
"" 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 ]
The value in the bottom-right cell is 3 - that's our Levenshtein Distance.
In practice, raw distance values are hard to compare across string pairs of different lengths. A distance of 3 between two 7-character strings is quite different from a distance of 3 between two 100-character strings. This is why we often use the normalized similarity:
1 - (distance / max(len(s1), len(s2))).
For our example, that gives us 1 - (3/7) = 0.5714, meaning the strings are about 57% similar. For a simple typo like "algorithm" vs "algoritm", the normalized similarity is 1 - (1/9) = 0.8889`- 89% similar, which intuitively makes sense.
The time complexity is O(m × n) where m and n are the string lengths, and with optimization, the space complexity can be reduced to O(min(m, n)) since we only need two rows of the matrix at any time. This makes Levenshtein excellent for short to medium strings names, cities, product titles but potentially slow for very long sequences like paragraphs or DNA strings.
2.2 Damerau-Levenshtein Distance
The standard Levenshtein Distance treats the transposition of two adjacent characters as two separate operations (a deletion and an insertion, or two substitutions). But research has shown that approximately 80% of real-world typographical errors fall into just four categories: wrong character (substitution), extra character (insertion), missing character (deletion), and two adjacent characters swapped (transposition). The Damerau-Levenshtein Distance addresses this by adding transposition as a fourth primitive operation, each costing just 1.
Consider transforming "flom" into "molf." Under standard Levenshtein, this requires multiple operations. Under Damerau-Levenshtein, it's simply one transposition of 'f' and 'm' a distance of just 1. This matters enormously in practice. When a user types "hte" instead of "the," a spell checker using Damerau-Levenshtein recognizes this as a single error, while standard Levenshtein sees two.
There's an important subtlety here: Damerau-Levenshtein has a restricted variant called OSA (Optimal String Alignment). The key difference is that OSA imposes a constraint that no substring may be edited more than once. In the full Damerau-Levenshtein, you can transpose two characters and then further edit the result for example, transforming "CA" to "ABC" takes just 2 operations (transpose CA → AC, then insert B). But under OSA, once you transpose CA → AC, you can't edit that substring again, so the path becomes CA → A → AB → ABC = 3 operations. The full Damerau-Levenshtein satisfies the triangle inequality (making it a true metric), while OSA does not. However, OSA runs in O(m × n) time compared to O(m × n × max(m, n)) for the unrestricted version, so the choice depends on whether you need the mathematical guarantees or prefer faster computation.
2.3 Indel Distance and Hamming Distance
Two other members of the edit distance family deserve brief mention. The Indel Distance only allows insertions and deletions no substitutions. Its name comes from the bioinformatics term "insertion/deletion." It's closely related to the Longest Common Subsequence (LCS): Indel = len(s1) + len(s2) - 2 × LCS_length. For "kitten" vs "sitting," the LCS length is 4, giving an Indel distance of 6 + 7–8 = 5.
The Hamming Distance is the simplest of all, it counts the number of positions where two equal-length strings differ. "karolin" vs "kathrin" has a Hamming distance of 3 (positions 3, 4, and 5 differ). It's only defined for strings of equal length, which limits its general applicability, but it's extremely useful for error-detecting codes, hash comparison, and fixed-length encoded data like barcodes.
3. Character Matching: Jaro and Jaro-Winkler
While edit distance algorithms count the operations needed for transformation, the Jaro family takes a fundamentally different approach. Developed by Matthew A. Jaro in 1989 for the U.S. Census Bureau's record linkage work, the Jaro Similarity is designed specifically for comparing short strings like names and addresses.
3.1 How Jaro Similarity Works
The algorithm operates in three steps. First, it defines a matching window :
characters from the two strings are considered matching only if they're identical and their positions differ by no more than floor(max(|s1|, |s2|) / 2) - 1. Second, it counts the transpositions matching characters that appear in different order in the two strings. Finally, these values are combined using the formula:
Jaro = (1/3) × (m/|s1| + m/|s2| + (m - t)/m)
where m is the number of matching characters and t is half the number of transpositions.
Let's work through "MARTHA" vs "MARHTA." The matching window is floor(6/2) - 1 = 2. All six characters match within this window, so m = 6. However, 'T' and 'H' are in different positions that's one transposition pair, giving t = 1. Plugging into the formula: Jaro = (1/3) × (6/6 + 6/6 + 5/6) = (1/3) × 2.833 = 0.9444.
