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Graphs Representation, BFS, DFS

Graphs Representation, BFS, DFS. Graph representations. Nodes or vertices Edges Weighted vs. unweighted Undirected vs. directed Multiple edges or not Self-loops or not. Degree. Vertex degree: edges for that vertex Undirected Degree is the number of edges leaving vertex

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Graphs Representation, BFS, DFS

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  1. GraphsRepresentation, BFS, DFS

  2. Graph representations • Nodes or vertices • Edges • Weighted vs. unweighted • Undirected vs. directed • Multiple edges or not • Self-loops or not

  3. Degree • Vertex degree: edges for that vertex • Undirected • Degree is the number of edges leaving vertex • Self loops count twice • Number of edges = 2*sum of vertex degrees • Directed • In-degree (incoming edge count) and out-degree (outgoing edge count) • Sum of all in-degrees equals sum of all out-degrees

  4. Traveling a graph • Path/walk/trail • Follow edges along vertices • Could repeat edge unless stated otherwise • Cycle/circuit/tour • Path that leads back to the original vertex • Usually assumes won’t follow same edge, but not always

  5. Storing graphs • Edge list • Keep a list of all edges • Store start, end, weight • Good if you need to work with all the edges at once • Kruskal’s MST algorithm • Not good at all if you want to know the adjacent edges for any one vertex

  6. Storing graphs • Adjacency list • Keep a list of edges in each vertex • Can be used for directed or undirected • Undirected – must keep consistent on both ends • Can keep weights per edge in the node list • Or, store pointer to a light edge structure • Good to find and iterate through edges for any vertex • Typical “default” implementation for a graph

  7. Storing graphs • Adjacency matrix • Store matrix indicating edges between vertices (rows/columns) • Directed: rows are from, columns are to • Value in the cell can be the weight of the edge • Bad if the graph is sparse, or too large • Can sometimes phrase graph calculations as matrix calculations this way; then, it can be more efficient to compute with

  8. Depth-First Search • Keep global list of visited T/F (or a number indicating which “tree” it is part of) • Recursive function: • Mark current node visited • For all adjacent edges: • If the node is unvisited, visit it • Basically, keep a stack of nodes that you want to visit • Forms the basis for several other operations

  9. Breadth-first search • Keep a “distance” for each node, initialized to infinity • Keep queue of nodes to process, start with intro node • Repeatedly pop a node off the queue • Go through list of adjacent nodes • If node is infinity away, then set its distance to current distance plus one and add to queue • Has a few good uses • Shortest path in an undirected graph

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