Question

I’m currently reading this paper and on page five, it discusses properties of binary heaps that it considers to be common knowledge. However, one of the points they make is something that I haven’t seen before and can’t make sense of. The authors claim that if you are given a balanced binary heap, you can list the elements of that heap in sorted order in O(log n) time per element using a standard breadth-first search. Here’s their original wording:

In a balanced heap, any new element can be
inserted in logarithmic time. We can list the elements of a heap in order by weight, taking logarithmic
time to generate each element, simply by using breadth first search.

I’m not sure what the authors mean by this. The first thing that comes to mind when they say “breadth-first search” would be a breadth-first search of the tree elements starting at the root, but that’s not guaranteed to list the elements in sorted order, nor does it take logarithmic time per element. For example, running a BFS on this min-heap produces the elements out of order no matter how you break ties:

             1
            / \
          10   100
         /  \
        11  12

This always lists 100 before either 11 or 12, which is clearly wrong.

Am I missing something? Is there a simple breadth-first search that you can perform on a heap to get the elements out in sorted order using logarithmic time each? Clearly you can do this by destructively modifying heap by removing the minimum element each time, but the authors’ intent seems to be that this can be done non-destructively.

Answer 1

You can get the elements out in sorted order by traversing the heap with a priority queue (which requires another heap!). I guess this is what he refers to as a “breadth first search”.

I think you should be able to figure it out (given your rep in algorithms) but basically the key of the priority queue is the weight of a node. You push the root of the heap onto the priority queue. Then:

while pq isn't empty
    pop off pq
    append to output list (the sorted elements)
    push children (if any) onto pq

I’m not really sure (at all) if this is what he was referring to but it vaguely fitted the description and there hasn’t been much activity so I thought I might as well put it out there.