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Notes: 03-09 More Graphs

·453 words·3 mins·

Last Time
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  • We defined graphs, vertices, and edges; directed and undirected.
  • We talked about graph representations: Adjacency Matrix, Adjacency List
  • We talked about algorithms to do graph traversals.

More Definitions
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  • Vertices are frequently called nodes.
  • Simple graphs: No self loops or duplicate edges.
    • Multigraphs: Self loops and duplicate edges allowed.
  • If there’s an edge u->v, then v is adjacent to and a neighbour of u.
  • The degree of a node is its number of neighbors (or out edges).
  • A walk is a sequence of vertices where each adjacent pair of vertices in the walk are adjacent in the graph.
  • A path is a walk that doesn’t visit any vertex multiple times.
  • A closed walk starts and ends at the same vertex.
  • A cycle is a closed path.
  • An acyclic graph contains no cycles.
  • A tree is a connected acyclic graph.
  • A subgraph of a graph G is a graph has a subset of the edges and a subset of the vertices in G.
  • A spanning tree of an undirected graph G is a tree that contains every vertex in G.

Traversals
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  • Do a depth-first search.
  • For a pre-order traversal, visit (e.g. print) the node before recursing.
  • For a post-order traversal, visit (e.g. print) the node after recursing.
  • This should be familiar from trees, although because we only recurse once and keep a “todo " data structure.

DFS
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DFS(v):
  mark v
  PreVisit(v)
  for each edge (v, w):
    if w is unmarked:
      w.parent = v
      DFS(w)
  PostVisit(v)

After, parent pointers give us a spanning tree.

Depth First Sequencing
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Preprocess(G):
  clock 0
PreVisit(v):
  clock = clock + 1
  v.pre clock
PostVisit(v):
  clock = clock + 1
  v.post = clock

Detecting Cycles
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For any edge u->v, if u.post < v.post then there’s a cycle.

Can we prove that?

Edge Types
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For a given DFS traversal, we end up with some different edge types:

  • Tree edges are the spanning tree defined by parent pointers.
  • Forward edges are edges that point down the tree.
  • Cross edges point across the tree.
  • Back edges point up the tree.

Topoligical Sort
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A topological ordering of a directed graph G is a total order on the vertices such that u < v for every edge u -> v . Less formally, a topological ordering arranges the vertices along a horizontal line so that all edges point from left to right. A topological ordering is clearly impossible if the graph G has a directed cycle—the rightmost vertex of the cycle would have an edge pointing to the left!

Just reverse a post-ordering.

Dynamic Programming
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This visits the dependency graph of the recurrence in topologically sorted order explicitly.

Memoization
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Memoization does the same thing implicitly.

Nat Tuck
Author
Nat Tuck