8_graphs.v3

# 8_graphs.v3 - CS161 Graph Algorithms David Kauchak •...

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Unformatted text preview: CS161 - Graph Algorithms David Kauchak • Types of graphs What is a graph? A graph is a set of vertices V and a set of edges ( u,v ) ∈ E where u,v ∈ V – undirected – directed – weighted Some special cases – tree - connected, undirected graph without any cycles – DAG (directed acyclic graph) - directed graph without any cycles – complete graph - a graph where every vertex is connected – bipartite graph - a graph where every vertex can be partitioned into two sets X and Y such that all edges connect a vertex u ∈ X and a vertex v ∈ Y • Examples – Transportation networks (flights, roads, etc.) – Communication networks – Web – Bayesian networks – Social networks – Circuit design • Representation 1 – Adjacency list Each vertex u ∈ V contains an adjacency list of the set of vertices v such that there exists an edge ( u,v ) ∈ E Example of undirected and directed – Adjacency matrix A graph is represented by a | V | x | V | matrix A such that a ij = braceleftBigg 1 if ( i,j ) ∈ E 0 otherwise Example of undirected and directed – Tradeoffs Dense vs. sparse graph Adjacency list: Space efficient, e.g. web Adjacency matrix: constant time lookup to see if an edge exists Best of both worlds? – Weighted graphs Adjacency list: store the weight in the adjacency list Adjacency matrix: a ij = braceleftBigg weight if ( i,j ) ∈ E otherwise • Graph algorithms – graph traversal (BFS, DFS) – Shortest path from a to b * unwieghted * weighted positive weights * negative/positive weights – All pair shortest paths – Are all nodes in the graph connected?...
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8_graphs.v3 - CS161 Graph Algorithms David Kauchak •...

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