Algorithm 1: Divide and conquer

1. Sort points along the x-coordinate. 2. Split the set of points into two equal-sized subsets by a vertical line $x=x_{mid}$. 3. Solve the problem recursively in the left and right subsets. This will give the left-side and right-side minimal distances $d_{Lmin}$ and $d_{Rmin}$, respectively. 4. Find the minimal distance $d_{LRmin}$ among the pair of points in which one point lies to the left of the dividing vertical and the second point lies to the right. 5. The final answer is the minimum among $d_{Lmin}$, $d_{Rmin}$, and $d_{LRmin}$.