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Operations on 2-4 Trees: Find, Insert, Delete, Catenate & Split

This document details the operations performed on 2-4 trees, a type of balanced search tree. Operations include finding an element, inserting a new node while maintaining balance, deleting nodes using established rules, concatenating two sorted lists, and splitting a list based on a specified value. 2-4 trees ensure that all leaves are at equal depth and can have 2, 3, or 4 children per internal node, providing efficient logarithmic time complexity for these operations. The provided guidelines outline the mechanics behind each operation for efficient data handling.

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Operations on 2-4 Trees: Find, Insert, Delete, Catenate & Split

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  1. 2-4 Trees

  2. Goal Keep sorted lists subject to the following operations: find(x,L) insert(x,L) delete(x,L) catenate(L1,L2) : Assumes that all items in L2 are greater than all items in L1. split(x,L) : returns two lists one with all item less than or equal to x and the other with all items greater than x.

  3. . . . . . . . . . . . . . . . . . 1 3 12 14 15 18 20 21 28 40 2-4 trees • Items are at the leaves. • All leaves are at the same distance from the root. • Every internal node has 2, 3, or 4 children.

  4. Insert . . . . . . . . . . . . . . . . . 1 3 12 14 15 18 20 21 28 40 16

  5. Insert (cont) . . . . . . . . . . . . . . . 1 3 12 14 15 16 18 20 21 28 40

  6. Insert (cont) . . . . . . . . . . . . . . . 1 3 12 14 15 16 18 20 21 28 40 23

  7. Insert (cont) . . . . . . . . . . . 1 3 12 14 15 16 18 20 21 23 28 40

  8. Insert (cont) . . . . . . . . . . . 1 3 12 14 15 16 18 20 21 23 28 40

  9. Insert (cont) . . . . . . . . . . . 1 3 12 14 15 16 18 20 21 23 28 40

  10. Insert -- definition Add the new leaf in its position. Say under a node v. (*) If the degree of v is 5 split v into a 2-node u and a 3-node w. If v was the root then create a new root r parent of u and w and stop. Replace v by u and w as children of p(v). Repeat (*) for v := p(v).

  11. Delete . . . . . . . . . . . 1 3 12 14 15 16 18 20 21 23 28 40

  12. Delete . . . . . . . . . . . 1 3 12 14 15 18 20 21 23 28 40

  13. Delete . . . . . . . . . . . 1 3 12 14 15 18 20 21 23 28 40

  14. Delete . . . . . . . . . . . 1 3 12 14 18 20 21 23 28 40

  15. Delete . . . . . . . . . . . 1 3 12 14 18 20 21 23 28 40

  16. Delete -- definition Remove the leaf. Let v be the parent of the removed leaf. (*) If the degree of v is one, and v is the root discard v. Otherwise (v is not a root), if v has a sibling w of degree 3 or 4, borrow a child from w to v and terminate. Otherwise, fuse v with its sibling to a degree 3 node and repeat (*) with the parent of v.

  17. Summary Delete and insert take O(log n) on the worst case. Theorem (homework) : A sequence of m delete and insert operations on an initial tree with n nodes takes O(m+n) time Can do catenate and split in a way similar to red-black trees.

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