L11 Balanced Trees. Alice E. Fischer. Fall Alice E. Fischer L11 Balanced Trees... 1/34 Fall / 34

Transcription

1 L11 Balaned Trees Alie E. Fisher Fall 2018 Alie E. Fisher L11 Balaned Trees... 1/34 Fall / 34

2 Outline 1 AVL Trees 2 Red-Blak Trees Insertion Insertion 3 B-Trees Alie E. Fisher L11 Balaned Trees... 2/34 Fall / 34

3 Outline Balaned Trees The Proble and General Solution AVL Trees B-Trees Red-Blak Trees Alie E. Fisher L11 Balaned Trees... 3/34 Fall / 34

4 Outline The Proble and General Solution The proble with Binary Searh Trees is that, when luk is bad, the tree an beoe very sparse and aess is not uh ore effiient than for a siple list. Statistially, an average BST is 27% less effiient than a balaned searh tree. That level of ineffiieny would be tolerable, but bad luk in the insertion order an ake it uh, uh worse. Balaned trees are bushy, and so lead to better searhing perforane. However, keeping a tree balaned an be both oplex and tie onsuing. Several kinds of balaned trees have been invented. Two are now in oon use, for different purposes. We will look at the insertion rules for B Trees and Red-Blak Trees. Alie E. Fisher L11 Balaned Trees... 4/34 Fall / 34

5 AVL Trees AVL Trees AVL trees were invented by two Soviet sientists, G.M. Adelson-Velskii and E.M. Landis, and published in This was the first kind of self-balaning trees to be invented. For any node, the heights of its left and right subtrees differ by at ost 1. The balane fator (-1, 0, or 1) is stored at the node or oputed eah tie it is needed. The basi operations (lookup, insertion, and deletion) take O(log n) tie. The balaning algoriths are oplex, onfusing, and tie onsuing. Red-Blak trees are easier and faster. Alie E. Fisher L11 Balaned Trees... 5/34 Fall / 34

6 AVL Trees AVL Balaning We diagra only one ase: the left leg of a node is 2 levels longer than the right leg, and the links go left, then right. Off by D Rotate; Still off by 2 5 Rotate again; Balaned A 4 4 D 4 B C 3 C 3 5 A B This also shows the balaning operation in the ase where both links go left. The right-then-left ase is the irror iage of this one. Alie E. Fisher L11 Balaned Trees... 6/34 Fall / 34 A B C D

7 Red-Blak Trees Definition Searh: Exatly like a Binary Searh Tree Insertion Balaning Alie E. Fisher L11 Balaned Trees... 7/34 Fall / 34

8 Red-Blak Trees In a Red-Blak Tree, the nodes are olored red or blak. An RBTree has these properties: The data is always in binary-searh-tree order. Every path fro root to leaf has the sae nuber of blak nodes on it. There are never two onseutive red nodes on a path. Thus, the length of the longest path fro root to leaf is no ore than double the length of the shortest path. When a node is inserted or deleted, we ust aintain three things: its red property, its blak property, and its binary-searh order of the data. If one of these properties is broken, then reoloring and rebalaning rules are applied to bring the tree bak in line with the RB rules. Alie E. Fisher L11 Balaned Trees... 8/34 Fall / 34

9 Exaple: a Red-Blak Tree Alie E. Fisher L11 Balaned Trees... 9/34 Fall / 34

10 Red-Blak Tree Appliations This is a very oon ipleentation of a balaned tree. The basi operations (lookup, insertion, and deletion) are O(log n) They give uh better balane, and therefore better searh perforane than a siple BSTree. They spend far less tie rebalaning the tree than the AVL algoriths, and balaning happens less often. RBTrees are used to ipleent the ap data struture in the C++ and Java libraries. Alie E. Fisher L11 Balaned Trees... 10/34 Fall / 34

