Even MORE Balanced Trees Computing 2 COMP s1

O PTIMISATION B ASED B ALANCED T REES So far we have seen local approaches Randomised trees…make poor performance unlikely Splay trees…reasonable amortized performance BUT both still have O(n) worst case. Ideally we want worst case to be O(log n) trees…use varying-sized nodes to assist balance red-black trees…isomorphic to 2-3-4, but with binary nodes

2-3-4 T REES trees have three kinds of nodes 2-nodes, with two children (same as normal BSTs) 3-nodes, two values and three children 4-nodes, three values and four children

2-3-4 T REES trees are ordered similarly to BSTs In a balanced tree: all leaves are at same distance from the root trees grow "upwards" from the leaves.

2-3-4 TREE IMPLEMENTATION : typedef struct node *TreeLink; struct node { int order; // 2, 3 or 4 Item data[3]; // items in node TreeLink child[4]; //links to subtrees }; //New nodes are always order 2 TreeLink newNode(Item item){ TreeLink n = malloc(sizeof(struct node)); n->order = 2; n->data[0] = it; n->child[0] = n->child[1] = NULL; return n; }

2-3-4 S EARCHING : Item search(Tree t, Key k){ if(t == NULL) return NULL; int i, int diff; int nItems = t->order-1; for(i = 0; i < nItems; i++){ diff = compare(k,keyOf(t->data[i]); if(diff <=0) break; } if(diff == 0) return t->data[i]; else return search(t->child[i],k); }

2-3-4 I NSERTION Insertion into a 2-node or 3-node:

S PLITTING A NODE IN A TREE : Insertion into a 4-node (requires a split):

S PLITTING THE ROOT :

B UILDING A TREE... 7 INSERTIONS :

E XERCISE : T REE INSERTION Insert S then U into this tree.

2-3-4 I NSERTION A LGORITHM Find leaf node where Item belongs (via search) if not full (i.e. order < 4) insert Item in this node, order++ if node is full (i.e. contains 3 Items) split into two 2-nodes as leaves promote middle element to parent insert item into appropriate leaf 2-node if parent is a 4-node continue split/promote upwards if promote to root, and root is a 4-node split root node and add new root

T OP D OWN I NSERTION To avoid propagation of splitting 4-nodes all the way up the tree we can use a top-down approach. Every time we encounter a 4-node on our way down the search path for insertion, we split it.

E XERCISE : T REE INSERTION Insert S then U into this tree using a top down approach

2-3-4 TREE SEARCHING COST ANALYSIS : Worst case determined by height h trees are always balanced ⇒ height is O(logN) worst case for height: all nodes are 2-nodes same case as for balanced BSTs, i.e. h ≅ log 2 N best case for depth: all nodes are 4-nodes balanced tree with branching factor 4, i.e. h ≅ log 4 N

S OME N OTES ON T REES Splitting the root is the only way depth increases. One variation: why stop at 4? Allow nodes to hold up between M/2 and M items. If each node is a disk-page, then we have a B-tree (databases). Implement similar ideas using just BST nodes → red-black trees.

R ED -B LACK T REES Red-black trees are a representation of trees using BST nodes. each node needs one extra value to encode link type but we no longer have to deal with different kinds of nodes Link types: red links... combine nodes to represent 3- and 4- nodes black links... analogous to "ordinary" BST links (child links) Advantages: standard BST search procedure works unmodified get benefits of tree self-balancing (although deeper)

R ED B LACK T REES Definition of a red-black tree: a BST in which each node is marked red or black no two red nodes appear consecutively on any path Balanced red-black tree: all paths from root to leaf have same number of black nodes Insertion algorithm ensures that balanced trees remain balanced Prevents O(n) behaviour

R EPRESENTING 3- AND 4- NODES IN RED - BLACK TREES :

R ED B LACK L INKS Colour allows us to distinguish links black = parent node is a "real"parent red = parent node is a neighbour

E QUIVALENT TREES ( ONE 2-3-4, ONE RED - BLACK ):

R ED - BLACK TREE IMPLEMENTATION : typedef enum {RED,BLACK} Colour; typedef struct Node *TreeLink; struct Node { Item data; // actual data Colour colour; // colour of link to // parent TreeLink left; // left subtree TreeLink right; // right subtree } ;

E XERCISE Draw the following tree as an equivalent red-black tree

S OLUTION

R ED B LACK T REE O PERATIONS Search is same as standard BST Insertion is more complex than for standard BSTs Need to balance only when we see 4-nodes If a node has 2 red children it is part of a 4-node. Low overhead as there are not usually many 4-nodes in a tree We can split 4-nodes in a top down manner, splitting any 4 nodes on the search path down the tree during insertion

S PLITTING A 4- NODE //Done while descending the tree if(isRed(currTree->L) && isRed(currTree->R)){ currTree->colour = RED; currTree->L-Colour = BLACK; currTree->R->Colour = BLACK; }

C HECKING SUBTREE AFTER I NSERTION Stage 1: in a red right subtree, and a successive leftward red link if(isRightSubtree &&isRed(currTree) && isRed(currTree->L)){ currTree = rotateR(currTree); }

C HECKING SUBTREE AFTER I NSERTION Stage 2: two successive red links in same direction = newly- created 4-node if(isRed(currTree->L) && isRed(currTree->L->L)){ currTree = rotateR(currTree); currTree->color = BLACK; currTree->color = RED; }

I NSERTION void Stinsert(Item item, ST st){ st->root = RBInsert(st->root,item,0); st->root->colour = BLACK; } TreeLink RBInsert(TreeLink t, Item item, int inRightSubtree){ Key currKey = Key(item); if(t == emptyTree){ //always part of an existing node return new(item, emptyTree,emtyTree,RED); } if(isRed(currTree->left) && isRed(currTree->right)){ currTree->colour = RED; currTree->left->Colour = BLACK; currTree->right->Colour = BLACK; }

I NSERTION if(less(currKey,Key(t->item))){ //insert in left currTree->left = RBinsert(currTree->left, item, 0); if(isRed(currTree && isRed(currTree->left) && inRightSubtree) currTree = rotateR(currTree); //Can’t be true if last if statement was true if(isRed(currTree->left) && isRed(currTree->left_left)){ currTree = rotateR(currTree); currTree->colour = BLACK; currTree->right->colour = RED; } } else { //insert into right

S IMPLE E XAMPLE Insert keys 1,3,2 1 becomes the root 3 becomes the right child of 1 and has a red link 1 \ 3 2 becomes the left child of 3 and has a red link 1 \ 3 (3 is red and a right child and has a red left child ) / so we rotate right at 3 2

S IMPLE EXAMPLE CONTINUED After rotating at 3 we get this 1 (Now 1 has a red right and a red right right link) \ so we rotate and change colours 2 1 now has a red parent but 2 does not \ 3 2 / \ 1 3

E XAMPLE OF S PLITTING Suppose we wish to insert 4 into 2 / \ is on the search path to insert 4 and has left and right red links so we split then keep inserting 2 / \ 1 3 \ 4

E XERCISE : R ED /B LACK T REE I NSERTION Show the result of inserting the following values 8,4,6,2,3

R ED -B LACK T REE C OST A NALYSIS Guaranteed O(log n) insertion, search and deletion Maximal height of a red-black tree 2 * log n Yay!