a searching algorithm in a circle: 1 unidirectional, 2- Bididirectional

[superbonzo]

Would it be possible to give me the formula? or show me where to get it from?

BTW, for big N,M, that's the avarage of the shortest path ( that is the euclidean distance ), that is, for a square region NxN :

(1/N^2) * integral_from{-N/2}to{N/2} integral_from{-N/2}to{N/2} sqrt( x^2 + y^2) dxdy

that is

(N/8) integral_from{-1}to{1} integral_from{-1}to{1} sqrt( x^2 + y^2) dxdy

that should equal N * ( sqrt(2) + arcsinh(1) )/6 = N * 0.382598...

EDIT: sorry, I misread OReuben's post; as it's defined that distance does NOT approach the euclidean distance at all, so the calculation above as referred to the OReuben's reply is wrong ... ( the calcaulation in itself is correct though )