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Problem 4-1. Extreme FIFO Queues [25 points] Design a data structure that maintains a FIFO queue of integers, supporting operations ENQUEUE, DEUEUE, and FIND-MIN, each in O(1) amortized time. In other words, any sequence of m op- erations should take time O(m). You may assume that, in any execution, all the items that get enqueued are distinct. (a) [5 points] Describe your data structure. Include clear invariants describing its key properties. Hint: Use an actual queue plus auxiliary data structure(s) for bookkeeping. (b) [5 points] Describe carefully, in words or pseudo-code, your ENQUEUE, DEQUEUE and FIND-MIN procedures. (c) [5 points] Prove that your operations give the right answers. Hint: You may want to prove that their correctness follows from your data structure invariants. In that case you should also sketch arguments for why the invariants hold. (d) [10 points] Analyze the time complexity: the worst-case cost for each operation, and the amortized cost of any sequence of m operations.

          Problem 4-1. Extreme FIFO Queues [25 points]
Design a data structure that maintains a FIFO queue of integers, supporting operations ENQUEUE,
DEUEUE, and FIND-MIN, each in O(1) amortized time. In other words, any sequence of m op-
erations should take time O(m). You may assume that, in any execution, all the items that get
enqueued are distinct.
(a) [5 points] Describe your data structure. Include clear invariants describing its key
properties. Hint: Use an actual queue plus auxiliary data structure(s) for bookkeeping.
(b) [5 points] Describe carefully, in words or pseudo-code, your ENQUEUE, DEQUEUE
and FIND-MIN procedures.
(c) [5 points] Prove that your operations give the right answers. Hint: You may want to
prove that their correctness follows from your data structure invariants. In that case
you should also sketch arguments for why the invariants hold.
(d) [10 points] Analyze the time complexity: the worst-case cost for each operation, and
the amortized cost of any sequence of m operations.

Problem 4-1. Extreme FIFO Queues [25 points]
Design a data structure that maintains a FIFO queue of integers, supporting operations ENQUEUE,
DEUEUE, and FIND-MIN, each in O(1) amortized time. In other words, any sequence of m op-
erations should take time O(m). You may assume that, in any execution, all the items that get
enqueued are distinct.
(a) [5 points] Describe your data structure. Include clear invariants describing its key
properties. Hint: Use an actual queue plus auxiliary data structure(s) for bookkeeping.
(b) [5 points] Describe carefully, in words or pseudo-code, your ENQUEUE, DEQUEUE
and FIND-MIN procedures.
(c) [5 points] Prove that your operations give the right answers. Hint: You may want to
prove that their correctness follows from your data structure invariants. In that case
you should also sketch arguments for why the invariants hold.
(d) [10 points] Analyze the time complexity: the worst-case cost for each operation, and
the amortized cost of any sequence of m operations.

Added by Kelly M.

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