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Algorithms Lecture 7: Graph Algorithms

The document provides a comprehensive analysis and design of algorithms related to graph theory, focusing on concepts such as directed and undirected graphs, acyclic and cyclic graphs, and techniques for graph traversal including breadth-first and depth-first search methods. It explains the significance of vertices and edges, the operations of searching and traversing nodes, and includes examples illustrating the application of these concepts. Additionally, it addresses cycle detection in directed graphs using depth-first traversal.

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Analysis and Design of Algorithms Graph Algorithms
Analysis and Design of Algorithms Graph Directed vs Undirected Graph Acyclic vs Cyclic Graph Backedge Search vs Traversal Breadth First Traversal Depth First Traversal Detect Cycle in a Directed Graph
Analysis and Design of Algorithms  Graph data structure consists of a finite set of vertices or nodes. The interconnected objects are represented by points termed as vertices, and the links that connect the vertices are called edges. A B C D E F Node or Vertices Edge
Analysis and Design of Algorithms  Vertex: Each node of the graph is represented as a vertex.  V={A,B,C,D,E,F}  Edge: Edge represents a path between two vertices or a line between two vertices.  E={AB,AC,BD,BE,CE,DE,DF,EF} A B C D E F Node or Vertices Edge
Analysis and Design of Algorithms A B C A B C Directed Graph Undirected Graph
Analysis and Design of Algorithms A B C Acyclic Graph Cyclic Graph A B C
Analysis and Design of Algorithms Backedge is an edge that is from node to itself (selfloop), or one to its ancestor. A B A
Analysis and Design of Algorithms Search: Look for a given node Traversal: Always visit all nodes
Analysis and Design of Algorithms Breadth First Traversal
Analysis and Design of Algorithms Breadth First Traversal (BFT) algorithm traverses all nodes on graph in a breadthward motion and uses a queue to remember to get the next vertex.
Analysis and Design of Algorithms Layers A B C D E F Layer1 Layer2 Layer3 Layer4
Analysis and Design of Algorithms  Visited=  Queue=  Print: A B C D E F 0 0 0 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue=  Print: A B C D E F 1 0 0 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= A  Print: A B C D E F 1 0 0 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue=  Print: A A B C D E F 1 0 0 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue=  Print: A A B C D E F 1 0 0 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue=  Print: A A B C D E F 1 0 0 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue=  Print: A A B C D E F 1 1 1 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= B C  Print: A A B C D E F 1 1 1 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= C  Print: A B A B C D E F 1 1 1 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= C  Print: A B A B C D E F 1 1 1 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= C  Print: A B A B C D E F 1 1 1 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= C  Print: A B A B C D E F 1 1 1 1 1 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= C D E  Print: A B A B C D E F 1 1 1 1 1 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= D E  Print: A B C A B C D E F 1 1 1 1 1 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= D E  Print: A B C A B C D E F 1 1 1 1 1 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= E  Print: A B C A B C D E F 1 1 1 1 1 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= E  Print: A B C D A B C D E F 1 1 1 1 1 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= E  Print: A B C D A B C D E F 1 1 1 1 1 0 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= E  Print: A B C D A B C D E F 1 1 1 1 1 1 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= E F  Print: A B C D A B C D E F 1 1 1 1 1 1 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= F  Print: A B C D E A B C D E F 1 1 1 1 1 1 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= F  Print: A B C D E A B C D E F 1 1 1 1 1 1 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue= F  Print: A B C D E A B C D E F 1 1 1 1 1 1 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue=  Print: A B C D E A B C D E F 1 1 1 1 1 1 A B C D E F
Analysis and Design of Algorithms  Visited=  Queue=  Print: A B C D E F A B C D E F 1 1 1 1 1 1 A B C D E F
Analysis and Design of Algorithms Breadth First on Tree
Analysis and Design of Algorithms
Analysis and Design of Algorithms  V={0,1,2,3,4,5}  E={01,02,13,14,24,34,35,45} 0 1 2 3 4 5
Analysis and Design of Algorithms
Analysis and Design of Algorithms  Another Example:
Analysis and Design of Algorithms Depth First Traversal
Analysis and Design of Algorithms Depth First Traversal (DFT) algorithm traverses a graph in a depthward motion and uses a stack to remember to get the next vertex to start a search.
