1Memoize(3pm)           Perl Programmers Reference Guide           Memoize(3pm)
2
3
4

NAME

6       Memoize - Make functions faster by trading space for time
7

SYNOPSIS

9               # This is the documentation for Memoize 1.03
10               use Memoize;
11               memoize('slow_function');
12               slow_function(arguments);    # Is faster than it was before
13
14       This is normally all you need to know.  However, many options are
15       available:
16
17               memoize(function, options...);
18
19       Options include:
20
21               NORMALIZER => function
22               INSTALL => new_name
23
24               SCALAR_CACHE => 'MEMORY'
25               SCALAR_CACHE => ['HASH', \%cache_hash ]
26               SCALAR_CACHE => 'FAULT'
27               SCALAR_CACHE => 'MERGE'
28
29               LIST_CACHE => 'MEMORY'
30               LIST_CACHE => ['HASH', \%cache_hash ]
31               LIST_CACHE => 'FAULT'
32               LIST_CACHE => 'MERGE'
33

DESCRIPTION

35       `Memoizing' a function makes it faster by trading space for time.  It
36       does this by caching the return values of the function in a table.  If
37       you call the function again with the same arguments, "memoize" jumps in
38       and gives you the value out of the table, instead of letting the
39       function compute the value all over again.
40
41       Here is an extreme example.  Consider the Fibonacci sequence, defined
42       by the following function:
43
44               # Compute Fibonacci numbers
45               sub fib {
46                 my $n = shift;
47                 return $n if $n < 2;
48                 fib($n-1) + fib($n-2);
49               }
50
51       This function is very slow.  Why?  To compute fib(14), it first wants
52       to compute fib(13) and fib(12), and add the results.  But to compute
53       fib(13), it first has to compute fib(12) and fib(11), and then it comes
54       back and computes fib(12) all over again even though the answer is the
55       same.  And both of the times that it wants to compute fib(12), it has
56       to compute fib(11) from scratch, and then it has to do it again each
57       time it wants to compute fib(13).  This function does so much
58       recomputing of old results that it takes a really long time to
59       run---fib(14) makes 1,200 extra recursive calls to itself, to compute
60       and recompute things that it already computed.
61
62       This function is a good candidate for memoization.  If you memoize the
63       `fib' function above, it will compute fib(14) exactly once, the first
64       time it needs to, and then save the result in a table.  Then if you ask
65       for fib(14) again, it gives you the result out of the table.  While
66       computing fib(14), instead of computing fib(12) twice, it does it once;
67       the second time it needs the value it gets it from the table.  It
68       doesn't compute fib(11) four times; it computes it once, getting it
69       from the table the next three times.  Instead of making 1,200 recursive
70       calls to `fib', it makes 15.  This makes the function about 150 times
71       faster.
72
73       You could do the memoization yourself, by rewriting the function, like
74       this:
75
76               # Compute Fibonacci numbers, memoized version
77               { my @fib;
78                 sub fib {
79                   my $n = shift;
80                   return $fib[$n] if defined $fib[$n];
81                   return $fib[$n] = $n if $n < 2;
82                   $fib[$n] = fib($n-1) + fib($n-2);
83                 }
84               }
85
86       Or you could use this module, like this:
87
88               use Memoize;
89               memoize('fib');
90
91               # Rest of the fib function just like the original version.
92
93       This makes it easy to turn memoizing on and off.
94
95       Here's an even simpler example: I wrote a simple ray tracer; the
96       program would look in a certain direction, figure out what it was
97       looking at, and then convert the `color' value (typically a string like
98       `red') of that object to a red, green, and blue pixel value, like this:
99
100           for ($direction = 0; $direction < 300; $direction++) {
101             # Figure out which object is in direction $direction
102             $color = $object->{color};
103             ($r, $g, $b) = @{&ColorToRGB($color)};
104             ...
105           }
106
107       Since there are relatively few objects in a picture, there are only a
108       few colors, which get looked up over and over again.  Memoizing
109       "ColorToRGB" sped up the program by several percent.
110

DETAILS

112       This module exports exactly one function, "memoize".  The rest of the
113       functions in this package are None of Your Business.
114
115       You should say
116
117               memoize(function)
118
119       where "function" is the name of the function you want to memoize, or a
120       reference to it.  "memoize" returns a reference to the new, memoized
121       version of the function, or "undef" on a non-fatal error.  At present,
122       there are no non-fatal errors, but there might be some in the future.
123
124       If "function" was the name of a function, then "memoize" hides the old
125       version and installs the new memoized version under the old name, so
126       that "&function(...)" actually invokes the memoized version.
127

