Lecture 7. Binary Search Trees / AVL Trees

Transcription

1 Lecture 7. Binary Searc Trees / AVL Trees T. H. Cormen, C. E. Leiserson and R. L. Rivest Introduction to Algoritms, 3rd Edition, MIT Press, 2009 Sungkyunkwan University Hyunseung Coo coo@skku.edu Copyrigt Networking Laboratory

2 Introduction Tree Single parent, multiple cildren Binary tree Tree wit 0 2 cildren per node Tree Binary Tree Algoritms Networking Laboratory 2/53

3 Introduction Searc trees support SEARCH, MINIMUM, MAXIMUM, PREDECESSOR, SUCCESSOR, INSERT, and DELETE Dictionary and priority queue Basic operations take time proportional to te eigt of te tree Complete binary tree: (lg n) Linear-cain tree: (n) Expected eigt of a randomly built binary searc tree: O(lg n) Algoritms Networking Laboratory 3/53

4 Trees Terminology Root no predecessor Leaf no successor Interior non-leaf Heigt distance from root to leaf Root node Interior nodes Heigt Leaf nodes Algoritms Networking Laboratory 4/53

5 Types of Binary Trees Degenerate only one cild Balanced mostly two cildren Complete always two cildren Degenerate Binary Tree Balanced Binary Tree Complete Binary Tree Algoritms Networking Laboratory 5/53

6 Binary Trees Properties Degenerate Heigt = O(n) for n nodes Similar to linear list Balanced Heigt = O( lg n ) for n nodes Useful for searces Degenerate Binary Tree Balanced Binary Tree Algoritms Networking Laboratory 6/53

7 Binary Searc Trees Key property Value at node Smaller values in left subtree Larger values in rigt subtree Example X > Y X < Z X Y Z Algoritms Networking Laboratory 7/53

8 Binary Searc Trees Binary Searc Tree (BST) Binary tree Represented by a linked data structure eac node is an object key + satellite data Pointers: left left cild, rigt rigt cild, p parent Binary searc tree property Let x be a node in a BST If y is a node in te left subtree of x, ten key[y] key[x] If y is a node in te rigt subtree of x, ten key[y] key[x] Algoritms Networking Laboratory 8/53

9 Binary Searc Trees Examples Binary Searc Trees Non-Binary Searc Tree Algoritms Networking Laboratory 9/53

10 Tree Traversal Tree Traversal A tecnique for processing te nodes of a tree in some order Preorder traversal Process all nodes of a tree by processing te root, ten recursively processing all subtrees Also known as prefix traversal Algoritms Networking Laboratory 10/53

11 Tree Traversal Inorder traversal Process all nodes of a tree by recursively processing te left subtree, ten processing te root, and finally te rigt subtree Algoritms Networking Laboratory 11/53

12 Tree Traversal Postorder traversal Process all nodes of a tree by recursively processing all subtree s, ten finally processing te root Also known as postfix traversal Algoritms Networking Laboratory 12/53

13 Tree Traversal b a c Tree Traversal Preorder Te key of te root of a subtree is printed before te values in its left subtree and tose in its rigt subtree a, b, d, e, c, f,, i, g Inorder Te key of te root of a subtree is printed between te values in its left subtree and tose in its rigt subtree d, b, e, a,, f, i, c, g Postorder Te key of te root of a subtree is printed after te values in its left subtree and tose in its rigt subtree d, e, b,, i, f, g, c, a d e f g i Algoritms Networking Laboratory 13/53

14 Computation Time of INORDER If x is te root of an n-node subtree, ten te call INORDER-TREE-WALK(x) takes (n) time Use te substitution metod T(n) = T(k) + T(n-k-1) + d; T(0)=c Sow tat T(n) = (n) by proving tat T(n) = (c+d)n+c For n=0 (c+d)*0 + c = c For n > 0 T(n) = T(k) + T(n-k-1) + d = ((c+d) *k+c)+((c+d) *(n-k-1)+c) + d = (c+d) *n+c-(c+d)+c+d = (c+d) *n + c Algoritms Networking Laboratory 14/53

15 Searcing Algoritms Networking Laboratory 15/53

16 Example Binary Searces Find ( 2 ) > 2, left 5 5 > 2, left > 2, left = 2, found = 2, found Algoritms Networking Laboratory 16/53

17 Example Binary Searces Find ( 25 ) < 25, rigt 5 5 < 25, rigt > 25, left > 25, left = 25, found > 25, left 10 < 25, rigt = 25, found 25 Algoritms Networking Laboratory 17/53

18 Practice Problems Suppose tat we ave numbers between 1 and 1000 in a binary searc tree and want to searc for te number 363. Wic of te following sequences could NOT be te sequence of nodes examined? a. 2, 252, 401, 398, 330, 344, 397, 363. b. 924, 220, 911, 244, 898, 258, 362, 363. c. 925, 202, 911, 240, 912, 245, 363. Algoritms Networking Laboratory 18/53

19 Minimum and Maximum Te element wit te minimum/maximum key can be found by following left/rigt cild pointers from te root until NIL is encountered Algoritms Networking Laboratory 19/53

20 Successor And Predecessor Assume distinct keys Te successor of a node x is te node wit te smallest key greater tan key[x] Rigt subtree nonempty successor is te leftmost node in te rigt subtree (Line 2) Oterwise, te successor is te lowest ancestor of x wose left cild is also an ancestor of x go up te tree until we encounter a node tat is te left cild of its parent (Lines 3-7) Algoritms Networking Laboratory 20/53

