[supernetworks_]

https://arxiv.org/abs/2504.17033

We give a deterministic O(mlog2/3n)-time algorithm for single-source shortest paths (SSSP) on directed graphs with real non-negative edge weights in the comparison-addition model. This is the first result to break the O(m+nlogn) time bound of Dijkstra's algorithm on sparse graphs, showing that Dijkstra's algorithm is not optimal for SSSP.

[hinkley]

log^2/3 might be the weirdest component I’ve ever seen in a complexity formula.

[jojomodding]

I'm continually amazed by the asymptotic complexity of union-find, which is O(alpha(n)), where alpha(x) is the inverse of the Ackermann function (and n the number of sets you union). In other words, O(6) or so as long as your inputs fit into the observable universe.

[hinkley]

There's definitely a divide on who sees what sort of algorithms. The subject of this article is in Graph Theory space, which a lot of us get even without trying (I dabbled in TSP for a while because it's a difficult distributed programming problem and I wanted to explore the space for that reason).

But if you're not implementing AI or game engines, some of the linear algebra space may be a road less traveled.

[bawolff]

I still think matrix multiplication's O(n^2.371339) is super weird.

[adgjlsfhk1]

Matrix multiplication definitely should be O(n^(2+o(1))).

[adgjlsfhk1]

for about a decade, integer multiplication was at n4^log*(n) where log* is the iterated logarithm.

Also the curently best factorization algorithm (GNFS) is at exp(k*log(n)^1/3log(log(n))^2/3).

Intro algorithms classes just tend to stay away from the really cursed runtimes since the normal ones are enough to traumatize the undergrads.

[hinkley]

Are BigInteger multiplications in logn² now or do they still have weird terms in them?

[adgjlsfhk1]

down to nlogn

[Sniffnoy]

Hey the asterisks in your reply got read as formatting so it's ended up messed-up.

[adgjlsfhk1]

oops. fixed.