Priority Queues and Heaps

15-211

Fundamental Data Structures and Algorithms

Klaus Sutner

January 29, 2004

Today:

Demo: intelligent math handbookPriority QueuesBinary heapsImplementation questions

Reading:

Chapter 20 in MAW. See also 6.7 and 6.8.

We want to simulate a chain of events Ei. Each event Ei is associated with a time ti.

No problem: sort the times, and run through them in order.

t1 < t2 < t3 < ... < tn

But what if an event can cause the creation of some new events in the future?

A Simulation

We can still sort the original events by time, but not the additional ones: we don't yet know what/when they are. We need to be able to dynamically - insert an event into a data structure, and - extract the next event.

And, of course, the operations should be cheap.

A Simulation

Suppose a robot wants to cross a street. No jaywalking, it can only move straight forward or backward.

Maximum speed of the robot is c.

Cars are coming from either side, but the robot has complete information about when and where they cross your line.

Problem: Find a way for the robot to get across without getting hit.

An Example: Street Crossing

How do we model the given information?

In space-time a car is just a rectangular box.

Movement of the robot is constrained to a light-cone.

More Street Crossing

Hugging an obstacle (not that unrealistic, just inflate the obstacles to account for safety buffer).

More Street Crossing

Note how obstacles may be “fused” together.

A Death Trap

The times when obstacles appear/disappear are clearly events.

But there are more: The times when the robot touches an obstacle.

Or when the light cones from both ends of a disappearing obstacle meet.

Note that these events have to be computed as we go along, there is no way to compute them all at the outset.

Street Crossing

Note how the first event happens before the first obstacle arrives! We need to go backwards in time, starting at the end of the last obstacle.

Time Travel

How does this translate into a data structure?

What are the crucial operations?

What would be a simple (not necessarily efficient) implementation?

Where is room for improvement?

Towards An Algorithm

Priority Queue Interface

Crucial Operations

insertfind-mindelete-min

Think about just inserting times (real numbers), but in reality there is associated information.

Some Applications

Event simulations

Shortest path computation

Huffman coding

Sorting (sort of)

Not a universal data structure (like hash tables) but irreplaceable in some cases.

PQ Interface

Requires total order on the elements of the universe.

Could be done with any traversable container: just run through and find the min.

We would like much better performance.

Binary search trees come to mind: min element is in the leftmost position.

• Linked list deleteMin O(1) O(N) insert O(N) O(1)

• Search trees All operations O(log N)

• Heaps avg (assume random) worst

deleteMin O(log N) O(log N) insert 2.6 O(log N)

special case: buildheap O(N) O(N)

i.e., insert*N

Possible priority queue implementations

deleteMin O(log N) O(log N) insert 2.6 O(log N)

special case: buildheap O(N) O(N)

i.e., insert*N

deleteMin O(log N) O(log N) insert 2.6 O(log N) buildheap O(N) O(N)

N inserts

A Digression:Perfect and Complete

Binary Trees

Perfect binary trees

2665 32

2665 266526

2665 32

2665 266526

2665 2665 2665 2665 2665 2665 2665 2665

2665 32

2665 266526

2665 2665 2665 2665 2665 2665 2665 2665

2665 2665 2665 2665 2665 2665 2665 2665 2665 2665 2665 2665 2665 2665 2665 2665

• How many nodes?

2665 266526

h=3N=15

• How many nodes?

In general: N = = 2h+1 - 1

Most of the nodes are leaves

2665 266526

h=3h=3N=15

A Serious Proof

Define PBT by structural induction:

- nil is a PBT and H(nil) = 0- (a,L,R) is a PBT whenever L and R are PBTs and H(L) = H(R).Moreover, H( (a,L,R) ) = H(L) + 1.

size(nil) = 0size( (a,L,R) ) = size(L) + size(R) + 1.

Then size(T) = 2H(T) -1.

Quiz Break

Red-green quiz

• In a perfect binary tree, what is the sum of the heights of the nodes?

Give a mathematical characterization, the prettier the better.

Give a tight upper bound (in big-O terms)

• What is the sum of the heights?

S = < N = O(N)

2665 266526

h=3N=15

Complete binary trees

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Complete binary trees

Representing complete binary trees

• Linked structures?

• Linked structures? No!

• Instead, use arrays!

• Arrays Parent at position i in the array Children at positions 2i and 2i+1

• Arrays (1-based) Parent at position i Children at 2i and 2i+1.

1 2 3 4 5 6 7 8 9 10

• Arrays (1-based) Parent at position i Children at 2i (and 2i+1).

1 2 3 4 5 6 7 8 9 10

public class BinaryHeap { private Comparable[] heap; private int size; public BinaryHeap(int capacity) { size=0; heap = new Comparable[capacity+1]; } . . .

• Arrays Parent at position i Children at 2i and 2i+1.

• Example: find the leftmost child int left=1;for(; left<size; left*=2);return heap[left/2];

• Example: find the rightmost child int right=1;for(; right<size; right=right*2+1);return heap[(right-1)/2];

Implementing PriorityQueues with Binary Heaps

Binary heaps: the invariant

• Representation invariants1. Structure property

•Complete binary tree (i.e. the elements of the heap are stored in positions 1…size of the array)

2. Heap order property•Parent keys less than children keys

• Representation invariant1. Structure property

•Complete binary tree • Hence: efficient compact representation

2. Heap order property•Parent keys less than children keys• Hence: rapid insert, findMin, and deleteMin

• O(log(N)) for insert and deleteMin• O(1) for findMin

The heap order property

• Each parent is less than each of its children.

