Graphs. Data Structures 1 Graphs

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1 Graphs Graph Applications Definitions Graph Representations(adjacency matrix/list) Graph ADT Graph Traversals: BFS, DFS Shortest Path Algorithms Minimum Spanning Tree Algorithms Data Structures Graphs

2 Graph Applications Communication Networks (Internet) Social networks. Geographic Maps (locations + distances) Transportation Networks (flow capacity) Scientific, familial, business taxonomies Data Structures 2 Graphs

3 Definitions and Terminology Graph: G = (V, E), set of vertices V, and a set of edges E connecting pairs of vertices. Sparse, Dense graphs. Complete Graph: Includes all possible edges. Digraph: Directed Graph, where edges are directed. Undirected Graph: Edges are undirected. Data Structures 3 Graphs

4 Definitions (Contd) Adjacency, Neighbors: vertices sharing an edge Incidence: Edge (u, v) is incident on verices u and v. Weighted Edges: Edges have weight. Path: Sequence of vertices v, v 2,.., v n connected by edges, of path length n. Simple Path: All vertices are distinct. Cycle: Path of length 3 or more connecting some vertex to itself. Acyclic: Graph with no cycles. Data Structures Graphs

5 Definitions (Contd) DAG: Directed Acyclic Graph. Simple Cycle: Path is simple. Subgraph: Select a subset V s of G s vertices and some edges of G. Connected: Graph that has at least one path from any vertex to any other. Connected Components: Maximally connected subgraphs of an undirected graph. Data Structures 5 Graphs

6 Graph: Connected Components An example graph with 3 connected components. Data Structures 6 Graphs

7 Graph Representations Two commonly used representations: Adjacency Matrix, Adjacency List Adjacency Matrix An V V 2D array (matrix), the dimensions representing a vertex v i v 0, v n Element (i, j) is a bit, which is, if an edge exists between v i and v j, else it is 0. Requires Θ(n 2 ) space. Data Structures 7 Graphs

8 Adjacency Matrix Representation Data Structures 8 Graphs

9 Graph Representation:Adjacency List An array of size V, each of which is a linked list. Position i of the array, stores all edges from vertex v i ; each node contains the terminating vertex, v j. Requires Θ( V + E ) storage. Data Structures 9 Graphs

10 Adjacency List Representation Data Structures 0 Graphs

11 Graph Implementation: ADT public interface Graph { / / Graph class ADT / I n i t i a l i z e the n The number of v e r t i c e s / public void I n i t ( i n t n ) ; The number of v e r t i c e s / public i n t n ( ) ; The c u r r e n t number of edges / public i n t e ( ) ; v s f i r s t neighbor / public i n t f i r s t ( i n t v ) ; v s next neighbor / public i n t next ( i n t v, i n t w ) ; / Set the weight f o r an i, j The v e r t i c e wght Edge weight / public void setedge ( i n t i, i n t j, i n t wght ) ; Data Structures Graphs

12 Graph Implementation: ADT / Delete an i, j The v e r t i c e s / public void deledge ( i n t i, i n t j ) ; / Determine i f an edge i s i n the i, j The v e r t i c e t r u e i f edge i, j has non zero weight / public boolean isedge ( i n t i, i n t j ) ; / Return an edge s i, j The v e r t i c e The weight of edge i, j, or zero / public i n t weight ( i n t i, i n t j ) ; / Set the mark value f o r a v The v a l The value to set / public void setmark ( i n t v, i n t v a l ) ; / Get the mark value f o r a v The The value of the mark / public i n t getmark ( i n t v ) ; Data Structures 2 Graphs

13 Graph Implementation Vertices are indices, edges are vertex pairs. Navigation: first(), next(). Edge functions: isedge, setedge(), deledge(). Weight(), Mark() functions Data Structures 3 Graphs

14 Graph Implementation: Adjacency Matrix first(int v), next(int v, int w): Must search a matrix row for the appropriate edge. setedge(int v, int w, Weight w), weight) : Set/Get the weight for a particular edge in the matrix. IsEdge(int v, int w): Tests if this edge exists. deledge(int v, int w): Deletes edge, if it exists. getmark(int v), setmark(int v, int val) : a mark array - useful in traversal algorithms Data Structures Graphs

