For the following set of points, describe how the closest-pair algorithm finds a closest pair of
points:
(3, 2), (2, 1), (2, 3), (1, 2), (3, 1), (2, 2), (1, 3), (3, −1), (5, −2)
To solve this numerically using the closest-pair algorithm (often implemented with a divide-and-conquer approach), we can break down the steps as follows: ### Step 1: Sort the points We first need to sort the points based on their x-coordinates and y-coordinates. The points given are: - (3, 2) - (2, 1) - (2, 3) - (1, 2) - (3, 1) - (2, 2) - (1, 3) - (3, −1) - (5, −2) Sorted by x-coordinates: - (1, 2) - (1, 3) - (2, 1) - (2, 2) - (2, 3) - (3, −1) - (3, 1) - (3, 2) - (5, −2) _________ ____ __________ ____ _________ ___ ______ ________.
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a) With the help of an example, explain the following:
i) Algorithm.
ii) Input and output for an algorithm.
iii) Running time of an algorithm.
b) Using Fig. 7.1 in page 147 of the book as the model, illustrate the operation of Partition on the array
Use Kruskal’s algorithm to find a minimal spanning tree in the graph given in Fig. 2.
a) Show the results of inserting the keys
The maximum subsequence sum problem is defined as follows: If are in Z, find the maximum value
for all
a) The maximum subsequence sum problem is defined as follows: If are in Z, find the maximum value
for all
We assume that the answer is 0 if all the
are negative or if the sum is empty. The following algorithm finds a solution to the problem. Here, we assume that
s are stored in the array A.
Maximum-Subsequence
1
2
3
4
5
6
7
8
State precisely a loop invariant for the for loop in line 2–8. Prove that your loop invariant holds and hence conclude that the algorithm works.
b) Analyse the algorithm and find an upper bound for the run time of the above algorithm.