17.6: Bounded heap

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A bounded heap is a heap that is limited to contain at most k elements. If you have n elements, you can keep track of the k largest elements like this:

Initially, the heap is empty. For each element, x:

Branch 1: If the heap is not full, add x to the heap.
Branch 2: If the heap is full, compare x to the smallest element in the heap. If x is smaller, it cannot be one of the largest k elements, so you can discard it.
Branch 3: If the heap is full and x is greater than the smallest element in the heap, remove the smallest element from the heap and add x.

Using a heap with the smallest element at the top, we can keep track of the largest k elements. Let’s analyze the performance of this algorithm. For each element, we perform one of:

Branch 1: Adding an element to the heap is \( O(\log{k}) \).
Branch 2: Finding the smallest element in the heap is \( O(1) \).
Branch 3: Removing the smallest element is \( O (\log{k}) \). Adding x is also \( O(\log{k}) \).

In the worst case, if the elements appear in ascending order, we always run Branch 3. In that case, the total time to process n elements is \( O (n \log{k}) \), which is linear in n.

In ListSorter.java you’ll find the outline of a method called topK that takes a List, a Comparator, and an integer k. It should return the k largest elements in the List in ascending order. Fill it in and then run ant ListSorterTest to confirm that it works.