Lecture No.5
1.11.2 Plane-sweep Algorithm
The question is whether we can make an significant improvement in the running time?
Here is an idea for how we might do it. We will sweep a vertical line across the plane
from left to right. As we sweep this line, we will build a structure holding the maximal
points lying to the left of the sweep line. When the sweep line reaches the rightmost point
of P, then we will have constructed the complete set of maxima. This approach of solving
geometric problems by sweeping a line across the plane is called plane sweep.
Although we would like to think of this as a continuous process, we need some way to
perform the plane sweep in discrete steps. To do this, we will begin by sorting the points
in increasing order of their x-coordinates. For simplicity, let us assume that no two points
have the same y-coordinate. (This limiting assumption is actually easy to overcome, but it
is good to work with the simpler version, and save the messy details for the actual
implementation.) Then we will advance the sweep-line from point to point in n discrete
steps. As we encounter each new point, we will update the current list of maximal points.
We will sweep a vertical line across the 2-d plane from left to right. As we sweep, we
will build a structure holding the maximal points lying to the left of the sweep line. When
the sweep line reaches the rightmost point of P, we will have the complete set of
maximal points. We will store the existing maximal points in a list The points that pi
dominates will appear at the end of the list because points are sorted by x-coordinate. We
will scan the list left to right. Every maximal point with y-coordinate less than pi will be
eliminated from computation. We will add maximal points onto the end of a list and
delete from the end of the list. We can thus use a stack to store the maximal points. The
point at the top of the stack will have the highest x-coordinate.
Here are a series of figures that illustrate the plane sweep. The figure also show the
content of the stack.
2 4 6 8 10 14 16 18 12
2
4
6
8
10
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