Class 33: Heaps and Heap Sort

Held: Monday, April 4, 2005

Summary: Today we consider the heap, an important implementation of priority queues. We also note how heaps can be used to derive a useful sorting algorithm.

Related Pages:

• EBoard.

Notes:

• The exam will be distributed on or before Friday and due the following Friday.
• Today is a lecture/recitation/discussion day.

Overview:

• Review: Priority Queues.
• Trees.
• Tree Terminology.
• Heaps.
• Implementing Heaps.

Priority Queues, Revisited

• You may recall that we've been working with priority queues, linear structures which provide the highest-priority-first rubric.
• We've come up with two implementations:
• A sorted list of values
• An unsorted array of values
• What are the running times for put and get?
• Can we do better?

A Binary Implementation

• We can use the divide and conquer to design this data structure.
  • In algorithms, we divide the problem up into parts.
  • In data structures, we can build structures that naturally have two halves (plus something that holds-together the two halves).
  • Each half will also have its own two halves, and so on and forth.
• This divided structure looks something like a tree (okay, an upside-down tree or a unisex family tree). Because each node has two subtrees, we call it a binary tree.
• We might express that technique with the following Java class

  public class BinaryTreeNode 
  {
  
    // +--------+--------------------------------------------------
    // | Fields |
    // +--------+
  
    /** The value associated with this node. */
    private Object value;
  
    /** 
     * Half of the remaining elements.  Set to null if there are
     * no other elements.
     */
    private BinaryTreeNode left;
  
    /**
     * The other half of the remaining elements.  Set to null
     * if there are no other elements.
     */
    private BinaryTreeNode right;
  
    /**
     * The comparator used to determine priorities.
     */
    private Comparator prioritize;
  
    // ...
  } // BinaryTreeNode
  

• Trees are common data structures. We'll revisit them a few times in the coming weeks.

Detour: Tree Terminology

• Heaps are a kind of tree. Hence, it is important that we consider some basic tree terminology
• The root is the top or beginning of the tree.
• A node is a part of the tree. (While this has the same name as the nodes we often use to implement trees and lists, you should think of it as independent of implementation.)
• Most nodes have one or more children.
• Each node other than the root has a parent.
• Nodes without children are called leaves.
• Nodes with children are called interior nodes.
• The level of a node is the number of steps from root to that node.
  • The root is at level 0.
  • The direct children of the root are at level 1.
  • The children of those nodes are at level 2.
  • ...
• The depth of a tree is the largest level of any node in the tree.
• The size of a tree is the number of nodes in the tree.
• In a binary tree, no node has more than two children.
  • The children are typically designated as left and right.
• In a complete tree, every level is full (all the interior nodes have the maximum number of children).
• We'll return to trees in the weeks to come. For today, we'll stick with the simple heaps we've just defined.

Implementation of Key Operations

• Here's a technique for deleting the highest-priority element:
  • Remove that element. This creates a hole at the root of the tree.
  • Put highest-priority root of a subtree in that hole, creating a hole in that subtree.
  • Recurse on the subtree.
• This also gives us a method for adding elements:
  • Find a hole at the bottom of the tree.
  • Put the element there.
  • If it has a higher priority than the value above it, swap them (again and again and again).
• How long do the operations take?
• How can we make the operations more efficient? (What properties of the tree help?)

An Introduction to Heaps

• The structures we just described are similar to a traditional implementation of priority queues known as the heap.
• Heaps are a particular form of binary tree designed to provide quick access to the highest-priority element in the tree.
• Unlike our structures, heaps must be balanced (it's part of the definition).
• A heap is
  • a binary tree,
  • that is nearly complete in that
    • at most one node has one child (the rest have zero or two)
    • the nodes on the last level are at the left-hand-side of the level (that is, if a node has one or two children, all of its left siblings have two children)
  • and that has the heap property: the value stored in each node is of higher priority (lower value) than the values stored below it.

The Heap Property

• An essential aspect of heaps is the heap property.
• We can talk about a global heap property (that the value stored at one node in the tree is of higher priority than anything stored below it.
• We can also speak about a local heap property (that the value stored at one node is of higher priority than the two values stored directly below it).
• If the local heap property holds everywhere in the tree, then the global heap property holds everywhere in the tree.
  • Consider the path; we can only be getting larger.

Examples

• Here are some heaps of varying sizes

      2     2   2    2      2       2      3
     / \       /    / \    / \     / \    / \
    3   7     3    3   7  3   7   3   7  3   3
   / \  |               /        / \
  9   7 8               9       9   7
  

• Here are some non-heaps. Can you tell why?

      2          2      2        2      2
     / \        / \    / \      /|\     |
    3   7      9   7  7   3    3 3 3    3
   /   / \    / \        / \           / \
  9   8   8  9   7      9   7         4   7
  

Heap Operations

• How do we modify heaps? Through insertion (put) and deletion (get).
• How do we do insertion while maintaining the two key properties?
  • We'll maintain one property (near-completeness) and then look to restore the other (heap order).
  • It turns out that we add and delete in slightly different ways than we suggested above if we want to maintain near-completeness.
• When we add an element to the heap, we just need to put the element at the end of the lowest level.
  • If that level is full, we put it at the left end of the next level.
• After removing the smallest element from the heap, we have a hole at the top of the heap. We put the last element in the last level in that hole (which may sound like an odd idea, but at least it maintains the near-completeness property).
• Both of these operations may violate the heap order (in fact, the second one is almost guaranteed to violate heap order), so we need to rearrange the tree.
• This rearrangement is often called percolating. When we put an element at the bottom, we percolate up. When we put an element at the top, we percolate down.

