Chapter 11 Multiway Search Trees

Transcription

1 Chater 11 Multiway Search Trees m-way Search Trees B-Trees B + -Trees C-C Tsai P.1 m-way Search Trees Definition: An m-way search tree is either emty or satisfies the following roerties: The root has at most m subtrees and the structure of each subtree has also at most m subtrees. Each node has u to m 1 airs and m children. For the height h, the maximum number of elements is m h -1. m = 2 => binary search tree. C-C Tsai P.2 1

2 Examles of m-way Search Tree A three-way search tree 20,40 10,15 25,30 45,50 A four-way search tree k < 10 10<k<30 30<k<35 k > 35 C-C Tsai P.3 Caacity of m-way Search Tree The maximum number of elements is m h -1. m = 2 m = 200 h = * h = * h = * C-C Tsai P.4 2

3 B-Trees Definition: A B-tree of order m is an m-way search tree that either is emty or satisfies the following roerties: The root has at least two children. All nodes other than the root node and external nodes have at least m/2 children. All external nodes are at the same level. 2-3 tree is B-tree of order tree is B-tree of order 4. C-C Tsai P Trees Each node can have a degree of 2 or 3. The number of elements is between 2 h 1 and 3 h 1, where h is the height of the tree. So if there are n elements, the height is between log 3 (n+1) and log 2 (n+1). Hence to search in a 2-3 tree, you need O(log n) time. < 40 > 40 External nodes are at the same level. C-C Tsai P.6 3

4 Search in A 2-3 Tree The search algorithm is similar to that for a BST (binary search tree). At each node (suose we are searching for x): k 1 k 2 Go this way if x < k 1 Go this way if x > k 1 and x < k 2 Go this way if x > k 2 C-C Tsai P.7 Insertion Into A 2-3 Tree Insert 70: First we search for 70. It is not in the tree, and the leaf node encountered during the search is node C. Since there is only one key there, we directly insert 70 into node C. C-C Tsai P.8 4

5 Insertion 30 into A 2-3 Tree Now we want to insert 30. The leaf we encounter is B. B is full. So we must create a new node D. The keys that will be concerned in this oeration are 10, 20 (in B) and 30 (to be inserted). Largest (30): ut into D. Smallest (10): remain in B. Median (20): insert into A, the arent of B. Add a link from A to D. median smallest largest C-C Tsai P. Insertion 60 into A 2-3 Tree We encounter leaf node C when we search for 60. C is full, so we: Create node E to hold max{70, 80, 60} = 80. Remain min{70,80,60} = 60 in C. The median, 70, will be inserted into A. But A is also full, so New node F will be created. F has children C (where 60 is in) and E (where 80 is in). 70 will be inserted to A A B D C E C-C Tsai P.10 5

6 But A is also full, so Create node F to hold max{20,40,70} = 70. F has children C and E. min{20,40,70} = 20 will remain in A. med{20,40,70} = 40 should be inserted into arent of A. But A has no arent, so create G to hold 40. G has children A and F. 70 will be A inserted to A E B D C C-C Tsai P.11 Slit A 3-Node Inserting y into a 3-node B causes a slit. A med will be inserted into A. A B x z B min max C D E F (F is a node that does not have a arent yet. ) G (new) min, max, and med are the minimum, maximum, and median of {x, y, z}, resectively. C-C Tsai P.12 6

7 Slit Observe that this attern reeats. arent() med (next y) arent() (next ) x z min max (next q) ch 1 () ch 2 () ch 3 () q q is initialized to be null. At that time is a leaf. ch 1 () ch 2 () ch 3 () The osition to insert the link to q deends on the situation. q C-C Tsai P.13 Slit Slit is simler when is the root. New root med x z min max ch 1 () ch 2 () ch 3 () q ch 1 () ch 2 () ch 3 () The osition to insert the link to q deends on the situation. q C-C Tsai P.14 7

8 Insertion Algorithm We are to insert key y in tree t. First, search for y in t. When you visit each node, ush it into a stack to facilitate finding arents later. Assume that y is not in t (otherwise we need not insert). Let be the leaf node we encountered in the search. So, if we o a node from the above stack, we ll obtain the arent of (assume that itself is not ushed into the stack). C-C Tsai P.15 Insertion Algorithm Initialize q to be null. If is a 2-node, then simly insert y into. Put q immediately to the right of y. That is, if w is originally in, then we have two cases: w y y w nil nil q=nil nil q=nil nil And we re done! C-C Tsai P.16 8

