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- 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 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.
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