Analysis and Design of Algorithms depthward motion A B C D E F
Analysis and Design of Algorithms  Visited= Stack=  Print: A B C D E F 0 0 0 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited= Stack=  Print: A B C D E F 0 0 0 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited= Stack=  Print: A B C D E F 1 0 0 0 0 0 A B C D E F
Analysis and Design of Algorithms  Visited= Stack=  Print: A B C D E F 1 0 0 0 0 0 A B C D E F A
Analysis and Design of Algorithms  Visited= Stack=  Print: A A B C D E F 1 0 0 0 0 0 A B C D E F A
Analysis and Design of Algorithms  Visited= Stack=  Print: A A B C D E F 1 0 0 0 0 0 A B C D E F A
Analysis and Design of Algorithms  Visited= Stack=  Print: A A B C D E F 1 1 0 0 0 0 A B C D E F A
Analysis and Design of Algorithms  Visited= Stack=  Print: A A B C D E F 1 1 0 0 0 0 A B C D E F B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B A B C D E F 1 1 0 0 0 0 A B C D E F B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B A B C D E F 1 1 0 0 0 0 A B C D E F B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B A B C D E F 1 1 0 1 0 0 A B C D E F B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B A B C D E F 1 1 0 1 0 0 A B C D E F D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D A B C D E F 1 1 0 1 0 0 A B C D E F D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D A B C D E F 1 1 0 1 0 0 A B C D E F D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D A B C D E F 1 1 0 1 0 0 A B C D E F D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D A B C D E F 1 1 0 1 1 0 A B C D E F D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D A B C D E F 1 1 0 1 1 0 A B C D E F E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E A B C D E F 1 1 0 1 1 0 A B C D E F E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E A B C D E F 1 1 0 1 1 0 A B C D E F E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E A B C D E F 1 1 0 1 1 0 A B C D E F E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E A B C D E F 1 1 0 1 1 1 A B C D E F E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E A B C D E F 1 1 0 1 1 1 A B C D E F F E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E F A B C D E F 1 1 0 1 1 1 A B C D E F F E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E F A B C D E F 1 1 0 1 1 1 A B C D E F E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E F A B C D E F 1 1 0 1 1 1 A B C D E F E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E F A B C D E F 1 1 0 1 1 1 A B C D E F E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E F A B C D E F 1 1 1 1 1 1 A B C D E F E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E F C A B C D E F 1 1 1 1 1 1 A B C D E F C E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E F C A B C D E F 1 1 1 1 1 1 A B C D E F E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E F C A B C D E F 1 1 1 1 1 1 A B C D E F E D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E F C A B C D E F 1 1 1 1 1 1 A B C D E F D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E F C A B C D E F 1 1 1 1 1 1 A B C D E F D B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E F C A B C D E F 1 1 1 1 1 1 A B C D E F B A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E F C A B C D E F 1 1 1 1 1 1 A B C D E F A
Analysis and Design of Algorithms  Visited= Stack=  Print: A B D E F C A B C D E F 1 1 1 1 1 1 A B C D E F
Analysis and Design of Algorithms Depth First on tree
Analysis and Design of Algorithms
Analysis and Design of Algorithms  V={0,1,2,3,4,5}  E={01,02,13,14,24,34,35,45} 0 1 2 3 4 5
Analysis and Design of Algorithms
Analysis and Design of Algorithms  Another Example:
Analysis and Design of Algorithms Given a directed graph, check whether the graph contains a cycle or not. Your function should return true if the given graph contains at least one cycle, else return false.
Analysis and Design of Algorithms Depth First Traversal can be used to detect cycle in a Graph. There is a cycle in a graph only if there is a back edge present in the graph.
Analysis and Design of Algorithms  Visited= Stack= A C B D 0 0 0 0 A B C D
Analysis and Design of Algorithms  Visited= Stack= A C B D 0 0 0 0 A B C D
Analysis and Design of Algorithms  Visited= Stack= A C B D 1 0 0 0 A B C D
Analysis and Design of Algorithms  Visited= Stack= A C B D 1 0 0 0 A B C D A
Analysis and Design of Algorithms  Visited= Stack= A C B D 1 0 0 0 A B C D A
Analysis and Design of Algorithms  Visited= Stack= A C B D 1 1 0 0 A B C D A
Analysis and Design of Algorithms  Visited= Stack= A C B D 1 1 0 0 A B C D B A
Analysis and Design of Algorithms  Visited= Stack= A C B D 1 1 0 0 A B C D B A
Analysis and Design of Algorithms  Visited= Stack= A C B D 1 1 1 0 A B C D B A
Analysis and Design of Algorithms  Visited= Stack= A C B D 1 1 1 0 A B C D C B A
Analysis and Design of Algorithms  Visited= Stack= A C B D 1 1 1 0 A B C D C B A
Analysis and Design of Algorithms  Visited= Stack= A C B D 1 1 1 0 A B C D C B A
Analysis and Design of Algorithms Graph contains cycle A C B D
Analysis and Design of Algorithms