Root/Documentation/prio_tree.txt

1The prio_tree.c code indexes vmas using 3 different indexes:
2    * heap_index = vm_pgoff + vm_size_in_pages : end_vm_pgoff
3    * radix_index = vm_pgoff : start_vm_pgoff
4    * size_index = vm_size_in_pages
5
6A regular radix-priority-search-tree indexes vmas using only heap_index and
7radix_index. The conditions for indexing are:
8    * ->heap_index >= ->left->heap_index &&
9        ->heap_index >= ->right->heap_index
10    * if (->heap_index == ->left->heap_index)
11        then ->radix_index < ->left->radix_index;
12    * if (->heap_index == ->right->heap_index)
13        then ->radix_index < ->right->radix_index;
14    * nodes are hashed to left or right subtree using radix_index
15      similar to a pure binary radix tree.
16
17A regular radix-priority-search-tree helps to store and query
18intervals (vmas). However, a regular radix-priority-search-tree is only
19suitable for storing vmas with different radix indices (vm_pgoff).
20
21Therefore, the prio_tree.c extends the regular radix-priority-search-tree
22to handle many vmas with the same vm_pgoff. Such vmas are handled in
232 different ways: 1) All vmas with the same radix _and_ heap indices are
24linked using vm_set.list, 2) if there are many vmas with the same radix
25index, but different heap indices and if the regular radix-priority-search
26tree cannot index them all, we build an overflow-sub-tree that indexes such
27vmas using heap and size indices instead of heap and radix indices. For
28example, in the figure below some vmas with vm_pgoff = 0 (zero) are
29indexed by regular radix-priority-search-tree whereas others are pushed
30into an overflow-subtree. Note that all vmas in an overflow-sub-tree have
31the same vm_pgoff (radix_index) and if necessary we build different
32overflow-sub-trees to handle each possible radix_index. For example,
33in figure we have 3 overflow-sub-trees corresponding to radix indices
340, 2, and 4.
35
36In the final tree the first few (prio_tree_root->index_bits) levels
37are indexed using heap and radix indices whereas the overflow-sub-trees below
38those levels (i.e. levels prio_tree_root->index_bits + 1 and higher) are
39indexed using heap and size indices. In overflow-sub-trees the size_index
40is used for hashing the nodes to appropriate places.
41
42Now, an example prio_tree:
43
44  vmas are represented [radix_index, size_index, heap_index]
45                 i.e., [start_vm_pgoff, vm_size_in_pages, end_vm_pgoff]
46
47level prio_tree_root->index_bits = 3
48-----
49                                                _
50  0 [0,7,7] |
51                              / \ |
52                      ------------------ ------------ | Regular
53                       / \ | radix priority
54  1 [1,6,7] [4,3,7] | search tree
55                  / \ / \ |
56             ------- ----- ------ ----- | heap-and-radix
57            / \ / \ | indexed
58  2 [0,6,6] [2,5,7] [5,2,7] [6,1,7] |
59            / \ / \ / \ / \ |
60  3 [0,5,5] [1,5,6] [2,4,6] [3,4,7] [4,2,6] [5,1,6] [6,0,6] [7,0,7] |
61           / / / _
62                  / / / _
63  4 [0,4,4] [2,3,5] [4,1,5] |
64           / / / |
65  5 [0,3,3] [2,2,4] [4,0,4] | Overflow-sub-trees
66          / / |
67  6 [0,2,2] [2,1,3] | heap-and-size
68             / / | indexed
69  7 [0,1,1] [2,0,2] |
70            / |
71  8 [0,0,0] |
72                                                  _
73
74Note that we use prio_tree_root->index_bits to optimize the height
75of the heap-and-radix indexed tree. Since prio_tree_root->index_bits is
76set according to the maximum end_vm_pgoff mapped, we are sure that all
77bits (in vm_pgoff) above prio_tree_root->index_bits are 0 (zero). Therefore,
78we only use the first prio_tree_root->index_bits as radix_index.
79Whenever index_bits is increased in prio_tree_expand, we shuffle the tree
80to make sure that the first prio_tree_root->index_bits levels of the tree
81is indexed properly using heap and radix indices.
82
83We do not optimize the height of overflow-sub-trees using index_bits.
84The reason is: there can be many such overflow-sub-trees and all of
85them have to be suffled whenever the index_bits increases. This may involve
86walking the whole prio_tree in prio_tree_insert->prio_tree_expand code
87path which is not desirable. Hence, we do not optimize the height of the
88heap-and-size indexed overflow-sub-trees using prio_tree->index_bits.
89Instead the overflow sub-trees are indexed using full BITS_PER_LONG bits
90of size_index. This may lead to skewed sub-trees because most of the
91higher significant bits of the size_index are likely to be 0 (zero). In
92the example above, all 3 overflow-sub-trees are skewed. This may marginally
93affect the performance. However, processes rarely map many vmas with the
94same start_vm_pgoff but different end_vm_pgoffs. Therefore, we normally
95do not require overflow-sub-trees to index all vmas.
96
97From the above discussion it is clear that the maximum height of
98a prio_tree can be prio_tree_root->index_bits + BITS_PER_LONG.
99However, in most of the common cases we do not need overflow-sub-trees,
100so the tree height in the common cases will be prio_tree_root->index_bits.
101
102It is fair to mention here that the prio_tree_root->index_bits
103is increased on demand, however, the index_bits is not decreased when
104vmas are removed from the prio_tree. That's tricky to do. Hence, it's
105left as a home work problem.
106
107
108

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