docs(frontend): adding a use-case for Levenshtein distance

closes #zama-ai/concrete-internal#750

docs/tutorials/see-all-tutorials.md

@@ -14,6 +14,7 @@
 * [Game of Life](../../frontends/concrete-python/examples/game_of_life/game_of_life.md)
 * [XOR distance](../../frontends/concrete-python/examples/xor_distance/xor_distance.md)
 * [SHA1 with Modules](../../frontends/concrete-python/examples/sha1/sha1.md)
+* [Levenshtein distance with Modules](../../frontends/concrete-python/examples/levenshtein_distance/levenshtein_distance.md)
 
 #### Blog tutorials
 

frontends/concrete-python/examples/levenshtein_distance/levenshtein_distance.md

@@ -0,0 +1,210 @@
+# Computing the Levenshtein distance in FHE
+
+## Levenshtein distance
+
+Levenshtein distance is a classical distance to compare two strings. Let's write strings a and b as
+vectors of characters, meaning a[0] is the first char of a and a[1:] is the rest of the string.
+Levenshtein distance is defined as:
+
+    Levenshtein(a, b) :=
+        length(a) if length(b) == 0, or
+        length(b) if length(a) == 0, or
+        Levenshtein(a[1:], b[1:]) if a[0] == b[0], or
+        1 + min(Levenshtein(a[1:], b), Levenshtein(a, b[1:]), Levenshtein(a[1:], b[1:]))
+
+More information can be found for example on the [Wikipedia page](https://en.wikipedia.org/wiki/Levenshtein_distance).
+
+## Computing the distance in FHE
+
+It can be interesting to compute this distance over encrypted data, for example in the banking sector.
+We show in [our code](levenshtein_distance.py) how to do that simply, with our FHE modules.
+
+Available options are:
+
+```
+usage: levenshtein_distance.py [-h] [--show_mlir] [--show_optimizer] [--autotest] [--autoperf] [--distance DISTANCE DISTANCE]
+                               [--alphabet {string,STRING,StRiNg,ACTG}] [--max_string_length MAX_STRING_LENGTH]
+
+Levenshtein distance in Concrete.
+
+optional arguments:
+  -h, --help            show this help message and exit
+  --show_mlir           Show the MLIR
+  --show_optimizer      Show the optimizer outputs
+  --autotest            Run random tests
+  --autoperf            Run benchmarks
+  --distance DISTANCE DISTANCE
+                        Compute a distance
+  --alphabet {string,STRING,StRiNg,ACTG}
+                        Setting the alphabet
+  --max_string_length MAX_STRING_LENGTH
+                        Setting the maximal size of strings
+```
+
+The different alphabets are:
+- string: non capitalized letters, ie `[a-z]*`
+- STRING: capitalized letters, ie `[A-Z]*`
+- StRiNg: non capitalized letters and capitalized letters
+- ACTG: `[ACTG]*`, for DNA analysis
+
+It is very easy to add a new alphabet in the code.
+
+The most important usages are:
+
+- `python levenshtein_distance.py --distance Zama amazing --alphabet StRiNg --max_string_length 7`: Compute the distance between
+strings "Zama" and "amazing", considering the chars of "StRiNg" alphabet
+
+```
+
+Running distance between strings 'Zama' and 'amazing' for alphabet StRiNg:
+
+    Computing Levenshtein between strings 'Zama' and 'amazing' - distance is 5, computed in 44.51 seconds
+
+Successful end
+```
+
+- `python levenshtein_distance.py --autotest`: Run random tests with the alphabet.
+
+```
+Making random tests with alphabet string
+Letters are abcdefghijklmnopqrstuvwxyz
+
+Computations in simulation
+
+    Computing Levenshtein between strings '' and '' - OK
+    Computing Levenshtein between strings '' and 'p' - OK
+    Computing Levenshtein between strings '' and 'vv' - OK
+    Computing Levenshtein between strings '' and 'mxg' - OK
+    Computing Levenshtein between strings '' and 'iuxf' - OK
