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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | 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. |