Balanced search trees. DS 2017/2018
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1 Balanced search trees. DS 2017/2018
2 Red-black trees Symmetric binary B-tree, Rudolf Bayer, The balancing is maintained by using a coloring of the nodes. The red-black trees are binary search trees that satisfy the following properties: 1. a node is colored with red or black; 2. the root and the leaf nodes (nil that belong to the structure) are colored with black; 3. if a node is red, then his both children are black; 4. the paths from a node to the boundary nodes have the same number of black nodes. FII, UAIC Lecture 10 DS 2017/ / 31
3 Red-black trees - example FII, UAIC Lecture 10 DS 2017/ / 31
4 Red-black trees Lemma: Any subtree of a red-black tree has at least 2 bh(v) 1 internal nodes, where: v the root of the subtree, bh(v) the number of black nodes found on a path from v down to a leaf, not including v; Proof. At class. FII, UAIC Lecture 10 DS 2017/ / 31
5 Red-black trees Theorem: A red-black tree with n internal nodes has the height h 2 log 2 (n + 1). Proof. According to property 3, n 2 h/2 1 h/2 log 2 (n + 1) h 2 log 2 (n + 1). FII, UAIC Lecture 10 DS 2017/ / 31
6 Red-black trees: operations Corollary: In a red-black tree with n nodes, the search operation has a time complexity of O(log n). FII, UAIC Lecture 10 DS 2017/ / 31
7 The insert operation Search for the insertion position and insert the new value as in the case of ordinary binary search trees. Color the new node with red. Restore the red-black properties by recoloring nodes and applying simple rotations. FII, UAIC Lecture 10 DS 2017/ / 31
8 The insert operation Property 1: satisfied. Property 2 satisfied (both children of the inserted node are nil). If the inserted node is the root recolor it in black. Property 4 satisfied (the new red node replaces a leaf). The property 3 may not hold - if the parent of the node is red. Move above this situation by recoloring the nodes until it can be repaired by rotation operations and recoloring. FII, UAIC Lecture 10 DS 2017/ / 31
9 The insert operation: the restauration of property 3 Case 1: the uncle of the inserted node is red Recolor the parent and the uncle in black and the grandfather in red. Case 2: the uncle of the inserted node is black and the inserted node is the right child of a left child Apply a simple left rotation between the current node and the parent node. Case 3: the uncle of the inserted node is black and the inserted node is the left child of a left child Apply a simple right rotation between the parent node and the grandfather node + recolor the parent node (in black) and the grandfather node (in red). Obs.: Apply similar operations for the symmetric case. FII, UAIC Lecture 10 DS 2017/ / 31
10 The insert operation - case 1 FII, UAIC Lecture 10 DS 2017/ / 31
11 The insert operation - Case 2 and 3 FII, UAIC Lecture 10 DS 2017/ / 31
12 Insertion example: node FII, UAIC Lecture 10 DS 2017/ / 31
13 Insertion CASE 1: recoloring FII, UAIC Lecture 10 DS 2017/ / 31
14 Insertion CASE 2: left rotation FII, UAIC Lecture 10 DS 2017/ / 31
15 Insertion CASE 3: right rotation + recoloring FII, UAIC Lecture 10 DS 2017/ / 31
16 Insertion The valid red-black tree FII, UAIC Lecture 10 DS 2017/ / 31
17 The insert operation: algorithm Consider that each node of the tree is a structure with the following fields: key: the useful information of the node; color: red / black; pred: the address of the parent node (null for the root); left: the address of the left child; right: the address of the right child. FII, UAIC Lecture 10 DS 2017/ / 31
18 The insert operation: algorithm Procedure insert(t, x) begin insbinarysearchtree(t, x) x color red while (x! = t and x pred color == red) do if (x pred == x pred pred left) then y x pred pred right if (y color == red) then Case 1 else if (x == x pred right) then Case 2 Case 3 else similarly to the branch then, but interchanging left with right t color black end FII, UAIC Lecture 10 DS 2017/ / 31
19 The insert operation: Case 1 x pred color black y color black x pred pred color red x x pred pred FII, UAIC Lecture 10 DS 2017/ / 31
20 The insert operation: Case 2 x x pred left-rotation(t, x) FII, UAIC Lecture 10 DS 2017/ / 31
21 The insert operation: Case 3 x pred color black x pred pred color red right-rotation(t, x pred pred) FII, UAIC Lecture 10 DS 2017/ / 31
22 The insert operation - example 2 FII, UAIC Lecture 10 DS 2017/ / 31
23 The insert operation - example 3 FII, UAIC Lecture 10 DS 2017/ / 31
24 The insert operation - example 3 FII, UAIC Lecture 10 DS 2017/ / 31
25 The delete operation Similarly to the delete operation from the ordinary binary search trees. Take into account that the null nodes belong to the structure. After deletion it is possible that the property 4 do not hold any more. Restore the properties of red-black trees by recoloring nodes and applying simple rotations. FII, UAIC Lecture 10 DS 2017/ / 31
26 The delete operation: the restoration of property 4 Case 1: Transform it into one of the cases 2), 3), 4) by a rotation. Case 2: The node for which the property is not satisfied is moved to the root with one level by recoloring a node. Case 3: Transform it into case 4) by an interchanging of colors and a rotation. Case 4: In this case restore the red-black tree property for the whole tree. FII, UAIC Lecture 10 DS 2017/ / 31
27 Deletion CASE 1 Caz 1: Transform it into one of the cases 2), 3), 4) by a rotation. v u A B y x C D z E F y v u A B x C D z E F FII, UAIC Lecture 10 DS 2017/ / 31
28 Deletion CASE 2 Case 2: The node for which the property is not satisfied is moved to the root with one level by recoloring a node. v u A B y x C D z E F v u A B y x C D z E F FII, UAIC Lecture 10 DS 2017/ / 31
29 Deletion CASE 3 Case 3: Transform it into case 4) by an interchanging of colors and a rotation. v u A B y x C D z E F v u A B x C y D z E F FII, UAIC Lecture 10 DS 2017/ / 31
30 Deletion CASE 4 Caz 4: In this case restore the red-black tree property for the whole tree. v u A B y x C D z E F y v u A B x C D z E F FII, UAIC Lecture 10 DS 2017/ / 31
31 Red-black trees The complexity of insert / delete algorithms: O(log n). Corollary: The class of red-black trees is O(log n) stable. FII, UAIC Lecture 10 DS 2017/ / 31
32 Red-black trees The complexity of insert / delete algorithms: O(log n). Corollary: The class of red-black trees is O(log n) stable. Applications: System symbol tables. Kernel Linux (Completely Fair Scheduler). FII, UAIC Lecture 10 DS 2017/ / 31
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