• Hence: Root is less than every other node. (obvious proof by induction)

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Operating with heaps

Representation invariant:

• All methods must: 1. Produce complete binary trees2. Guarantee the heap order property

• All methods may assume1. The tree is initially complete binary2. The heap order property holds

Constructor method

• All methods must: 1. Produce complete binary trees

•Trivially true

2. Guarantee the heap order property•Also trivially true

• This is the base case

findMin ()

• The codepublic boolean isEmpty() { return size == 0;

}public Comparable findMin() { if(isEmpty()) return null; return heap[1]; }

• Does not change the tree Trivially preserves the invariant

insert (Comparable x)

• Process1. Create a “hole” at the next tree cell for x.

heap[size+1]This preserves the completeness of the tree.

2. Percolate the hole up the tree until the

heap order property is satisfied.This assures the heap order property is satisfied.

insert (Comparable x)

• Process1. Create a “hole” at the next tree cell for x.

heap[size+1]This preserves the completeness of the treeassuming it was complete to begin with.

2. Percolate the hole up the tree until the

heap order property is satisfied.This assures the heap order property is satisfied assuming it held at the outset.

Percolation up

public void insert(Comparable x) throws Overflow

{ if(isFull()) throw new Overflow(); int hole = ++size; for(; hole>1 && x.compareTo(heap[hole/2])<0;

hole/=2) heap[hole] = heap[hole/2]; heap[hole] = x; }

Percolation up

• Bubble the hole up the tree until the heap order property is satisfied.

hole = 11HOP false

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14 Not really there...

Percolation up

hole = 11 hole = 5HOP false HOP false

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14 6819

Percolation up

hole = 5 hole = 2HOP false HOP true done

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14 6819

Percolation up

public void insert(Comparable x) throws Overflow

{ if(isFull()) throw new Overflow(); int hole = ++size; for(; hole>1 && x.compareTo(heap[hole/2])<0;

hole/=2) heap[hole] = heap[hole/2]; heap[hole] = x; }

Integer divisionInteger divisionInteger division

deleteMin()

/*** Remove the smallest item from the priority queue.* @return the smallest item, or null, if empty.*/public Comparable deleteMin( ){ if(isEmpty()) return null; Comparable min = heap[1]; heap[1] = heap[size--]; percolateDown(1); return min; }

Temporarily place last

element at top

Grab min element

Percolation down

• Bubble the transplanted leaf value down the tree until the heap order property is satisfied.

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Percolation down

21 6819

Percolation down

31 6819

21 6819

3 4 done

percolateDown(int hole)

private void percolateDown( int hole ){ int child = hole; Comparable tmp = heap[hole]; for( ; hole*2 <= size; hole=child ) { child = hole * 2;

if(child!=size && heap[child+1].compareTo(heap[child]) < 0) child++;

if(array[child].compareTo(tmp) < 0) heap[hole] = heap[child]; else break; } heap[hole] = tmp;}

Is there a right child?

Bubble up if smaller than tmp

Initially 1

Select smaller child

Finally, place the orphan

Start at left child

Exit loop if not bubbling

deleteMin ()

• Observe that both components of the representation invariant are preserved by deleteMin.1. Completeness

• The last cell ( heap[size] ) is vacated, providing the value to percolate down.

• This assures that the tree remains complete.

2. Heap order property

deleteMin ()

2. Heap order property

deleteMin ()

2. Heap order property• The percolation algorithm assures that the

orphaned value is relocated to a suitable position.

buildHeap()

• Start with complete (unordered) tree

• Starting from bottom, repeatedly call percolateDown()

Void buildHeap () { for (int i = size/2; i>0; i--) percolateDown(i);}

buildHeap() performance

• At each iteration, have to do work proportional to the height of the current node

• Therefore, total running time is bounded by the sum of the heights of all of the nodes

Another Digression:Invariants

Representation Invariants

• Necessary for a clear understand an algorithm or piece of code.

• The most useful kind of comment you can make in a piece of code!

(Invariants)

Plus ça change, plus c’est la même chose.

• The role of an induction hypothesis

(Invariants and induction)

• Induction hypothesis What you are allowed to assume

• At the start • About the result values of a recursive call• About the object state when a method is

called What you must deliver

• At the end • Of the result values when the recursive call

returns• About the object state when the method

returns

• At the start of a loop iteration• About the result values of a recursive call• About the object state when a method is

• At the end of the loop iteration• Of the result values when the recursive call

returns

• • About the result values of a recursive call• About the object state when a method is

• • Of the result values when the recursive call

returns

(Invariants)

Plus ça change, plus c’est la même chose

• The role of an induction hypothesis

• Invariants in programs Loop invariants Recursion invariants Representation invariants

(Representation invariant)

• What must always be true of the data structure when an operation completes.

• Theorem: Suppose each constructor assures the representation

invariant is initially correct.

Suppose each method preserves the representation invariant, assuming it is true initially.

Then the representation invariant will always be true at the completion of each method.

(Representation invariant)

(Representation invariants)

Assuming the code is not concurrent!

Heapsort

• Obviously we can use a priority queue to sort . . . just insert all the keys then do repeated deleteMins

• However we can take advantage of the fact that the heap always uses the first part of the array to do this with no extra space.

• This is called heapsort.

Heapsort

• Reverse the sense of the heap order (largest at the root)

• Start from position 0 in the array (children are 2i+1 and 2i+2)

• Call it percDown(a, i, len)

Public static void heapsort(Comparable[] a) {

for(int i = a.length/2; i>=0; i--)

percDown(a, i, a.length);

for (int j = a.length-1; j>0; j--) {

swapReferences(a, 0, j);

percDown(a, 0, j);

Heapsort invariant

• At the start of the loop over j: a[j]…a[a.length-1] are sorted a[0]…a[j-1] are a heap

Homework 3

On Blackboard.

You’re all working on it already. Right?

Priority Queues and Heaps

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