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Chapter 16. Graphs. Outline. Graph Categories Digraph Connectedness of Digraph Adjacency Matrix, Set vertexInfo Object Breadth-First Search Algorithm Depth first Search Algorithm. Strong Components Graph G and Its Transpose G T Shortest-Path Dijkstra Minimum-Path Algorithm

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Chapter 16 l.jpg

Chapter 16

Graphs


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Outline

  • Graph Categories

  • Digraph

  • Connectedness of Digraph

  • Adjacency Matrix, Set

  • vertexInfo Object

  • Breadth-First Search Algorithm

  • Depth first Search Algorithm

  • Strong Components

  • Graph G and Its Transpose GT

  • Shortest-Path

  • Dijkstra Minimum-Path Algorithm

  • Minimum Spanning Tree

  • Prim’s algorithm

  • Kruskal’s algorithm


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Graph Terminology

  • A graph G=(V, E) consists of a set V of vertices and a set E of edges that connect pair of vertices.

    • V={v1, v2, …, vn}

    • E={e1, e2, …, em}

    • An edge e E is a pair (vi, vj)

  • A subgraph Gs=(Vs, Es) is a subset of the vertices and edges, where Vs V, Es E

  • Two vertices, vi, vj are adjacent if and only if there is an edge e=(vi, vj) E

  • A path in a graph is a sequence of vertices v0, v1, v2, …, vk such that (vi, vI+1) E

    • Simple Path: each vertex occur only once

    • Cycle: there is a vertex appearing more than once on a path


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GraphTerminology

  • A graph is connected if each pair of vertices have a path between them

  • A complete graph is a connected graph in which each pair of vertices are linked by an edge


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Directed Graphs

  • Graph with ordered edges are called directedgraphs or digraphs, otherwise it is undirected.

  • The number of edges that emanate from a vertex v is called the out-degree of v

  • The number of edges that terminate on vertex is called the in-degree of v


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Directed Acyclic Graph (DAG)

  • A directed graph that has no cycle is called a directed acyclic graph (DAG)

    • Directed path

    • Directed cycle: a directed path of length 2 or more that connects a vertex to itself

  • A weighted digraph is a directed graph that associates values with the edges.

    • A weight edge e=(vi, vj, wt)


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Connectedness of Digraph

  • Strongly connected if for each pair of vertices vi and vj, there is a path P(vi, vj)

  • Weakly connected if for each pair of vertices vi and vj, there is either a path P(vi, vj) or a path P(vj,vi).


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The Graph Class

  • Access the properties of a graph

  • Add or delete vertices and edges

  • Update the weight of an edge

  • Identify the list of adjacent vertices


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Representation of Graphs

  • Adjacency matrix

  • Adjacent set


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Adjacency Matrix

  • An n by n matrix, called an adjacency matrix, identifies the edges. An entry in row i and column j corresponds to the edge e = (vi, vj). Its value is the weight of the edge, or -1 if the edge does not exist.


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Adjacency Set

  • For each vertex, an element in the adjacent set is a pair consisting of the adjacent vertex and the weight of the edge.


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vertexInfo Object

  • A vertexInfo object consists of seven data members. The first two members, called vtxMapLoc and edges, identify the vertex in the map and its adjacency set.


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Vertex Map and Vector vInfo

  • To store the vertices in a graph, we provide a map<T,int> container, called vtxMap, where a vertex name is the key of type T. The int field of a map object is an index into a vector of vertexInfo objects, called vInfo. The size of the vector is initially the number of vertices in the graph, and there is a 1-1 correspondence between an entry in the map and a vertexInfo entry in the vector



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Graph Traversal Algorithms

  • Breadth-First Visit(Search): visits vertices in the order of their path length from a starting vertex. (generalizes the level-order scan in a binary tree)

  • Depth-First Visit(Search): traverses vertices of a graph by making a series of recursive function calls that follow paths through the graph. (emulate the postorder scan in a binary tree)

BFS: A,B,C,G,D,E,F

DFS (reverse order of processing): A,C,B,D,F,G,E





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Implementation of BFS algorithm

  • uses a queue to store the vertices awaiting a visit temporally.

