Introduce new alpha_inc pass, to avoid artificial duplicates.

ChangeLog

@@ -1,5 +1,44 @@
 2002-11-17  Bruno Haible  <bruno@clisp.org>
 
+	Avoid artificial duplicates.
+	* src/keyword.h (KeywordExt::init_selchars_tuple): New declaration.
+	(KeywordExt::init_selchars_multiset): Renamed from
+	KeywordExt::init_selchars.
+	(KeywordExt::init_selchars_low): New declaration.
+	* src/keyword.cc (KeywordExt::init_selchars_low): Renamed from
+	KeywordExt::init_selchars. Add alpha_inc argument. Remove sorting.
+	(KeywordExt::init_selchars_tuple): New method.
+	(KeywordExt::init_selchars_multiset): New method, replaces
+	KeywordExt::init_selchars.
+	* src/search.h (Search::init_selchars_tuple): Renamed from
+	Search::init_selchars.
+	(Search::count_duplicates_tuple): Renamed from Search::count_duplicates.
+	(Search::init_selchars_multiset, Search::count_duplicates_multiset,
+	Search::find_alpha_inc): New declarations.
+	(Search::_alpha_inc): New field.
+	(Search::_alpha_size, Search::_occurrences, Search::_asso_values,
+	Search::_determined): Make non-const.
+	* src/search.cc (Search::Search): Don't initialize _key_positions,
+	_alpha_size, _occurrences, _asso_values, _determined here.
+	(Search::init_selchars_tuple): Renamed from Search::init_selchars.
+	(Search::count_duplicates_tuple): Renamed from Search::count_duplicates.
+	(Search::find_positions): Update.
+	(Search::init_selchars_multiset, Search::count_duplicates_multiset,
+	Search::find_alpha_inc): New methods.
+	(Search::prepare): Move preprepare, find_positions calls away.
+	Initialize _alpha_size, _occurrences, _asso_values, _determined here.
+	(Search::optimize): Call preprepare, find_positions here. Initialize
+	_key_positions here.
+	(Search::~Search): Deallocate _alpha_inc.
+	* src/output.cc (Output::Output): Add alpha_inc argument.
+	(Output::output_hash_function): Use _alpha_inc.
+	* src/output.h (Output::Output): Add alpha_inc argument.
+	(Output::_alpha_inc): New field.
+	* src/main.cc (main): Pass _alpha_inc from Search to Output.
+	* tests/chill.exp: Update.
+	* doc/gperf.texi (Algorithmic Details): Remove description of
+	artificial duplicates.
+
 	* src/keyword.h (KeywordExt::_selchars): Change type to
 	'const unsigned int *'.
 	* src/keyword.cc (sort_char_set): Change argument type to

NEWS

@@ -31,6 +31,9 @@ New in 2.8:
   computed depending on the set of keywords.
 * If the input file is given by name, the output file will now contain
   #line directives referring to the input file.
+* Some keyword sets containing permutations, like { "xy", "yx", "xz", "zx" }
+  or { "abc", "acb", "bca", "cab" }, are now handled by gperf without
+  requiring the option -D.
 * Bug fixes.
 
 New in 2.7.2:

doc/gperf.texi

@@ -993,7 +993,7 @@ through a search that minimizes the number of byte positions.
 @itemx --duplicates
 @cindex Duplicates
 Handle keywords whose selected byte sets hash to duplicate values.
-Duplicate hash values can occur for three reasons:
+Duplicate hash values can occur for two reasons:
 
 @itemize @bullet
 @item
@@ -1003,15 +1003,6 @@ However, frequently only a very small number of duplicates occur, and
 the majority of keywords still require one probe into the table.  To
 overcome this problem, the option @samp{-m 50} should be used.
 
