3 * This program is free software; you can redistribute it and/or modify
4 * it under the terms of the GNU General Public License as published by
5 * the Free Software Foundation; either version 2 of the License, or
6 * (at your option) any later version.
8 * This program is distributed in the hope that it will be useful,
9 * but WITHOUT ANY WARRANTY; without even the implied warranty of
10 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
11 * GNU General Public License for more details.
13 * You should have received a copy of the GNU General Public License along
14 * with this program; if not, write to the Free Software Foundation, Inc.,
15 * 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA.
16 * http://www.gnu.org/copyleft/gpl.html
20 * @author Aaron Schulz
24 * Multi-datacenter aware caching interface
26 * All operations go to the local cache, except the delete()
27 * and touchCheckKey(), which broadcast to all clusters.
28 * This class is intended for caching data from primary stores.
29 * If the get() method does not return a value, then the caller
30 * should query the new value and backfill the cache using set().
31 * When the source data changes, the delete() method should be called.
32 * Since delete() is expensive, it should be avoided. One can do so if:
33 * - a) The object cached is immutable; or
34 * - b) Validity is checked against the source after get(); or
35 * - c) Using a modest TTL is reasonably correct and performant
36 * Consider using getWithSetCallback() instead of the get()/set() cycle.
38 * Instances of this class must be configured to point to a valid
39 * PubSub endpoint, and there must be listeners on the cache servers
40 * that subscribe to the endpoint and update the caches.
42 * Broadcasted operations like delete() and touchCheckKey() are done
43 * synchronously in the local cluster, but are relayed asynchronously.
44 * This means that callers in other datacenters will see older values
45 * for a however many milliseconds the datacenters are apart. As with
46 * any cache, this should not be relied on for cases where reads are
47 * used to determine writes to source (e.g. non-cache) data stores.
49 * All values are wrapped in metadata arrays. Keys use a "WANCache:" prefix
50 * to avoid collisions with keys that are not wrapped as metadata arrays. The
51 * prefixes are as follows:
52 * - a) "WANCache:v" : used for regular value keys
53 * - b) "WANCache:s" : used for temporarily storing values of tombstoned keys
54 * - c) "WANCache:t" : used for storing timestamp "check" keys
59 class WANObjectCache
{
60 /** @var BagOStuff The local cluster cache */
62 /** @var string Cache pool name */
64 /** @var EventRelayer */
68 protected $lastRelayError = self
::ERR_NONE
;
70 /** Seconds to tombstone keys on delete() */
71 const HOLDOFF_TTL
= 10;
72 /** Seconds to keep dependency purge keys around */
73 const CHECK_KEY_TTL
= 31536000; // 1 year
74 /** Seconds to keep lock keys around */
77 /** Idiom for set()/getWithSetCallback() TTL */
79 /** Idiom for getWithSetCallback() callbacks to avoid calling set() */
80 const TTL_UNCACHEABLE
= -1;
82 /** Cache format version number */
85 /** Fields of value holder arrays */
86 const FLD_VERSION
= 0;
91 /** Possible values for getLastError() */
92 const ERR_NONE
= 0; // no error
93 const ERR_NO_RESPONSE
= 1; // no response
94 const ERR_UNREACHABLE
= 2; // can't connect
95 const ERR_UNEXPECTED
= 3; // response gave some error
96 const ERR_RELAY
= 4; // relay broadcast failed
98 const VALUE_KEY_PREFIX
= 'WANCache:v:';
99 const STASH_KEY_PREFIX
= 'WANCache:s:';
100 const TIME_KEY_PREFIX
= 'WANCache:t:';
102 const PURGE_VAL_PREFIX
= 'PURGED:';
105 * @param array $params
106 * - cache : BagOStuff object
108 * - relayer : EventRelayer object
110 public function __construct( array $params ) {
111 $this->cache
= $params['cache'];
112 $this->pool
= $params['pool'];
113 $this->relayer
= $params['relayer'];
117 * @return WANObjectCache Cache that wraps EmptyBagOStuff
119 public static function newEmpty() {
120 return new self( array(
121 'cache' => new EmptyBagOStuff(),
123 'relayer' => new EventRelayerNull( array() )
128 * Fetch the value of a key from cache
130 * If passed in, $curTTL is set to the remaining TTL (current time left):
131 * - a) INF; if the key exists, has no TTL, and is not expired by $checkKeys
132 * - b) float (>=0); if the key exists, has a TTL, and is not expired by $checkKeys
133 * - c) float (<0); if the key is tombstoned or existing but expired by $checkKeys
134 * - d) null; if the key does not exist and is not tombstoned
136 * If a key is tombstoned, $curTTL will reflect the time since delete().