9 Insertion Algorithm If is a 3-node, then slit. arent() med (next y) arent() (next ) x z min max (next q) nil nil nil q=nil nil nil nil q=nil Then, let = arent(), q be the new node holding max, and y = med. We ll now consider the insertion of the new y into the new. C-C Tsai P.17 Insertion Algorithm In the remaining rocess, if is a 2-node, then simly insert y into, and udate the links as: w y y w a b q a q b And we re done! C-C Tsai P.18

10 Insertion Algorithm If is a 3-node, then slit. Then we ll continue to insert the new y into the new. arent() med (next y) arent() (next ) x z min max (next q) ch 1 () ch 2 () ch 3 () q ch 1 () ch 2 () ch 3 () q The osition to insert the link to q deends on the situation. C-C Tsai P.1 Insertion Algorithm If (3-node) is the root, then the slit is done in the manner as stated before. We re done after this. New root med x z min max ch 1 () ch 2 () ch 3 () q ch 1 () ch 2 () ch 3 () The osition to insert the link to q deends on the situation. q C-C Tsai P.20 10

11 Correctness of Insertion Note that, all keys in art B, including y and keys in q, lie between u and v. Because we followed the middle link of arent() when we did the search in the examle below, the inut key (to be inserted) falls between u and v. Besides the (inut) key to insert, all keys in B were originally there and fall between u and v. u v arent()?? y to be inserted in ch 1 () ch 2 () ch 3 () q A B C-C Tsai P.21 C Correctness of Insertion So the global relationshi is ok. As to the local relationshi among the keys, the insertion actions clearly maintain such roerly. u v arent() w y ch 1 () ch 2 () q You should use induction as well as these observations to give a more rigorous roof. and q are always 2-3 trees after each iteration. C-C Tsai P.22 11

12 Time Comlexity of Insertion At each level, the algorithm takes O(1) time. There are O(log n) levels. So insertion takes O(log n) time. C-C Tsai P.23 Deletion From A 2-3 Tree Deletion of any element can be transformed into deletion of a leaf element. To delete 50, we relace 50 by 60 or 20. Then delete corresondingly the leaf element 60 or is the leftmost leaf element in the right subtree of is the rightmost leaf element in the left subtree of Use the algorithm resented later to delete 20 in the leaf. C-C Tsai P.24 12

13 Deletion From A 2-3 Tree Delete 70 (in C). This case is straightforward, as the resulting C is non-emty. C-C Tsai P.25 Deletion From A 2-3 Tree Delete 0 (in D). This is also simle; a shift of 5 in D suffices. C-C Tsai P.26 13

14 Deletion From A 2-3 Tree Delete 60 (in C). C becomes emty. Left sibling of C is a 3-node. Hence (rotation): Move 50 from A to C. Move 20 from B to A. C-C Tsai P.27 Deletion From A 2-3 Tree Delete 5 (from D): D becomes emty. Its left sibling C is a 2-node, hence (combine): Move 80 from A to C. Delete D. C-C Tsai P.28 14

15 Deletion From A 2-3 Tree Delete 50 (in C). Simly shift. C-C Tsai P.2 Deletion From A 2-3 Tree Delete 10 (in B): B becomes emty. The right sibling of B is a 2-node, hence (combine): Move 20 from A to B. Move 80 from C to B. The arent A, which is also the root, is emty. Hence simly let B be the new root. C-C Tsai P.30 15

16 Rotation and Combine When a deletion in node leaves emty, then: Let r be the arent of. If is the left child of r, then let q be the right sibling of. Otherwise, let q be the left sibling of. If q is a 2-node, then rotation. If q is a 3-node, then combine. C-C Tsai P.31 Rotation If is the left child of r: (? means don t care) Observe the correctness. C-C Tsai P.32 16

17 Rotation If is the middle child of r. C-C Tsai P.33 Rotation If is the right child of r. C-C Tsai P.34 17

18 Combine If is the left child of r: Case 1: If r is a 2-node. r becomes emty, so we set to be r, and continue to consider to rotate/combine the new. If r is a root, then let become the new root. C-C Tsai P.35 Combine If is the left child of r. Case 2: If r is a 3-node. C-C Tsai P.36 18

19 Combine If is the middle child of r: Case 1: If r is a 2-node. Continue to handle the emty r as before. r w r q y y w a b c a b c C-C Tsai P.37 Combine If is the middle child of r: Case 2: If r is a 3-node. r w x r x q y d y w d a b c a b c C-C Tsai P.38 1

20 Combine If is the right child of r: r w x r w a q y a y x b c d b c d C-C Tsai P.3 Correctness of Deletion Observe that, if a combine results in a new emty node, then that node must have the following aearance (r with one tail): r r will become in the next iteration. In the (left-hand side) ictures we ve seen, has the above aearance. So alicable. We begin with being a leaf. At that time, the children of are all null. So rotation/combine as illustrated in the revious figures are also alicable. Correctness of other arts should be clear. C-C Tsai P.40 20