-@item
-Since the @code{gperf} generated hash function treats the bytes at
-different byte positions with equal weight, keywords that are permutations
-of each other can lead to the same hash function value if they are not
-disambiguated by the set of selected byte positions.  Sometimes even this
-is not possible; for example, the keyword set @{"xy", "yx", "xz", "zx"@}
-will always lead to duplicates, regardless how the selected byte positions
-are chosen.  You can use the option @samp{-D} to handle this rare case.
-
 @item
 Sometimes a set of keywords may have the same names, but possess different
 attributes.  With the -D option @code{gperf} treats all these keywords as

src/keyword.cc

@@ -57,8 +57,9 @@ static inline void sort_char_set (unsigned int *base, int len)
    Furthermore we sort the selchars array, to ease detection of duplicates
    later.
  */
-void
+
-KeywordExt::init_selchars (bool use_all_chars, const Positions& positions)
+unsigned int *
+KeywordExt::init_selchars_low (bool use_all_chars, const Positions& positions, const unsigned int *alpha_inc)
 {
   const char *k = _allchars;
   unsigned int *key_set =
@@ -69,7 +70,10 @@ KeywordExt::init_selchars (bool use_all_chars, const Positions& positions)
     /* Use all the character positions in the KEY.  */
     for (int i = _allchars_length; i > 0; k++, i--)
       {
-        *ptr = static_cast<unsigned char>(*k);
+        unsigned int c = static_cast<unsigned char>(*k);
+        if (alpha_inc)
+          c += alpha_inc[k-_allchars];
+        *ptr = c;
         ptr++;
       }
   else
@@ -81,24 +85,45 @@ KeywordExt::init_selchars (bool use_all_chars, const Positions& positions)
 
       for (int i; (i = iter.next ()) != PositionIterator::EOS; )
         {
+          unsigned int c;
           if (i == Positions::LASTCHAR)
             /* Special notation for last KEY position, i.e. '$'.  */
-            *ptr = static_cast<unsigned char>(_allchars[_allchars_length - 1]);
+            c = static_cast<unsigned char>(_allchars[_allchars_length - 1]);
           else if (i <= _allchars_length)
-            /* Within range of KEY length, so we'll keep it.  */
+            {
-            *ptr = static_cast<unsigned char>(_allchars[i - 1]);
+              /* Within range of KEY length, so we'll keep it.  */
+              c = static_cast<unsigned char>(_allchars[i - 1]);
+              if (alpha_inc)
+                c += alpha_inc[i - 1];
+            }
           else
             /* Out of range of KEY length, so we'll just skip it.  */
             continue;
+          *ptr = c;
           ptr++;
         }
     }
 
-  /* Sort the KEY_SET items alphabetically.  */
-  sort_char_set (key_set, ptr - key_set);
-
   _selchars = key_set;
   _selchars_length = ptr - key_set;
+
+  return key_set;
+}
+
+void
+KeywordExt::init_selchars_tuple (bool use_all_chars, const Positions& positions)
+{
+  init_selchars_low (use_all_chars, positions, NULL);
+}
+
+void
+KeywordExt::init_selchars_multiset (bool use_all_chars, const Positions& positions, const unsigned int *alpha_inc)
+{
+  unsigned int *selchars =
+    init_selchars_low (use_all_chars, positions, alpha_inc);
+
+  /* Sort the selchars elements alphabetically.  */
+  sort_char_set (selchars, _selchars_length);
 }
 
 /* Deletes selchars.  */

src/keyword.h

@@ -67,8 +67,10 @@ struct KeywordExt : public Keyword
   KeywordExt *          _duplicate_link;
 
   /* Methods depending on the keyposition list.  */
-  /* Initializes selchars and selchars_length.  */
+  /* Initializes selchars and selchars_length, without reordering.  */
-  void                  init_selchars (bool use_all_chars, const Positions& positions);
+  void                  init_selchars_tuple (bool use_all_chars, const Positions& positions);
+  /* Initializes selchars and selchars_length, with reordering.  */
+  void                  init_selchars_multiset (bool use_all_chars, const Positions& positions, const unsigned int *alpha_inc);
   /* Deletes selchars.  */
   void                  delete_selchars ();
 
