Trac #28182: Linear algorithm for squares in words

This ticket implement a linear time algorithm to retrieve all the
distinct squares in a word.

URL: https://trac.sagemath.org/28182
Reported by: enadeau
Ticket author(s): Émile Nadeau
Reviewer(s): Nadia Lafrenière, Travis Scrimshaw

src/doc/en/reference/references/index.rst

@@ -1813,6 +1813,14 @@ REFERENCES:
 .. [DS1994] J. Dalbec and B. Sturmfels. Invariant methods in discrete and computational geometry,
             chapter Introduction to Chow forms, pages 37-58. Springer Netherlands, 1994.
 
+.. [DS2004] Dan Gusfield, Jens Stoye,
+            Linear time algorithms for finding and representing all the tandem repeats in a string,
+            Journal of Computer and System Sciences,
+            Volume 69, Issue 4,
+            2004,
+            Pages 525-546,
+            https://doi.org/10.1016/j.jcss.2004.03.004.
+
 .. [Du2001] \I. Duursma, "From weight enumerators to zeta functions", in
             Discrete Applied Mathematics, vol. 111, no. 1-2,
             pp. 55-73, 2001.

src/sage/combinat/words/finite_word.py

@@ -229,7 +229,6 @@
 from sage.rings.all import Integer, Infinity, ZZ, QQ
 from sage.sets.set import Set
 
-
 class FiniteWord_class(Word_class):
     def __str__(self):
         r"""
@@ -268,7 +267,6 @@ def __str__(self):
             sage: print(w)
             012340123401234012340123401234012340123401234012340123401234
         """
-        global word_options
         if word_options['display'] == 'string':
             ls = word_options['letter_separator']
             letters = [str(_) for _ in self]
@@ -290,7 +288,6 @@ def _repr_(self):
             sage: Word(range(100))._repr_()
             'word: 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,...'
         """
-        global word_options
         if word_options['old_repr']:
             if word_options['truncate'] and \
                     self.length() > word_options['truncate_length']:
@@ -4062,7 +4059,7 @@ def lyndon_factorization(self):
             while k < i:
                 F.append(k + j - i + 1)
                 k = k + j - i
-        return Factorization([self[F[i]:F[i+1]] for i in range(len(F)-1)])
+        return Factorization([self[F[l]:F[l+1]] for l in range(len(F)-1)])
 
     def inversions(self):
         r"""
@@ -6864,14 +6861,27 @@ def is_square_free(self):
             sage: Word('3211').is_square_free()
             False
         """
-        L = self.length()
-        if L < 2:
-            return True
-        for start in range(0, L-1):
-            for end in range(start+2, L+1, 2):
-                if self[start:end].is_square():
-                    return False
-        return True
+        from sage.combinat.words.suffix_trees import DecoratedSuffixTree
+        T = DecoratedSuffixTree(self)
+        return T.square_vocabulary() == [(0, 0)]
+
+    def squares(self):
+        r"""
+        Returns a set of all distinct squares of ``self``.
+
+        EXAMPLES::
+
+            sage: sorted(Word('cacao').squares())
+            [word: , word: caca]
+            sage: sorted(Word('1111').squares())
+            [word: , word: 11, word: 1111]
+            sage: w = Word('00110011010')
+            sage: sorted(w.squares())
+            [word: , word: 00, word: 00110011, word: 01100110, word: 1010, word: 11]
+        """
+        from sage.combinat.words.suffix_trees import DecoratedSuffixTree
+        T = DecoratedSuffixTree(self)
+        return set(T.square_vocabulary(output='word'))
 
     def is_cube(self):
         r"""