Thief and Police

There is an $N$ by $M$ grid. A thief is trying to escape from a policeman. The thief is currently in cell $(x,y)$ in the grid, while the policeman is in cell $(a,b)$. Cells are numbered from $1$ to $N$, top to bottom, and $1$ to $M$, left to right. Both of them are at different cells initially. Both of them take turns alternatively. First it is the thief's turn, then the police's, and so on. Th