linux/Documentation/prio_tree.txt
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   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.
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