Documentation/prio_tree.txt

The prio_tree.c code indexes vmas using 3 different indexes:
	* heap_index  = vm_pgoff + vm_size_in_pages : end_vm_pgoff
	* radix_index = vm_pgoff : start_vm_pgoff
	* size_index = vm_size_in_pages

A regular radix-priority-search-tree indexes vmas using only heap_index and
radix_index. The conditions for indexing are:
	* ->heap_index >= ->left->heap_index &&
		->heap_index >= ->right->heap_index
	* if (->heap_index == ->left->heap_index)
		then ->radix_index < ->left->radix_index;
	* if (->heap_index == ->right->heap_index)
		then ->radix_index < ->right->radix_index;
	* nodes are hashed to left or right subtree using radix_index
	  similar to a pure binary radix tree.

A regular radix-priority-search-tree helps to store and query
intervals (vmas). However, a regular radix-priority-search-tree is only
suitable for storing vmas with different radix indices (vm_pgoff).

Therefore, the prio_tree.c extends the regular radix-priority-search-tree
to handle many vmas with the same vm_pgoff. Such vmas are handled in
2 different ways: 1) All vmas with the same radix _and_ heap indices are
linked using vm_set.list, 2) if there are many vmas with the same radix
index, but different heap indices and if the regular radix-priority-search
tree cannot index them all, we build an overflow-sub-tree that indexes such
vmas using heap and size indices instead of heap and radix indices. For
example, in the figure below some vmas with vm_pgoff = 0 (zero) are
indexed by regular radix-priority-search-tree whereas others are pushed
into an overflow-subtree. Note that all vmas in an overflow-sub-tree have
the same vm_pgoff (radix_index) and if necessary we build different
overflow-sub-trees to handle each possible radix_index. For example,
in figure we have 3 overflow-sub-trees corresponding to radix indices
0, 2, and 4.

In the final tree the first few (prio_tree_root->index_bits) levels
are indexed using heap and radix indices whereas the overflow-sub-trees below
those levels (i.e. levels prio_tree_root->index_bits + 1 and higher) are
indexed using heap and size indices. In overflow-sub-trees the size_index
is used for hashing the nodes to appropriate places.

Now, an example prio_tree:

  vmas are represented [radix_index, size_index, heap_index]
                 i.e., [start_vm_pgoff, vm_size_in_pages, end_vm_pgoff]

level  prio_tree_root->index_bits = 3
-----
												_
  0			 				[0,7,7]					 |
  							/     \					 |
				      ------------------       ------------			 |     Regular
  				     /					   \			 |  radix priority
  1		 		[1,6,7]					  [4,3,7]		 |   search tree
  				/     \					  /     \		 |
			 -------       -----			    ------       -----		 |  heap-and-radix
			/		    \			   /		      \		 |      indexed
  2		    [0,6,6]	 	   [2,5,7]		[5,2,7]		    [6,1,7]	 |
		    /     \		   /     \		/     \		    /     \	 |
  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]	 |
		   /			   /		       /		   		_
                  /		          /		      /					_
  4	      [0,4,4]		      [2,3,5]		   [4,1,5]				 |
  		 /			 /		      /					 |
  5	     [0,3,3]		     [2,2,4]		  [4,0,4]				 |  Overflow-sub-trees
  		/			/							 |
  6	    [0,2,2]		    [2,1,3]							 |    heap-and-size
  	       /		       /							 |       indexed
  7	   [0,1,1]		   [2,0,2]							 |
  	      /											 |
  8	  [0,0,0]										 |
  												_

Note that we use prio_tree_root->index_bits to optimize the height
of the heap-and-radix indexed tree. Since prio_tree_root->index_bits is
set according to the maximum end_vm_pgoff mapped, we are sure that all
bits (in vm_pgoff) above prio_tree_root->index_bits are 0 (zero). Therefore,
we only use the first prio_tree_root->index_bits as radix_index.
Whenever index_bits is increased in prio_tree_expand, we shuffle the tree
to make sure that the first prio_tree_root->index_bits levels of the tree
is indexed properly using heap and radix indices.

We do not optimize the height of overflow-sub-trees using index_bits.
The reason is: there can be many such overflow-sub-trees and all of
them have to be suffled whenever the index_bits increases. This may involve
walking the whole prio_tree in prio_tree_insert->prio_tree_expand code
path which is not desirable. Hence, we do not optimize the height of the
heap-and-size indexed overflow-sub-trees using prio_tree->index_bits.
Instead the overflow sub-trees are indexed using full BITS_PER_LONG bits
of size_index. This may lead to skewed sub-trees because most of the
higher significant bits of the size_index are likely to be be 0 (zero). In
the example above, all 3 overflow-sub-trees are skewed. This may marginally
affect the performance. However, processes rarely map many vmas with the
same start_vm_pgoff but different end_vm_pgoffs. Therefore, we normally
do not require overflow-sub-trees to index all vmas.

From the above discussion it is clear that the maximum height of
a prio_tree can be prio_tree_root->index_bits + BITS_PER_LONG.
However, in most of the common cases we do not need overflow-sub-trees,
so the tree height in the common cases will be prio_tree_root->index_bits.

It is fair to mention here that the prio_tree_root->index_bits
is increased on demand, however, the index_bits is not decreased when
vmas are removed from the prio_tree. That's tricky to do. Hence, it's
left as a home work problem.