Linear Time Algorithm to Find MST?

Question

Given 2 Algorithms for a graph G=(V,E):

One:

Sort edges from lowest to highest weight.
Set T = {}
for each edge e in the previous order, check if e Union T doesn't have any cycles. If yes, Add e to T.
Return T if it's a spanning Tree.

Two:

Sort edges from highest to lowest weight.
Set T = E
for each edge e in the previous order, check if T{e} is connected graph. If yes, Remove e from T.
Return T if it's a spanning Tree.

Do both algorithms return Minimum Spanning Tree for sure? If not I would like to see a counter example.

Your first looks like Kruskal's algorithm; the second looks like reverse-delete. Neither is linear time. — jhnc, Jul 31 '22 at 08:17
What do you call linear time ? There are two independent input sizes. — , Jul 31 '22 at 08:41
OP, these algorithms are not linear-time in the models where computer scientists care about the open problem of finding a linear-time MST algorithm because in those models, sorting is Omega(n log n). — David Eisenstat, Jul 31 '22 at 11:57

score 1 · Accepted Answer · answered Jul 31 '22 at 13:08

Both of these algorithms will find an MST if it exists. The first is Kruskal's algorithm, and the second can be proven equivalent pretty easily.

Neither of them are linear time unless the weights are constrained somehow, because they start with an O(N log N) edge sort.

Disregarding the sort, the remainder of Kruskal's algorithm is very close to linear time, because it uses a disjoint set data structure to check for connectivity.

The second algorithm doesn't have a similarly quick and straightforward implementation -- anything fast is going to be more difficult than using Kruskal's algorithm instead.

score 0 · Answer 2 · answered Jul 31 '22 at 08:45

0

None of the algorithms is guaranteed to build a tree because E might not be simply connected.

answered Jul 31 '22 at 08:45

Linear Time Algorithm to Find MST?

2 Answers2