138 * The timestamp of $key will be checked against the last-purge timestamp
139 * of each of $checkKeys. Those $checkKeys not in cache will have the last-purge
140 * initialized to the current timestamp. If any of $checkKeys have a timestamp
141 * greater than that of $key, then $curTTL will reflect how long ago $key
142 * became invalid. Callers can use $curTTL to know when the value is stale.
143 * The $checkKeys parameter allow mass invalidations by updating a single key:
144 * - a) Each "check" key represents "last purged" of some source data
145 * - b) Callers pass in relevant "check" keys as $checkKeys in get()
146 * - c) When the source data that "check" keys represent changes,
147 * the touchCheckKey() method is called on them
149 * For keys that are hot/expensive, consider using getWithSetCallback() instead.
151 * @param string $key Cache key
152 * @param mixed $curTTL Approximate TTL left on the key if present [returned]
153 * @param array $checkKeys List of "check" keys
154 * @return mixed Value of cache key or false on failure
156 final public function get( $key, &$curTTL = null, array $checkKeys = array() ) {
158 $values = $this->getMulti( array( $key ), $curTTLs, $checkKeys );
159 $curTTL = isset( $curTTLs[$key] ) ?
$curTTLs[$key] : null;
161 return isset( $values[$key] ) ?
$values[$key] : false;
165 * Fetch the value of several keys from cache
167 * @see WANObjectCache::get()
169 * @param array $keys List of cache keys
170 * @param array $curTTLs Map of (key => approximate TTL left) for existing keys [returned]
171 * @param array $checkKeys List of "check" keys
172 * @return array Map of (key => value) for keys that exist
174 final public function getMulti(
175 array $keys, &$curTTLs = array(), array $checkKeys = array()
180 $vPrefixLen = strlen( self
::VALUE_KEY_PREFIX
);
181 $valueKeys = self
::prefixCacheKeys( $keys, self
::VALUE_KEY_PREFIX
);
182 $checkKeys = self
::prefixCacheKeys( $checkKeys, self
::TIME_KEY_PREFIX
);
184 // Fetch all of the raw values
185 $wrappedValues = $this->cache
->getMulti( array_merge( $valueKeys, $checkKeys ) );
186 $now = microtime( true );
188 // Get/initialize the timestamp of all the "check" keys
189 $checkKeyTimes = array();
190 foreach ( $checkKeys as $checkKey ) {
191 $timestamp = isset( $wrappedValues[$checkKey] )
192 ? self
::parsePurgeValue( $wrappedValues[$checkKey] )
194 if ( !is_float( $timestamp ) ) {
195 // Key is not set or invalid; regenerate
196 $this->cache
->add( $checkKey,
197 self
::PURGE_VAL_PREFIX
. $now, self
::CHECK_KEY_TTL
);
201 $checkKeyTimes[] = $timestamp;
204 // Get the main cache value for each key and validate them
205 foreach ( $valueKeys as $vKey ) {
206 if ( !isset( $wrappedValues[$vKey] ) ) {
207 continue; // not found
210 $key = substr( $vKey, $vPrefixLen ); // unprefix
212 list( $value, $curTTL ) = $this->unwrap( $wrappedValues[$vKey], $now );
213 if ( $value !== false ) {
214 $result[$key] = $value;
215 foreach ( $checkKeyTimes as $checkKeyTime ) {
216 // Force dependant keys to be invalid for a while after purging
217 // to reduce race conditions involving stale data getting cached
218 $safeTimestamp = $checkKeyTime + self
::HOLDOFF_TTL
;
219 if ( $safeTimestamp >= $wrappedValues[$vKey][self
::FLD_TIME
] ) {
220 $curTTL = min( $curTTL, $checkKeyTime - $now );
225 $curTTLs[$key] = $curTTL;
232 * Set the value of a key from cache
234 * Simply calling this method when source data changes is not valid because
235 * the changes do not replicate to the other WAN sites. In that case, delete()
236 * should be used instead. This method is intended for use on cache misses.