+    Computing Levenshtein between strings 'k' and '' - OK
+    Computing Levenshtein between strings 'p' and 'g' - OK
+    Computing Levenshtein between strings 'v' and 'ky' - OK
+    Computing Levenshtein between strings 'f' and 'uoq' - OK
+    Computing Levenshtein between strings 'f' and 'kwfj' - OK
+    Computing Levenshtein between strings 'ut' and '' - OK
+    Computing Levenshtein between strings 'pa' and 'g' - OK
+    Computing Levenshtein between strings 'bu' and 'sx' - OK
+    Computing Levenshtein between strings 'is' and 'diy' - OK
+    Computing Levenshtein between strings 'fz' and 'unda' - OK
+    Computing Levenshtein between strings 'sem' and '' - OK
+    Computing Levenshtein between strings 'dbr' and 'o' - OK
+    Computing Levenshtein between strings 'dgj' and 'hk' - OK
+    Computing Levenshtein between strings 'ejb' and 'tfo' - OK
+    Computing Levenshtein between strings 'afa' and 'ygqo' - OK
+    Computing Levenshtein between strings 'lhcc' and '' - OK
+    Computing Levenshtein between strings 'uoiu' and 'u' - OK
+    Computing Levenshtein between strings 'tztt' and 'xo' - OK
+    Computing Levenshtein between strings 'ufsa' and 'mil' - OK
+    Computing Levenshtein between strings 'uuzl' and 'dzkr' - OK
+
+Computations in FHE
+
+    Computing Levenshtein between strings '' and '' - OK in 1.29 seconds
+    Computing Levenshtein between strings '' and 'p' - OK in 0.26 seconds
+    Computing Levenshtein between strings '' and 'vv' - OK in 0.26 seconds
+    Computing Levenshtein between strings '' and 'mxg' - OK in 0.22 seconds
+    Computing Levenshtein between strings '' and 'iuxf' - OK in 0.22 seconds
+    Computing Levenshtein between strings 'k' and '' - OK in 0.22 seconds
+    Computing Levenshtein between strings 'p' and 'g' - OK in 1.09 seconds
+    Computing Levenshtein between strings 'v' and 'ky' - OK in 1.93 seconds
+    Computing Levenshtein between strings 'f' and 'uoq' - OK in 3.09 seconds
+    Computing Levenshtein between strings 'f' and 'kwfj' - OK in 3.98 seconds
+    Computing Levenshtein between strings 'ut' and '' - OK in 0.25 seconds
+    Computing Levenshtein between strings 'pa' and 'g' - OK in 1.90 seconds
+    Computing Levenshtein between strings 'bu' and 'sx' - OK in 3.52 seconds
+    Computing Levenshtein between strings 'is' and 'diy' - OK in 5.04 seconds
+    Computing Levenshtein between strings 'fz' and 'unda' - OK in 6.53 seconds
+    Computing Levenshtein between strings 'sem' and '' - OK in 0.22 seconds
+    Computing Levenshtein between strings 'dbr' and 'o' - OK in 2.78 seconds
+    Computing Levenshtein between strings 'dgj' and 'hk' - OK in 4.92 seconds
+    Computing Levenshtein between strings 'ejb' and 'tfo' - OK in 7.18 seconds
+    Computing Levenshtein between strings 'afa' and 'ygqo' - OK in 9.25 seconds
+    Computing Levenshtein between strings 'lhcc' and '' - OK in 0.22 seconds
+    Computing Levenshtein between strings 'uoiu' and 'u' - OK in 3.52 seconds
+    Computing Levenshtein between strings 'tztt' and 'xo' - OK in 6.45 seconds
+    Computing Levenshtein between strings 'ufsa' and 'mil' - OK in 9.11 seconds
+    Computing Levenshtein between strings 'uuzl' and 'dzkr' - OK in 12.01 seconds
+
+Successful end
+```
+
+- `python levenshtein_distance.py --autoperf`: Benchmark with random strings, for the different alphabets.
+
+```
+
+Typical performances for alphabet ACTG, with string of maximal length:
+
+    Computing Levenshtein between strings 'CGGA' and 'GCTA' - OK in 4.77 seconds
+    Computing Levenshtein between strings 'TTCC' and 'CAAG' - OK in 4.45 seconds