3.2 The Winkler Extension
n 1990, William E. Winkler extended Jaro's work with a key insight: when two strings share a common prefix, they're more likely to refer to the same entity. This is based on the practical observation that people rarely misspell the first few characters of a name. The Jaro-Winkler formula adds a bonus proportional to the common prefix length (up to 4 characters):
Jaro-Winkler = Jaro + (L × P × (1 - Jaro))
where L is the common prefix length (max 4) and P is a scaling factor (typically 0.1).
For "MARTHA" vs "MARHTA," the common prefix is "MAR" (L = 3). So Jaro-Winkler = 0.9444 + (3 × 0.1 × (1–0.9444)) = 0.9444 + 0.0167 = 0.9611. The prefix bonus pushed the score from 94.4% to 96.1%. This might seem like a small difference, but at scale when you're comparing millions of name pairs it meaningfully improves ranking accuracy.
Jaro-Winkler is the go-to metric for name matching, address comparison, and customer record deduplication. If your problem involves comparing names or short identifiers where the beginning of the string is typically more reliable, Jaro-Winkler is almost certainly better than Levenshtein.
4. Sequence-Based Metrics: LCS and Alignment Algorithms
Some string similarity problems require looking at the longest shared structure between two strings rather than counting edits. This is where sequence-based metrics shine.
4.1 Longest Common Subsequence (LCS)
The LCS finds the longest sequence of characters that appears in both strings in the same order but not necessarily contiguously. This distinction is important. For "ABCBDAB" and "BDCABA," the LCS is "BCBA" (length 4). The characters B, C, B, and A appear in both strings in the same relative order, even though they're not adjacent in either string. The LCS distance is then max(|s1|, |s2|) - LCS_length.
There's a related but distinct concept: the Longest Common Substring (LCSstr), which requires the matching characters to be contiguous. For "the quick brown fox" vs "the quick red fox," the LCS subsequence would be much longer than the LCSstr substring ("the quick "), because the subsequence can skip over non-matching sections while the substring cannot.
LCS computation uses dynamic programming and runs in O(m × n) time, but modern implementations use bit-parallel techniques (processing 64 or 128 characters at once using integer bitmasks) to achieve dramatic speedups on practical inputs.
4.2 Smith-Waterman and Needleman-Wunsch
These two algorithms come from bioinformatics, where they were designed for aligning DNA and protein sequences, but they're equally powerful for general string comparison.
Needleman-Wunsch (1970) performs global alignment: it aligns two entire sequences from start to finish, maximizing the overall similarity score. It uses a scoring system where matches earn positive points, mismatches earn negative points, and gaps (insertions/deletions) incur a penalty. The algorithm fills a dynamic programming matrix and traces back through it to find the optimal alignment. It's ideal when comparing sequences of similar length that you expect to be similar throughout.
Smith-Waterman (1981) performs local alignment: instead of aligning entire sequences, it finds the most similar region within two sequences. The key difference in implementation is that negative scores in the matrix are reset to zero, which allows the algorithm to "restart" and find locally optimal alignments. The result is the highest-scoring subsequence alignment, which is perfect for finding conserved domains in proteins or matching a short query against a long document.
The distinction matters in practice. If you're comparing "the quick brown fox" to "fox brown quick the," Needleman-Wunsch will give a global score reflecting the overall rearrangement, while Smith-Waterman will identify "fox" or "brown" as strongly matching local regions regardless of the surrounding text.
5. Set-Based Metrics: Looking at Token Overlap
All the algorithms we've discussed so far are sequence-sensitive, the order of characters or tokens matters. But in many real-world scenarios, order doesn't matter at all. "Hello World" and "World Hello" mean the same thing. "New York City" and "City York New" refer to the same place. Set-based metrics address this by treating strings as bags of tokens (typically words) and measuring the overlap between these sets.
5.1 Jaccard Similarity
The Jaccard Index, perhaps the most intuitive set similarity metric, measures the size of the intersection divided by the size of the union of two sets:
Jaccard(A, B) = |A ∩ B| / |A ∪ B|
For "hello world" vs "world hello," both token sets are {"hello", "world"}, so the intersection and union are identical Jaccard = 1.0, a perfect match. For "the cat sat" vs "the dog sat," the intersection is {"the", "sat"} (size 2) and the union is {"the", "cat", "sat", "dog"} (size 4), giving Jaccard = 0.5. Half the words are shared, so we get 50% similarity perfectly intuitive.
In practice, many implementations use multiset (Counter) semantics, where word frequency matters. Instead of simple sets, they use intersection = Σ min(count_A(x), count_B(x)) and union = Σ max(count_A(x), count_B(x)). This is important when the same word appears multiple times, as in "the the cat" vs "the dog."