@@ -78,6 +80,9 @@ struct KeywordExt : public Keyword
 
   /* Data members used by the output routines.  */
   int                   _final_index;
+
+private:
+  unsigned int *        init_selchars_low (bool use_all_chars, const Positions& positions, const unsigned int *alpha_inc);
 };
 
 /* An abstract factory for creating Keyword instances.

src/main.cc

@@ -106,6 +106,7 @@ main (int argc, char *argv[])
                           searcher._max_key_len,
                           searcher._min_key_len,
                           searcher._key_positions,
+                          searcher._alpha_inc,
                           searcher._total_duplicates,
                           searcher._alpha_size,
                           searcher._occurrences,

src/output.cc

@@ -88,8 +88,8 @@ Output::Output (KeywordExt_List *head, const char *struct_decl,
                 const char *verbatim_code, const char *verbatim_code_end,
                 unsigned int verbatim_code_lineno,
                 int total_keys, int max_key_len, int min_key_len,
-                const Positions& positions, int total_duplicates,
+                const Positions& positions, const unsigned int *alpha_inc,
-                int alpha_size, const int *occurrences,
+                int total_duplicates, int alpha_size, const int *occurrences,
                 const int *asso_values)
   : _head (head), _struct_decl (struct_decl),
     _struct_decl_lineno (struct_decl_lineno), _return_type (return_type),
@@ -102,7 +102,7 @@ Output::Output (KeywordExt_List *head, const char *struct_decl,
     _verbatim_code_lineno (verbatim_code_lineno),
     _total_keys (total_keys),
     _max_key_len (max_key_len), _min_key_len (min_key_len),
-    _key_positions (positions),
+    _key_positions (positions), _alpha_inc (alpha_inc),
     _total_duplicates (total_duplicates), _alpha_size (alpha_size),
     _occurrences (occurrences), _asso_values (asso_values)
 {
@@ -521,9 +521,14 @@ Output::output_hash_function () const
               option[NOLENGTH] ? "len" : "hval");
 
       for (int i = _max_key_len; i > 0; i--)
-        printf ("      case %d:\n"
+        {
-                "        hval += asso_values[%sstr[%d]];\n",
+          printf ("      case %d:\n"
-                i, char_to_index, i - 1);
+                  "        hval += asso_values[%sstr[%d]",
+                  i, char_to_index, i - 1);
+          if (_alpha_inc[i - 1])
+            printf ("+%u", _alpha_inc[i - 1]);
+          printf ("];\n");
+        }
 
       printf ("        break;\n"
               "    }\n"
@@ -560,13 +565,21 @@ Output::output_hash_function () const
               && _key_positions[0] == 1
               && _key_positions[1] == Positions::LASTCHAR)
             /* Optimize special case of "-k 1,$".  */
-            printf ("asso_values[%sstr[len - 1]] + asso_values[%sstr[0]]",
+            {
-                    char_to_index, char_to_index);
+              printf ("asso_values[%sstr[len - 1]] + asso_values[%sstr[0]",
+                      char_to_index, char_to_index);
+              if (_alpha_inc[0])
+                printf ("+%u", _alpha_inc[0]);
+              printf ("]");
+            }
           else
             {
               for (; key_pos != Positions::LASTCHAR; )
                 {
-                  printf ("asso_values[%sstr[%d]]", char_to_index, key_pos - 1);
+                  printf ("asso_values[%sstr[%d]", char_to_index, key_pos - 1);
+                  if (_alpha_inc[key_pos - 1])
+                    printf ("+%u", _alpha_inc[key_pos - 1]);
+                  printf ("]");
                   if ((key_pos = iter.next ()) != PositionIterator::EOS)
                     printf (" + ");
                   else
@@ -601,8 +614,11 @@ Output::output_hash_function () const
                   for ( ; i >= key_pos; i--)
                     printf ("      case %d:\n", i);
 