15 Graph Implementation: Adjacency Matrix class Graphm implements Graph { / / Graph : Adjacency m a t r i x private i n t [ ] [ ] matrix ; / / The edge matrix private i n t numedge ; / / Number of edges public i n t [ ] Mark ; / / The mark array public Graphm ( ) { public Graphm ( i n t n ) { / / Constructor I n i t ( n ) ; public void I n i t ( i n t n ) { Mark = new i n t [ n ] ; m a t r i x = new i n t [ n ] [ n ] ; numedge = 0; public i n t n ( ) { return Mark. length ; / / # of v e r t i c e s public i n t e ( ) { return numedge ; / / # of edges public i n t f i r s t ( i n t v ) { / / Get v s f i r s t neighbor for ( i n t i =0; i <Mark. length ; i ++) i f ( m a trix [ v ] [ i ]!= 0) return i ; return Mark. length ; / / No edge f o r t h i s vertex Data Structures 5 Graphs

16 Graph Implementation: Adjacency Matrix public i n t next ( i n t v, i n t w) { / / Get v s next edge for ( i n t i =w+; i <Mark. length ; i ++) i f ( m a trix [ v ] [ i ]!= 0) return i ; return Mark. length ; / / No next edge ; / / Set edge weight public void setedge ( i n t i, i n t j, i n t wt ) { a s s e r t wt!=0 : Cannot set weight to 0 ; i f ( m a t r ix [ i ] [ j ] == 0) numedge++; m a t r i x [ i ] [ j ] = wt ; public void deledge ( i n t i, i n t j ) { / / Delete edge ( i, j ) i f ( m a t r ix [ i ] [ j ]!= 0) numedge ; m a t r i x [ i ] [ j ] = 0; public boolean isedge ( i n t i, i n t j ) / / I s ( i, j ) an edge? { return matrix [ i ] [ j ]!= 0; public i n t weight ( i n t i, i n t j ) { / / Return edge weight return matrix [ i ] [ j ] ; / / Get and set marks public void setmark ( i n t v, i n t v a l ) { Mark [ v ] = v a l ; public i n t getmark ( i n t v ) { return Mark [ v ] ; Data Structures 6 Graphs

17 Graph Implementation: Adjacency List Use an extension of the List class; allocate the required number of lists as an array of lists. first(int v), next(int v, int w): first or next element corresponding to index v ; setedge(int i, int j, int w) : Set the weight for an existing edge, or insert a new edge. IsEdge(int v, int w): Tests if this edge exists. deledge(int v, int w): Deletes edge, if it exists. setmark(int v, int val) : a Mark array is marked - useful in traversal algorithms int getmark(int v): returns mark value of v. Data Structures 7 Graphs

18 Graph Implementation: Adjacency List class Graphl implements Graph { private GraphList [ ] vertex ; / / The vertex l i s t private i n t numedge ; / / Number of edges public i n t [ ] Mark ; / / The mark array public Graphl ( ) { public Graphl ( i n t n ) / / Constructor { I n i t ( n ) ; public void I n i t ( i n t n ) { Mark = new i n t [ n ] ; v e r t e x = new GraphList [ n ] ; for ( i n t i =0; i <n ; i ++) v e r t e x [ i ] = new GraphList ( ) ; numedge = 0; public i n t n ( ) { return Mark. length ; / / # of v e r t i c e s public i n t e ( ) { return numedge ; / / # of edges Data Structures 8 Graphs

19 Graph Implementation: Adjacency List public i n t f i r s t ( i n t v ) { / / Get v s f i r s t neighbor i f ( v e r t ex [ v ]. length ( ) == 0) return Mark. length ; / / No neighbor v e r t e x [ v ]. movetostart ( ) ; Edge i t = v ertex [ v ]. getvalue ( ) ; return i t. vertex ( ) ; public i n t next ( i n t v, i n t w) { / / Get next neighbor Edge i t = null ; i f ( isedge ( v, w) ) { v e r t e x [ v ]. next ( ) ; i t = vertex [ v ]. getvalue ( ) ; i f ( i t!= null ) return i t. vertex ( ) ; else return Mark. length ; Data Structures 9 Graphs

