list_sort.c

  1// SPDX-License-Identifier: GPL-2.0
  2#include <linux/compiler.h>
  3#include <linux/export.h>
  4#include <linux/list_sort.h>
  5#include <linux/list.h>
  6
  7/*
  8 * Returns a list organized in an intermediate format suited
  9 * to chaining of merge() calls: null-terminated, no reserved or
 10 * sentinel head node, "prev" links not maintained.
 11 */
 12__attribute__((nonnull(2,3,4)))
 13static struct list_head *merge(void *priv, list_cmp_func_t cmp,
 14				struct list_head *a, struct list_head *b)
 15{
 16	struct list_head *head, **tail = &head;
 17
 18	for (;;) {
 19		/* if equal, take 'a' -- important for sort stability */
 20		if (cmp(priv, a, b) <= 0) {
 21			*tail = a;
 22			tail = &a->next;
 23			a = a->next;
 24			if (!a) {
 25				*tail = b;
 26				break;
 27			}
 28		} else {
 29			*tail = b;
 30			tail = &b->next;
 31			b = b->next;
 32			if (!b) {
 33				*tail = a;
 34				break;
 35			}
 36		}
 37	}
 38	return head;
 39}
 40
 41/*
 42 * Combine final list merge with restoration of standard doubly-linked
 43 * list structure.  This approach duplicates code from merge(), but
 44 * runs faster than the tidier alternatives of either a separate final
 45 * prev-link restoration pass, or maintaining the prev links
 46 * throughout.
 47 */
 48__attribute__((nonnull(2,3,4,5)))
 49static void merge_final(void *priv, list_cmp_func_t cmp, struct list_head *head,
 50			struct list_head *a, struct list_head *b)
 51{
 52	struct list_head *tail = head;
 53	u8 count = 0;
 54
 55	for (;;) {
 56		/* if equal, take 'a' -- important for sort stability */
 57		if (cmp(priv, a, b) <= 0) {
 58			tail->next = a;
 59			a->prev = tail;
 60			tail = a;
 61			a = a->next;
 62			if (!a)
 63				break;
 64		} else {
 65			tail->next = b;
 66			b->prev = tail;
 67			tail = b;
 68			b = b->next;
 69			if (!b) {
 70				b = a;
 71				break;
 72			}
 73		}
 74	}
 75
 76	/* Finish linking remainder of list b on to tail */
 77	tail->next = b;
 78	do {
 79		/*
 80		 * If the merge is highly unbalanced (e.g. the input is
 81		 * already sorted), this loop may run many iterations.
 82		 * Continue callbacks to the client even though no
 83		 * element comparison is needed, so the client's cmp()
 84		 * routine can invoke cond_resched() periodically.
 85		 */
 86		if (unlikely(!++count))
 87			cmp(priv, b, b);
 88		b->prev = tail;
 89		tail = b;
 90		b = b->next;
 91	} while (b);
 92
 93	/* And the final links to make a circular doubly-linked list */
 94	tail->next = head;
 95	head->prev = tail;
 96}
 97
 98/**
 99 * list_sort - sort a list
100 * @priv: private data, opaque to list_sort(), passed to @cmp
101 * @head: the list to sort
102 * @cmp: the elements comparison function
103 *
104 * The comparison function @cmp must return > 0 if @a should sort after
105 * @b ("@a > @b" if you want an ascending sort), and <= 0 if @a should
106 * sort before @b *or* their original order should be preserved.  It is
107 * always called with the element that came first in the input in @a,
108 * and list_sort is a stable sort, so it is not necessary to distinguish
109 * the @a < @b and @a == @b cases.
110 *
111 * The comparison function must adhere to specific mathematical properties
112 * to ensure correct and stable sorting:
113 * - Antisymmetry: cmp(@a, @b) must return the opposite sign of
114 * cmp(@b, @a).
115 * - Transitivity: if cmp(@a, @b) <= 0 and cmp(@b, @c) <= 0, then
116 * cmp(@a, @c) <= 0.
117 *
118 * This is compatible with two styles of @cmp function:
119 * - The traditional style which returns <0 / =0 / >0, or
120 * - Returning a boolean 0/1.
121 * The latter offers a chance to save a few cycles in the comparison
122 * (which is used by e.g. plug_ctx_cmp() in block/blk-mq.c).
123 *
124 * A good way to write a multi-word comparison is::
125 *
126 *	if (a->high != b->high)
127 *		return a->high > b->high;
128 *	if (a->middle != b->middle)
129 *		return a->middle > b->middle;
130 *	return a->low > b->low;
131 *
132 *
133 * This mergesort is as eager as possible while always performing at least
134 * 2:1 balanced merges.  Given two pending sublists of size 2^k, they are
135 * merged to a size-2^(k+1) list as soon as we have 2^k following elements.
136 *
137 * Thus, it will avoid cache thrashing as long as 3*2^k elements can
138 * fit into the cache.  Not quite as good as a fully-eager bottom-up
139 * mergesort, but it does use 0.2*n fewer comparisons, so is faster in
140 * the common case that everything fits into L1.
