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For 1), both of Verlet list and cell list can be used. For Verlet list, you needn't update the list in each particle move until after several Monte Carlo move that the particle moves out of the Verlet sphere. For the cell list, you can use the doubly linked list method to update the cell list. For 2) The complexity of $N \times N$ matrix is $O(N^2)$ which is ...