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   1 The 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
   6 A regular radix-priority-search-tree indexes vmas using only heap_index and
   7 radix_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
  17 A regular radix-priority-search-tree helps to store and query
  18 intervals (vmas). However, a regular radix-priority-search-tree is only
  19 suitable for storing vmas with different radix indices (vm_pgoff).
  20
  21 Therefore, the prio_tree.c extends the regular radix-priority-search-tree
  22 to handle many vmas with the same vm_pgoff. Such vmas are handled in
  23 2 different ways: 1) All vmas with the same radix _and_ heap indices are
  24 linked using vm_set.list, 2) if there are many vmas with the same radix
  25 index, but different heap indices and if the regular radix-priority-search
  26 tree cannot index them all, we build an overflow-sub-tree that indexes such
  27 vmas using heap and size indices instead of heap and radix indices. For
  28 example, in the figure below some vmas with vm_pgoff = 0 (zero) are
  29 indexed by regular radix-priority-search-tree whereas others are pushed
  30 into an overflow-subtree. Note that all vmas in an overflow-sub-tree have
  31 the same vm_pgoff (radix_index) and if necessary we build different
  32 overflow-sub-trees to handle each possible radix_index. For example,
  33 in figure we have 3 overflow-sub-trees corresponding to radix indices
  34 0, 2, and 4.
  35
  36 In the final tree the first few (prio_tree_root->index_bits) levels
  37 are indexed using heap and radix indices whereas the overflow-sub-trees below
  38 those levels (i.e. levels prio_tree_root->index_bits + 1 and higher) are
  39 indexed using heap and size indices. In overflow-sub-trees the size_index
  40 is used for hashing the nodes to appropriate places.
  41
  42 Now, 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
  47 level  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
  74 Note that we use prio_tree_root->index_bits to optimize the height
  75 of the heap-and-radix indexed tree. Since prio_tree_root->index_bits is
  76 set according to the maximum end_vm_pgoff mapped, we are sure that all
  77 bits (in vm_pgoff) above prio_tree_root->index_bits are 0 (zero). Therefore,
  78 we only use the first prio_tree_root->index_bits as radix_index.
  79 Whenever index_bits is increased in prio_tree_expand, we shuffle the tree
  80 to make sure that the first prio_tree_root->index_bits levels of the tree
  81 is indexed properly using heap and radix indices.
  82
  83 We do not optimize the height of overflow-sub-trees using index_bits.
  84 The reason is: there can be many such overflow-sub-trees and all of
  85 them have to be suffled whenever the index_bits increases. This may involve
  86 walking the whole prio_tree in prio_tree_insert->prio_tree_expand code
  87 path which is not desirable. Hence, we do not optimize the height of the
  88 heap-and-size indexed overflow-sub-trees using prio_tree->index_bits.
  89 Instead the overflow sub-trees are indexed using full BITS_PER_LONG bits
  90 of size_index. This may lead to skewed sub-trees because most of the
  91 higher significant bits of the size_index are likely to be 0 (zero). In
  92 the example above, all 3 overflow-sub-trees are skewed. This may marginally
  93 affect the performance. However, processes rarely map many vmas with the
  94 same start_vm_pgoff but different end_vm_pgoffs. Therefore, we normally
  95 do not require overflow-sub-trees to index all vmas.
  96
  97 From the above discussion it is clear that the maximum height of
  98 a prio_tree can be prio_tree_root->index_bits + BITS_PER_LONG.
  99 However, in most of the common cases we do not need overflow-sub-trees,
 100 so the tree height in the common cases will be prio_tree_root->index_bits.
 101
 102 It is fair to mention here that the prio_tree_root->index_bits
 103 is increased on demand, however, the index_bits is not decreased when
 104 vmas are removed from the prio_tree. That's tricky to do. Hence, it's
 105 left as a home work problem.
 106
 107