238 * @param string $key Cache key
239 * @param mixed $value
240 * @param integer $ttl Seconds to live [0=forever]
241 * @return bool Success
243 final public function set( $key, $value, $ttl = 0 ) {
244 $key = self
::VALUE_KEY_PREFIX
. $key;
245 $wrapped = $this->wrap( $value, $ttl );
247 $func = function ( $cache, $key, $cWrapped ) use ( $wrapped ) {
248 return ( is_string( $cWrapped ) )
249 ?
false // key is tombstoned; do nothing
253 return $this->cache
->merge( $key, $func, $ttl, 1 );
257 * Purge a key from all clusters
259 * This instantiates a hold-off period where the key cannot be
260 * written to avoid race conditions where dependent keys get updated
261 * with a stale value (e.g. from a DB slave). This is implemented by
262 * storing a special "tombstone" value at the cache key that this
263 * class recognizes; get() calls will return false for the key.
265 * This should only be called when the underlying data (being cached)
266 * changes in a significant way. If called twice on the same key, then
267 * the last TTL takes precedence.
269 * @param string $key Cache key
270 * @param integer $ttl How long to block writes to the key [seconds]
271 * @return bool True if the item was purged or not found, false on failure
273 final public function delete( $key, $ttl = self
::HOLDOFF_TTL
) {
274 $key = self
::VALUE_KEY_PREFIX
. $key;
275 // Avoid indefinite key salting for sanity
276 $ttl = max( $ttl, 1 );
277 // Update the local cluster immediately
278 $ok = $this->cache
->set( $key, self
::PURGE_VAL_PREFIX
. microtime( true ), $ttl );
279 // Publish the purge to all clusters
280 return $this->relayPurge( $key, $ttl ) && $ok;
284 * Fetch the value of a timestamp "check" key
286 * The key will be *initialized* to the current time if not set,
287 * so only call this method if this behavior is actually desired
289 * The timestamp can be used to check whether a cached value is valid.
290 * Callers should not assume that this returns the same timestamp in
291 * all datacenters due to relay delays.
293 * The level of staleness can roughly be estimated from this key, but
294 * if the key was evicted from cache, such calculations may show the
295 * time since expiry as ~0 seconds.
297 * Note that "check" keys won't collide with other regular keys
300 * @return float UNIX timestamp of the key
302 final public function getCheckKeyTime( $key ) {
303 $key = self
::TIME_KEY_PREFIX
. $key;
305 $time = self
::parsePurgeValue( $this->cache
->get( $key ) );
306 if ( $time === false ) {
307 // Casting assures identical floats for the next getCheckKeyTime() calls
308 $time = (string)microtime( true );
309 $this->cache
->add( $key, self
::PURGE_VAL_PREFIX
. $time, self
::CHECK_KEY_TTL
);
310 $time = (float)$time;
317 * Purge a "check" key from all clusters, invalidating keys that use it
319 * This should only be called when the underlying data (being cached)
320 * changes in a significant way, and it is impractical to call delete()
321 * on all keys that should be changed. When get() is called on those
322 * keys, the relevant "check" keys must be supplied for this to work.
324 * The "check" key essentially represents a last-modified field.
325 * It is set in the future a few seconds when this is called, to
326 * avoid race conditions where dependent keys get updated with a
327 * stale value (e.g. from a DB slave).
329 * This is typically useful for keys with static names or some cases
330 * dynamically generated names where a low number of combinations exist.
331 * When a few important keys get a large number of hits, a high cache
332 * time is usually desired as well as lockTSE logic. The resetCheckKey()
333 * method is less appropriate in such cases since the "time since expiry"
334 * cannot be inferred.
