Kropki Sudoku

Kropki Sudoku is a variant of the Sudoku puzzle. You must enter a number in each cell so that all neighbourhood conditions are met. Kropki is Polish and means “dots”.

Another name for Kropki Sudoku is Dot Sudoku. The origins of Kropki Sudoku, including its inventor and first publication, remain unknown.
Rules: In each cell of the NxN grid, enter a number in the range 1–N. Each number must appear exactly once in each row, column, and block.
Additional rules: A white circle between two cells indicates that one of the two numbers in the two cells must be exactly 1 greater than the other.
A black circle between two cells indicates that one of the two numbers must be exactly twice the other. If there is no circle between two cells, none of the additional rules may apply.