Question: 1. (10 marks) Consider an abstract data type that consists of a set S of positive integers upon which the following two operations can be

1. (10 marks) Consider an abstract data type that consists of

1. (10 marks) Consider an abstract data type that consists of a set S of positive integers upon which the following two operations can be performed: DELETE(S, i): Delete integer i from the set S. If i S. there is no effect. SUCCESSOR(S, i): Return the successor of integer i in S, i.e. min{j ES|ji. Ifi has no successor in S, i.e. if i 2 max S, then return 0. Note that it is not necessary for i to be in S. Initially, S is a set of consecutive integers from a to b. For example, if a 5, b = 10 then ini- tially S -15,6,7,8,9, 10. After some DELETEO operations, suppose that S 6,8, 10) then SUCCESSOR(S, 14) 0, SUCCESSOR(S, 3) 6 and SUCCESSOR(S, 6) SUCCESSOR(S, 7)- 8. In this question you will describe a data structure with O(( answering the following questions. amortized cost per operation by (d) Provide an algorithm for SUCCESSOR that takes O(a(n)) amortized time

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