-                  printf ("        hval += asso_values[%sstr[%d]];\n",
+                  printf ("        hval += asso_values[%sstr[%d]",
                           char_to_index, key_pos - 1);
+                  if (_alpha_inc[key_pos - 1])
+                    printf ("+%u", _alpha_inc[key_pos - 1]);
+                  printf ("];\n");
 
                   key_pos = iter.next ();
                 }

src/output.h

@@ -51,6 +51,7 @@ public:
                                 int total_keys,
                                 int max_key_len, int min_key_len,
                                 const Positions& positions,
+                                const unsigned int *alpha_inc,
                                 int total_duplicates,
                                 int alpha_size,
                                 const int *occurrences,
@@ -121,6 +122,8 @@ private:
   int const             _min_key_len;
   /* Key positions.  Only to be used if !options[ALLCHARS].  */
   Positions const       _key_positions;
+  /* Adjustments to add to bytes add specific key positions.  */
+  const unsigned int * const _alpha_inc;
   /* Total number of duplicate hash values. */
   int const             _total_duplicates;
   /* Minimum hash value for all keywords. */

src/search.cc

@@ -31,15 +31,44 @@
 #include "options.h"
 #include "hash-table.h"
 
+/* The most general form of the hash function is
+
+      hash (keyword) = sum (asso_values[keyword[i] + alpha_inc[i]] : i in Pos)
+
+   where Pos is a set of byte positions,
+   each alpha_inc[i] is a nonnegative integer,
+   each asso_values[c] is a nonnegative integer.
+
+   Theorem 1: If all keywords are different, there is a set Pos such that
+   all tuples (keyword[i] : i in Pos) are different.
+
+   Theorem 2: If all tuples (keyword[i] : i in Pos) are different, there
+   are nonnegative integers alpha_inc[i] such that all multisets
+   {keyword[i] + alpha_inc[i] : i in Pos} are different.
+
+   Theorem 3: If all multisets selchars[keyword] are different, there are
+   nonnegative integers asso_values[c] such that all hash values
+   sum (asso_values[c] : c in selchars[keyword]) are different.
+
+   Based on these three facts, we find the hash function in three steps:
+
+   Step 1 (Finding good byte positions):
+   Find a set Pos, as small as possible, such that all tuples
+   (keyword[i] : i in Pos) are different.
+
+   Step 2 (Finding good alpha increments):
+   Find nonnegative integers alpha_inc[i], as many of them as possible being
+   zero, and the others being as small as possible, such that all multisets
+   {keyword[i] + alpha_inc[i] : i in Pos} are different.
+
+   Step 3 (Finding good asso_values):
+   Find asso_values[c] such that all hash (keyword) are different.
+ */
+
 /* -------------------- Initialization and Preparation --------------------- */
 
 Search::Search (KeywordExt_List *list)
-  : _head (list),
+  : _head (list)
-    _key_positions (option.get_key_positions()),
-    _alpha_size (option[SEVENBIT] ? 128 : 256),
-    _occurrences (new int[_alpha_size]),
-    _asso_values (new int[_alpha_size]),
-    _determined (new bool[_alpha_size])
 {
 }
 
@@ -77,12 +106,14 @@ Search::preprepare ()
     }
 }
 
+/* ---------------------- Finding good byte positions ---------------------- */
+
 /* Initializes each keyword's _selchars array.  */
 void
-Search::init_selchars (bool use_all_chars, const Positions& positions) const
+Search::init_selchars_tuple (bool use_all_chars, const Positions& positions) const
 {
   for (KeywordExt_List *temp = _head; temp; temp = temp->rest())
-    temp->first()->init_selchars(use_all_chars, positions);
+    temp->first()->init_selchars_tuple(use_all_chars, positions);
 }
 
 /* Deletes each keyword's _selchars array.  */
@@ -95,29 +126,31 @@ Search::delete_selchars () const
 
