note07-1x2 - Graph Algorithms Many problems in CS can be...

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Graph Algorithms Many problems in CS can be modeled as graph problems . Algorithms for solving graph problems are fundamental to the field of algorithm design. Definition A graph G = ( V , E ) consists of a vertex set V and an edge set E . | V | = n and | E | = m . Each edge e = ( x , y ) E is an unordered pair of vertices. If ( u , v ) E , we say v is a neighbor of u . The degree deg ( u ) of a vertex u is the number of edges incident to u . c ± Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 2 / 64 Graph Algorithms Fact ± v V deg ( v ) = 2 m This is because, for each e = ( u , v ) , e is counted twice in the sum, once for deg ( v ) and once for deg ( u ) . c ± Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 3 / 64
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Directed Graphs Definition If the two end vertices of e are ordered, the edge is directed , and we write e = x y . If all edges are directed, then G is a directed graph. The in-degree deg in ( u ) of a vertex u is the number of edges that are directed into u . The out-degree deg out ( u ) of a vertex u is the number of edges that are directed from u . c ± Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 4 / 64 Directed Graphs Fact ± v V deg in ( v ) = ± v V deg out ( v ) = m This is because, for each e = ( u v ) , e is counted once ( deg in ( v ) ) in the sum of in-degrees, and once ( deg out ( u ) ) in the sum of out-degrees. c ± Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 5 / 64
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The numbers n ( = | V | ) and m ( = | E | ) are two important parameters to describe the size of a graph. It is easy to see 0 m n 2 . It is also easy to show: if G is a tree (namely undirected, connected graph with no cycles), them m = n - 1 . If m is close to n , we say G is sparse . If m is close to n 2 , we say G is dense . Because n and m are rather independent to each other, we usually use both parameters to describe the runtime of a graph algorithm. Such as O ( n + m ) or O ( n 1 / 2 m ). c ± Xin He (University at Buffalo) CSE 431/531 Algorithm Analysis and Design 6 / 64 Graph Representations We mainly use two graph representations. Adjacency Matrix Representation We use a 2D array A [ 1 .. n , 1 .. n ] to represent G = ( V , E ) : A [ i , j ] = ± 1 if ( v i , v j ) E 0 if ( v i , v j ) ±∈ E Sometimes, there are other information associated with the edges. For example, each edge e = ( v i , v j ) may have a weight w ( e ) = w ( v i , v j ) (for example, MST). In this case, we set A [ i , j ] = w ( v i , v j ) . For undirected graph, A is always symmetric . The Adjacency Matrix Representation for directed graph is similar. A [ i , j ] = 1 (or w ( v i , v j ) if G has edge weights) iff v i v j E . For directed graphs,
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note07-1x2 - Graph Algorithms Many problems in CS can be...

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