11 Insertion Balaning Rules: Insertion Cases 1 4 Case 1: N is the root of the tree. Paint it blak. Case 2: P is blak. No ation is needed beause we have not hanged the nuber of blak nodes in a tree. Case 3: P and U are both red. Repaint both of the blak and olor G red. Repeat fro G. We have not hanged the nuber of blak nodes in any path on the tree. Case 4: P is red but U is blak. Also, there is a zig-zag (N is the right hild of P and P is the left hild of G). Rotate N and P to reove the zigzag.. 1 Move N to the left link of G and ove P to the left-link of N. 2 Fix the 2-red-in-a-row proble using Case 5, with the labels of N and P interhanged. If the zig-zag goes the other way, reverse the words left and right in Case 4. Alie E. Fisher L11 Balaned Trees... 11/34 Fall / 34

12 Insertion Balaning Proess: Insertion Case 5 Case 5: P is red but U is blak. Also, N is the left hild of P and P is the left hild of G. Do a right-rotation on G. 1 Move G to the to the right link of P. 2 Move P s forer right hild to the left link of G. 3 Reolor P blak and G red. N 3 G 50 P 10 U Right Rotation Reolor P 10 N 3 N 3 G 50 P U G U Alie E. Fisher L11 Balaned Trees... 12/34 Fall / 34

13 Insertion Building a Red-Blak Tree: Steps 1 3 We will build a red-blak tree with nodes,, z, a, b, d, k. 1 Create a blak root and put the first key value in it, with two pointers to null. 2 Put the seond key value to the left or right of the root node. Color it red, and give it two pointers to null. 3 The third key added will ause a rebalaning ation if it is on the sae side of the tree as the seond node. 4 Manage it aording to the olor of its nearby nodes: parent, grandparent, and unle. Alie E. Fisher L11 Balaned Trees... 13/34 Fall / 34

14 Insertion Exaple: Building a Red-Blak Tree. Suppose we inserted letters into a red-blak tree. Here are the first three insertions: Insertion order:,, z, a, b, Case 1: Root is blak. Case 2: New node is red, parent is blak. z Alie E. Fisher L11 Balaned Trees... 14/34 Fall / 34

15 Insertion Step 2: the fourth insertion. Insert a. Fixup is needed. z Case 3: P and U are red. Color both blak, and olor G red. P G z U a N a Alie E. Fisher L11 Balaned Trees... 15/34 Fall / 34

16 Insertion The 4th insertion ontinued... Now repeat, starting with G. Case 1: fix the olor of the root. This adds one blak node to all paths. G P z U z N a a Alie E. Fisher L11 Balaned Trees... 16/34 Fall / 34

17 Insertion The 5th insertion. Now insert b. Case 4: P is red but U is blak. Zig-zag. P a G b U N z Rotate N and P to reove Zig-zag. Swap labels of N and P. Repeat was P now N was N now P a G b U z Alie E. Fisher L11 Balaned Trees... 17/34 Fall / 34

18 Insertion The 5th insertion ontinued... Now do Case 5: P is red, G is blak. First, rotate. Tree is now balaned, but olors are still wrong. P G b U z N a P b G U z N a Alie E. Fisher L11 Balaned Trees... 18/34 Fall / 34

19 Insertion Finishing the 5th insertion. Finishing Case 5. Reverse olors of P and G. N a P b G U z N a P b G z U Alie E. Fisher L11 Balaned Trees... 19/34 Fall / 34

20 Insertion The 6th insertion: d Case 3: P and U are both red. Repaint P and U blak. Repaint G red. Repeat fro G, that is, relabel G as N, et. Case 2: P is blak, nothing to do. U a G b P z a N b P z d N d We still have the sae nuber of blak nodes on every path (3, ounting the nil node), so the tree is still blak-balaned. Alie E. Fisher L11 Balaned Trees... 20/34 Fall / 34

21 Insertion The 7th insertion: k Case 5: P is red, U is blak, no zig-zag. Rotate P up and G down. b z b z a G U d k P N a P G d k N Alie E. Fisher L11 Balaned Trees... 21/34 Fall / 34

22 Insertion Finishing the 7th insertion. After the rotation, reolor P blak and G red. Tree is balaned. N d z a b P k G N d z a b P k G Alie E. Fisher L11 Balaned Trees... 22/34 Fall / 34

