Assume I have a polygon chain implementation which is backed by a key-value store which stores the position of a point inside the chain as key and the point itself as value. So a polygon chain of the points $p_1\rightarrow p_2\rightarrow p_3$ is stored as $\{0\Rightarrow p_1,1\Rightarrow p_2,2\Rightarrow p_3\}$. The key-value store is implemented with a data structure which allows the operations split and join in $O(\log n)$ time.

I want to join the chain $P = p_1\rightarrow p_2\rightarrow p_3$ and the chain $Q = q_1->q_2->q_3$ to a chain $p_1\rightarrow p_2\rightarrow p_3\rightarrow q_1\rightarrow q_2\rightarrow q_3$.

Is there a way to join two polygon chains in logarithmic time without updating the keys? If I do a simple join, I get a problem with duplicated keys, so the keys of $Q$ have to be updated before the join, but this destroys my logarithmic time approach as I need linear time for updating the keys.

Am I wrong here already?

If not, is there a way to keep the logarithmic time?

  • $\begingroup$ After merging indexes (keys) of points that was in $q$ should be increased on $|p|$, suchwise if your store indexes (keys) explicitly you need at least $O(|q|)$ time to update it. I can recommend use treap with implicit indexes (treap is example of randomized binary trees). List implementation based on treap have $O(log N)$ time-complexity for add/select/delete element and for concatenate/split lists operations. $\endgroup$
    – knok16
    Jul 23, 2015 at 20:58

1 Answer 1


You are asking for a data structure for sequences (aka vectors or arrays) that permits efficient concatenation. I suggest using any standard binary tree data structure, e.g., AVL tree, red-black tree, splay tree, treap, etc. They support $O(\lg n)$ time concatenation and $O(\lg n)$ time lookups. The differences between them come down to details related to implementation complexity and constant-factor differences in running time.

With these algorithms, the keys are not stored explicitly. Instead, they are implicit. To lookup key $i$, you search for the $i$th leaf from the left, but the key $i$ is never stored anywhere. Thus, when concatenating two trees, there is no need to renumber the keys. You can find the $i$th leaf in $O(\lg n)$ time: augment the data structure so that each node keeps a count of the number of leafs underneath it; then it's clear how to traverse the tree during a lookup (whether to go left or right at each node is an easy decision, given the information stored in each node).

  • $\begingroup$ no, my main question is the neccesarity of updating keys before a join-operation. the polygon chain itself is already implemented as a binary tree which allows join in logarithmic time $\endgroup$ Jul 24, 2015 at 7:08
  • $\begingroup$ @ManuelJain, I've updated my answer to explain that aspect: see the 2nd paragraph of my answer. $\endgroup$
    – D.W.
    Jul 24, 2015 at 16:55

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