
(a)
To show that there will always be a point where maximum overlap is an endpoint of one of the segments.
(a)

Explanation of Solution
Given Information: A point of maximum overlap in a set of intervals is a point with the largest number of intervals in the set that overlap it.
Explanation:
Consider that there is no point of maximum overlap in an endpoint of a segment. The maximum overlap occurs in the interior of m segments. Here, the point P is the intersection of those m points.
There must be another point
Hence, it is proved that the there is always a point where maximum overlap has an endpoint of the segment.
(b)
To show that there will always be a point where maximum overlap is an endpoint of one of the segments.
(b)

Explanation of Solution
Explanation:
Consider a balanced binary tree of endpoints. For inserting the interval, it is necessary to insert the endpoints separately. Consider the endpoints as e . For left endpoint e , increase the value of e by 1 and for right endpoint e , decrease the overlap by 1.
For multiple endpoints with same value, insert the left endpoints with the value before the right endpoints with the value.
Consider that
Where
Here, each node x store the new node that includes the endpoints
For bottom up approach to satisfy the conditions of red black tree following conditions must be hold:
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