AN O(n log n) ALGORITHM FOR INPUT-OR-OUTPUT TEST IN DISJUNCTIVE SCHEDULING

of O(n log n) algorithm Step1(initial procedure) Initialize a binary tree BT . Step2(update procedure) If possible, update BT . Else, exit the algorithm. Step3(test procedure) Test candidate pairs of jobs. Go to Step2. Let Lpair(v) be a set of values fpair(i, j) of v. The initial procedure is below. Initial procedure Step1 Put indices to jobs so that d1 < d2 < · · · < dn. Step2 Construct a binary tree BT that has n leaves {l1, l2, . . . , ln}. Let head = min{ri | i ∈ J}. Each leaf li corresponds to a time interval [head, di]. Step3 Set Linterval(li) = finterval(head, di), ∀i ∈ J . Step4 Set Linterval(v) = min{Linterval(l) | l ∈ dleaf(v)}, ∀v ∈ V . Step5 Set Lpair(v) = ∅ for all v ∈ V . The time complexity of the initial procedure is O(n) except for the sorting. The required memory space is O(n), since each vertex v ∈ V has just one label Linterval(v). Figure 4 shows an instance of the disjunctive scheduling problem.