chapter 9 priority queues, heaps, and graphs. 2 goals describe a priority queue at the logical level...
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Chapter 9
Priority Queues, Heaps, and Graphs
2
Goals
• Describe a priority queue at the logical level and implement a priority queue as a list
• Describe the shape and order property of a heap and implement a heap in a nonlinked tree representation in an array
• Implement a priority queue as a heap
• Compare the implementations of a priority queue using a heap, a linked list, and a binary search tree
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Goals
• Define the following terms related to graphs:
Directed graph Complete graph
Undirected graphWeighted graph
Vertex Adjacency matrix
Edge Adjacency list
Path• Implement a graph using an adjacency
matrix to represent the edges
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Goals
• Explain the difference between a depth-first and a breadth-first search and implement these searching strategies using stacks and queues for auxiliary storage
• Implement a shortest-path operation, using a priority queue to access the edge with the minimum weight
• Describe a set at the logical level and implement a set both explicitly and implicitly
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Priority Queue
Priority QueueAn ADT in which only the item with the highest priority can be accessed
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Priority Queue
Items on a priority queue are made up of
<data, priority> pairs
Can you think of how a priority queue might be implemented as
An unsorted list?
An array-based sorted list?
A linked sorted list?
A binary search tree?
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Heaps
HeapA complete binary tree, each of whose elements contains a value that is greater than or equal to the value of each of its children
Remember?Complete tree
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Heaps
Heap with letters 'A' .. 'J'
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Heaps
Another heap with letters 'A' .. 'J'
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Heaps
C
A T
treePtr
50
20
18
30
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Are these heaps?
treePtr
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Heaps
70
60
40 30
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8 10
treePtr
Is this a heap?
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Heaps
70
60
40 30
12
8
treePtr
Can we use this fact to implement an efficient PQ?
Whereis the
largestelement?
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Heaps
We have immediate access to highest priority item BUT if we remove it, the structure isn't a heap
Can we "fix" the problem efficiently?
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Heaps
Shape propertyis restored. Canorder property be restored?
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Heaps
How?
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Heaps
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Heaps
template<class ItemType>
struct HeapType
{
void reheapDown(int root, int bottom);
…
ItemType* elements; // Dynamic array
int numElements;
}
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Heaps
70
0
60
1
40
3
30
4
12
2
8
5
root
Numbernodes
leftto
rightby level
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Heaps
70
0
60
1
40
3
30
4
12
2
8
5
tree[ 0 ]
[ 1 ]
[ 2 ]
[ 3 ]
[ 4 ]
[ 5 ]
[ 6 ]
70
60
12
40
30
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elements
Use number as index
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Heaps
ReapDown(heap, root, bottom)
if heap.elements[root] is not a leaf
Set maxChild to index of child with larger value
if heap.elements[root] < heap.elements[maxChild]
Swap(heap.elements[root], heap.elements[maxChild])
ReheapDown(heap, maxChild, bottom)
What variables do we need?
Where are a node's children?
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Heaps
// IMPLEMENTATION OF RECURSIVE HEAP MEMBER FUNCTIONSvoid HeapType<ItemType>::ReheapDown(int root, int bottom)// Pre: ItemType overloads the relational operators// root is the index of the node that may violate the // heap order property// Post: Heap order property is restored between root and // bottom
{ int maxChild ; int rightChild ; int leftChild ;
leftChild = root * 2 + 1 ; rightChild = root * 2 + 2 ;
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Heaps if (leftChild <= bottom) // ReheapDown continued { if (leftChild == bottom) maxChild = leftChld; else { if (elements[leftChild] <= elements[rightChild]) maxChild = rightChild; else maxChild = leftChild; } if (elements[root] < elements[maxChild]) { Swap(elements[root], elements[maxChild]); ReheapDown(maxChild, bottom); } }}
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Heaps
What other operations might our HeapType need?
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Heaps
Add K: Where must the new node be put?
Now what?
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Heaps
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void HeapType::ReheapUp(int root, int bottom)// Pre: ItemType overloads the relational operators// bottom is the index of the node that may violate// the heap order property. The order property is // satisfied from root to next-to-last node.// Post:Heap order property is restored between root and// bottom{ int parent; if (bottom > root) { parent = ( bottom - 1 ) / 2; if (elements[parent] < elements[bottom]) { Swap(elements[parent], elements[bottom]); ReheapUp(root, parent); } }}
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Heaps
How can heaps be used to implement Priority Queues?
