As illustrated in the figure below, you are walking along the X axis back and forth and there are n balloons floating above you. The balloons were left by kids who were here earlier. One thing you know is that if two balloons overlap horizontally, then they cannot belong to the same kid (as a kid only releases one balloon at a time). For example, in the figure, the leftmost balloons with labels 1 and 2 cannot belong to the same kid, since their horizontal ranges (along X) marked by their left and right boundaries overlap.

Given balloons each described as a center location and a radius (), you need to infer the minimum number of kids who were here earlier. For example, in the example in the figure, there were at least 3 kids, and each balloon is labeled by the corresponding kid number. Design an efficient algorithm to determine the minimum number of kids, and an assignment of kid number to each balloon. There can be many assignments and you only need to find one of them. Prove its correctness and analyze its complexity.