  • At each iterative step, the algorithm deletes a vertex from the queue, mark it as visited, and then inserts it into visitSet, the set of visited vertices.

  • The step concludes by placing all unvisited neighbors of the vertex in the queue.

  • Each vertex is associate a color from WHITE, GRAY, BLACK.

    • Unvisited: (Initially) WHITE

    • In the process of being searched: (enter into queue) GRAY

    • Visited: (remove from queue) BLACK

  • Time complexity: O(|V|)+O(|E|)


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Depth-First Search Algorithm

Discovery order: A, B, D, E, F, G, C

Finishing order: E, G, F, D, B, C, A


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F

G

E

Depth-First Search… (Cont.)

A

B

C

D

Discovery order & finishing order?


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Implementing the DFS

  • dfsVisit()

    • Includes four arguments: a graph, a WHITE starting vertex, the list of visited vertices in reverse order of finishing times, and Boolean variable for checking cycle

    • Search only vertices that are reachable from the starting vertex

  • dfs()

    • Takes two arguments: a graph and a list

    • Repeatedly call dfsVisit()

  • Time complexity: O(|V|)+O(|E|)

dfsList: descending/reverse order of their visit (finish-time) order


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Graph Traversal Applications

  • Acyclic graphs

  • Topological sort

  • Strongly connected component


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Acyclic graphs

  • A graph is acyclic if it contains no cycles

  • Function acyclic() determine if the graph is acyclic

    • Back edge (v,w): current vertex v and a neighbor vertex w with color gray


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Topological sort

  • Topological order: if P(v,w) is a path from v to w, then v must occur before w in the list.

  • If graph is acyclic, defList produce a topological sort of the vertices

  • Implementation, performance, & applications


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A

C

B

E

D

F

G

Strong Components

  • A strongly connected component of a graph G is a maximal set of vertices SC in G that are mutually accessible.

Components

A, B, C

D, F, G

E


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Graph G and Its Transpose GT

  • The transpose has the same set of vertices V as graph G but a new edge set ET consisting of the edges of G but with the opposite direction.


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Finding Strong Components

  • Algorithm

    • Step 1. Execute dfs() for graph

    • Step 2. Generate the transpose graph, GT

    • Step 3. Execute a series of dfsVisit( ) calls for vertices using the order of the elements in dfsList obtained in step 1 as the starting vertices.


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Finding Strong Components

  • Example, Verification, Implementation, performance


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Graph-Minimization Algorithms

  • Shortest-Path Algorithm

  • Dijkstra’s Minimum-Path Algorithm

  • Minimum-Spanning Tree Algorithms


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Shortest-Path algorithm

  • Find a shortest path (minimum number of edges) from a given vertex vs to every other vertex in a directed graph

  • The shortest-path algorithm includes a queue that indirectly stores the vertices, using the corresponding vInfo index.

  • Each iterative step removes a vertex from the queue and searches its adjacency set to locate all of the unvisited neighbors and add them to the queue.


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Shortest-Path Example

  • Example: Find the shortest path from F to C.


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Shortest-Path algorithm

  • Implementation

  • Performance

    • Similar to BFS search, O(|V|+|E|)


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Dijkstra Minimum-Path Algorithm

  • Find the minimum weight (minimum sum of edge weight) from a given vertex to every other vertex in a weighted directed graph

  • Use a minimum priority queue to store the vertices, using minInfo class, which contains a vInfo index and the pathWeight value

  • At each iterative step, the priority queque identifies a new vertex whose pathWeight value is the minimum path weight from the starting vertex


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minInfo(

A

,0)

priority queue

Dijkstra Minimum-Path Algorithm From A to D Example


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Dijkstra Minimum-Path Algorithm

  • The Dijkastra algorithm is a greedy algorithm

    • at each step, make the best choice available.