OPTIONS

129       There are some optional options you can pass to "memoize" to change the
130       way it behaves a little.  To supply options, invoke "memoize" like
131       this:
132
133               memoize(function, NORMALIZER => function,
134                                 INSTALL => newname,
135                                 SCALAR_CACHE => option,
136                                 LIST_CACHE => option
137                                );
138
139       Each of these options is optional; you can include some, all, or none
140       of them.
141
142   INSTALL
143       If you supply a function name with "INSTALL", memoize will install the
144       new, memoized version of the function under the name you give.  For
145       example,
146
147               memoize('fib', INSTALL => 'fastfib')
148
149       installs the memoized version of "fib" as "fastfib"; without the
150       "INSTALL" option it would have replaced the old "fib" with the memoized
151       version.
152
153       To prevent "memoize" from installing the memoized version anywhere, use
154       "INSTALL => undef".
155
156   NORMALIZER
157       Suppose your function looks like this:
158
159               # Typical call: f('aha!', A => 11, B => 12);
160               sub f {
161                 my $a = shift;
162                 my %hash = @_;
163                 $hash{B} ||= 2;  # B defaults to 2
164                 $hash{C} ||= 7;  # C defaults to 7
165
166                 # Do something with $a, %hash
167               }
168
169       Now, the following calls to your function are all completely
170       equivalent:
171
172               f(OUCH);
173               f(OUCH, B => 2);
174               f(OUCH, C => 7);
175               f(OUCH, B => 2, C => 7);
176               f(OUCH, C => 7, B => 2);
177               (etc.)
178
179       However, unless you tell "Memoize" that these calls are equivalent, it
180       will not know that, and it will compute the values for these
181       invocations of your function separately, and store them separately.
182
183       To prevent this, supply a "NORMALIZER" function that turns the program
184       arguments into a string in a way that equivalent arguments turn into
185       the same string.  A "NORMALIZER" function for "f" above might look like
186       this:
187
188               sub normalize_f {
189                 my $a = shift;
190                 my %hash = @_;
191                 $hash{B} ||= 2;
192                 $hash{C} ||= 7;
193
194                 join(',', $a, map ($_ => $hash{$_}) sort keys %hash);
195               }
196
197       Each of the argument lists above comes out of the "normalize_f"
198       function looking exactly the same, like this:
199
200               OUCH,B,2,C,7
201
202       You would tell "Memoize" to use this normalizer this way:
203
204               memoize('f', NORMALIZER => 'normalize_f');
205
206       "memoize" knows that if the normalized version of the arguments is the
207       same for two argument lists, then it can safely look up the value that
208       it computed for one argument list and return it as the result of
209       calling the function with the other argument list, even if the argument
210       lists look different.
211
212       The default normalizer just concatenates the arguments with character
213       28 in between.  (In ASCII, this is called FS or control-\.)  This
214       always works correctly for functions with only one string argument, and
215       also when the arguments never contain character 28.  However, it can
216       confuse certain argument lists:
217
218               normalizer("a\034", "b")
219               normalizer("a", "\034b")
220               normalizer("a\034\034b")
221
222       for example.
223
224       Since hash keys are strings, the default normalizer will not
225       distinguish between "undef" and the empty string.  It also won't work
226       when the function's arguments are references.  For example, consider a
227       function "g" which gets two arguments: A number, and a reference to an
228       array of numbers:
229
230               g(13, [1,2,3,4,5,6,7]);
231
232       The default normalizer will turn this into something like
233       "13\034ARRAY(0x436c1f)".  That would be all right, except that a
234       subsequent array of numbers might be stored at a different location
235       even though it contains the same data.  If this happens, "Memoize" will
236       think that the arguments are different, even though they are
237       equivalent.  In this case, a normalizer like this is appropriate:
238
239               sub normalize { join ' ', $_[0], @{$_[1]} }
240
241       For the example above, this produces the key "13 1 2 3 4 5 6 7".
242
243       Another use for normalizers is when the function depends on data other
244       than those in its arguments.  Suppose you have a function which returns
245       a value which depends on the current hour of the day:
246
247               sub on_duty {
248                 my ($problem_type) = @_;
249                 my $hour = (localtime)[2];
250                 open my $fh, "$DIR/$problem_type" or die...;
251                 my $line;
252                 while ($hour-- > 0)
253                   $line = <$fh>;
254                 }
255                 return $line;
256               }
257
258       At 10:23, this function generates the 10th line of a data file; at 3:45
259       PM it generates the 15th line instead.  By default, "Memoize" will only
260       see the $problem_type argument.  To fix this, include the current hour
261       in the normalizer:
262
263               sub normalize { join ' ', (localtime)[2], @_ }
264
265       The calling context of the function (scalar or list context) is
266       propagated to the normalizer.  This means that if the memoized function
267       will treat its arguments differently in list context than it would in
268       scalar context, you can have the normalizer function select its
269       behavior based on the results of "wantarray".  Even if called in a list
270       context, a normalizer should still return a single string.
271
272   "SCALAR_CACHE", "LIST_CACHE"
273       Normally, "Memoize" caches your function's return values into an
274       ordinary Perl hash variable.  However, you might like to have the
275       values cached on the disk, so that they persist from one run of your
276       program to the next, or you might like to associate some other
277       interesting semantics with the cached values.
278
279       There's a slight complication under the hood of "Memoize": There are
280       actually two caches, one for scalar values and one for list values.
281       When your function is called in scalar context, its return value is
282       cached in one hash, and when your function is called in list context,
283       its value is cached in the other hash.  You can control the caching
284       behavior of both contexts independently with these options.
285
