Before revision:  We describe the first algorithm for this problem that returns, with high probability, a (1 + \epsilon)-approximation to the optimal transportation map in O(n \:\text{poly (1 / \epsilon) \:\text{polylog ) time.
After revision:  We describe the first algorithm for this problem that returns, with high probability, a (1 + \epsilon)-approximation to the optimal transportation map in O(n \:\text{poly (1 / \epsilon) \, \mathrm{polylog ) time.