141 *
142 *
143 * The merging is controlled by "count", the number of elements in the
144 * pending lists.  This is beautifully simple code, but rather subtle.
145 *
146 * Each time we increment "count", we set one bit (bit k) and clear
147 * bits k-1 .. 0.  Each time this happens (except the very first time
148 * for each bit, when count increments to 2^k), we merge two lists of
149 * size 2^k into one list of size 2^(k+1).
150 *
151 * This merge happens exactly when the count reaches an odd multiple of
152 * 2^k, which is when we have 2^k elements pending in smaller lists,
153 * so it's safe to merge away two lists of size 2^k.
154 *
155 * After this happens twice, we have created two lists of size 2^(k+1),
156 * which will be merged into a list of size 2^(k+2) before we create
157 * a third list of size 2^(k+1), so there are never more than two pending.
158 *
159 * The number of pending lists of size 2^k is determined by the
160 * state of bit k of "count" plus two extra pieces of information:
161 *
162 * - The state of bit k-1 (when k == 0, consider bit -1 always set), and
163 * - Whether the higher-order bits are zero or non-zero (i.e.
164 *   is count >= 2^(k+1)).
165 *
166 * There are six states we distinguish.  "x" represents some arbitrary
167 * bits, and "y" represents some arbitrary non-zero bits:
168 * 0:  00x: 0 pending of size 2^k;           x pending of sizes < 2^k
169 * 1:  01x: 0 pending of size 2^k; 2^(k-1) + x pending of sizes < 2^k
170 * 2: x10x: 0 pending of size 2^k; 2^k     + x pending of sizes < 2^k
171 * 3: x11x: 1 pending of size 2^k; 2^(k-1) + x pending of sizes < 2^k
172 * 4: y00x: 1 pending of size 2^k; 2^k     + x pending of sizes < 2^k
173 * 5: y01x: 2 pending of size 2^k; 2^(k-1) + x pending of sizes < 2^k
174 * (merge and loop back to state 2)
175 *
176 * We gain lists of size 2^k in the 2->3 and 4->5 transitions (because
177 * bit k-1 is set while the more significant bits are non-zero) and
178 * merge them away in the 5->2 transition.  Note in particular that just
179 * before the 5->2 transition, all lower-order bits are 11 (state 3),
180 * so there is one list of each smaller size.
181 *
182 * When we reach the end of the input, we merge all the pending
183 * lists, from smallest to largest.  If you work through cases 2 to
184 * 5 above, you can see that the number of elements we merge with a list
185 * of size 2^k varies from 2^(k-1) (cases 3 and 5 when x == 0) to
186 * 2^(k+1) - 1 (second merge of case 5 when x == 2^(k-1) - 1).
187 */
188__attribute__((nonnull(2,3)))
189void list_sort(void *priv, struct list_head *head, list_cmp_func_t cmp)
190{
191	struct list_head *list = head->next, *pending = NULL;
192	size_t count = 0;	/* Count of pending */
193
194	if (list == head->prev)	/* Zero or one elements */
195		return;
196
197	/* Convert to a null-terminated singly-linked list. */
198	head->prev->next = NULL;
199
200	/*
201	 * Data structure invariants:
202	 * - All lists are singly linked and null-terminated; prev
203	 *   pointers are not maintained.
204	 * - pending is a prev-linked "list of lists" of sorted
205	 *   sublists awaiting further merging.
206	 * - Each of the sorted sublists is power-of-two in size.
207	 * - Sublists are sorted by size and age, smallest & newest at front.
208	 * - There are zero to two sublists of each size.
209	 * - A pair of pending sublists are merged as soon as the number
210	 *   of following pending elements equals their size (i.e.
211	 *   each time count reaches an odd multiple of that size).
212	 *   That ensures each later final merge will be at worst 2:1.
213	 * - Each round consists of:
214	 *   - Merging the two sublists selected by the highest bit
215	 *     which flips when count is incremented, and
216	 *   - Adding an element from the input as a size-1 sublist.
217	 */
218	do {
219		size_t bits;
220		struct list_head **tail = &pending;
221
222		/* Find the least-significant clear bit in count */
223		for (bits = count; bits & 1; bits >>= 1)
224			tail = &(*tail)->prev;
225		/* Do the indicated merge */
226		if (likely(bits)) {
227			struct list_head *a = *tail, *b = a->prev;
228
229			a = merge(priv, cmp, b, a);
230			/* Install the merged result in place of the inputs */
231			a->prev = b->prev;
232			*tail = a;
233		}
234
235		/* Move one element from input list to pending */
236		list->prev = pending;
237		pending = list;
238		list = list->next;
239		pending->next = NULL;
240		count++;
241	} while (list);
242
243	/* End of input; merge together all the pending lists. */
244	list = pending;
245	pending = pending->prev;
246	for (;;) {
247		struct list_head *next = pending->prev;
248
249		if (!next)
250			break;
251		list = merge(priv, cmp, pending, list);
252		pending = next;
253	}
254	/* The final merge, rebuilding prev links */
255	merge_final(priv, cmp, head, pending, list);
256}
257EXPORT_SYMBOL(list_sort);