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Question 3. We have seen how to compute the closest pair of points in a plane using a divide-and-conquer algorithm. In the algorithm, the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ was taken to be $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$. In geometry, we also use other distances. One such distance is the distance $D_{special}$ is $D_{special} = ((x_1, y_1), (x_2, y_2)) = |x_1 - x_2| + |y_1 - y_2|$. Consider the following computational problem Input: Given a list $L$ of $n$ points in the plane such that no two points have the same $x$-coordinate. Assume $n > 1$. Output: Distinct points $(x_1, y_1)$ and $(x_2, y_2)$ from the list $L$ such that $(x_1, y_1)$ and $(x_2, y_2)$ have the smallest $D_{special}$ distance amongst all pairs of distinct points in $L$. Example: Let $L$ consist of the pairs $(1, 2), (5, 5) (-5, 2)$. The output of the algorithm should be $(1, 2), (-5, 2)$. Requirement: Give an algorithm that solves the above problem in $O(n \log n)$ running time. You must explain why your algorithm is correct and attains the stated time bound.

          Question 3.
We have seen how to compute the closest pair of points in a plane using a divide-and-conquer algorithm. In
the algorithm, the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ was taken to be
$\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$. In geometry, we also use other distances. One such distance is the distance
$D_{special}$ is $D_{special} = ((x_1, y_1), (x_2, y_2)) = |x_1 - x_2| + |y_1 - y_2|$.
Consider the following computational problem
Input: Given a list $L$ of $n$ points in the plane such that no two points have the same $x$-coordinate. Assume
$n > 1$.
Output: Distinct points $(x_1, y_1)$ and $(x_2, y_2)$ from the list $L$ such that $(x_1, y_1)$ and $(x_2, y_2)$ have the
smallest $D_{special}$ distance amongst all pairs of distinct points in $L$.
Example: Let $L$ consist of the pairs $(1, 2), (5, 5) (-5, 2)$. The output of the algorithm should be
$(1, 2), (-5, 2)$.
Requirement: Give an algorithm that solves the above problem in $O(n \log n)$ running time. You must
explain why your algorithm is correct and attains the stated time bound.

Question 3.
We have seen how to compute the closest pair of points in a plane using a divide-and-conquer algorithm. In
the algorithm, the distance between two points (x1, y1) and (x2, y2) was taken to be
√((x1 - x2)^2 + (y1 - y2)^2). In geometry, we also use other distances. One such distance is the distance
Dspecial is Dspecial = ((x1, y1), (x2, y2)) = |x1 - x2| + |y1 - y2|.
Consider the following computational problem
Input: Given a list L of n points in the plane such that no two points have the same x-coordinate. Assume
n > 1.
Output: Distinct points (x1, y1) and (x2, y2) from the list L such that (x1, y1) and (x2, y2) have the
smallest Dspecial distance amongst all pairs of distinct points in L.
Example: Let L consist of the pairs (1, 2), (5, 5) (-5, 2). The output of the algorithm should be
(1, 2), (-5, 2).
Requirement: Give an algorithm that solves the above problem in O(n log n) running time. You must
explain why your algorithm is correct and attains the stated time bound.

Added by Richard A.

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