> And when one doesn't do it right, one will just end-up visiting each cell anyway.
And I think there is no algorithm which can guaranty otherwise.
consider this input
2
2 2
2 2 2
2 2 2 2
. . . . . 2s in between
2 2 . . 2 1 last row
no matter which top-down search you apply, you can move the 1 in a way that it's only found at last try.
Which means all cells needed to be visited.°
update
°) NB: complexity big O() calculations are about worst case (if not stated otherwise )
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