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Graph Algorithms: What is a Graph, Traversals, and Applications

Learn about graphs, their properties, and how to perform breadth-first search (BFS) and depth-first search (DFS) traversals. Discover applications of DFS such as topological sort and finding strongly connected components.

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Graph Algorithms: What is a Graph, Traversals, and Applications

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  1. Graph Algorithms What is a graph? V - vertices E µ V x V - edges directed / undirected • Representation: • adjacency matrix • adjacency lists Why graphs?

  2. Graph Algorithms Graph properties: • connected • cyclic • … Tree – a connected acyclic (undirected) graph

  3. Graph Traversals Objective: list all vertices reachable from a given vertex s

  4. Breadth-first search (BFS) Finds all vertices “reachable” from a starting vertex. Byproduct: computes distances from the starting vertex to every vertex • BFS ( G=(V,E), s ) • seen[v]=false, dist[v]=1 for every vertex v • beg=1; end=2; Q[1]=s; seen[s]=true; dist[s]=0; • while (beg<end) do • head=Q[beg]; • for every u s.t. (head,u) is an edge and • not seen[u] do • Q[end]=u; dist[u]=dist[head]+1; • seen[u]=true; end++; • beg++;

  5. Depth-first search (DFS) Finds all vertices “reachable” from a starting vertex, in a different order than BFS. • DFS-RUN ( G=(V,E), s ) • seen[v]=false for every vertex v • DFS(s) • DFS(v) • seen[v]=true • for every neighbor u of v • if not seen[u] then DFS(u)

  6. Applications of DFS: topological sort Def: A topological sort of a directed graph is an order of vertices such that every edge goes from “left to right.”

  7. Applications of DFS: topological sort Def: A topological sort of a directed graph is an order of vertices such that every edge goes from “left to right.”

  8. Applications of DFS: topological sort • TopSort ( G=(V,E) ) • for every vertex v • seen[v]=false • fin[v]=1 • time=0 • for every vertex s • if not seen[s] then • DFS(s) • DFS(v) • seen[v]=true • for every neighbor u of v • if not seen[u] then • DFS(u) • time++ • fin[v]=time (and output v)

  9. Applications of DFS: topological sort • TopSort ( G=(V,E) ) • for every vertex v • seen[v]=false • fin[v]=1 • time=0 • for every vertex s • if not seen[s] then • DFS(s) • DFS(v) • seen[v]=true • for every neighbor u of v • if not seen[u] then • DFS(u) • time++ • fin[v]=time (and output v) What if the graph contains a cycle?

  10. Appl.DFS: strongly connected components Vertices u,v are in the same strongly connected component if there is a (directed) path from u to v and from v to u.

  11. Appl.DFS: strongly connected components Vertices u,v are in the same strongly connected component if there is a (directed) path from u to v and from v to u. How to find strongly connected components ?

  12. Appl.DFS: strongly connected components • STRONGLY-CONNECTED COMPONENTS ( G=(V,E) ) • for every vertex v • seen[v]=false • fin[v]=1 • time=0 • for every vertex s • if not seen[s] then • DFS(G,s) (the finished-time version) • compute G^T by reversing all arcs of G • sort vertices by decreasing finished time • seen[v]=false for every vertex v • for every vertex v do • if not seen[v] then • output vertices seen by DFS(v)

  13. Many other applications of D/BFS • DFS • find articulation points • find bridges • BFS • e.g. sokoban

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