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 i th 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 operation INSERT (i) is represented by the value of i 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, . . . ,I_m, E, I_m + 1, where each E represents a single EXTRACT-MIN call and each Ij represents a (possibly empty) sequence of INSERT calls. For each subsequence I_j, we initially place the keys inserted by these operations into a set K_j, which is empty if I_j is empty.

We then do the following:

OFF-LINE-MINIMUM (m, n)

b. Argue that the array extracted returned by OFF-LINE-MINIMUM is correct.

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.

1 for i = 1 to n determine j such that i e K; if j + m + 1 extracted[j] = i let I be the smallest value greater than j for which set K; exists 2 3 4 5 K1 = K; U K1, destroying K; 7 return extracted