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. | |