CS 240: Lecture 20 - Heaps Continued

Dear students:

Today we continue our discussion of heaps. First we'll examine how they can be used to implement a very pleasant sorting algorithm that uses little time and little space. Then we'll discuss the complexity of the various priority queue and heap operations.

Heap Sort

When we last left sorts, we had two reasonable, linearithmic algorithms: merge sort and quick sort. Neither is perfect: merge sort requires extra space and quick sort can degenerate to quadratic time. With heaps, we can achieve a very fine sort that requires no extra space and has a linearithmic worst case complexity.

Heap sort starts by taking the unsorted array and imposing a max-heap structure on it. That means we need to get larger items above smaller items. We have two choices for how to implement this. We could perform this task:

for each node from top to bottom
  percolate node up

Or this one:

for each node from bottom to top
  percolate node down

Which approach is better? The items at the bottom of the tree already satisfy the heap property and won't need to percolate down. The second approach will be less work. We can just skip over the bottom row completely.

Once the heap is formed, we repeatedly remove the maximum. Instead of throwing it into a new sorted collection, which would give heap sort the same blemish as merge sort, we place it at the end of the original array. After all items have been removed, the array contains the sorted list.

Time Complexity

Suppose we want to implement the priority queue ADT. We have many data structures we could use. Let's consider our choices and their computational complexities by filling out this table:

data structure insert removeMin peekMin
sorted array ? ? ?
unbalanced BST ? ? ?
AVL tree ? ? ?
heap ? ? ?

The insert operation takes in a priority-item pair. The removeMin operation always removes and returns the item with the lowest priority value. The peekMin operation returns it but does not remove it.

Heapify

Heaps have an advantage over balanced BSTs that is not reflected in the table above. Suppose you have an array filled with values in an arbitrary order, and you want to turn it into one of these collections. How many operations will it take? Sorting it will take \(~O(n \log_2 n)~\) operations. Populating an unbalanced BST will take \(~O(n^2)~\) operations. Populating a balanced BST will take \(~O(n \log_2 n)~\) operations.

What about populating a heap? A first analysis might observe that there are \(~n~\) values and inserting each requires \(~O(\log_2 n)~\) steps. Thus the complexity is \(~O(n \log_2 n)~\). This is true, but it's unnecessarily pessimistic. A more nuanced analysis can give you a tighter upper bound.

Imagine the un-heapified array drawn in tree form. The most costly tree to turn into a heap is a complete one, so imagine a complete tree. How many items are in it? This table matches the number of levels to the number of nodes:

height nodes
0 1
1 3
2 7
3 15
4 31
\(h\) \(2^{h + 1} - 1\)

Now let's consider the cost of getting each of these items into place so that we have a heap. To keep things concrete, let's examine a tree of height 5, which will have \(~2^6 - 1 = 63~\) nodes. This table reports the maximum number of swaps that might need to be performed at each level:

level nodes swaps per node total swaps
\(5\) \(32\) \(0\) \(32 \times 0 = 0\)
\(4\) \(16\) \(1\) \(16 \times 1 = 16\)
\(3\) \(8\) \(2\) \(8 \times 2 = 16\)
\(2\) \(4\) \(3\) \(4 \times 3 = 12\)
\(1\) \(2\) \(4\) \(2 \times 4 = 8\)
\(0\) \(1\) \(5\) \(1 \times 5 = 5\)

All told, we'll need to perform 57 swaps to get the array into a heap. There are only 63 nodes. This suggests that \(~O(n \log_2 n)~\) is an exaggeration. But we need to generalize our calculation to prove this. In a tree with \(~h~\) levels, a level requiring \(~p~\) swaps per node will have \(~2^{h - p}~\) nodes and will therefore incur at most \(~2^{h - p} \times p~\) swaps. Let's sum up the costs across all levels and apply some identities to arrive at a useful expression of the total cost:

$$\begin{align} \sum_{p = 0}^h 2^{h - p} \times p &= \sum \frac{2^h}{2^p} \times p \\ &= 2^h \times \sum \frac{p}{2^p} \\ \end{align}$$

The summation term is interesting. It expands to this series:

$$\sum_{p = 0}^{h} \frac{p}{2^p} = \frac{0}{1} + \frac{1}{2} + \frac{2}{4} + \frac{3}{8} + \frac{4}{16} + \ldots$$

In fact, this is a geometric series, and we can reduce it down to a simpler expression with some sleight of hand. First we consider the sum, the half sum, and their difference:

$$\begin{align} S &= \frac{1}{2} + \frac{2}{4} + \frac{3}{8} + \frac{4}{16} + \ldots \\ \frac{S}{2} &= \frac{1}{4} + \frac{2}{8} + \frac{3}{16} + \frac{4}{32} + \ldots \\ S - \frac{S}{2} &= \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \ldots \\ \end{align}$$

The sequence represented by \(~S - \frac{S}{2}~\) approaches \(~1~\) as \(~n~\) increases. That means when can reduce the summation term \(~S~\) to a simple number:

$$\begin{align} S - \frac{S}{2} = 1 \\ \frac{S}{2} = 1 \\ S = 2 \\ \end{align}$$

We now rewrite our total number of swaps:

$$\begin{align} 2^h \times \sum \frac{p}{2^p} &\le 2^h \times S \\ &= 2^h \times 2 \\ &= 2^{h + 1} \\ \end{align}$$

Earlier we said that \(~n = 2^{h + 1} - 1~\). Our total number of swaps is one less than \(~n~\), which means that building a heap is \(~O(n)~\). Heaps win.

We achieve this performance because we do the least work per node in the levels where the most nodes live. If we had implemented the heapify operation in reverse, starting at the root and percolating upward, we would be doing the least work per node in the levels where the fewest nodes live. It's mildly disturbing that a subtly different approach to the task can lead to a slower complexity class.

Space Complexity

Let's consider also how much space these data structures consume.

data structure incidental space
sorted array ?
unbalanced BST ?
AVL tree ?
heap ?

TODO

You've got a few things to do before we meet again:

See you next time!

Sincerely,

P.S. It's time for a haiku:

Is speed what matters? What about nice sort? Or green sort? Who wrote this sort sort?