23 Insertion Deletion fro a Red-Blak Tree The steps in deletion are: 1 Find N, the node that ontains the data to delete. 2 Find L. the leaf node that is N s predeessor (or suessor). 3 Move the data fro L to N 4 Delete L fro the tree. 5 Reolor and possibly rotate: see next slide. Alie E. Fisher L11 Balaned Trees... 23/34 Fall / 34

24 Insertion Readjusting after a Deletion. We ust aintain three things: the red property of the tree, the blak property, and the binary-searh order of the data. Case 1. The deleted node, D, was red. No further ation is needed. Case 2. D was blak and it s parent node, P, is red. Color the P blak. Done. Case 3. D was blak and P is blak. Now the nuber of blak nodes on this path is too sall. Continue reursively up the tree doing the sae kind of reoloring and possibly rotation that are done for inserting a node. Alie E. Fisher L11 Balaned Trees... 24/34 Fall / 34

25 B-Trees B-Trees In a B-Tree (1962), all paths fro the root to a leaf are the sae length. All values are added at a leaf. Leaves all have 0 hild-links A node that stores n values will have n + 1 links to hild nodes. An interior node an have a variable nuber of links to hild nodes, within a pre-defined range. A 2-3 tree has either 2 or 3 hildren. When a node is full, and a value ust be inserted in that loation, the node splits in half. This ay ause its parent node to split, reursively. When the root splits, another level is added to the tree. The basi operations (lookup, insertion, and deletion) are O(log n) Alie E. Fisher L11 Balaned Trees... 25/34 Fall / 34

26 B-Trees B-Tree Appliations A B-tree is optiized for systes that read and write large bloks of data. BTrees are used to ipleent the indexing struture used with databases. Alie E. Fisher L11 Balaned Trees... 26/34 Fall / 34

27 B-Trees Building a 2-3 Tree: Steps 1 3 Initially, the root of a 2-3 Tree is NULL. The gold olor indiates a node added at this step. Step 1: Create first node. Step 2: Node is full. Step 3: Split, add a level Any value an be added now without ausing a split. Alie E. Fisher L11 Balaned Trees... 27/34 Fall / 34

28 B-Trees Building a 2-3 Tree: Steps 4 6 Step 4: Leaf fills up. Step 5: Split, add a node Step 6: Node fills up Alie E. Fisher L11 Balaned Trees... 28/34 Fall / 34

29 B-Trees Building a 2-3 Tree: Step 7 Now, adding any value less than 40 will ause nodes to split at 2 levels. Step 7a: insert 10 Split, add node. Push id-value up. 20 Step 7b: insert 10 Split at next level, ake a new nood and new root Alie E. Fisher L11 Balaned Trees... 29/34 Fall / 34

30 B-Trees Building a 2-3 Tree: Step 8 Step 8: Node fills up Now adding any value over 60 will ause a split. Alie E. Fisher L11 Balaned Trees... 30/34 Fall / 34

31 B-Trees Building a 2-3 Tree: Step 9 Step 9: Add 100, split node. Move id-value up Alie E. Fisher L11 Balaned Trees... 31/34 Fall / 34

32 B-Trees Building a 2-3 Tree: Step 10 Step 10: Add 90, Node fills up Alie E. Fisher L11 Balaned Trees... 32/34 Fall / 34

33 B-Trees Building a 2-3 Tree: Step 11 Step 11: Add 95. Corret node is full, so borrow spae fro brother. Push index value down, push node s left value up, and put the 95 in the vaant spot Now adding any value over 60 will ause a split at the leaf, another split at level 2, and the root node will fill up. Alie E. Fisher L11 Balaned Trees... 33/34 Fall / 34

34 B-Trees B-Tree Variations In a B-Tree, all paths fro the root to a leaf are the sae length. We illustrated the insertion algorith for a 2-3 tree, where a non-leaf node always has either 2 or 3 hildren. Other node-shapes are ore useful: in a 4-6 tree, a node an store up to 5 values. When it splits, both new nodes will have 3 values. The ore values stored in a node, the less often the splitting happens. A B+-tree stores opies of the keys in the internal nodes, but all data is stored in leaf nodes. This is the kind of B-tree that is useful for a database. Alie E. Fisher L11 Balaned Trees... 34/34 Fall / 34