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author | Ian Campbell <ian.campbell@xensource.com> |
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date | Wed Apr 11 14:15:44 2007 +0100 (2007-04-11) |

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1 The prio_tree.c code indexes vmas using 3 different indexes:

2 * heap_index = vm_pgoff + vm_size_in_pages : end_vm_pgoff

3 * radix_index = vm_pgoff : start_vm_pgoff

4 * size_index = vm_size_in_pages

6 A regular radix-priority-search-tree indexes vmas using only heap_index and

7 radix_index. The conditions for indexing are:

8 * ->heap_index >= ->left->heap_index &&

9 ->heap_index >= ->right->heap_index

10 * if (->heap_index == ->left->heap_index)

11 then ->radix_index < ->left->radix_index;

12 * if (->heap_index == ->right->heap_index)

13 then ->radix_index < ->right->radix_index;

14 * nodes are hashed to left or right subtree using radix_index

15 similar to a pure binary radix tree.

17 A regular radix-priority-search-tree helps to store and query

18 intervals (vmas). However, a regular radix-priority-search-tree is only

19 suitable for storing vmas with different radix indices (vm_pgoff).

21 Therefore, the prio_tree.c extends the regular radix-priority-search-tree

22 to handle many vmas with the same vm_pgoff. Such vmas are handled in

23 2 different ways: 1) All vmas with the same radix _and_ heap indices are

24 linked using vm_set.list, 2) if there are many vmas with the same radix

25 index, but different heap indices and if the regular radix-priority-search

26 tree cannot index them all, we build an overflow-sub-tree that indexes such

27 vmas using heap and size indices instead of heap and radix indices. For

28 example, in the figure below some vmas with vm_pgoff = 0 (zero) are

29 indexed by regular radix-priority-search-tree whereas others are pushed

30 into an overflow-subtree. Note that all vmas in an overflow-sub-tree have

31 the same vm_pgoff (radix_index) and if necessary we build different

32 overflow-sub-trees to handle each possible radix_index. For example,

33 in figure we have 3 overflow-sub-trees corresponding to radix indices

34 0, 2, and 4.

36 In the final tree the first few (prio_tree_root->index_bits) levels

37 are indexed using heap and radix indices whereas the overflow-sub-trees below

38 those levels (i.e. levels prio_tree_root->index_bits + 1 and higher) are

39 indexed using heap and size indices. In overflow-sub-trees the size_index

40 is used for hashing the nodes to appropriate places.

42 Now, an example prio_tree:

44 vmas are represented [radix_index, size_index, heap_index]

45 i.e., [start_vm_pgoff, vm_size_in_pages, end_vm_pgoff]

47 level prio_tree_root->index_bits = 3

48 -----

49 _

50 0 [0,7,7] |

51 / \ |

52 ------------------ ------------ | Regular

53 / \ | radix priority

54 1 [1,6,7] [4,3,7] | search tree

55 / \ / \ |

56 ------- ----- ------ ----- | heap-and-radix

57 / \ / \ | indexed

58 2 [0,6,6] [2,5,7] [5,2,7] [6,1,7] |

59 / \ / \ / \ / \ |

60 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] |

61 / / / _

62 / / / _

63 4 [0,4,4] [2,3,5] [4,1,5] |

64 / / / |

65 5 [0,3,3] [2,2,4] [4,0,4] | Overflow-sub-trees

66 / / |

67 6 [0,2,2] [2,1,3] | heap-and-size

68 / / | indexed

69 7 [0,1,1] [2,0,2] |

70 / |

71 8 [0,0,0] |

72 _

74 Note that we use prio_tree_root->index_bits to optimize the height

75 of the heap-and-radix indexed tree. Since prio_tree_root->index_bits is

76 set according to the maximum end_vm_pgoff mapped, we are sure that all

77 bits (in vm_pgoff) above prio_tree_root->index_bits are 0 (zero). Therefore,

78 we only use the first prio_tree_root->index_bits as radix_index.

79 Whenever index_bits is increased in prio_tree_expand, we shuffle the tree

80 to make sure that the first prio_tree_root->index_bits levels of the tree

81 is indexed properly using heap and radix indices.

83 We do not optimize the height of overflow-sub-trees using index_bits.

84 The reason is: there can be many such overflow-sub-trees and all of

85 them have to be suffled whenever the index_bits increases. This may involve

86 walking the whole prio_tree in prio_tree_insert->prio_tree_expand code

87 path which is not desirable. Hence, we do not optimize the height of the

88 heap-and-size indexed overflow-sub-trees using prio_tree->index_bits.

89 Instead the overflow sub-trees are indexed using full BITS_PER_LONG bits

90 of size_index. This may lead to skewed sub-trees because most of the

91 higher significant bits of the size_index are likely to be be 0 (zero). In

92 the example above, all 3 overflow-sub-trees are skewed. This may marginally

93 affect the performance. However, processes rarely map many vmas with the

94 same start_vm_pgoff but different end_vm_pgoffs. Therefore, we normally

95 do not require overflow-sub-trees to index all vmas.

97 From the above discussion it is clear that the maximum height of

98 a prio_tree can be prio_tree_root->index_bits + BITS_PER_LONG.

99 However, in most of the common cases we do not need overflow-sub-trees,

100 so the tree height in the common cases will be prio_tree_root->index_bits.

102 It is fair to mention here that the prio_tree_root->index_bits

103 is increased on demand, however, the index_bits is not decreased when

104 vmas are removed from the prio_tree. That's tricky to do. Hence, it's

105 left as a home work problem.