Talk:2-3-4 tree

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[edit] Move article?

Shouldn't this article be at 2-3-4 tree instead, with the plural linking to the singular? Enochlau 20:34, 1 Jun 2005 (UTC)

Agreed. I'll put up a request for a page move. —Ghakko 30 June 2005 09:26 (UTC)

[edit] Draft

I'm planning to expand the article to include a walkthough of all the 2-3-4 tree operations, a la red-black tree. I'll put the draft-in-progress below for comment. —Ghakko 13:41, 27 July 2005 (UTC)

[edit] Operations

[edit] Lookup

To find an element, we look for it in the root; if it's not there, we descend into the child whose interval the element is in and repeat. If we encounter an empty child, the element is not present in the tree.

[edit] Insertion

Don't mean to be harsh, but this isn't a 2-3-4-tree. Over Christmas I will have a go at improving this.

  • Split 4-nodes on way down (splitting root node heightens tree by one).

You don't do this, instead you put the middle element into the node above and point it to the other two elements and 4 children of this node. Only the root node can add a level like this.

    • Image:2-3-4 tree split.png
  • Insertion is at leaf.
    • 2-node leaf insertion (2 cases).
      • Image:2-3-4 tree 2-node insertion 1.png Image:2-3-4 tree 2-node insertion 2.png
    • 3-node leaf insertion (3 cases).
      • Image:2-3-4 tree 3-node insertion 1.png Image:2-3-4 tree 3-node insertion 2.png Image:2-3-4 tree 3-node insertion 3.png
    • 4-node leaf insertion (4 cases).

You can never actually insert at a 4-node, because it would have been split by the node before. This is trivally wrong as it generates 5-nodes.

      • Image:2-3-4 tree 4-node insertion 1.png Image:2-3-4 tree 4-node insertion 2.png Image:2-3-4 tree 4-node insertion 3.png Image:2-3-4 tree 4-node insertion 4.png

131.111.8.104 03:40, 29 November 2005 (UTC)

[edit] Deletion

  • Find node to delete, then find in-order successor if any. (Any 4-nodes encountered should be split first, to reduce the number of special cases.)
  • Delete from successor, heeding the following cases:
    • 3-node? Shrink to 2-node.
    • 2-node at root? Tree becomes empty.
    • 2-node with sibling, sibling is a 3-node? Redistribute.
    • Otherwise? Merge.
      • Merge leaves hole in parent? Recurse, treating parent as new deletee.

[edit] Analysis

The height of a 2-3-4 tree with n \,\! elements is O(\log {n}) \,\!.

  • Worst case is a tree of 2-nodes.
  • Suppose the height of the tree is h \,\!.
  • n \geq 1 + 2 + \ldots + 2^{h - 1}
  • n \geq 2^{h} - 1
  • \log_2({n + 1}) \geq h

[edit] References

Need citation for original paper.

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