2. (20 points) Alice writes n distinct integers on a blackboard, and picks a positive integer...

 

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2. (20 points) Alice writes n distinct integers on a blackboard, and picks a positive integer K. She then allows Bob to make moves, each of which consist of the following steps. 1. Identify two integers x and y on the blackboard which differ by at most K, i.e. |xy|K. 2. Erase the smaller of the two chosen integers. Bob's task is to make moves in this way until he is no longer able to do so. Note that in some cases, Bob may be unable to make even a single move. Design an algorithm which runs in O(n log n) time and finds the longest sequence of moves. If there are several sequences of maximum length, you may find any of them. 2. (20 points) Alice writes n distinct integers on a blackboard, and picks a positive integer K. She then allows Bob to make moves, each of which consist of the following steps. 1. Identify two integers x and y on the blackboard which differ by at most K, i.e. |xy|K. 2. Erase the smaller of the two chosen integers. Bob's task is to make moves in this way until he is no longer able to do so. Note that in some cases, Bob may be unable to make even a single move. Design an algorithm which runs in O(n log n) time and finds the longest sequence of moves. If there are several sequences of maximum length, you may find any of them.