Editing Borůvka's algorithm (section)

== Other algorithms ==

Other algorithms for this problem include [[Prim's algorithm]] and [[Kruskal's algorithm]]. Fast parallel algorithms can be obtained by combining Prim's algorithm with Borůvka's.<ref>{{cite journal|last1=Bader|first1=David A.|last2=Cong|first2=Guojing|title=Fast shared-memory algorithms for computing the minimum spanning forest of sparse graphs|journal=Journal of Parallel and Distributed Computing|date=2006|volume=66|issue=11|pages=1366–1378|doi=10.1016/j.jpdc.2006.06.001|citeseerx=10.1.1.129.8991|s2cid=2004627}}</ref>

A faster randomized minimum spanning tree algorithm based in part on Borůvka's algorithm due to Karger, Klein, and Tarjan runs in expected {{math|O(''E'')}} time.<ref>{{cite journal|last1=Karger|first1=David R.|last2=Klein|first2=Philip N.|last3=Tarjan|first3=Robert E.|title=A randomized linear-time algorithm to find minimum spanning trees|journal=Journal of the ACM|date=1995|volume=42|issue=2|pages=321–328|doi=10.1145/201019.201022|citeseerx=10.1.1.39.9012|s2cid=832583}}</ref>  The best known (deterministic) minimum spanning tree algorithm by [[Bernard Chazelle]] is also based in part on Borůvka's and runs in {{math|O(''E'' α(''E'',''V''))}} time, where α is the [[Ackermann function#Inverse|inverse Ackermann function]].<ref>{{Cite journal|last=Chazelle|first=Bernard|title=A minimum spanning tree algorithm with inverse-Ackermann type complexity|journal=J. ACM|volume=47|year=2000|issue=6|pages=1028–1047|url=http://www.cs.princeton.edu/~chazelle/pubs/mst.pdf|doi=10.1145/355541.355562|citeseerx=10.1.1.115.2318|s2cid=6276962}}</ref> These randomized and deterministic algorithms combine steps of Borůvka's algorithm, reducing the number of components that remain to be connected, with steps of a different type that reduce the number of edges between pairs of components.