# Question

The off-line minimum problem asks us to maintain a dynamic set T of elements from the domain {1, 2, ..., n} under the operations INSERT and EXTRACT-MIN. We are given a sequence S of n INSERT and m EXTRACT-MIN calls, where each key in {1, 2, ..., n} is inserted exactly once. We wish to determine which key is returned by each EXTRACT-MIN call. Specifically, we wish to fill in an array extracted[1 m], where for i = 1, 2, ..., m, extracted[i] is the key returned by the ith EXTRACT-MIN call. The problem is "off-line" in the sense that we are allowed to process the entire sequence S before determining any of the returned keys.

a. In the following instance of the off-line minimum problem, each INSERT is represented by a number and each EXTRACT-MIN is represented by the letter E: 4, 8, E, 3, E, 9, 2, 6, E, E, E, 1, 7, E, 5.

Fill in the correct values in the extracted array.

To develop an algorithm for this problem, we break the sequence S into homogeneous subsequences. That is, we represent S by I1, E, I2, E, I3, ..., Im, E, Im+1, where each E represents a single EXTRACT-MIN call and each Ij represents a (possibly empty) sequence of INSERT calls. For each subsequence Ij, we initially place the keys inserted by these operations into a set Kj , which is empty if Ij is empty. We then do the following.

OFF-LINE-MINIMUM (m, n)

c. Describe how to implement OFF-LINE-MINIMUM efficiently with a disjoint-set data structure. Give a tight bound on the worst-case running time of your implementation.

a. In the following instance of the off-line minimum problem, each INSERT is represented by a number and each EXTRACT-MIN is represented by the letter E: 4, 8, E, 3, E, 9, 2, 6, E, E, E, 1, 7, E, 5.

Fill in the correct values in the extracted array.

To develop an algorithm for this problem, we break the sequence S into homogeneous subsequences. That is, we represent S by I1, E, I2, E, I3, ..., Im, E, Im+1, where each E represents a single EXTRACT-MIN call and each Ij represents a (possibly empty) sequence of INSERT calls. For each subsequence Ij, we initially place the keys inserted by these operations into a set Kj , which is empty if Ij is empty. We then do the following.

OFF-LINE-MINIMUM (m, n)

c. Describe how to implement OFF-LINE-MINIMUM efficiently with a disjoint-set data structure. Give a tight bound on the worst-case running time of your implementation.

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