Multi-way Search Trees. (Multi-way Search Trees) Data Structures and Programming Spring / 25
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1 Multi-way Search Trees (Multi-way Search Trees) Data Structures and Programming Spring / 25
2 Multi-way Search Trees Each internal node of a multi-way search tree T: has at least two children contains d - 1 items, where d is the number of children d-nodes contains 2 pseudo-items: k0 =, k d = Children of each internal node are between items all keys in the subtree rooted at the child fall between keys of those items (Multi-way Search Trees) Data Structures and Programming Spring / 25
3 Multi-way Searching (Multi-way Search Trees) Data Structures and Programming Spring / 25
4 2-4 Trees (or Trees) Nodes may contain 1, 2 or 3 items. A node with k items has k + 1 children All leaves are on same level. 2 h 1 n 4 h 1 log 4 (n + 1) h log 2 (n + 1). (Multi-way Search Trees) Data Structures and Programming Spring / 25
5 Insertion Find the appropriate leaf. If there is only one or two items, just add to leaf. If no room, move middle item to parent and split remaining two items among two children. (Multi-way Search Trees) Data Structures and Programming Spring / 25
6 Insertion (Cont d) Split & move middle element to parent (Multi-way Search Trees) Data Structures and Programming Spring / 25
7 Removal First : find the key with a simple multi-way search If the item to delete is in an internal node, reduce to the case where item is at the bottom of the tree by: Find item which precedes it in in-order traversal which one? Swap them Remove the item Not enough items in the node Underflow! Pull an item from the parent, replace it with an item from a sibling - transfer Still not good enough! What happens if siblings are 2-nodes? Could we just pull one item from the parent? (Multi-way Search Trees) Data Structures and Programming Spring / 25
8 Removal (Cont d) Remove 3 move 10 into the subtree move 25 into the parent (Multi-way Search Trees) Data Structures and Programming Spring / 25
9 Removal (Cont d) If siblings are 2-nodes (i.e. contain only one key) cannot steal from them Do node merging Remove 3 move 10 into the subtree merge 10 with 25 (Multi-way Search Trees) Data Structures and Programming Spring / 25
10 More on removal What if parent is a 2-node Propagate underflow up the tree Delete 9 (Multi-way Search Trees) Data Structures and Programming Spring / 25
11 2-4 Trees 2-4 trees are easy to maintain Insertion and deletion take O(log n) Balanced trees Binary Tree Representation f 2-4 Trees => Red-Black Trees (Multi-way Search Trees) Data Structures and Programming Spring / 25
12 One-Pass Operations No bottom-to-top pass. Can pipeline inserts. Can pipeline deletes from leaf nodes. Bottom-up pass is triggered when new pair is inserted into a 4-node leaf. Split 4-nodes on the way down so you never insert into a 4-node leaf! Look before you leap! If the node you are about to move to is a 4-node, split it into two 2-nodes. Then move to a 2-node. (Multi-way Search Trees) Data Structures and Programming Spring / 25
13 Cases For 4-node Move The 4-node we attempt to move to may be: The root. Child of a 2-node. Child of a 3-node. It cannot be the child of a 4-node, because we will never be at a 4-node. (Multi-way Search Trees) Data Structures and Programming Spring / 25
14 Root Is A 4-node Height of tree increases by 1. Compare with y and then move to x or z. (Multi-way Search Trees) Data Structures and Programming Spring / 25
15 4-node Left Child of 2-node No change in height of subtree. Compare with x and then move to w or y. 4-node right child of 2-node is similar. (Multi-way Search Trees) Data Structures and Programming Spring / 25
16 4-node Left Child of 3-node No change in height of subtree. Compare with w and then move to v or x. 4-node middle or right child of 3-node is similar. (Multi-way Search Trees) Data Structures and Programming Spring / 25
17 Top-Down Delete Bottom-up pass is triggered when deletion is from a 2-node leaf. Look before you leap! May start at a 2-node root but may not be at any other 2-node. If the node you are about to move to is a 2-node, make it a 3-node or 4-node. Then move to the 3-node 4-node. (Multi-way Search Trees) Data Structures and Programming Spring / 25
18 Cases For 2-node Move Moving to a 2-node root is permitted. No other move to a 2-node is permitted. Other attempts to move to a 2-node may be classified as below. The 2-nodes nearest sibling is also a 2-node. The 2-nodes nearest sibling is a 3-node. The 2-nodes nearest sibling is a 4-node. In each of the preceding cases, the 2-nodes parent may be a 2-node root, a 3-node, or a 4-node. (Multi-way Search Trees) Data Structures and Programming Spring / 25
19 Root Is 2-node Leaf Delete root. Tree becomes empty. (Multi-way Search Trees) Data Structures and Programming Spring / 25
20 Moving To 2-node Whose Nearest Sibling Is 2-node Current node is 2-node => at root. Height decreases by 1. Reapply moving rules before you move down. (Multi-way Search Trees) Data Structures and Programming Spring / 25
21 Moving To 2-node Whose Nearest Sibling Is 2-node Current node is 3-node. Moving to v. No change in height of subtree. Moving to middle or right child is similar. Current node is 4-node is also similar. (Multi-way Search Trees) Data Structures and Programming Spring / 25
22 Moving To 2-node Whose Nearest Sibling Is 3-node Current node is 2-node => at root. Moving to w. No change in height of subtree. Moving to right child 2-node is similar. (Multi-way Search Trees) Data Structures and Programming Spring / 25
23 Moving To 2-node Whose Nearest Sibling Is 3-node Current node is 3-node. Moving to w. No change in height of subtree. Moving to middle or right child 2-node is similar. Current node is 4-node is also similar. (Multi-way Search Trees) Data Structures and Programming Spring / 25
24 Moving To 2-node Whose Nearest Sibling Is 4-node Current node is 2-node => at root. Moving to u. No change in height of subtree. Moving to right child 2-node is similar. (Multi-way Search Trees) Data Structures and Programming Spring / 25
25 Moving To 2-node Whose Nearest Sibling Is 4-node Current node is 3-node. Moving to u. No change in height of subtree. Moving to middle or right child 2-node is similar. Current node is 4-node is also similar. (Multi-way Search Trees) Data Structures and Programming Spring / 25
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