[PATCH] prio_tree: add Documentation/prio_tree.txt

Add prio_tree.c documentation.
Signed-off-by: Rajesh Venkatasubramanian <vrajesh@umich.edu>
Signed-off-by: Andrew Morton <akpm@osdl.org>
Signed-off-by: Linus Torvalds <torvalds@osdl.org>

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.
+
+