Graphs 1.5 - Adjacency Matrix Structure Edge list structure...

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Unformatted text preview: Adjacency Matrix Structure Edge list structure Augmented vertex objects a v b u Integer key (index) associated with vertex w 2D-array adjacency array Reference to edge object for adjacent vertices 0 u 1 0 Null for nonnonadjacent vertices 0 a CSE 2011 Prof. J. Elder - 21 - 2 1 Ø 1 2 v w 2 Ø Ø Ø Ø b Last Updated: 4/1/10 10:16 AM Asymptotic Performance (assuming collections V and E represented as doubly-linked lists) |V| vertices, |E| edges no parallel edges no self-loops Bounds are “big-Oh” Edge List Adjacency List Adjacency Matrix Space |V|+|E| |V|+|E| |V|2 incidentEdges(v) areAdjacent (v, w) insertVertex(o) |E| |E| 1 deg(v) min(deg(v), deg(w)) 1 |V| 1 |V|2 insertEdge(v, w, o) 1 1 1 |E| deg(v) |V|2 1 1 1 removeVertex(v) removeEdge(e) CSE 2011 Prof. J. Elder - 22 - Last Updated: 4/1/10 10:16 AM Graph Search Algorithms CSE 2011 Prof. J. Elder - 23 - Last Updated: 4/1/10 10:16 AM Depth First Search (DFS) Idea: Continue searching “deeper” into the graph, until we get stuck. If all the edges leaving v have been explored we “backtrack” to the vertex from which v was discovered. Analogous to Euler tour for trees Used to help solve many graph problems, including Nodes that are reachable from a specific node v Topological sorts Detection of cycles Extraction of strongly connected components CSE 2011 Prof. J. Elder - 24 - Last Updated: 4/1/10 10:16 AM Depth-First Search The DFS algorithm is similar to a classic strategy for exploring a maze We mark each intersection, corner and dead end (vertex) visited We mark each corridor (edge ) traversed We keep track of the path back to the entrance (start vertex) by means of a rope (recursion stack) CSE 2011 Prof. J. Elder - 25 - Last Updated: 4/1/10 10:16 AM ...
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