Why would the movement speed of units negatively affect the computational speed of pathing algorithms?
In the Civ4 AI, units explore a certain multiple of their speed to find a target.
The faster they move, the larger the area they have to search to find targets they could attack in a round or two.
I can see that their A* might be poorly written or inefficient (hopefully the phrase 'an astonishing new engine built from the ground-up' means this has been fixed!) but I just don't see how increasing a units move to 2 tiles from 1 would affect the time it takes to find a path.
Because they have to search a larger radius of area for valid paths.
Reducing the number of choices at each node on the path on the other hand would clearly have a significant effect since the number of computations for n choices at each node on a path of length x is more closely related to x^n than x*n
What you care about is how many tiles there are in a radius of n away from a location.
Now, as unit velocity goes up, n goes up.
As the branching factor of the game goes up, the base goes up.
On a square map, the number of squares within distance n is (2n+1)^2 = 4n^2 + 4n + 1
On a hex map, the number of squares within distance n is ...
http://en.wikipedia.org/wiki/Hexagonal_number
(2n+1)(2(2n+1)-1)
=(2n+1)(4n+1)
=8n^2+6n+1
That isn't quite right. It should be 1 7 19 37 (+6, +12, +18) or so, as opposed to 1 9 25 49. (+8,+16,+24, etc).
But in any case, it will end up also being an O(n^2) number. Hexagonal ends up being about 3/4 the volume at a given radius, but with unit velocity being faster, the search space blowup from a 50% to 100% boost in speed (for the same number of turn lookahead) is a factor of 2 to 4.
Worst case pathing situations (where you have problems reaching your target) end up having to search a volume whose radius is roughly proportional to n. The fact that one is a hexagonal tiled, and the other is square tiled, won't matter much.