+    Computing Levenshtein between strings 'TGAG' and 'CATC' - OK in 4.38 seconds
+
+Typical performances for alphabet string, with string of maximal length:
+
+    Computing Levenshtein between strings 'tsyl' and 'slTz' - OK in 13.76 seconds
+    Computing Levenshtein between strings 'rdfu' and 'qbam' - OK in 12.89 seconds
+    Computing Levenshtein between strings 'ngoz' and 'fxGw' - OK in 12.88 seconds
+
+Typical performances for alphabet STRING, with string of maximal length:
+
+    Computing Levenshtein between strings 'OjgB' and 'snQc' - OK in 23.94 seconds
+    Computing Levenshtein between strings 'UXWO' and 'rVgF' - OK in 23.69 seconds
+    Computing Levenshtein between strings 'NsBT' and 'IFuC' - OK in 23.40 seconds
+
+Typical performances for alphabet StRiNg, with string of maximal length:
+
+    Computing Levenshtein between strings 'ImNJ' and 'zyUB' - OK in 23.71 seconds
+    Computing Levenshtein between strings 'upAT' and 'XfWs' - OK in 23.52 seconds
+    Computing Levenshtein between strings 'HVXJ' and 'dQvr' - OK in 23.73 seconds
+
+Successful end
+
+```
+
+## Complexity analysis
+
+Let's analyze a bit the complexity of the function `levenshtein_fhe` in FHE. We can see that the
+function cannot apply `if`'s as in the clear function `levenshtein_clear`: it has to compute the two
+branches (the one for the True, and the one for the False), and finally compute an `fhe.if_then_else`
+of the two possible values. This slowdown is not specific to Concrete, it is by nature of FHE, where
+encrypted conditions imply such a trick.
+
+Another interesting part is the impact of the choice of the alphabet: in `run`, we are going to
+compare two chars of the alphabet, and return an encrypted boolean to code for the equality / inequality
+of these two chars. This is basically done with a single programmable bootstrapping (PBS) of `w+1`
+bits, where `w` is the floored log2 value of the number of chars in the alphabet. For example, for
+the 'string' alphabet, which has 26 letters, `w = 5` and so we use a signed 6-bit value as input of a
+table lookup. For the larger 'StRiNg' alphabet, that's a signed 7-bit PBS. For small DNA alphabet 'ACTG',
+it's only signed 3-bit PBS.
+
+## Benchmarks on hpc7a
+
+The benchmarks were done using Concrete 2.7 on `hpc7a` machine on AWS, and give:
+
+```
+
+Typical performances for alphabet ACTG, with string of maximal length:
+
+    Computing Levenshtein between strings 'CGGA' and 'GCTA' - OK in 4.77 seconds
+    Computing Levenshtein between strings 'TTCC' and 'CAAG' - OK in 4.45 seconds
+    Computing Levenshtein between strings 'TGAG' and 'CATC' - OK in 4.38 seconds
+
+Typical performances for alphabet string, with string of maximal length:
+
+    Computing Levenshtein between strings 'tsyl' and 'slTz' - OK in 13.76 seconds
+    Computing Levenshtein between strings 'rdfu' and 'qbam' - OK in 12.89 seconds
+    Computing Levenshtein between strings 'ngoz' and 'fxGw' - OK in 12.88 seconds
+
+Typical performances for alphabet STRING, with string of maximal length:
+
+    Computing Levenshtein between strings 'OjgB' and 'snQc' - OK in 23.94 seconds
+    Computing Levenshtein between strings 'UXWO' and 'rVgF' - OK in 23.69 seconds
+    Computing Levenshtein between strings 'NsBT' and 'IFuC' - OK in 23.40 seconds
+
+Typical performances for alphabet StRiNg, with string of maximal length:
+
+    Computing Levenshtein between strings 'ImNJ' and 'zyUB' - OK in 23.71 seconds
+    Computing Levenshtein between strings 'upAT' and 'XfWs' - OK in 23.52 seconds
+    Computing Levenshtein between strings 'HVXJ' and 'dQvr' - OK in 23.73 seconds
+
+Successful end
+```