20 Graph Implementation: Adjacency List / / Store edge weight public void setedge ( i n t i, i n t j, i n t weight ) { a s s e r t weight!= 0 : May not set weight to 0 ; Edge curredge = new Edge ( j, weight ) ; i f ( isedge ( i, j ) ) { / / Edge already e x i s t s i n graph v e r t e x [ i ]. remove ( ) ; v e r t e x [ i ]. i n s e r t ( curredge ) ; else { / / Keep neighbors sorted by vertex index numedge++; for ( vertex [ i ]. movetostart ( ) ; vertex [ i ]. currpos ( ) < vertex [ i ]. length ( ) ; vertex [ i ]. next ( ) ) i f ( vertex [ i ]. getvalue ( ). vertex ( ) > j ) break ; v e r t e x [ i ]. i n s e r t ( curredge ) ; public void deledge ( i n t i, i n t j ) / / Delete edge { i f ( isedge ( i, j ) ) { vertex [ i ]. remove ( ) ; numedge ; Data Structures 20 Graphs

21 Graph Implementation: Adjacency List public boolean isedge ( i n t v, i n t w) { / / I s ( i, j ) an edge? Edge i t = v ertex [ v ]. getvalue ( ) ; i f ( ( i t!= null ) && ( i t. vertex ( ) == w) ) return true ; for ( v e r tex [ v ]. movetostart ( ) ; v e rtex [ v ]. currpos ( ) < vertex [ v ]. length ( ) ; v e rtex [ v ]. next ( ) ) / / Check whole l i s t i f ( v ertex [ v ]. getvalue ( ). vertex ( ) == w) return true ; return false ; public i n t weight ( i n t i, i n t j ) { / / Return weight of edge i f ( isedge ( i, j ) ) return vertex [ i ]. getvalue ( ). weight ( ) ; return 0; / / Set and get marks public void setmark ( i n t v, i n t v a l ) { Mark [ v ] = v a l ; public i n t getmark ( i n t v ) { return Mark [ v ] ; Data Structures 2 Graphs

22 Graph Traversals Goal is to visit the nodes of a graph. Need a start vertex. Problems: graph may be disconnected, and contain cycles. Use a mark bit to identify visited nodes. void graphtraverse ( Graph G) { for ( v =0; v<g. n ( ) ; v ++) G. setmark ( v, UNVISITED ) ; / / I n i t i a l i z e mark b i t s for ( v =0; v<g. n ( ) ; v ++) i f (G. getmark ( v ) == UNVISITED ) dotraverse (G, v ) ; Data Structures 22 Graphs

23 Depth First Search (DFS) When a node v is visited, recursively visit all of its neighbors. Effect is to follow one branch all the way through and then back up and follow another branch, etc. A A D D C C B E B E F F Data Structures 23 Graphs

24 DFS Implementation s t a t i c void DFS( Graph G, i n t v ) { / / Depth f i r s t search P r e V i s i t (G, v ) ; / / Take a p p r o p r i a t e a c t i o n G. setmark ( v, VISITED ) ; for ( i n t w = G. f i r s t ( v ) ; w < G. n ( ) ; w = G. next ( v, w) ) i f (G. getmark (w) == UNVISITED ) DFS(G, w ) ; P o s t V i s i t (G, v ) ; / / Take a p p r o p r i a t e a c t i o n Data Structures 2 Graphs

25 Breadth First Search(BFS) Examine all vertices connected to a vertex v, before visiting vertices deeper (further away). Equivalent to level traversal of a tree. A A D D C C B E B E F F Data Structures 25 Graphs

26 BFS Implementation / / Breadth f i r s t ( queue based ) search s t a t i c void BFS( Graph G, i n t s t a r t ) { Queue<Integer> Q = new AQueue<Integer >(G. n ( ) ) ; Q. enqueue ( s t a r t ) ; G. setmark ( s t a r t, VISITED ) ; while (Q. l e n g t h ( ) > 0) { / / Process each v e r t e x on Q i n t v = Q. dequeue ( ) ; P r e V i s i t (G, v ) ; / / Take a p p r o p r i a t e a c t i o n for ( i n t w = G. f i r s t ( v ) ; w < G. n ( ) ; w = G. next ( v, w) ) i f (G. getmark (w) == UNVISITED ) / / Put neighbors on { G. setmark (w, VISITED ) ; Q. enqueue (w ) ; P o s t V i s i t (G, v ) ; / / Take a p p r o p r i a t e a c t i o n Data Structures 26 Graphs

27 Data Structures 27 Graphs

Data Structure Lecture#23: Graphs (Chapter 11) U Kang Seoul National University

Data Structure Lecture#23: Graphs (Chapter 11) U Kang Seoul National University U Kang 1 In This Lecture Basic terms and definitions of graphs How to represent graphs Graph traversal methods U Kang 2 Graphs