In the normal case of a chord ring the big O notation of the look up is O(logn) because of long haul pointers of the Finger Table (or Routing Table).
In this question what if the Finger Table has a limit of size 3
(1) I would like to ask the Big O notation of the look up AND (2) I would like to know the formula for the 1st, 2nd and 3rd Index of a Finger Table.
O(log(N^2/3))
Because you will divide the chord into 3 equal parts and so does every node's neighbour.
Thus, Maximally you will go through 2/3's of the chord for each node.
Draw 2-3 nodes on a circular diagram and it will be easier to understand.
You haven't specified what entries are put in the finger table. The answer depends on that, so the question is not answerable. A lookup might take as many as $O(N)$ steps if you choose the finger table poorly (e.g., you select the next 3 nodes in the circle), or might take as few as $O(N^{1/4})$ steps if you choose the finger table well (e.g., you select the nodes at distance $N^{1/4}$, $N^{1/2}$, and $N^{3/4}$).
