CS 61B: Lecture 25
Wednesday, March 19, 2014
Today's reading: Goodrich & Tamassia, Sections 8.1-8.3.
PRIORITY QUEUES
===============
A priority queue, like a dictionary, contains _entries_ that each consist of
a key and an associated value. However, whereas a dictionary is used when we
want to be able to look up arbitrary keys, a priority queue is used to
prioritize entries. Define a total order on the keys (e.g. alphabetical
order). You may identify or remove the entry whose key is the lowest (but no
other entry). This limitation helps to make priority queues fast. However, an
entry with any key may be inserted at any time.
For concreteness, let's use Integer objects as our keys. The main operations:
- insert() adds an entry to the priority queue;
- min() returns the entry with the minimum key; and
- removeMin() both removes and returns the entry with the minimum key.
key
--------- | --------- ---------
|4: womp| v |4: womp| | |
|7: gong|-insert(5, hoot)->|7: gong|-removeMin()->|7: gong|->min()
| | ^ |5: hoot| | |5: hoot| |
--------- | --------- v --------- v
value (4, womp) (5, hoot)
Priority queues are most commonly used as "event queues" in simulations. Each
value on the queue is an event that is expected to take place, and each key
is the time the event takes place. A simulation operates by removing
successive events from the queue and simulating them. This is why most
priority queues return the minimum, rather than maximum, key: we want to
simulate the events that occur first first.
public interface PriorityQueue {
public int size();
public boolean isEmpty();
Entry insert(Object key, Object value);
Entry min();
Entry removeMin();
}
See page 340 of Goodrich & Tamassia for how they implement an "Entry".
Binary Heaps: An Implementation of Priority Queues
---------------------------------------------------
A _complete_binary_tree_ is a binary tree in which every row is full, except
possibly the bottom row, which is filled from left to right as in the
illustration below. Just the keys are shown; the associated values are
omitted.
2 index: 0 1 2 3 4 5 6 7 8 9 10
/ \
/ \ ------------------------------------------------
5 3 | | 2 | 5 | 3 | 9 | 6 | 11 | 4 | 17 | 10 | 8 |
/ \ / \ ------------------------------------------------
9 6 11 4 ^
/ \ / |
17 10 8 \--- array index 0 intentionally left empty.
A _binary_heap_ is a complete binary tree whose entries satisfy the
_heap-order_property_: no child has a key less than its parent's key.
Observe that every subtree of a binary heap is also a binary heap, because
every subtree is complete and satisfies the heap-order property.
Because it is complete, a binary heap can be stored compactly as an array of
entries. We map tree nodes to array indices with _level_numbering_, which
places the root at index 1 and orders the remaining nodes by a level-order
traversal of the tree.
Observe that if a node's index is i, its children's indices are 2i and 2i+1,
and its parent's index is floor(i/2). Hence, no node needs to store explicit
references to its parent or children. (Array index 0 is left empty to make the
indexing work out nicely. If we instead put the root at index 0, node i's
children are at indices 2i+1 and 2i+2, and its parent is at floor([i-1]/2).)
We can use either an array-based or a node-and-reference-based tree data
structure, but the array representation tends to be faster (by a significant
constant factor) because there is no need to read and write the references that
connect nodes to each other, cache performance is better, and finding the last
node in the level order is easier.
Just like in hash tables, either each tree node has two references (one for the
key, and one for the value), or each node references an "Entry" object (see
page 340 of Goodrich and Tamassia).
Let's look at how we implement priority queue operations with a binary heap.
[1] Entry min();
The heap-order property ensures that the entry with the minimum key is always
at the top of the heap. Hence, we simply return the entry at the root node.
If the heap is empty, return null or throw an exception.
[2] Entry insert(Object k, Object v);
Let x be the new entry (k, v), whose key is k and whose value is v. We place
the new entry x in the bottom level of the tree, at the first free spot from
the left. (If the bottom level is full, start a new level with x at the far
left.) In an array-based implementation, we place x in the first free location
in the array (excepting index 0).
