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)))
merge(void * priv,list_cmp_func_t cmp,struct list_head * a,struct list_head * b)13 static 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)))
merge_final(void * priv,list_cmp_func_t cmp,struct list_head * head,struct list_head * a,struct list_head * b)49 static 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)))
list_sort(void * priv,struct list_head * head,list_cmp_func_t cmp)189 void 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 }
257 EXPORT_SYMBOL(list_sort);
258