 /* Count the duplicate keywords that occur with a given set of positions.  */
 unsigned int
-Search::count_duplicates (const Positions& positions) const
+Search::count_duplicates_tuple (const Positions& positions) const
 {
-  init_selchars (false, positions);
+  init_selchars_tuple (option[ALLCHARS], positions);
 
   unsigned int count = 0;
-  Hash_Table representatives (_total_keys, option[NOLENGTH]);
+  {
-  for (KeywordExt_List *temp = _head; temp; temp = temp->rest())
+    Hash_Table representatives (_total_keys, option[NOLENGTH]);
-    {
+    for (KeywordExt_List *temp = _head; temp; temp = temp->rest())
-      KeywordExt *keyword = temp->first();
+      {
-      if (representatives.insert (keyword))
+        KeywordExt *keyword = temp->first();
-        count++;
+        if (representatives.insert (keyword))
-    }
+          count++;
+      }
+  }
 
   delete_selchars ();
 
   return count;
 }
 
+/* Find good key positions.  */
+
 void
 Search::find_positions ()
 {
-  /* Determine good key positions.  */
-
   /* 1. Find positions that must occur in order to distinguish duplicates.  */
   Positions mandatory;
 
@@ -159,7 +192,7 @@ Search::find_positions ()
   int imax = (_max_key_len < Positions::MAX_KEY_POS
               ? _max_key_len : Positions::MAX_KEY_POS);
   Positions current = mandatory;
-  unsigned int current_duplicates_count = count_duplicates (current);
+  unsigned int current_duplicates_count = count_duplicates_tuple (current);
   for (;;)
     {
       Positions best;
@@ -170,7 +203,7 @@ Search::find_positions ()
           {
             Positions tryal = current;
             tryal.add (i);
-            unsigned int try_duplicates_count = count_duplicates (tryal);
+            unsigned int try_duplicates_count = count_duplicates_tuple (tryal);
 
             /* We prefer 'try' to 'best' if it produces less duplicates,
                or if it produces the same number of duplicates but with
@@ -203,7 +236,7 @@ Search::find_positions ()
           {
             Positions tryal = current;
             tryal.remove (i);
-            unsigned int try_duplicates_count = count_duplicates (tryal);
+            unsigned int try_duplicates_count = count_duplicates_tuple (tryal);
 
             /* We prefer 'try' to 'best' if it produces less duplicates,
                or if it produces the same number of duplicates but with
@@ -243,7 +276,7 @@ Search::find_positions ()
                     tryal.remove (i2);
                     tryal.add (i3);
                     unsigned int try_duplicates_count =
-                      count_duplicates (tryal);
+                      count_duplicates_tuple (tryal);
 
                     /* We prefer 'try' to 'best' if it produces less duplicates,
                        or if it produces the same number of duplicates but with
@@ -269,18 +302,141 @@ Search::find_positions ()
   _key_positions = current;
 }
 