286       The argument to "LIST_CACHE" or "SCALAR_CACHE" must either be one of
287       the following four strings:
288
289               MEMORY
290               FAULT
291               MERGE
292               HASH
293
294       or else it must be a reference to an array whose first element is one
295       of these four strings, such as "[HASH, arguments...]".
296
297       "MEMORY"
298           "MEMORY" means that return values from the function will be cached
299           in an ordinary Perl hash variable.  The hash variable will not
300           persist after the program exits.  This is the default.
301
302       "HASH"
303           "HASH" allows you to specify that a particular hash that you supply
304           will be used as the cache.  You can tie this hash beforehand to
305           give it any behavior you want.
306
307           A tied hash can have any semantics at all.  It is typically tied to
308           an on-disk database, so that cached values are stored in the
309           database and retrieved from it again when needed, and the disk file
310           typically persists after your program has exited.  See "perltie"
311           for more complete details about "tie".
312
313           A typical example is:
314
315                   use DB_File;
316                   tie my %cache => 'DB_File', $filename, O_RDWR|O_CREAT, 0666;
317                   memoize 'function', SCALAR_CACHE => [HASH => \%cache];
318
319           This has the effect of storing the cache in a "DB_File" database
320           whose name is in $filename.  The cache will persist after the
321           program has exited.  Next time the program runs, it will find the
322           cache already populated from the previous run of the program.  Or
323           you can forcibly populate the cache by constructing a batch program
324           that runs in the background and populates the cache file.  Then
325           when you come to run your real program the memoized function will
326           be fast because all its results have been precomputed.
327
328           Another reason to use "HASH" is to provide your own hash variable.
329           You can then inspect or modify the contents of the hash to gain
330           finer control over the cache management.
331
332       "TIE"
333           This option is no longer supported.  It is still documented only to
334           aid in the debugging of old programs that use it.  Old programs
335           should be converted to use the "HASH" option instead.
336
337                   memoize ... ['TIE', PACKAGE, ARGS...]
338
339           is merely a shortcut for
340
341                   require PACKAGE;
342                   { tie my %cache, PACKAGE, ARGS...;
343                     memoize ... [HASH => \%cache];
344                   }
345
346       "FAULT"
347           "FAULT" means that you never expect to call the function in scalar
348           (or list) context, and that if "Memoize" detects such a call, it
349           should abort the program.  The error message is one of
350
351                   `foo' function called in forbidden list context at line ...
352                   `foo' function called in forbidden scalar context at line ...
353
354       "MERGE"
355           "MERGE" normally means that the memoized function does not
356           distinguish between list and sclar context, and that return values
357           in both contexts should be stored together.  Both "LIST_CACHE =>
358           MERGE" and "SCALAR_CACHE => MERGE" mean the same thing.
359
360           Consider this function:
361
362                   sub complicated {
363                     # ... time-consuming calculation of $result
364                     return $result;
365                   }
366
367           The "complicated" function will return the same numeric $result
368           regardless of whether it is called in list or in scalar context.
369
370           Normally, the following code will result in two calls to
371           "complicated", even if "complicated" is memoized:
372
373               $x = complicated(142);
374               ($y) = complicated(142);
375               $z = complicated(142);
376
377           The first call will cache the result, say 37, in the scalar cache;
378           the second will cach the list "(37)" in the list cache.  The third
379           call doesn't call the real "complicated" function; it gets the
380           value 37 from the scalar cache.
381
382           Obviously, the second call to "complicated" is a waste of time, and
383           storing its return value is a waste of space.  Specifying
384           "LIST_CACHE => MERGE" will make "memoize" use the same cache for
385           scalar and list context return values, so that the second call uses
386           the scalar cache that was populated by the first call.
387           "complicated" ends up being called only once, and both subsequent
388           calls return 3 from the cache, regardless of the calling context.
389
390       List values in scalar context
391
392       Consider this function:
393
394           sub iota { return reverse (1..$_[0]) }
395
396       This function normally returns a list.  Suppose you memoize it and
397       merge the caches:
398
399           memoize 'iota', SCALAR_CACHE => 'MERGE';
400
401           @i7 = iota(7);
402           $i7 = iota(7);
403
404       Here the first call caches the list (1,2,3,4,5,6,7).  The second call
405       does not really make sense. "Memoize" cannot guess what behavior "iota"
406       should have in scalar context without actually calling it in scalar
407       context.  Normally "Memoize" would call "iota" in scalar context and
408       cache the result, but the "SCALAR_CACHE => 'MERGE'" option says not to
409       do that, but to use the cache list-context value instead. But it cannot
410       return a list of seven elements in a scalar context. In this case $i7
411       will receive the first element of the cached list value, namely 7.
412
413       Merged disk caches
414
415       Another use for "MERGE" is when you want both kinds of return values
416       stored in the same disk file; this saves you from having to deal with
417       two disk files instead of one.  You can use a normalizer function to
418       keep the two sets of return values separate.  For example:
419
420               tie my %cache => 'MLDBM', 'DB_File', $filename, ...;
421
422               memoize 'myfunc',
423                 NORMALIZER => 'n',
424                 SCALAR_CACHE => [HASH => \%cache],
425                 LIST_CACHE => 'MERGE',
426               ;
427
428               sub n {
429                 my $context = wantarray() ? 'L' : 'S';
430                 # ... now compute the hash key from the arguments ...
431                 $hashkey = "$context:$hashkey";
432               }
433
434       This normalizer function will store scalar context return values in the
435       disk file under keys that begin with "S:", and list context return
436       values under keys that begin with "L:".
437

