MC 302 Graph Theory Tuesday, 112304

1
MC 302 Graph TheoryTuesday, 11/23/04

• Today
• Graph representation and algorithms
• Do Worksheet 9
• Adjacency matrices and counting walks
• Graph Searching DFS and BFS

• Todays reading exercises
• 7.3 1, 7ab (Modify Mathematica file)
• 8.1 9 (both BFS and DFS vertices are visited
  alphabetically)
  

2
Cool Property of the Adjacency Matrix

• If A aij is the adjacency matrix, then aij
  1 iff vi and vj are adjacent.
  
• Equivalently, aij 1 iff there is a walk of
  length 1 from vi to vj. (Remember Length of a
  walk is the of edges.)
  
• Let bij ai1a1j ai2a2j ainanj
• bij obtained by multiplying corresponding entries
  of row i and column j.
  
• Matrix B bij is called the A x A, or A2.
• Cool observation Entry bij number of walks of
  length 2 from vi to vj!

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Counting Walks with the Adjacency Matrix
A

• Theorem. If A is the adjacency matrix of G, then
  entry i, j of Ak equals the number of walks of
  length k from vi to vj.
  
• Proof. By induction (see text).
• Example See MC302041123.nb

A3
A2
A3
4
Graph Searching

• Many graph theory questions can be answered by
  doing an organized traversal of the graph.
  
• The two main graph search algorithms
• Depth-first Search
• Breadth-first Search.

5
Depth-First Search

• Basic Idea
• Travel as far as you can from the starting vertex
  to new vertices
  
• When you cant travel any further, back up along
  path to first vertex adjacent to a new vertex.
  
• Repeat until all vertices have been visited.
• Result of DFS
• Every edge of G is labeled as either a tree
  edge or a back edge.
  
• If G is connected, the tree edges form a spanning
  tree called a depth-first spanning tree.
  (Otherwise they form a spanning forest).

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Depth-First Search Algorithm

• Label all vertices and edges with label
  unvisited.
  
• Pick arbitrary vertex v and label it visited.
• If v has an unvisited neighbor w, label w
  visited and label edge v-w tree edge. Well
  also call v the parent of w. Replace v by w and
  repeat this step. Else go to next step.
• If all vs neighbors are visited, label any
  unvisited incident edges back edges. If v has a
  parent x, replace v by x and go to previous step.
  Else go to next step.
• If any vertex y is unvisited, label y visited,
  and go to step 3. Otherwise, algorithm is
  completed.

7
Properties of Depth-First Search

• If implemented with adjacency matrix, runs in
  time O(n2)
  
• If implemented with adjacency list, runs in time
  O(n e)
  
• In general, runs in time proportional to size of
  data structure used to represent graph
  
• Can be used for Finding spanning forest cycle
  detection finding path between vertices u and v
  testing for connectedness testing bipartiteness
  bridge detection cut point detection

8
Breadth-First Search

• Basic Idea
• Visit an initial vertex v, then visit all of its
  neighbors.
  
• Repeat this with each neighbor of v, until all
  vertices have been visited
  
• Result of BFS
• If G is connected, the tree edges form a spanning
  tree called a breadth-first spanning tree.
  (Otherwise they form a spanning forest).
  
• If v is the root of a BFS spanning tree T, then
  the path from v to w in T is a shortest path from
  v to w in G.

9
Breadth-First Search Algorithm and Properties

• Algorithm
• Pick arbitrary vertex v and number it vertex 1.
  Set count 1, current 1.
  
• For each unnumbered neighbor w of v, increment
  count and assign that number to w. Assign label
  tree edge to edge v-w.
  
• If count n, algorithm terminates. Otherwise,
  increment current and repeat from step 2.

• Same complexity as DFS.
• Can be used for Shortest path between u and v
  single source shortest path All pairs shortest
  path