12. (Staircase / Pareto-optimal points) Let P be a set of n points in a 2-dimensional...

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12. ("Staircase / Pareto-optimal points") Let P be a set of n points in a 2-dimensional plane. A point pe P is Pareto-optimal if no other point is both above and to the right of p. The sorted sequence of Pareto-optimal points describes a top-right staircase with the interior points of P below and to the left of the staircase. We want to compute this staircase. Describe an algorithm to compute the staircase of P in O(nh) time, where h is the number of Pareto-optimal points. Describe a divide-and-conquer algorithm to compute the staircase of P in O(n log n) time. Describe an algorithm to compute the staircase of P in O(n log h) time, where h is the number of Pareto-optimal points. Finally, suppose the points in P are already given in sorted order from left to right. Describe an algorithm to compute the staircase of P in O(n) time. 12. ("Staircase / Pareto-optimal points") Let P be a set of n points in a 2-dimensional plane. A point pe P is Pareto-optimal if no other point is both above and to the right of p. The sorted sequence of Pareto-optimal points describes a top-right staircase with the interior points of P below and to the left of the staircase. We want to compute this staircase. Describe an algorithm to compute the staircase of P in O(nh) time, where h is the number of Pareto-optimal points. Describe a divide-and-conquer algorithm to compute the staircase of P in O(n log n) time. Describe an algorithm to compute the staircase of P in O(n log h) time, where h is the number of Pareto-optimal points. Finally, suppose the points in P are already given in sorted order from left to right. Describe an algorithm to compute the staircase of P in O(n) time.