336 * Note that "check" keys won't collide with other regular keys
338 * @see WANObjectCache::get()
340 * @param string $key Cache key
341 * @return bool True if the item was purged or not found, false on failure
343 final public function touchCheckKey( $key ) {
344 $key = self
::TIME_KEY_PREFIX
. $key;
345 // Update the local cluster immediately
346 $ok = $this->cache
->set( $key,
347 self
::PURGE_VAL_PREFIX
. microtime( true ), self
::CHECK_KEY_TTL
);
348 // Publish the purge to all clusters
349 return $this->relayPurge( $key, self
::CHECK_KEY_TTL
) && $ok;
353 * Delete a "check" key from all clusters, invalidating keys that use it
355 * This is similar to touchCheckKey() in that keys using it via
356 * getWithSetCallback() will be invalidated. The differences are:
357 * a) The timestamp will be deleted from all caches and lazily
358 * re-initialized when accessed (rather than set everywhere)
359 * b) Thus, dependent keys will be known to be invalid, but not
360 * for how long (they are treated as "just" purged), which
361 * effects any lockTSE logic in getWithSetCallback()
362 * The advantage is that this does not place high TTL keys on every cache
363 * server, making it better for code that will cache many different keys
364 * and either does not use lockTSE or uses a low enough TTL anyway.
366 * This is typically useful for keys with dynamically generated names
367 * where a high number of combinations exist.
369 * Note that "check" keys won't collide with other regular keys
371 * @see WANObjectCache::touchCheckKey()
372 * @see WANObjectCache::get()
374 * @param string $key Cache key
375 * @return bool True if the item was purged or not found, false on failure
377 final public function resetCheckKey( $key ) {
378 $key = self
::TIME_KEY_PREFIX
. $key;
379 // Update the local cluster immediately
380 $ok = $this->cache
->delete( $key );
381 // Publish the purge to all clusters
382 return $this->relayDelete( $key ) && $ok;
386 * Method to fetch/regenerate cache keys
388 * On cache miss, the key will be set to the callback result,
389 * unless the callback returns false. The arguments supplied are:
390 * (current value or false, &$ttl)
391 * The callback function returns the new value given the current
392 * value (false if not present). Preemptive re-caching and $checkKeys
393 * can result in a non-false current value. The TTL of the new value
394 * can be set dynamically by altering $ttl in the callback (by reference).
396 * Usually, callbacks ignore the current value, but it can be used
397 * to maintain "most recent X" values that come from time or sequence
398 * based source data, provided that the "as of" id/time is tracked.
400 * Usage of $checkKeys is similar to get()/getMulti(). However,
401 * rather than the caller having to inspect a "current time left"
402 * variable (e.g. $curTTL, $curTTLs), a cache regeneration will be
403 * triggered using the callback.
405 * The simplest way to avoid stampedes for hot keys is to use
406 * the 'lockTSE' option in $opts. If cache purges are needed, also:
407 * a) Pass $key into $checkKeys
408 * b) Use touchCheckKey( $key ) instead of delete( $key )
409 * Following this pattern lets the old cache be used until a
410 * single thread updates it as needed. Also consider tweaking
411 * the 'lowTTL' parameter.
415 * $key = wfMemcKey( 'cat-recent-actions', $catId );
416 * // Function that derives the new key value given the old value
417 * $callback = function( $cValue, &$ttl ) { ... };
418 * // Get the key value from cache or from source on cache miss;
419 * // try to only let one cluster thread manage doing cache updates
420 * $opts = array( 'lockTSE' => 5, 'lowTTL' => 10 );
421 * $value = $cache->getWithSetCallback( $key, $callback, 60, array(), $opts );
426 * $key = wfMemcKey( 'cat-state', $catId );
427 * // The "check" keys that represent things the value depends on;
428 * // Calling touchCheckKey() on them invalidates "cat-state"
429 * $checkKeys = array(
430 * wfMemcKey( 'water-bowls', $houseId ),
431 * wfMemcKey( 'food-bowls', $houseId ),
432 * wfMemcKey( 'people-present', $houseId )
434 * // Function that derives the new key value
435 * $callback = function() { ... };
436 * // Get the key value from cache or from source on cache miss;
437 * // try to only let one cluster thread manage doing cache updates
438 * $opts = array( 'lockTSE' => 5, 'lowTTL' => 10 );
439 * $value = $cache->getWithSetCallback( $key, $callback, 60, $checkKeys, $opts );
442 * @see WANObjectCache::get()
444 * @param string $key Cache key
445 * @param callable $callback Value generation function
446 * @param integer $ttl Seconds to live for key updates. Special values are:
447 * - WANObjectCache::TTL_NONE : cache forever
448 * - WANObjectCache::TTL_UNCACHEABLE : do not cache at all
449 * @param array $checkKeys List of "check" keys
450 * @param array $opts Options map:
451 * - lowTTL : consider pre-emptive updates when the current TTL (sec)
452 * of the key is less than this. It becomes more likely
453 * over time, becoming a certainty once the key is expired.