5.2 Sørensen-Dice, Cosine, Tversky, and Overlap
The Sørensen-Dice coefficient is closely related to Jaccard but gives more weight to the intersection: Dice = 2|A∩B| / (|A| + |B|). For the "the cat sat" vs "the dog sat" example, Dice = 2×2 / (3+3) = 0.667, compared to Jaccard's 0.5. Dice is always greater than or equal to Jaccard, and the relationship is Dice = 2 × Jaccard / (1 + Jaccard).
Cosine Similarity takes a different approach entirely. Instead of treating strings as sets, it converts them into term frequency vectors and measures the cosine of the angle between these vectors. For "data science is great" vs "science data is wonderful," we construct vectors over the vocabulary {data, science, is, great, wonderful}. String 1 becomes [1,1,1,1,0] and string 2 becomes [1,1,1,0,1]. The dot product is 3, each vector has magnitude 2, so Cosine = 3/4 = 0.75. Cosine's key advantage over Jaccard is that it considers term frequency (how often a word appears) and is normalized by vector magnitude, making it less sensitive to document length which is why it's the dominant metric in information retrieval and document classification.
The Tversky Index generalizes Jaccard with two asymmetry parameters (α, β): Tversky = |A∩B| / (|A∩B| + α|A\B| + β|B\A|). Setting α = β = 1 gives Jaccard; α = β = 0.5 gives Dice. The asymmetry is useful when comparing a search query (A) against a candidate document (B): you might want to penalize query terms missing from the candidate more heavily than extra terms in the candidate.
The Overlap Coefficient normalizes by the smaller set: |A∩B| / min(|A|, |B|). This measures whether the smaller set is a subset of the larger one. For "hello" vs "hello world test," the overlap is 1/1 = 1.0 - the smaller set is entirely contained in the larger one.
6. Fuzzy Matching: Combining Algorithms for Practical Use
While the individual algorithms above are powerful, real-world string matching often requires combining them intelligently. This is exactly what fuzzy matching libraries do.
The basic ratio computes character-level normalized similarity on a 0–100 scale. For "fuzzy wuzzy" vs "wuzzy fuzzy," it returns just 45.45 because at the character level, these strings are quite different; the characters don't align well positionally.
Token Sort Ratio solves this by tokenizing both strings, sorting the tokens alphabetically, rejoining them, and then computing the ratio. Both "fuzzy wuzzy" and "wuzzy fuzzy" become "fuzzy wuzzy" after sorting, yielding a perfect score of 100.0. This makes it ideal for scenarios where the same information appears in different order like "Ahmet Yılmaz" vs "Yılmaz Ahmet."
Token Set Ratio goes further by decomposing strings into three components: common tokens, tokens unique to string 1, and tokens unique to string 2. It then computes ratios between various combinations of these components and returns the maximum. This handles cases where one string has extra words that don't disqualify it from matching. "The quick brown fox" vs "the quick fox jumped" shares {"quick", "the", "fox"} as common tokens, with "brown" and "jumped" as extras. The ratio between just the common tokens will be very high.
Partial Ratio uses a sliding window approach. It takes the shorter string and slides it across the longer string, computing the ratio at each position and returning the maximum. For "test" vs "this is a test string," the 4-character window slides across positions until it lands on "test" within the longer string yielding a score of 100.0. This is invaluable when searching for whether one string is a substring of another, a common pattern in search and autocomplete.
WRatio (Weighted Ratio) is the most sophisticated fuzz function. It examines the length ratio between the two strings and automatically selects the best strategy. If the strings are similar in length, it uses ratio or token_sort_ratio. If one is much shorter than the other, it switches to partial_ratio. It also considers token_set_ratio for reordered content and applies weights to the results. WRatio embodies the "just give me the best answer" philosophy call one function and get an intelligent similarity score regardless of the input pattern.
QRatio (Quick Ratio) is the minimalist counterpart: it lowercases both strings and returns the basic ratio. No token sorting, no partial matching just speed.
There is no single "best" string similarity algorithm - the right choice depends entirely on your problem. Here's a practical decision guide.
For simple typo detection in short strings, Levenshtein is the most direct and well-understood option. For name matching where the beginning of the string is typically reliable, Jaro-Winkler's prefix bonus gives it a clear edge. When word order varies but the content is the same, Token Sort Ratio eliminates order sensitivity. When you need substring matching is one string contained within another? Partial Ratio's sliding window is purpose-built for the task. For document-level similarity where word frequency matters more than position, Cosine Similarity is the standard. And when you're dealing with transposition errors like "hte" instead of "the," Damerau-Levenshtein recognizes these as single-cost operations. If you're unsure which algorithm to use, WRatio is a strong default it automatically adapts to the input. But understanding the individual algorithms lets you make informed choices when performance or accuracy in specific scenarios matters.