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Priority Queuesclass FullPQ(){};class EmptyPQ(){};template<class ItemType>class PQType{public: PQType(int); ~PQType(); void MakeEmpty(); bool IsEmpty() const; bool IsFull() const; void Enqueue(ItemType newItem); void Dequeue(ItemType& item);private: int length; HeapType<ItemType> items; int maxItems;};
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Priority Queuestemplate<class ItemType>PQType<ItemType>::PQType(int max){ maxItems = max; items.elements = new ItemType[max]; length = 0;}template<class ItemType>void PQType<ItemType>::MakeEmpty(){ length = 0;}template<class ItemType>PQType<ItemType>::~PQType(){ delete [] items.elements;}
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Priority Queues
DequeueSet item to root element from queueMove last leaf element into root positionDecrement lengthitems.ReheapDown(0, length-1)
EnqueueIncrement lengthPut newItem in next available positionitems.ReheapUp(0, length-1)
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Priority Queues
template<class ItemType>void PQType<ItemType>::Dequeue(ItemType& item){ if (length == 0) throw EmptyPQ(); else { item = items.elements[0]; items.elements[0] = items.elements[length-1]; length--; items.ReheapDown(0, length-1); }}
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Priority Queues
template<class ItemType>void PQType<ItemType>::Enqueue(ItemType newItem){ if (length == maxItems) throw FullPQ(); else { length++; items.elements[length-1] = newItem; items.ReheapUp(0, length-1); }}
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Comparison of Priority Queue Implementations
Enqueue Dequeue
Heap O(log2N) O(log2N)
Linked List O(N) O(N)
Binary Search Tree
Balanced O(log2N) O(log2N)
Skewed O(N) O(N)
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Graphs
GraphA data structure that consists of a set of models and a set of edges that relate the nodes to each otherVertexA node in a graphEdge (arc) A pair of vertices representing a connection between two nodes in a graphUndirected graphA graph in which the edges have no directionDirected graph (digraph) A graph in which each edge is directed from one vertex to another (or the same) vertex
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Graphs
Formally a graph G is defined as follows
G = (V,E)where
V(G) is a finite, nonempty set of vertices
E(G) is a set of edges (written as pairs of vertices)
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Graphs
Vertex
Edge or arc
UndirectedGraphshave noarrows on theedges
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Graphs
Directed edge
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Graphs
Whatother
structureis this
?
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Graphs
Adjacent verticesTwo vertices in a graph that are connected by an edgePathA sequence of vertices that connects two nodes in a graphComplete graphA graph in which every vertex is directly connected to every other vertexWeighted graphA graph in which each edge carries a value
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Graphs
How many edges ina directed graph with
N vertices?
How many edges in anundirected graph with
N vertices?
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Graphs
Weight
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Graph Algorithms
Depth-first search algorithmVisit all the nodes in a branch to its deepest point before moving up Breadth-first search algorithmVisit all the nodes on one level before going to the next level Single-source shortest-path algorithmAn algorithm that displays the shortest path from a designated starting node to every other node in the graph
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Graphs
Operations: The usual suspects
IsFull
IsEmpty
Initialize
Insert requires two operations
Add vertex
Add Edge
Others? Let's see.
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Graphs
DepthFirstSearchSet found to falsestack.Push(startVertex)do
stack.Pop(vertex)if vertex = endVertex
Write final vertexSet found to true
elseWrite this vertexPush all adjacent vertices onto stack
while !stack.IsEmpty() AND !foundif (!found)
Write "Path does not exist"
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Graphs
Try depth first search from Austin
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Graphs
Do you see a problem? Do you have a solution?
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Graphs
We need to "mark" a vertex as visited
ClearMarks
MarkVertex(VertexType, vertex)
Boolean IsMarked(VertexType, vertex)
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GraphsBreadthFirstSearchSet found to falsequeue.Enqueue(startVertex)do
queue.Dequeue(vertex)if vertex = endVertex
Write final vertexSet found to true
elseif (vertex is not marked)
Mark vertexWrite vertexGet a queue of outgoing vertices from vertexwhile (vertexQ is not empty)
vertexQ.Dequeue(Item)if (item is not marked)
queue.Enqueue(item)while queue IsEmpty() AND !foundif (!found)
Write "Path does not exist"
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Graphs
Try depth breadth search from Austin
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Graphs
DFS uses a stack; BFS uses a queue
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Graphs
BFS result
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Graphs
What other operation does the Graph ADT need?
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Array-Based Implementation
Adjacency MatrixFor a graph with N nodes, an N by N table that shows the existence (and weights) of all edges in the graphMapping ArrayAn array that maps vertex names into array indexesMarkingAssociate a Boolean variable with each vertex: true if visited, false if not yet visited
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Adjacency Matrix for Flight Connections
Where could marking variable be kept?
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Linked Implementation
Adjacency List
A linked list that identifies all the vertices to which a particular vertex is connected; each vertex has its own adjacency list
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Adjacency List Representation of Graphs
Wherecould
amarkingvariable
go?
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Sets
Set
An unordered collection of distinct values (items or components), chosen from the possible values of a single data type, called the component or base type
Base type
The type of the items in the set
Cardinality
The number of items in a setSubsetA set contained within another set
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SetsUniversal setA set that contains all the values of the base typeEmpty setA set with no membersSet unionA set made up of all the items in either setsSet intersectionA set made up of all the items in both setsSet differenceA set made up of all the items in the first set that are not in the second set
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Sets
GivenSetA = {X, Y, Z, A, B}SetB = {A, C, W, Z}
What is SetA union SetB?What is SetA intersection SetB?What is SetA - SetB (difference)?Give a subset of SetAGive a proper subset of SetB
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Sets
Beware – At the Logical Level sets can not contain
duplicates – Inserting a duplicate item into a set does
not change the set– Sets are not ordered
This does not mean that we cannot implement a set using an ordered structure with duplicates
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Sets
Explicit implementation
Each item in the base type has a representation in each instance of a set; the representation is either true (item is in the set) or false (item is not in the set)
Bit Vector
An array of Boolean flags, with each item in the base type mapped to a specific flag (index) Algorithms use Boolean operations
To what is space for each set instance proportional?
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Sets
If sets A and B are represented as bit vectors, what Boolean operation would implement set union?
Set intersection?
Set difference?
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Sets
Implicit implementation (List)The items in an instance of a set are on a listthat represents the set; those items that are not on the list are not in the set Algorithms use ADT List operations
To what is space for each set instance proportional?
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Sets
If sets A and B are represented as lists, what list operations would implement set union?
Set intersection?
Set difference?
Which list implementation would be more efficient?