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Efficient Graph Representation Methods and Algorithms

Learn about representing graphs with nodes and edges, weighted vs unweighted graphs, directed vs undirected graphs, and storing graph data using adjacency lists or matrices. Explore BFS and DFS algorithms for traversing graphs efficiently and the use of graph structures in various applications.

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Efficient Graph Representation Methods and Algorithms

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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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