OTHER FACILITIES

439   "unmemoize"
440       There's an "unmemoize" function that you can import if you want to.
441       Why would you want to?  Here's an example: Suppose you have your cache
442       tied to a DBM file, and you want to make sure that the cache is written
443       out to disk if someone interrupts the program.  If the program exits
444       normally, this will happen anyway, but if someone types control-C or
445       something then the program will terminate immediately without
446       synchronizing the database.  So what you can do instead is
447
448           $SIG{INT} = sub { unmemoize 'function' };
449
450       "unmemoize" accepts a reference to, or the name of a previously
451       memoized function, and undoes whatever it did to provide the memoized
452       version in the first place, including making the name refer to the
453       unmemoized version if appropriate.  It returns a reference to the
454       unmemoized version of the function.
455
456       If you ask it to unmemoize a function that was never memoized, it
457       croaks.
458
459   "flush_cache"
460       "flush_cache(function)" will flush out the caches, discarding all the
461       cached data.  The argument may be a function name or a reference to a
462       function.  For finer control over when data is discarded or expired,
463       see the documentation for "Memoize::Expire", included in this package.
464
465       Note that if the cache is a tied hash, "flush_cache" will attempt to
466       invoke the "CLEAR" method on the hash.  If there is no "CLEAR" method,
467       this will cause a run-time error.
468
469       An alternative approach to cache flushing is to use the "HASH" option
470       (see above) to request that "Memoize" use a particular hash variable as
471       its cache.  Then you can examine or modify the hash at any time in any
472       way you desire.  You may flush the cache by using "%hash = ()".
473