    • Usually, greedy algorithms produce locally optimum results, but not globally optimal

    • Dijkstra algorithm produces a globally optimum result

  • Implementation

  • Running-Time Analysis

    • O(|V|+|E|log|E|)


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Minimum Spanning Tree Algorithms

  • Minimum Spanning Tree (For Undirected Graph)

    • Tree: a connected graph with no cycles.

    • Spanning Tree: a tree which contains all vertices in G.

      • Note: Connected graph with n vertices and exactly n – 1 edges is Spanning Tree.

    • Minimum Spanning Tree: a spanning Tree with minimum total weight



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Prim’s Algorithm

  • Basic idea: Start from vertex 1 and let T  Ø (T will contain all edges in the S.T.); the next edge to be included in T is the minimum cost edge(u, v), s.t. u is in the tree and v is not.



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Implementation of Prim’s algorithm:

  • Using a priority queque of miniInfo objects to store information

    • Color: White not in tree, Black in tree

    • Data Value: the weight of the minimum edge that would connect the vertex to an existing vertex in tree

    • parent

  • Each iterative steps similar to Dijkstra algorithm


Prim s minimum spanning tree algorithm example l.jpg

A

12

2

B

C

5

8

7

D

(a)

Prim’s Minimum Spanning-Tree Algorithm example:


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Prim’s Minimum Spanning Tree Algorithm

  • Implementation

  • Running-Time Analysis

    • O(|V|+|E|log|E|)


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10

1

2

30

50

25

45

4

3

40

35

20

5

6

55

15

  • Kruskal’s Algorithm

  • Basic idea: Don’t care if T is a tree or not in the intermediate stage, as long as the including of a new edge will not create a cycle, we include the minimum cost edge

Example:

Step 1: Sort all of edges

(1,2) 10 √

(3,6) 15 √

(4,6) 20 √

(2,6) 25 √

(1,4) 30 × reject create cycle

(3,5) 35 √


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2

3

4

1

6

4

3

1

2

6

4

6

2

3

1

5

1

2

1

3

6

2

Step 2: T


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  • How to check: adding an edge will create a cycle or not?

  • If Maintain a set for each group

  • (initially each node represents a set)

  • Ex: set1set2set3

  •  new edge

  • from different groups  no cycle created

  • Data structure to store sets so that:

    • The group number can be easily found, and

    • Two sets can be easily merged

1

2

3

6

4

5

2

6


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Kruskal’s algorithm

While (T contains fewer than n-1 edges) and (E   ) do

Begin

Choose an edge (v,w) from E of lowest cost;

Delete (v,w) from E;

If (v,w) does not create a cycle in T

then add (v,w) to T

else discard (v,w);

End;

Time complexity: O(|V|+|E|log|E|)


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Summary Slide 1

§- Undirected and Directed Graph (digraph)

- Both types of graphs can be either weighted or nonweighted.

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Summary Slide 2

§- Breadth-First, bfs()

- locates all vertices reachable from a starting vertex

- can be used to find the minimum distance from a starting vertex to an ending vertex in a graph.

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Summary Slide 3

§- Depth-First search, dfs()

- produces a list of all graph vertices in the reverse order of their finishing times.

- supported by a recursive depth-first visit function, dfsVisit()

- an algorithm can check to see whether a graph is acyclic (has no cycles) and can perform a topological sort of a directed acyclic graph (DAG)

- forms the basis for an efficient algorithm that finds the strong components of a graph

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Summary Slide 4

§-Dijkstra's algorithm

- if weights, uses a priority queue to determine a path from a starting to an ending vertex, of minimum weight

- This idea can be extended to Prim's algorithm, which computes the minimum spanning tree in an undirected, connected graph.

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Summary Slide 5

§-Minimum Spanning Tree algorithm

- Prim’s algorithm

- Kruskal’s Algorithm

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