Adding to Heaps

• To add an element to a heap we,
  • put the element at the end of the lowest level and
  • percolate up.
• Percolating up is fairly simple
  • At each step, we compare the percolated node to its parent.
  • If the percolated node is smaller (violating the heap property), we swap the two and continue up the tree.
  • Otherwise, we're done, since the heap property is no longer violated.
• When we do the swap,
  • the subtree that contains the old parent is clearly in heap order (the old parent was an ancestor to all the nodes in that subtree, and therefore smaller) and
  • the present node is clearly smaller than both of its new subtrees (it's smaller than the old parent, and the old parent was smaller than everything else below it).
• Eventually, we stop (either because we no longer violate heap property or because we reach the root).
• Here's an example, based on inserting the values 5, 6, 4, 4, 7, 2
  • Initial tree:

        5
    

  • Insert 6, no need to swap.

        5
       /
      6
    

  • Insert 4, swap with root.

        5         4
       / \  to   / \
      6   4     6   5
    

  • Insert 4, swap once

        4         4
       / \       / \
      6  5  to  4   5
     /         /
    4         6
    

  • Insert 7, no need to swap.

        4
       / \
      4   5
     / \
    6   7
    

  • Insert 2, need to percolate up to root.

        4           4          2
       / \         / \        / \
      4   5  to   4   2 to   4   4
     / \  |      / \  |     / \  |
    6   7 2     6   7 5    6   7 5
    

• How much time does this take? Well, the depth of a complete binary tree with n nodes is O(log₂n), and the algorithm may require swapping up from leaf to root, so the running time is also O(log₂n).

Deleting Elements from Heaps

• To delete the smallest element of the heap we need to
  • remove and remember the root;
  • put the last element of the last level at the root;
  • percolate down;
  • return the old root.
• Percolating an element down is slightly more difficult than percolating up, since there are two possible subtrees to move to. As you might guess, you must swap with the root of the smaller subtree and then continue within that subtree.
  • Why don't we worry about increasing the size of the wrong subtree? (As we did in building binary search trees.) Because we're not changing the size of the subtrees. We're swapping elements, but not adding them.
• In some sense, deletion reverses the process of insertion (delete last element in the heap vs. insert last element in heap; percolate down vs. percolate up).
• Here's a sample case of removal of least element.

      2          ?          5          3          3
     / \        / \        / \        / \        / \
    3   4  to  3   4  to  3   4  to  5   4  to  4   4
   / \  |     / \  |     / \        / \        / \
  6   4 5    6   4 5    6   4      6   4      6   5
  

• What's the running time? O(log₂n) again.

Implementing Heaps with Arrays

• When considering lists and other structures, we found ways to implement the structures with both arrays and nodes.
  • It seems likely that we can implement heaps with special nodes (as we did at the beginning of class).
  • Can we also implement heaps with arrays?
• It turns out to be relatively easy to implement binary heaps and other binary trees, particularly complete binary trees with arrays.
• How?
  • Assume we have a complete binary tree in that every interior (nonleaf) node has exactly two children.
  • Number the nodes, starting at the top and working across each level. (The root is node 0, its left child is node 1, the root's right child is node 2, node 1's left child is node 3, node 1's right child is node 4, ...).

           0
         /   \
        1     2
       / \   / \
      3   4 5   6
     / \
    7   8
    

  • This numbering gives you the positions in the array for each element.
  • (If you don't want to build complete trees and are willing to waste space, you can store a special value to represent nothing at this position.)
• This provides a very convenient way of figuring out where children belong.
  • The root of the tree is in location 0.
  • The left child of an element stored at location i can be found in location 2*i+1.
  • The right child of an element stored at location i can be found in location 2*i+2 (also representable as 2*(i+1)).
• The parent of an element stored at location i can be found at location floor((i-1)/2).
• Can we prove all this? Yes, but that's an exercise for another day.
• These properties make it simple to move the cursor around the tree and to get values.
• Note that we have an interesting double indirection here.
  • We've decided to implement priority queues with this divide-and-conquer structure.
  • We've decided to implement this divide-and-conquer structure with arrays.
• That is, we've given an implementation of an implementation of a data structure.

Heap Sort

• We can use the heap structure to provide a fairly simple and quick sorting algorithm. To sort a set of n elements,
  • insert them into a heap, one-by-one.
  • remove them from the heap in order.
• What's the running time?
  • There are n insertions. Each takes O(log₂n) time by our prior analysis.
  • There are n delete least operations. Each takes O(log₂n) by our prior analysis.
• Hence, we've developed yet another O(nlog₂n) sorting algorithm.
• Can we do this in place (provided, of course, that our original information was in an array)? You'll need to think about it.
• Most people implement heap sort with array-based heaps. Some even define heap sort completely in terms of the array operations, and forget the origins.
• Here's the core of heap sort.

      for (int i = 1; i < stuff.length; ++i) {
        percolateUp(stuff[i]);
      } // for
  

• You still need to extract the values from the heap.