CAVEATS

475       Memoization is not a cure-all:
476
477       •   Do not memoize a function whose behavior depends on program state
478           other than its own arguments, such as global variables, the time of
479           day, or file input.  These functions will not produce correct
480           results when memoized.  For a particularly easy example:
481
482                   sub f {
483                     time;
484                   }
485
486           This function takes no arguments, and as far as "Memoize" is
487           concerned, it always returns the same result.  "Memoize" is wrong,
488           of course, and the memoized version of this function will call
489           "time" once to get the current time, and it will return that same
490           time every time you call it after that.
491
492       •   Do not memoize a function with side effects.
493
494                   sub f {
495                     my ($a, $b) = @_;
496                     my $s = $a + $b;
497                     print "$a + $b = $s.\n";
498                   }
499
500           This function accepts two arguments, adds them, and prints their
501           sum.  Its return value is the numuber of characters it printed, but
502           you probably didn't care about that.  But "Memoize" doesn't
503           understand that.  If you memoize this function, you will get the
504           result you expect the first time you ask it to print the sum of 2
505           and 3, but subsequent calls will return 1 (the return value of
506           "print") without actually printing anything.
507
508       •   Do not memoize a function that returns a data structure that is
509           modified by its caller.
510
511           Consider these functions:  "getusers" returns a list of users
512           somehow, and then "main" throws away the first user on the list and
513           prints the rest:
514
515                   sub main {
516                     my $userlist = getusers();
517                     shift @$userlist;
518                     foreach $u (@$userlist) {
519                       print "User $u\n";
520                     }
521                   }
522
523                   sub getusers {
524                     my @users;
525                     # Do something to get a list of users;
526                     \@users;  # Return reference to list.
527                   }
528
529           If you memoize "getusers" here, it will work right exactly once.
530           The reference to the users list will be stored in the memo table.
531           "main" will discard the first element from the referenced list.
532           The next time you invoke "main", "Memoize" will not call
533           "getusers"; it will just return the same reference to the same list
534           it got last time.  But this time the list has already had its head
535           removed; "main" will erroneously remove another element from it.
536           The list will get shorter and shorter every time you call "main".
537
538           Similarly, this:
539
540                   $u1 = getusers();
541                   $u2 = getusers();
542                   pop @$u1;
543
544           will modify $u2 as well as $u1, because both variables are
545           references to the same array.  Had "getusers" not been memoized,
546           $u1 and $u2 would have referred to different arrays.
547
548       •   Do not memoize a very simple function.
549
550           Recently someone mentioned to me that the Memoize module made his
551           program run slower instead of faster.  It turned out that he was
552           memoizing the following function:
553
554               sub square {
555                 $_[0] * $_[0];
556               }
557
558           I pointed out that "Memoize" uses a hash, and that looking up a
559           number in the hash is necessarily going to take a lot longer than a
560           single multiplication.  There really is no way to speed up the
561           "square" function.
562
563           Memoization is not magical.
564

PERSISTENT CACHE SUPPORT

566       You can tie the cache tables to any sort of tied hash that you want to,
567       as long as it supports "TIEHASH", "FETCH", "STORE", and "EXISTS".  For
568       example,
569
570               tie my %cache => 'GDBM_File', $filename, O_RDWR|O_CREAT, 0666;
571               memoize 'function', SCALAR_CACHE => [HASH => \%cache];
572
573       works just fine.  For some storage methods, you need a little glue.
574
575       "SDBM_File" doesn't supply an "EXISTS" method, so included in this
576       package is a glue module called "Memoize::SDBM_File" which does provide
577       one.  Use this instead of plain "SDBM_File" to store your cache table
578       on disk in an "SDBM_File" database:
579
580               tie my %cache => 'Memoize::SDBM_File', $filename, O_RDWR|O_CREAT, 0666;
581               memoize 'function', SCALAR_CACHE => [HASH => \%cache];
582
583       "NDBM_File" has the same problem and the same solution.  (Use
584       "Memoize::NDBM_File instead of plain NDBM_File.")
585
586       "Storable" isn't a tied hash class at all.  You can use it to store a
587       hash to disk and retrieve it again, but you can't modify the hash while
588       it's on the disk.  So if you want to store your cache table in a
589       "Storable" database, use "Memoize::Storable", which puts a hashlike
590       front-end onto "Storable".  The hash table is actually kept in memory,
591       and is loaded from your "Storable" file at the time you memoize the
592       function, and stored back at the time you unmemoize the function (or
593       when your program exits):
594
595               tie my %cache => 'Memoize::Storable', $filename;
596               memoize 'function', SCALAR_CACHE => [HASH => \%cache];
597
598               tie my %cache => 'Memoize::Storable', $filename, 'nstore';
599               memoize 'function', SCALAR_CACHE => [HASH => \%cache];
600
601       Include the `nstore' option to have the "Storable" database written in
602       `network order'.  (See Storable for more details about this.)
603
604       The "flush_cache()" function will raise a run-time error unless the
605       tied package provides a "CLEAR" method.
606

