O(1) algorithm to determine if node is descendant of another node in a multiway tree?

Question

Imagine the following tree:

    A
   / \\
  B   C
 / \\   \\
D   E   F

I\'m looking for a way to query if for example F is a descendant of A (n

Answer 1

Traversing any tree will require "depth-of-tree" steps. Therefore if you maintain balanced tree structure it is provable that you will need O(log n) operations for your lookup operation. From what I understand your tree looks special and you can not maintain it in a balanced way, right? So O(n) will be possible. But this is bad during creation of the tree anyways, so you will probably die before you use the lookup anyway...

Depending on how often you will need that lookup operation compared to insert, you could decide to pay during insert to maintain an extra data structure. I would suggest a hashing if you really need amortized O(1). On every insert operation you put all parents of a node into a hashtable. By your description this could be O(n) items on a given insert. If you do n inserts this sounds bad (towards O(n^2)), but actually your tree can not degrade that bad, so you probably get an amortized overall hastable size of O(n log n). (actually, the log n part depends on the degration-degree of your tree. If you expect it to be maximal degraed, don't do it.)

So, you would pay about O(log n) on every insert, and get hashtable efficiency O(1) for a lookup.