454 * - lockTSE : if the key is tombstoned or expired (by $checkKeys) less
455 * than this many seconds ago, then try to have a single
456 * thread handle cache regeneration at any given time.
457 * Other threads will try to use stale values if possible.
458 * If, on miss, the time since expiration is low, the assumption
459 * is that the key is hot and that a stampede is worth avoiding.
460 * - tempTTL : when 'lockTSE' is set, this determines the TTL of the temp
461 * key used to cache values while a key is tombstoned.
462 * This avoids excessive regeneration of hot keys on delete() but
463 * may result in stale values.
464 * @return mixed Value to use for the key
466 final public function getWithSetCallback(
467 $key, $callback, $ttl, array $checkKeys = array(), array $opts = array()
469 $lowTTL = isset( $opts['lowTTL'] ) ?
$opts['lowTTL'] : min( 10, $ttl );
470 $lockTSE = isset( $opts['lockTSE'] ) ?
$opts['lockTSE'] : -1;
471 $tempTTL = isset( $opts['tempTTL'] ) ?
$opts['tempTTL'] : 5;
473 // Get the current key value
475 $cValue = $this->get( $key, $curTTL, $checkKeys ); // current value
476 $value = $cValue; // return value
478 // Determine if a regeneration is desired
479 if ( $value !== false && $curTTL > 0 && !$this->worthRefresh( $curTTL, $lowTTL ) ) {
483 $isTombstone = ( $curTTL !== null && $value === false );
484 // Assume a key is hot if requested soon after invalidation
485 $isHot = ( $curTTL !== null && $curTTL <= 0 && abs( $curTTL ) <= $lockTSE );
489 // Acquire a cluster-local non-blocking lock
490 if ( $this->cache
->lock( $key, 0, self
::LOCK_TTL
) ) {
491 // Lock acquired; this thread should update the key
493 } elseif ( $value !== false ) {
494 // If it cannot be acquired; then the stale value can be used
499 if ( !$locked && ( $isTombstone ||
$isHot ) ) {
500 // Use the stash value for tombstoned keys to reduce regeneration load.
501 // For hot keys, either another thread has the lock or the lock failed;
502 // use the stash value from the last thread that regenerated it.
503 $value = $this->cache
->get( self
::STASH_KEY_PREFIX
. $key );
504 if ( $value !== false ) {
509 if ( !is_callable( $callback ) ) {
510 throw new InvalidArgumentException( "Invalid cache miss callback provided." );
513 // Generate the new value from the callback...
514 $value = call_user_func_array( $callback, array( $cValue, &$ttl ) );
515 // When delete() is called, writes are write-holed by the tombstone,
516 // so use a special stash key to pass the new value around threads.
517 if ( $value !== false && ( $isHot ||
$isTombstone ) && $ttl >= 0 ) {
518 $this->cache
->set( self
::STASH_KEY_PREFIX
. $key, $value, $tempTTL );
522 $this->cache
->unlock( $key );
525 if ( $value !== false && $ttl >= 0 ) {
526 // Update the cache; this will fail if the key is tombstoned
527 $this->set( $key, $value, $ttl );
534 * Get the "last error" registered; clearLastError() should be called manually
535 * @return int ERR_* constant for the "last error" registry
537 final public function getLastError() {
538 if ( $this->lastRelayError
) {
539 // If the cache and the relayer failed, focus on the later.
540 // An update not making it to the relayer means it won't show up
541 // in other DCs (nor will consistent re-hashing see up-to-date values).
542 // On the other hand, if just the cache update failed, then it should
543 // eventually be applied by the relayer.
