(a)
To finds the correct values in the extracted array of sequence
(a)
Explanation of Solution
The EXTRACT-MIN
The INSERT algorithm is used to insert the value after the minimum value in the off-line minimum problem.
The algorithm takes the sequence of number and generates the corresponding values. The correct values in the extracted array are given in table below:
| INDEX | VALUE |
| 1 | 4 |
| 2 | 3 |
| 3 | 2 |
| 4 | 6 |
| 5 | 8 |
| 6 | 1 |
The algorithm picks the minimum from the sequence by considering the sequence and then it remove that value then add to the extracted array and the remaining keys are added to the old sequence by using insert algorithm.
(b)
To argue the array extracted returned by OFF-LINE-MINIMUM is correct.
(b)
Explanation of Solution
The OFF-LINE-MINIMUM algorithm find the minimum number from the available sequence by using the extract min and then it add the remaining number to the sequence using insert operation.
The algorithm pick the smallest element from the available element then removed it from the sequence then it checks that the extracted value is minimum or not and combine the set of elements that needs to inserted in the sequence that is remaining elements other than min is combined to the original sequence.
Every iteration of the algorithm is extracted the minimum one by one and added to the extracted array until the last is extracted and stored to the array.
Therefore, the output returns from the algorithm is correct and have finite sequence.
(c)
To describe the implementation of OFF-LINE-MINIMUM algorithm for disjoint set data structure and also gives a tight bound of the implementation.
(c)
Explanation of Solution
The algorithm pick the smallest element from the available element then removed it from the sequence then it checks that the extracted value is minimum or not and combine the set of elements that needs to inserted in the sequence that is remaining elements other than min is combined to the original sequence.
The implementation of the disjoint set of data structure through OFF-LINE-MINIMUM algorithm is given below:
Step 1: Select the disjoint sets.
Step 2: Check that
Step 3: Then add all the values of
Step 4: Find the appropriate value of
Step 5: For every
After removing
For the every value of
Therefore, the total running time of the implementation of algorithm for disjoint set is
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