Editing Kruskal's algorithm (section)

== Complexity ==

For a graph with {{mvar|E}} edges and {{mvar|V}} vertices, Kruskal's algorithm can be shown to run in time {{math|''O''(''E'' log ''E'')}} time, with simple data structures. This time bound is often written instead as {{math|''O''(''E'' log ''V'')}}, which is equivalent for graphs with no isolated vertices, because for these graphs {{math|''V''/2 ≤ ''E'' < ''V''<sup>2</sup>}} and the logarithms of {{mvar|V}} and {{mvar|E}} are again within a constant factor of each other.

To achieve this bound, first sort the edges by weight using a [[comparison sort]] in {{math|''O''(''E'' log ''E'')}} time. Once sorted, it is possible to loop through the edges in sorted order in constant time per edge. Next, use a [[disjoint-set data structure]], with a set of vertices for each component, to keep track of which vertices are in which components. Creating this structure, with a separate set for each vertex, takes {{mvar|V}} operations and {{math|''O''(''V'')}} time. The final iteration through all edges performs two find operations and possibly one union operation per edge. These operations take [[amortized time]] {{math|''O''(''α''(''V''))}} time per operation, giving worst-case total time {{math|''O''(''E'' ''α''(''V''))}} for this loop, where {{mvar|α}} is the extremely slowly growing [[inverse Ackermann function]]. This part of the time bound is much smaller than the time for the sorting step, so the total time for the algorithm can be simplified to the time for the sorting step.

In cases where the edges are already sorted, or where they have small enough integer weight to allow [[integer sorting]] algorithms such as [[counting sort]] or [[radix sort]] to sort them in linear time, the disjoint set operations are the slowest remaining part of the algorithm and the total time is {{math|''O''(''E'' ''α''(''V''))}}.