Consider the following variant of two-dimensional range reporting queries. We keep a set of two-dimensional points...

 

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Consider the following variant of two-dimensional range reporting queries. We keep a set of two-dimensional points P in a data structure. All points have positive coordi- nates. For a query range Q = [a, b] [0, c], we must find a lowest point p E P Q and report it. In other words, we must find a point p such that p.y is as small as possible, and p.y c and a p.x b. Here p.x and p.y denote the x- and y-coordinates of a point p. If PnQ is empty, report NULL. Describe a data structure that answers queries described above in O(log n) time and uses space O(n). Consider the following variant of two-dimensional range reporting queries. We keep a set of two-dimensional points P in a data structure. All points have positive coordi- nates. For a query range Q = [a, b] [0, c], we must find a lowest point p E P Q and report it. In other words, we must find a point p such that p.y is as small as possible, and p.y c and a p.x b. Here p.x and p.y denote the x- and y-coordinates of a point p. If PnQ is empty, report NULL. Describe a data structure that answers queries described above in O(log n) time and uses space O(n).