+/* --------------------- Finding good alpha increments --------------------- */
+
+/* Initializes each keyword's _selchars array.  */
+void
+Search::init_selchars_multiset (bool use_all_chars, const Positions& positions, const unsigned int *alpha_inc) const
+{
+  for (KeywordExt_List *temp = _head; temp; temp = temp->rest())
+    temp->first()->init_selchars_multiset(use_all_chars, positions, alpha_inc);
+}
+
+/* Count the duplicate keywords that occur with the given set of positions
+   and a given alpha_inc[] array.  */
+unsigned int
+Search::count_duplicates_multiset (const unsigned int *alpha_inc) const
+{
+  init_selchars_multiset (option[ALLCHARS], _key_positions, alpha_inc);
+
+  unsigned int count = 0;
+  {
+    Hash_Table representatives (_total_keys, option[NOLENGTH]);
+    for (KeywordExt_List *temp = _head; temp; temp = temp->rest())
+      {
+        KeywordExt *keyword = temp->first();
+        if (representatives.insert (keyword))
+          count++;
+      }
+  }
+
+  delete_selchars ();
+
+  return count;
+}
+
+/* Find good _alpha_inc[].  */
+
+void
+Search::find_alpha_inc ()
+{
+  /* The goal is to choose _alpha_inc[] such that it doesn't introduce
+     artificial duplicates.  */
+  unsigned int duplicates_goal = count_duplicates_tuple (_key_positions);
+
+  /* Start with zero increments.  This is sufficient in most cases.  */
+  unsigned int *current = new unsigned int [_max_key_len];
+  for (int i = 0; i < _max_key_len; i++)
+    current[i] = 0;
+  unsigned int current_duplicates_count = count_duplicates_multiset (current);
+
+  if (current_duplicates_count > duplicates_goal)
+    {
+      /* Look which _alpha_inc[i] we are free to increment.  */
+      unsigned int nindices;
+      if (option[ALLCHARS])
+        nindices = _max_key_len;
+      else
+        {
+          /* Ignore Positions::LASTCHAR.  Remember that since Positions are
+             sorted in decreasing order, Positions::LASTCHAR comes last.  */
+          nindices = (_key_positions.get_size() == 0
+                      || _key_positions[_key_positions.get_size() - 1]
+                         != Positions::LASTCHAR
+                      ? _key_positions.get_size()
+                      : _key_positions.get_size() - 1);
+        }
+
+      unsigned int indices[nindices];
+      if (option[ALLCHARS])
+        for (unsigned int j = 0; j < nindices; j++)
+          indices[j] = j;
+      else
+        {
+          PositionIterator iter (_key_positions);
+          for (unsigned int j = 0; j < nindices; j++)
+            {
+              int key_pos = iter.next ();
+              if (key_pos == PositionIterator::EOS
+                  || key_pos == Positions::LASTCHAR)
+                abort ();
+              indices[j] = key_pos - 1;
+            }
+        }
+
+      /* Perform several rounds of searching for a good alpha increment.
+         Each round reduces the number of artificial collisions by adding
+         an increment in a single key position.  */
+      unsigned int best[_max_key_len];
+      unsigned int tryal[_max_key_len];
+      do
+        {
+          /* An increment of 1 is not always enough.  Try higher increments
+             also.  */
+          for (unsigned int inc = 1; ; inc++)
+            {
+              unsigned int best_duplicates_count = UINT_MAX;
+
+              for (unsigned int j = 0; j < nindices; j++)
+                {
+                  memcpy (tryal, current, _max_key_len * sizeof (unsigned int));
+                  tryal[indices[j]] += inc;
+                  unsigned int try_duplicates_count =
+                    count_duplicates_multiset (tryal);
+
+                  /* We prefer 'try' to 'best' if it produces less
+                     duplicates.  */
+                  if (try_duplicates_count < best_duplicates_count)
+                    {
+                      memcpy (best, tryal, _max_key_len * sizeof (unsigned int));
+                      best_duplicates_count = try_duplicates_count;
+                    }
+                }
+
+              /* Stop this round when we got an improvement.  */
+              if (best_duplicates_count < current_duplicates_count)
+                {
+                  memcpy (current, best, _max_key_len * sizeof (unsigned int));
+                  current_duplicates_count = best_duplicates_count;
+                  break;
+                }
+            }
+        }
+      while (current_duplicates_count > duplicates_goal);
+    }
+
+  _alpha_inc = current;
+}
+
+/* ------------------------------------------------------------------------- */
+
 void
 Search::prepare ()
 {
   KeywordExt_List *temp;
 
-  preprepare ();
-
-  if (!option[POSITIONS])
-    find_positions ();
-
   /* Initialize each keyword's _selchars array.  */
-  init_selchars (option[ALLCHARS], _key_positions);
+  init_selchars_multiset(option[ALLCHARS], _key_positions, _alpha_inc);
 
   /* Check for duplicates, i.e. keywords with the same _selchars array
      (and - if !option[NOLENGTH] - also the same length).
@@ -357,7 +513,16 @@ Search::prepare ()
         }
     }
 
