Algorithm 1.

Fast algorithm to find a Hamiltonian loop.

for i in range(m): //row

//x direction communication distance

dx[i][0] = 1 if i == 0 else 0

//y direction communication distance

dy[i][0] = 0 if i == 0 else -1

if n == 2:

dx[i][n - 1] = -1 if i == m-1 else 0

dy[i][n - 1] = -1 if i == m-1 else 0

else:

dx[i][n - 1] = 0 if i % 2 ==0 else -1

dy[i][n - 1] = 1 if i % 2 ==0 else 0

for j in range(1,n-1): //column

if j = =1 and i != m - 1:

dx[i][j] = 1 if i % 2 ==0 else 0

dy[i][j] = 0 if i % 2 ==0 else 1

else:

dx[i][j] = 1 if i % 2 ==0 else 1

dy[i][j] = 0