EXPIRATION SUPPORT

608       See Memoize::Expire, which is a plug-in module that adds expiration
609       functionality to Memoize.  If you don't like the kinds of policies that
610       Memoize::Expire implements, it is easy to write your own plug-in module
611       to implement whatever policy you desire.  Memoize comes with several
612       examples.  An expiration manager that implements a LRU policy is
613       available on CPAN as Memoize::ExpireLRU.
614

BUGS

616       The test suite is much better, but always needs improvement.
617
618       There is some problem with the way "goto &f" works under threaded Perl,
619       perhaps because of the lexical scoping of @_.  This is a bug in Perl,
620       and until it is resolved, memoized functions will see a slightly
621       different "caller()" and will perform a little more slowly on threaded
622       perls than unthreaded perls.
623
624       Some versions of "DB_File" won't let you store data under a key of
625       length 0.  That means that if you have a function "f" which you
626       memoized and the cache is in a "DB_File" database, then the value of
627       "f()" ("f" called with no arguments) will not be memoized.  If this is
628       a big problem, you can supply a normalizer function that prepends "x"
629       to every key.
630

MAILING LIST

632       To join a very low-traffic mailing list for announcements about
633       "Memoize", send an empty note to "mjd-perl-memoize-request@plover.com".
634

AUTHOR

636       Mark-Jason Dominus ("mjd-perl-memoize+@plover.com"), Plover Systems co.
637
638       See the "Memoize.pm" Page at http://perl.plover.com/Memoize/ for news
639       and upgrades.  Near this page, at http://perl.plover.com/MiniMemoize/
640       there is an article about memoization and about the internals of
641       Memoize that appeared in The Perl Journal, issue #13.  (This article is
642       also included in the Memoize distribution as `article.html'.)
643
644       The author's book Higher-Order Perl (2005, ISBN 1558607013, published
645       by Morgan Kaufmann) discusses memoization (and many other topics) in
646       tremendous detail. It is available on-line for free.  For more
647       information, visit http://hop.perl.plover.com/ .
648
649       To join a mailing list for announcements about "Memoize", send an empty
650       message to "mjd-perl-memoize-request@plover.com".  This mailing list is
651       for announcements only and has extremely low traffic---fewer than two
652       messages per year.
653
655       Copyright 1998, 1999, 2000, 2001, 2012  by Mark Jason Dominus
656
657       This library is free software; you may redistribute it and/or modify it
658       under the same terms as Perl itself.
659

THANK YOU

661       Many thanks to Florian Ragwitz for administration and packaging
662       assistance, to John Tromp for bug reports, to Jonathan Roy for bug
663       reports and suggestions, to Michael Schwern for other bug reports and
664       patches, to Mike Cariaso for helping me to figure out the Right Thing
665       to Do About Expiration, to Joshua Gerth, Joshua Chamas, Jonathan Roy
666       (again), Mark D. Anderson, and Andrew Johnson for more suggestions
667       about expiration, to Brent Powers for the Memoize::ExpireLRU module, to
668       Ariel Scolnicov for delightful messages about the Fibonacci function,
669       to Dion Almaer for thought-provoking suggestions about the default
670       normalizer, to Walt Mankowski and Kurt Starsinic for much help
671       investigating problems under threaded Perl, to Alex Dudkevich for
672       reporting the bug in prototyped functions and for checking my patch, to
673       Tony Bass for many helpful suggestions, to Jonathan Roy (again) for
674       finding a use for "unmemoize()", to Philippe Verdret for enlightening
675       discussion of "Hook::PrePostCall", to Nat Torkington for advice I
676       ignored, to Chris Nandor for portability advice, to Randal Schwartz for
677       suggesting the '"flush_cache" function, and to Jenda Krynicky for being
678       a light in the world.
679
680       Special thanks to Jarkko Hietaniemi, the 5.8.0 pumpking, for including
681       this module in the core and for his patient and helpful guidance during
682       the integration process.
683

POD ERRORS

685       Hey! The above document had some coding errors, which are explained
686       below:
687
688       Around line 755:
689           You forgot a '=back' before '=head3'
690
691       Around line 804:
692           =back without =over
693
694
695
696perl v5.34.1                      2022-03-15                      Memoize(3pm)
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