21 TREE-SUCCESSOR(X) Algoritms Networking Laboratory 21/53

22 Examples Algoritms Networking Laboratory 22/53

23 Insertion Trace down te tree until a leaf Sould be inserted in te left subtree Sould be inserted in te rigt subtree Algoritms Networking Laboratory 23/53

24 Insertion Example Insert ( 20 ) < 20, rigt 30 > 20, left 25 > 20, left Insert 20 on left 20 Algoritms Networking Laboratory 24/53

25 Insertion Example 12 Insert an item wit key 13 into a BST Algoritms Networking Laboratory 25/53

26 Deletion Consider tree cases for te node z to be deleted z as no cildren: modify p[z] to replace z wit NIL as its cild z as only one cild: splice out z by making a new link between its cild and its parent z as two cildren: splice z s successor y, wic as no left cild and replace z s key and satellite data wit y s key and satellite Algoritms Networking Laboratory 26/53

27 Deletion Example Algoritms Networking Laboratory 27/53

28 Deletion Example Algoritms Networking Laboratory 28/53

29 TREE-DELETE Algoritm Algoritms Networking Laboratory 29/53

30 TREE-DELETE Algoritm TREE-DELETE Algoritm Lines 1-3: determines a node y to splice out (z itself or z s successor) Lines 4-6: x is set to te non-nil cild of y, or to NIL if y as no cildren Lines 7-13: splice out y by modifying pointers in p[y] and x Special care for wen x=nil or wen y is te root Lines 14-16: if te successor of z was te node spliced out, y s key and satellite data are moved to z, overwriting te previous key and satellite data Line 17: return node y for recycle it via te free list Algoritms Networking Laboratory 30/53

31 Example Deletion (Leaf) Delete ( 25 ) < 25, rigt 30 > 25, left 25 = 25, delete Algoritms Networking Laboratory 31/53

32 Example Deletion (Internal Node) Delete ( 10 ) Replacing 10 wit smallest value in rigt subtree Deleting leaf Resulting tree Algoritms Networking Laboratory 32/53

33 Example Deletion (Internal Node) Delete ( 10 ) Replacing 10 wit largest value in left subtree Replacing 5 wit largest value in left subtree Deleting leaf Algoritms Networking Laboratory 33/53

34 AVL Trees AVL Trees (Adelson-Velskii and Landis) BST : all values in T 1 x all values in T 2 x T 1 T 2 T1, T2 are AVL trees and eigt(t1) - eigt(t2) 1 Algoritms Networking Laboratory 34/53

35 AVL Trees AVL Tree Property: It is a BST in wic te eigts of te left and rigt subtrees of te root differ by at most 1 and in wic te rigt and left subtrees are also AVL trees Example: Select a AVL tree! OK NO Algoritms Networking Laboratory 35/53

36 Insertion Very muc like as in BST wit operations to maintain eigt balancerotation code associated wit eac node -2, -1, 0, 1, Algoritms Networking Laboratory 36/53

37 Rotation 4 Types Single Left Rotation Double Left Rotation Single Rigt Rotation Double Rigt Rotation SLR, DLR : left ( +2 ) SRR, DRR : rigt ( -2 ) Algoritms Networking Laboratory 37/53

38 Rotation Case Case Algoritms Networking Laboratory 38/53

39 Rotation Case 1 (a) C -2 A 0-1 A C 0 T 2 T 3 SRR T 1 T 2 T 3 T Algoritms Networking Laboratory 39/53

40 Rotation Case 1 (a)` 0 A T 2 C -2 T 3 SRR -1 A 1 T 1 T 2 T 1 C -1 T 3-1 Algoritms Networking Laboratory 40/53

41 Rotation Case 1 (b) +1 T 1 +1 A T 2 C -2 T 4 B -1 0 T DRR 0 T 1 +1 A B 0 T 2 T C 0 T 4 +1 Algoritms Networking Laboratory 41/53

42 Rotation Case 2 (a) A +2 C 0 or 1 A 0 C -1 or 0 T 1 T 2 SLR T 3 T 3 or +1 T 1 T 2 or +1 Algoritms Networking Laboratory 42/53

43 Rotation Case 2 (b) T 1 A +2 C -1 0 A B 0 C 0 B DLR T 2 T 4 T 1 T 2 T 3 T 4 T 3 DLR = SRR at C + SLR at A DRR = SLR at A + SRR at C Algoritms Networking Laboratory 43/53

44 Example SLR DLR Algoritms Networking Laboratory 44/53 7

45 Example DLR SRR Algoritms Networking Laboratory 45/53

46 Example SRR DLR Algoritms Networking Laboratory 46/53

47 Example Algoritms Networking Laboratory 47/53

48 Insertion Strategy Insert at leaf Keep going upward and update code from leaf to parent if Stop wen Rotate wen Algoritms Networking Laboratory 48/53

49 Insertion Strategy Left Rigt Double Single Algoritms Networking Laboratory 49/53

50 Deletion Strategy Delete te node as in BST deletion Keep going upward and update code from leaf to parent if Stop wen Rotate wen Algoritms Networking Laboratory 50/53

51 Example Delete SLR 8 12 Delete Algoritms Networking Laboratory 51/53

52 Example Delete SRR DRR 7 Algoritms Networking Laboratory 52/53

53 Example SRR DRR Algoritms Networking Laboratory 53/53