Multi-Way Search Tree

Transcription

1 Multi-Way Search Tree A multi-way search tree is an ordered tree such that Each internal node has at least two and at most d children and stores d -1 data items (k i, D i ) Rule: Number of children = 1 + number of data items in a node k 1 D 1 k 2 D 2... k d-1 D d-1 child 1 child 2 child child d d is the degree or order of the tree

2 Multi-Way Search Tree A multi-way search tree is an ordered tree such that Each internal node has at least two and at most d children and stores d -1 data items (k i, D i ) An internal node storing keys k 1 k 2 k d-1 has d children v 1 v 2 v d such that k 1 k 2... k d-1 < k 1 > k 1 < k 2 > k 2 < k > k d-2 < k d-1 > k d-1 (2,4) Trees 2

3 Multi-Way Search Tree A multi-way search tree is an ordered tree such that Each internal node has at least two and at most d children and stores d -1 data items (k i, D i ) An internal node storing keys k 1 k 2 k d-1 has d children v 1 v 2 v d such that By convenience we add sentinel keys k 0 = - and k d = k 0 = - k 1 k 2... k d-1 k d = < k 1 > k 1 < k 2 > k 2 < k > k d-2 < k d-1 > k d-1 (2,4) Trees

4 Multi-Way Search Tree A multi-way search tree is an ordered tree such that Each internal node has at least two and at most d children and stores d -1 data items (k i, D i ) An internal node storing keys k 1 k 2 k d-1 has d children v 1 v 2 v d such that By convenience we add sentinel keys k 0 = - and k d = The leaves store no items and serve as placeholders (2,4) Trees 4

5 Multi-Way Search Tree (2,4) Trees 5

6 Multi-Way Search Tree Not a multiway search tree (2,4) Trees 6

7 Multi-Way Inorder Traversal We can extend the notion of inorder traversal from binary trees to multi-way search trees An inorder traversal of a multi-way search tree visits the keys in increasing order Inorder traversal: 2, 6, 8, 11, 15, 24, 27, 0, 2 (2,4) Trees 7

8 Data Structures for Multi-Way Search Trees k 0 = - k 1 k 2... k d-1 k d = child 1 child 2 child child d node null D 1 D 2... D d-1 null data - k 1 k 2... k d-1 keys ch1 ch 2 ch ch d children d-1 (2,4) Trees 8

9 Data Structures for Multi-Way Search Trees k 0 = - k 1 k 2... k d-1 k d = child 1 child 2 child child d null D 1 D 2... D d-1 null data - k 1 k 2... k d-1 keys ch1 ch 2 ch ch d children d-1 Secondary data structures (2,4) Trees 9

10 Multi-Way Searching Similar to search in a binary search tree Example: search for (2,4) Trees 10

11 Multi-Way Searching Similar to search in a binary search tree Example: search for (2,4) Trees 11

12 Multi-Way Searching Algorithm get(r,k) In: Root r of a multiway search tree, key k Out: data for key k or null if k not in tree if r is a leaf then return null else { } Use binary search to find the index i such that r.keys[i] k < r.keys[i+1] if k = r.keys[i] then return r.data[i] else return get(r.child[i],k) (2,4) Trees 12

13 Multi-Way Searching Algorithm get(r,k) In: Root r of a multiway search tree, key k Out: data for key k or null if k not in tree if r is a leaf then return null else { } Use binary search to find the index i such that r.keys[i] k < r.keys[i+1] if k = r.keys[i] then return r.data[i] else return get(r,r.child[i]) c operations Ignoring recursive calls: c 1 log d + c 2 operations (2,4) Trees 1

14 Time Complexity of get Operation c1 + c2 log d c1 + c2 log d c1 + c2 log d... c f(n) = c + (c1 + c2 log d) height of tree is O(log d height of tree) (2,4) Trees 14

15 Smallest Operation c c c k s... c f(n) = c height of tree is O(height of tree) (2,4) Trees 15

16 Successor Operation smallest value in subtree successor (27) (2,4) Trees 16

17 Successor Operation successor (17) (2,4) Trees 17

18 Successor Operation Time complexity: get + smallest: O(log d height), or get + travel up: O(log d height) so time complexity is O(log d height) (2,4) Trees 18

19 Put Operation insert (51) Assume degree = 4 (2,4) Trees 19

20 Put Operation Time Complexity: Find node to store new data: O(log d height) Add data to node: O(d) time complexity: O(d+log d height) (2,4) Trees 20

21 Remove Operation remove(50) (2,4) Trees 21

22 Remove Operation remove(50) (2,4) Trees 22

23 Remove Operation remove(50) (2,4) Trees 2

24 Remove Operation remove(27) (2,4) Trees 24

25 Remove Operation remove(27) (2,4) Trees 25

26 Remove Operation Time complexity: get + smallest + remove data + delete 2 nodes: O(d+log d height) (2,4) Trees 26