Question

can we perform graph lookup( A and B is connected or not?) and connect( connect A and B) in log time using a single array?

I learned some algo like:

quick find (linear time in connect and constant time in lookup) – single array

quick union(constant time connect and average n/2 in lookup) – single array

weighted union(log time in lookup and constant connect time) but this algorithm requires 2 array, one for nodes and another for number of nodes connected for each node.

I am asking just for curiosity. Is it possible to get weighted union sort of complexity using a single array?

Answer 1

You can use Disjoint-Set. It works as a forest, but you can always use an array to implement this forest, with an array of roots, instead of the root field per node.

The armotized complexity of this implementation is sub-logarithmic, and is often labled as O(log*(n))