544 return $this->lastRelayError
;
547 $code = $this->cache
->getLastError();
549 case BagOStuff
::ERR_NONE
:
550 return self
::ERR_NONE
;
551 case BagOStuff
::ERR_NO_RESPONSE
:
552 return self
::ERR_NO_RESPONSE
;
553 case BagOStuff
::ERR_UNREACHABLE
:
554 return self
::ERR_UNREACHABLE
;
556 return self
::ERR_UNEXPECTED
;
561 * Clear the "last error" registry
563 final public function clearLastError() {
564 $this->cache
->clearLastError();
565 $this->lastRelayError
= self
::ERR_NONE
;
569 * Do the actual async bus purge of a key
571 * This must set the key to "PURGED:<UNIX timestamp>"
573 * @param string $key Cache key
574 * @param integer $ttl How long to keep the tombstone [seconds]
575 * @return bool Success
577 protected function relayPurge( $key, $ttl ) {
578 $event = $this->cache
->modifySimpleRelayEvent( array(
581 'val' => 'PURGED:$UNIXTIME$',
582 'ttl' => max( $ttl, 1 ),
583 'sbt' => true, // substitute $UNIXTIME$ with actual microtime
586 $ok = $this->relayer
->notify( "{$this->pool}:purge", $event );
588 $this->lastRelayError
= self
::ERR_RELAY
;
595 * Do the actual async bus delete of a key
597 * @param string $key Cache key
598 * @return bool Success
600 protected function relayDelete( $key ) {
601 $event = $this->cache
->modifySimpleRelayEvent( array(
606 $ok = $this->relayer
->notify( "{$this->pool}:purge", $event );
608 $this->lastRelayError
= self
::ERR_RELAY
;
615 * Check if a key should be regenerated (using random probability)
617 * This returns false if $curTTL >= $lowTTL. Otherwise, the chance
618 * of returning true increases steadily from 0% to 100% as the $curTTL
619 * moves from $lowTTL to 0 seconds. This handles widely varying
620 * levels of cache access traffic.
622 * @param float $curTTL Approximate TTL left on the key if present
623 * @param float $lowTTL Consider a refresh when $curTTL is less than this
626 protected function worthRefresh( $curTTL, $lowTTL ) {
627 if ( $curTTL >= $lowTTL ) {
629 } elseif ( $curTTL <= 0 ) {
633 $chance = ( 1 - $curTTL / $lowTTL );
635 return mt_rand( 1, 1e9
) <= 1e9
* $chance;
639 * Do not use this method outside WANObjectCache
641 * @param mixed $value
642 * @param integer $ttl [0=forever]
645 protected function wrap( $value, $ttl ) {
647 self
::FLD_VERSION
=> self
::VERSION
,
648 self
::FLD_VALUE
=> $value,
649 self
::FLD_TTL
=> $ttl,
650 self
::FLD_TIME
=> microtime( true )
655 * Do not use this method outside WANObjectCache
657 * @param array|string|bool $wrapped
658 * @param float $now Unix Current timestamp (preferrable pre-query)
659 * @return array (mixed; false if absent/invalid, current time left)
661 protected function unwrap( $wrapped, $now ) {
662 // Check if the value is a tombstone
663 $purgeTimestamp = self
::parsePurgeValue( $wrapped );
664 if ( is_float( $purgeTimestamp ) ) {
665 // Purged values should always have a negative current $ttl
666 $curTTL = min( -0.000001, $purgeTimestamp - $now );
667 return array( false, $curTTL );
670 if ( !is_array( $wrapped ) // not found
671 ||
!isset( $wrapped[self
::FLD_VERSION
] ) // wrong format
672 ||
$wrapped[self
::FLD_VERSION
] !== self
::VERSION
// wrong version
674 return array( false, null );
677 if ( $wrapped[self
::FLD_TTL
] > 0 ) {
678 // Get the approximate time left on the key
679 $age = $now - $wrapped[self
::FLD_TIME
];
680 $curTTL = max( $wrapped[self
::FLD_TTL
] - $age, 0.0 );
682 // Key had no TTL, so the time left is unbounded
686 return array( $wrapped[self
::FLD_VALUE
], $curTTL );
691 * @param string $prefix
694 protected static function prefixCacheKeys( array $keys, $prefix ) {
696 foreach ( $keys as $key ) {
697 $res[] = $prefix . $key;
704 * @param string $value String like "PURGED:<timestamp>"
705 * @return float|bool UNIX timestamp or false on failure
707 protected static function parsePurgeValue( $value ) {
709 if ( is_string( $value ) &&
710 preg_match( '/^' . self
::PURGE_VAL_PREFIX
. '([^:]+)$/', $value, $m )