integer - A fast algorithm for minimum spanning trees when edge lengths are constrained?

Question

Suppose that you have a directed graph with nonnegative, integer edge lengths that are in the range 0 to U - 1, inclusive. What is the fastest algorithm for computing a minimum spanning tree of this graph? We can still use existing minimum spanning tree algorithms, such as Kruskal's algorithm O(m log n)) or Prim's algorithm (O(m + n log n)). However, for cases where U is small, I think it should be possible to do much better this.

Are there any algorithms that are competitive with more traditional MST algorithms that are able to exploit the fact that the edge lengths are constrained to be in some range?

Thanks!

Answer 1

Fredman–Willard gave an O(m + n) algorithm for integer edge lengths on the unit-cost RAM.

That's arguably not much of an improvement: without the restriction on edge lengths (i.e., the lengths are an opaque data type that supports only comparisons), Chazelle gave an O(m alpha(m, n) + n) algorithm (alpha is the inverse Ackermann function) and Karger–Klein–Tarjan gave a randomized O(m + n) algorithm.

I don't think Darren's idea leads to a O(m + n + U)-time algorithm. Jarnik ("Prim") does not use its priority queue monotonically, so buckets may be scanned multiple times; Kruskal requires a disjoint-set data structure, which cannot be O(m + n).