Of course, the new entry's key may violate the heap-order property. We correct
this by having the entry bubble up the tree until the heap-order property is
satisfied. More precisely, we compare x's key with its parent's key.
While x's key is less, we exchange x with its parent, then repeat the test with
x's new parent. Continue until x's key is greater than or equal to its parent,
or x reaches the root. For instance, if we insert an entry whose key is 2:
2 2 2 2
/ \ / \ / \ / \
/ \ / \ / \ / \
5 3 5 3 5 3 2 3
/ \ / \ => / \ / \ => / \ / \ => / \ / \
9 6 11 4 9 6 11 4 9 2 11 4 9 5 11 4
/ \ / / \ / \ / \ / \ / \ / \
17 10 8 17 10 8 2 17 10 8 6 17 10 8 6
As this example illustrates, a heap can contain multiple entries with the same
key. (After all, in a typical simulation, we can't very well outlaw multiple
events happening at the same time.)
When we finish, is the heap-order property satisfied? p x
Yes, if the heap-order property was satisfied before the / \ / \
insertion. Let's look at a typical exchange of x with a s x => s p
parent p (right) during the insertion operation. Since /\ /\ /\ /\
the heap-order property was satisfied before the insertion, l r l r
we know that p <= s (where s is x's sibling), p <= l, and
p <= r (where l and r are x's children). We swap only if x < p, which implies
that x < s; after the swap, x is the parent of s. After the swap, p is the
parent of l and r. All other relationships in the subtree rooted at x are
unchanged, so after the swap, the tree rooted at x has the heap-order property.
For maximum speed, don't put x at the bottom of the tree and bubble it up.
Instead, bubble a hole up the tree, then fill in x. This modification saves
the time that would be spent setting a sequence of references to x that are
going to change anyway.
insert() returns an Entry object representing (k, v).
[3] Entry removeMin();
If the heap is empty, return null or throw an exception. Otherwise, begin by
removing the entry at the root node and saving it for the return value. This
leaves a gaping hole at the root. We fill the hole with the last entry in the
tree (which we call "x"), so that the tree is still complete.
It is unlikely that x has the minimum key. Fortunately, both subtrees rooted
at the root's children are heaps, and thus the new mimimum key is one of these
two children. We bubble x down the heap until the heap-order property is
satisfied, as follows. We compare x's key with both its children's keys.
While x has a child whose key is smaller, swap x with the child having the
minimum key, then repeat the test with x's new children. Continue until x is
less than or equal to its children, or reaches a leaf.
Consider running removeMin() on our original tree.
2 8 3 3
/ \ / \ / \ / \
/ \ / \ / \ / \
5 3 5 3 5 8 5 4
/ \ / \ => / \ / \ => / \ / \ => / \ / \
9 6 11 4 9 6 11 4 9 6 11 4 9 6 11 8
/ \ / / \ / \ / \
17 10 8 17 10 17 10 17 10
Above, the entry bubbled all the 1 4 2
way to a leaf. This is not / \ / \ / \
always the case, as the / \ / \ / \
example at right shows. 2 3 => 2 3 => 4 3
/ \ / \ / \ / / \ /
9 6 11 4 9 6 11 9 6 11
For maximum speed, don't put x at the root and bubble it down. Instead, bubble
a hole down the tree, then fill in x.
Running Times
-------------
There are other, less efficient ways we could implement a priority queue than
using a heap. For instance, we could use a list or array, sorted or unsorted.
The following table shows running times for all, with n entries in the queue.
