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See the example in Figure 1. Your answer should start with the words: Assume that such an algorithm that runs in better time exists, then one could use this algorithm to Explain exactly how this algorithm would be used. Assume that such an algorithm, which constructs T , exists that runs in better time, i. Note that T is not given, and the previous question shows that you cannot construct T in the required time.

First take O n time to convert the integers into 2-digits numbers base n. Give a simple, linear-time algorithm that solves the selection problem for an arbitrary order statistic. For example, Median A, 1, n runs in O n. You cannot alter the code of Median. As such, this gives the following psuedocode for Smallest 2based upon underlying logic of Quicksort with the exception that only onerecursive call rather than two are made: From each well, a spur pipeline is to be connected directly to the main pipeline along a shortest path either north or south , as shown in Figure 9.

Given x- and y-coordinates of the wells, how should the professor pick the optimal location of the main pipeline the one that minimizes the total length of the spurs?

Show that the optimal location can be determined in linear time. The optimal y-coordinate for Professor Olays east-west oil pipeline is as follows: Proof We examine various cases.

In each case, we will start out with thepipeline at a particular y-coordinate and see what happens when we move it. We start with the case in which n is even. Let us start with the pipelinesomewhere on or between the two oil wells whose y-coordinates are the lowerand upper medians.

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