7. Putting It Into Practice with Python: The HyperFuzz Library
Understanding the theory is essential, but ultimately we need to apply these algorithms in code. This is where HyperFuzz [link] comes in - an open-source library that implements all of the algorithms discussed in this article in Rust and exposes them to Python through PyO3. Because all computation happens in native Rust, HyperFuzz achieves dramatically better performance than pure Python implementations while maintaining a clean, Pythonic API.
Getting started is straightforward:
`
pip install hyperfuzz
The distance module provides all the edit distance and character matching metrics:
from hyperfuzz import distance
Levenshtein - the classic edit distance
distance.levenshtein_distance("kitten", "sitting") → 3
distance.levenshtein_normalized_similarity("kitten", "sitting") → 0.5714
distance.levenshtein_normalized_similarity("algorithm", "algoritm") → 0.8889
Jaro & Jaro-Winkler - character matching with prefix bonus
distance.jaro_similarity("MARTHA", "MARHTA") → 0.9444
distance.jaro_winkler_similarity("MARTHA", "MARHTA") → 0.9611
Damerau-Levenshtein vs OSA - transposition handling
distance.damerau_levenshtein_distance("CA", "ABC") → 2
distance.osa_distance("CA", "ABC") → 3
Other distance metrics
distance.indel_distance("kitten", "sitting") → 5
distance.lcs_seq_distance("kitten", "sitting") → 3
Notice how damerau_levenshtein_distance("CA", "ABC") returns 2 while osa_distance("CA", "ABC") returns 3. This is exactly the theoretical difference we discussed earlier: Damerau-Levenshtein allows multiple edits to the same substring (transpose CA→AC, then insert B), while OSA does not.
The fuzz module provides the complete family of fuzzy ratios:
python
from hyperfuzz import fuzz
# Basic ratio - character-level similarity (0-100 scale)
fuzz.ratio("fuzzy wuzzy", "wuzzy fuzzy") # → 45.45
fuzz.ratio("hello", "hello") # → 100.0
# Token Sort - eliminates word order sensitivity
fuzz.token_sort_ratio("fuzzy wuzzy", "wuzzy fuzzy") # → 100.0
fuzz.token_sort_ratio("New York City", "City York New") # → 100.0
# Token Set - handles extra words gracefully
fuzz.token_set_ratio("the cat", "the cat sat on mat") # → 100.0
# Partial Ratio - substring matching via sliding window
fuzz.partial_ratio("test", "this is a test string") # → 100.0
# WRatio - automatic best-strategy selection
fuzz.w_ratio("New York City", "new york city") # → 100.0
# QRatio - quick lowercase ratio
fuzz.q_ratio("hello world", "hello world") # → 100.0
To understand why the basic ratio for "fuzzy wuzzy" vs "wuzzy fuzzy" is just 45.45, consider what happens at the character level: the strings don't align well position-by-position. The 'f' in position 0 of string 1 corresponds to 'w' in string 2, and so on. But token_sort_ratio sorts the tokens alphabetically first both strings become "fuzzy wuzzy" and then the ratio is a trivial 100.0.
Similarly, partial_ratio("test", "this is a test string") returns 100.0 because the algorithm slides a 4-character window across the longer string. When the window reaches position 10 aligning with "test" inside "this is a test string" it finds a perfect match.
python
from hyperfuzz import (
jaccard_similarity,
sorensen_dice_similarity,
cosine_similarity,
overlap_similarity,
tversky_similarity,
)
# Jaccard — intersection over union of token sets
jaccard_similarity("hello world", "world hello") # → 1.0
jaccard_similarity("the cat sat", "the dog sat") # → 0.5
# Sørensen-Dice — more weight on intersection
sorensen_dice_similarity("the cat sat", "the dog sat") # → 0.6667
# Cosine — vector-based similarity, frequency-aware
cosine_similarity("data science", "science data") # → 1.0
# Overlap — subset detection
overlap_similarity("hello", "hello world test") # → 1.0
# Tversky — asymmetric similarity
tversky_similarity("the cat", "the cat sat on a mat")
The Jaccard result for "the cat sat" vs "the dog sat" is 0.5 because the token sets share {"the", "sat"} (2 tokens) out of a union {"the", "cat", "sat", "dog"} (4 tokens): 2/4 = 0.5. The Dice coefficient for the same pair is 0.6667 because it uses a different formula: 2×2 / (3+3) = 4/6. Both are correct they simply weight the intersection differently.