Binary Heap Sorted List/Array Unsorted List/Array
min() Theta(1) Theta(1) Theta(n)
insert()
worst-case Theta(log n) * Theta(n) Theta(1) *
best-case Theta(1) * Theta(1) * Theta(1) *
removeMin()
worst-case Theta(log n) Theta(1) ** Theta(n)
best-case Theta(1) Theta(1) ** Theta(n)
* If you're using an array-based data structure, these running times assume
that you don't run out of room. If you do, it will take Theta(n) time to
allocate a larger array and copy the entries into it. However, if you
double the array size each time, the _average_ running time will still be
as indicated.
** Removing the minimum from a sorted array in constant time is most easily
done by keeping the array always sorted from largest to smallest.
In a binary heap, min's running time is clearly in Theta(1).
insert() puts an entry x at the bottom of the tree and bubbles it up. At each
level of the tree, it takes O(1) time to compare x with its parent and swap if
indicated. An n-node complete binary tree has height floor(log2 n). In the
worst case, x will bubble all the way to the top, taking Theta(log n) time.
Similarly, removeMin() may cause an entry to bubble all the way down the heap,
taking Theta(log n) worst-case time.
Bottom-Up Heap Construction
---------------------------
Suppose we are given a bunch of randomly ordered entries, and want to make a
heap out of them. We could insert them one by one in O(n log n) time, but
there's a faster way. We define one more heap operation.
[4] void bottomUpHeap();
First, we make a complete tree out of the entries, in any order. (If we're
using an array representation, we just throw all the entries into an array.)
Then we work backward from the last internal node (non-leaf node) to the root
node, in reverse order in the array or the level-order traversal. When we
visit a node this way, we bubble its entry down the heap as in removeMin().
Before we bubble an entry down, we know (inductively) that its two child
subtrees are heaps. Hence, by bubbling the entry down, we create a larger heap
rooted at the node where that entry started.
+-+
9 9 9 |2|
/ \ / \ / \ /-\
/ \ / \-+ +-/ \ / \
4 7 => 4 |2| => |2| 2 => 4 2
/ \ / \ / \ /-\ /-\ / \ / \ / \
2 8 2 6 2 8 7 6 4 8 7 6 9 8 7 6
The running time of bottomUpHeap is tricky to derive. If each internal node
bubbles all the way down, then the running time is proportional to the sum of
the heights of all the nodes in the tree. Page 371 of Goodrich and Tamassia
has a simple and elegant argument showing that this sum is less than n, where n
is the number of entries being coalesced into a heap. Hence, the running time
is in Theta(n), which beats inserting n entries into a heap individually.
Postscript: Other Types of Heaps (not examinable)
---------------------------------
Binary heaps are not the only heaps in town. Several important variants are
called "mergeable heaps", because it is relatively fast to combine two
mergeable heaps together into a single mergeable heap. We will not describe
these complicated heaps in CS 61B, but it's worthwhile for you to know they
exist in case you ever need one.
The best-known mergeable heaps are called "binomial heaps," "Fibonacci heaps,"
"skew heaps," and "pairing heaps." Fibonacci heaps have another remarkable
property: if you have a reference to an arbitrary node in a Fibonacci heap,
you can decrease its key in constant time. (Pairing heaps are suspected of
having the same property, but nobody knows for sure.) This operation is used
frequently by Dijkstra's algorithm, an important algorithm for finding the
shortest path in a graph. The following running times are all worst-case.
Binary Binomial Skew Pairing Fibonacci
insert() O(log n) O(log n) O(1) O(log n) * O(1)
removeMin() O(log n) O(log n) O(log n) O(log n) O(log n)
merge() O(n) O(log n) O(1) O(log n) * O(1)
decreaseKey() O(log n) O(log n) O(log n) O(log n) * O(1)
* Conjectured to be O(1), but nobody has proven or disproven it.
The time bounds given here for skew heaps, pairing heaps, and Fibonacci heaps
are "amortized" bounds, not worst case bounds. This means that, if you start
from an empty heap, any sequence of operations will take no more than the given
time bound on average, although individual operations may occasionally take
longer. We'll discuss amortized analysis near the end of the semester.