Question

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 operations 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.

Answer 1

To maintain a FIFO queue of integers with O(1) amortized time for operations, use a combination of an actual queue and a min heap. The ENQUEUE procedure adds integers to the queue and updates the minimum value if necessary. The DEQUEUE procedure removes the first integer and updates the minimum value.

Step-by-step explanation:

To design a data structure that maintains a FIFO queue of integers and supports the operations ENQUEUE, DEQUEUE, and FIND-MIN in O(1) amortized time, we can use a combination of an actual queue and a min heap as an auxiliary data structure.

Invariants:

The actual queue stores the integers in the order they were enqueued.

The min heap keeps track of the minimum value in the queue.

ENQUEUE Procedure:

1. Add the integer to the actual queue.

2. If the integer is smaller than the minimum value in the min heap, update the minimum value.

DEQUEUE Procedure:

1. Remove the first integer from the actual queue.

2. If the integer being removed is the minimum value in the min heap, update the minimum value.

FIND-MIN Procedure:

Return the minimum value stored in the min heap.

Correctness Proof:

- The invariants ensure that the ENQUEUE procedure correctly maintains the order of the queue and updates the minimum value.

- The DEQUEUE procedure removes the first integer from the queue and updates the minimum value if necessary, which doesn't violate the invariants. By always removing the first integer, the FIFO property is preserved.

Time Complexity:

All three procedures have a worst-case cost of O(1) amortized time.

The ENQUEUE and DEQUEUE procedures may update the minimum value, which can be done in O(log n) time for the min heap, where n is the number of integers in the queue.

The amortized cost of any sequence of m operations is O(m).