python
from hyperfuzz import (
smith_waterman_score,
needleman_wunsch_score,
lcs_str_similarity,
)
# Smith-Waterman - best local alignment score
smith_waterman_score("the quick brown fox", "the quick red fox")
# Needleman-Wunsch - global alignment score
needleman_wunsch_score("ACGT", "ACGT")
# Longest Common Substring similarity
lcs_str_similarity("the quick brown fox", "the quick red fox")
When you need to compare thousands or millions of string pairs, calling functions one by one becomes a bottleneck not because of algorithm speed, but because of Python function call overhead. HyperFuzz solves this with batch variants that use Rust's Rayon thread pool for native multi-core parallelism:
python
from hyperfuzz import jaccard_similarity_batch
pairs = [
("hello world", "world hello"),
("foo bar", "baz qux"),
("test", "test"),
("Python", "Rust"),
("New York", "new york"),
]
scores = jaccard_similarity_batch(pairs) # → [1.0, 0.0, 1.0, 0.0, 1.0]
The batch functions release Python's GIL (Global Interpreter Lock) through PyO3, allowing Rust threads to run at full efficiency across all CPU cores. For large scale deduplication or matching tasks, this can mean orders-of-magnitude speedups compared to sequential Python loops.
7.1 Real-World Example: Financial Transaction Matching
Let's put everything together with a practical example. Imagine you work at a fintech company and need to match bank transaction descriptions against your accounting system records. The bank might record a transaction as "AMAZON MARKETPLACE EU SARL" while your accounting system has "Amazon Marketplace EU."
python
from hyperfuzz import fuzz, jaccard_similarity
bank_record = "AMAZON MARKETPLACE EU SARL"
accounting = "Amazon Marketplace EU"
print(f"ratio: {fuzz.ratio(bank_record, accounting):.1f}")
print(f"token_sort_ratio: {fuzz.token_sort_ratio(bank_record, accounting):.1f}")
print(f"token_set_ratio: {fuzz.token_set_ratio(bank_record, accounting):.1f}")
print(f"partial_ratio: {fuzz.partial_ratio(bank_record, accounting):.1f}")
print(f"w_ratio: {fuzz.w_ratio(bank_record, accounting):.1f}")
print(f"jaccard: {jaccard_similarity(bank_record, accounting):.4f}")
In this scenario, token_set_ratio and partial_ratio will produce the highest scores. Token Set Ratio recognizes that "AMAZON", "MARKETPLACE", and "EU" are common tokens the extra "SARL" is isolated as a difference but doesn't penalize the core match. Partial Ratio slides the shorter string across the longer one and finds that "Amazon Marketplace EU" is almost entirely contained within "AMAZON MARKETPLACE EU SARL." The basic ratio, by contrast, will be lower because the raw character-level alignment is thrown off by the extra word and case differences.
Understanding which metric to use isn't just academic it directly affects whether your system correctly matches 95% or 99.5% of transactions. In production, many systems run multiple metrics in parallel and use the results as features in a scoring model, rather than relying on a single threshold from a single metric.
Conclusion
String similarity algorithms are indispensable tools in modern software engineering. As we've seen throughout this article, the landscape is rich and varied: Levenshtein and its variants measure edit distance at the character level. Jaro-Winkler excels at name matching with its prefix bonus. Jaccard and Cosine compare token sets and frequency vectors, ignoring word order entirely. Smith-Waterman and Needleman-Wunsch bring the power of sequence alignment from bioinformatics. And the fuzzy matching family ratio, token sort/set, partial ratio, WRatio intelligently combines these primitives into practical, ready-to-use tools.
The key takeaway is that there is no universal "best" algorithm. The right choice depends on your data and your problem. But understanding how each algorithm works what it measures, what it ignores, and where it excels empowers you to make informed decisions that directly impact the accuracy and performance of your systems.
Libraries like HyperFuzz make these algorithms accessible in Python while delivering native Rust performance. Whether you're deduplicating customer records, building a search engine, matching financial transactions, or analyzing biological sequences, the combination of Python's ease of use and Rust's computational speed gives you the best of both worlds.
- Project Repo : https://github.com/ademakdogan/hyperfuzz
- Github : https://github.com/ademakdogan
- Linkedin : https://www.linkedin.com/in/adem-akdo%C4%9Fan-948334177/
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