+  /* Compute _alpha_size, the upper bound on the indices passed to
+     asso_values[].  */
+  unsigned int max_alpha_inc = 0;
+  for (int i = 0; i < _max_key_len; i++)
+    if (max_alpha_inc < _alpha_inc[i])
+      max_alpha_inc = _alpha_inc[i];
+  _alpha_size = (option[SEVENBIT] ? 128 : 256) + max_alpha_inc;
+
   /* Compute the occurrences of each character in the alphabet.  */
+  _occurrences = new int[_alpha_size];
   memset (_occurrences, 0, _alpha_size * sizeof (_occurrences[0]));
   for (temp = _head; temp; temp = temp->rest())
     {
@@ -366,6 +531,10 @@ Search::prepare ()
       for (int count = keyword->_selchars_length; count > 0; ptr++, count--)
         _occurrences[*ptr]++;
     }
+
+  /* Memory allocation.  */
+  _asso_values = new int[_alpha_size];
+  _determined = new bool[_alpha_size];
 }
 
 /* ---------------- Reordering the Keyword list (optional) ----------------- */
@@ -878,6 +1047,11 @@ void
 Search::optimize ()
 {
   /* Preparations.  */
+  preprepare ();
+  _key_positions = option.get_key_positions();
+  if (!option[POSITIONS])
+    find_positions ();
+  find_alpha_inc ();
   prepare ();
   if (option[ORDER])
     reorder ();
@@ -1025,4 +1199,5 @@ Search::~Search ()
     }
   delete[] _asso_values;
   delete[] _occurrences;
+  delete[] _alpha_inc;
 }

src/search.h

@@ -40,15 +40,26 @@ private:
   void                  preprepare ();
 
   /* Initializes each keyword's _selchars array.  */
-  void                  init_selchars (bool use_all_chars, const Positions& positions) const;
+  void                  init_selchars_tuple (bool use_all_chars, const Positions& positions) const;
   /* Deletes each keyword's _selchars array.  */
   void                  delete_selchars () const;
 
   /* Count the duplicate keywords that occur with a given set of positions.  */
-  unsigned int          count_duplicates (const Positions& positions) const;
+  unsigned int          count_duplicates_tuple (const Positions& positions) const;
 
+  /* Find good key positions.  */
   void                  find_positions ();
 
+  /* Initializes each keyword's _selchars array.  */
+  void                  init_selchars_multiset (bool use_all_chars, const Positions& positions, const unsigned int *alpha_inc) const;
+
+  /* Count the duplicate keywords that occur with the given set of positions
+     and a given alpha_inc[] array.  */
+  unsigned int          count_duplicates_multiset (const unsigned int *alpha_inc) const;
+
+  /* Find good _alpha_inc[].  */
+  void                  find_alpha_inc ();
+
   void                  prepare ();
 
   /* Computes the sum of occurrences of the _selchars of a keyword.  */
@@ -112,19 +123,22 @@ public:
   /* User-specified or computed key positions.  */
   Positions             _key_positions;
 
+  /* Adjustments to add to bytes add specific key positions.  */
+  unsigned int *        _alpha_inc;
+
   /* Total number of duplicates that have been moved to _duplicate_link lists
      (not counting their representatives which stay on the main list).  */
   int                   _total_duplicates;
 
   /* Size of alphabet.  */
-  int const             _alpha_size;
+  int                   _alpha_size;
 
   /* Counts occurrences of each key set character.
      _occurrences[c] is the number of times that c occurs among the _selchars
      of a keyword.  */
-  int * const           _occurrences;
+  int *                 _occurrences;
   /* Value associated with each character. */
-  int * const           _asso_values;
+  int *                 _asso_values;
 
 private:
 
@@ -132,7 +146,7 @@ private:
   int                   _list_len;
 
   /* Vector used during Search::reorder().  */
-  bool * const          _determined;
+  bool *                _determined;
 
   /* Exclusive upper bound for every _asso_values[c].  A power of 2.  */
   int                   _asso_value_max;