21 Time Comlexity of Deletion At each level: O(1) time. Rotation/combine need O(1) time. #levels: O(log n). Total: O(log n) time. C-C Tsai P.41 B + -Trees Definition: A B + -tree of order m is a tree that either is emty or satisfies the following roerties: All data nodes are the same level and are leaves. Data nodes contain elements only. The index nodes define a B-tree of order m; each index node has keys but not elements. Remaining nodes have following structure: j a 0 k 1 a 1 k 2 a 2 k j a j j = number of keys in node. a i is a ointer to a subtree. k i <= smallest key in subtree a i and > largest in a i-1. C-C Tsai P.42 21

22 B + -Tree vs B-Tree B + -tree is a close cousin of the B-tree, but some differences: Data nodes and index nodes corresond to internal nodes and external nodes, resectively, of a B-tree. Data nodes stores elements and index nodes store keys (not elements) and ointers. Data nodes are linked together, in left to right order, to form a doubly linked list. C-C Tsai P.43 Examle of a B + -Tree index nodes: store keys and ointers leaf/data nodes: store elements C-C Tsai P.44 22

23 B + -Tree Search Search the key: key = 5 Search the key: 6 <= key <= 20 C-C Tsai P.45 B + -Tree Insert Insert 3 C-C Tsai P.46 23

24 B+-Tree Insert Insert a air with key = 2. New air goes into a 3-node. C-C Tsai P.47 Insert Into A 3-node Insert new air so that the keys are in ascending order. Slit into two nodes Insert smallest key in new node and ointer to this new node into arent. 1 2 C-C Tsai P

25 Insert Insert an index entry 2 lus a ointer into arent. C-C Tsai P.4 Insert Now, insert a air with key = 18. C-C Tsai P.50 25

26 Insert Now, insert a air with key = 18. Insert an index entry17 lus a ointer into arent. C-C Tsai P.51 Insert Now, insert a air with key = 18. Insert an index entry17 lus a ointer into arent. C-C Tsai P.52 26

27 Insert Now, insert a air with key = 7. C-C Tsai P.53 Delete Delete air with key = 16. Note: delete air is always in a leaf. C-C Tsai P.54 27

28 Delete Delete air with key = 16. Note: delete air is always in a leaf. C-C Tsai P.55 Delete Delete air with key = 1. Get >= 1 from sibling and udate arent key. C-C Tsai P.56 28

29 Delete Delete air with key = 1. Get >= 1 from sibling and udate arent key. C-C Tsai P.57 Delete Delete air with key = 2. Merge with sibling, delete in-between key in arent. C-C Tsai P.58 2

30 Delete Delete air with key = 3. Get >= 1 from sibling and udate arent key. C-C Tsai P.5 Delete Delete air with key =. Merge with sibling, delete in-between key in arent. C-C Tsai P.60 30

31 Delete C-C Tsai P.61 Delete Delete air with key = 6. Merge with sibling, delete in-between key in arent. C-C Tsai P.62 31

32 Delete Index node becomes deficient. Get >= 1 from sibling, move last one to arent, get arent key. C-C Tsai P.63 Delete Delete. Merge with sibling, delete in-between key in arent. C-C Tsai P.64 32

33 Delete Index node becomes deficient. Merge with sibling and in-between key in arent. C-C Tsai P.65 Delete Index node becomes deficient. It s the root; discard. C-C Tsai P.66 33

34 B*-Trees Root has between 2 and 2 * floor((2m 2)/3) + 1 children. Remaining nodes have between ceil((2m 1)/3) and m children. All external/failure nodes are on the same level. C-C Tsai P.67 Insert When insert node is overfull, check adjacent sibling. If adjacent sibling is not full, move a dictionary air from overfull node, via arent, to nonfull adjacent sibling. If adjacent sibling is full, slit overfull node, adjacent full node, and in-between air from arent to get three nodes with floor((2m 2)/3), floor((2m 1)/3), floor(2m/3) airs lus two additional airs for insertion into arent. C-C Tsai P.68 34

35 Delete When combining, must combine 3 adjacent nodes and 2 in-between airs from arent. Total # airs involved = 2 * floor((2m-2)/3) + [floor((2m-2)/3) 1] + 2. Equals 3 * floor((2m-2)/3) + 1. Combining yields 2 nodes and a air that is to be inserted into the arent. m mod 3 = 0 => nodes have m 1 airs each. m mod 3 = 1 => one node has m 1 airs and the other has m 2. m mod 3 = 2 => nodes have m 2 airs each. C-C Tsai P.6 35