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Dawn J' Lawrie

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Euler path. Classic problem studied by Leonhard Euler and presented to Russian Academy in ... A graph has a Euler tour if and only if it is connected and all ... – PowerPoint PPT presentation

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Title: Dawn J' Lawrie


1
Announcements
  • Programming Assignment 3
  • Code due tonight at midnight
  • Print out and Analysis due Friday beginning of
    class
  • Today's Office Hours
  • 11-12 and 2-3 PM

2
Nontrivial Graph-Processing
  • Lecture 17

3
Simple Path
  • What is a simple path?
  • Given what we have studied so far, can we answer
    the question
  • Does a simple path exist between v and w?

4
Find a Specific Simple Path
  • Use Depth-First-Search
  • Solve the recursive problem
  • Assume I know a path from v to t, is there a path
    from t to w?

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5
Simple path search
  • template ltclass GraphTypegt
  • class SPath
  • private
  • const GraphType graph
  • vector ltboolgt visited
  • vector ltintgt path
  • int length
  • bool found

6
Simple path search
  • bool searchR(int v, int w, int depth)
  • if (v w) // base case
  • pathdepth v
  • length depth1
  • return true
  • visitedv true
  • typename GraphadjIterator adj(graph, v)
  • for (int t adj.beg() !adj.end() t
    adj.nxt())
  • if (!visitedt)
  • if (searchR(t, w, depth1))
  • pathdepth v
  • return true
  • return false

7
Simple path search
  • public
  • SPath(const Graph G, int v, int w)
  • graph(G), visited(G.V(), false), path(G.V(),
    -1),
  • length(0)
  • found searchR(v,w,0)
  • bool exists() const
  • return found
  • void displayPath() const
  • for (int i 0 i lt length i)
  • if (i ! 0)
  • cout ltlt " - "
  • cout ltlt pathi
  • cout ltlt endl

8
Hamiltonian Path
  • Given two vertices, is there a simple path
    connecting them that visits every vertex in the
    graph exactly once?
  • Hamiltonian Tour Is there a cycle that visits
    every vertex in the graph exactly oncegt

9
Hamiltonian Path
  • bool searchR(int v, int w, int depth)
  • if (v w) // base case
  • pathdepth v
  • length depth1
  • return true // remove
    line
  • return (length graph.V()) // new line
  • visitedv true
  • typename GraphadjIterator adj(graph, v)
  • for (int t adj.beg() !adj.end() t
    adj.nxt())
  • if (!visitedt)
  • if (searchR(t, w, depth1))
  • pathdepth v
  • return true
  • visitedv false // new line
  • return false

10
Euler path
  • Classic problem studied by Leonhard Euler and
    presented to Russian Academy in 1735

11
Euler path
  • Is there a path connecting two given vertices
    that used each edge in the graph exactly once?
  • Not necessarily a simple path
  • A graph has a Euler tour if and only if it is
    connected and all its vertices are of even degree

12
Proof
  • Must be connected
  • By definition a tour is a path connecting each
    pair of vertices
  • Vertex v must be of even degree
  • When we traverse the tour (starting from anywhere
    else), we enter v on one edge and leave on a
    different edge
  • Number of edges incident on the vertex must be
    twice the number of visits

13
Proof of converse
  • Consider any connected graph that has more than
    one edge, with all vertices of even degree
  • Starting at any vertex v, we follow and remove
    any edge
  • Continue until arriving at a vertex that has no
    more edges
  • Process terminates
  • Outcomes
  • We must end up back a v because v is the only
    vertex of odd degree

14
Converse Proof continued
  • We may not have a full tour
  • All the vertices in the group that remain have
    even degree but may not be connected
  • Each connected component is a Euler tour by
    inductive hypothesis
  • Consider these detours

15
Finding a Euler Path
  • Cannot implement a direct recursive method
    because it would be as slow as the algorithm for
    finding Hamiltonian Path
  • Is there a linear time solution?

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16
Linear-time Euler path
  • template ltclass Graphgt
  • int EPathltGraphgttour(int v)
  • bool finished false
  • while (!finished)
  • typename GraphadjInterator adj(graph, v)
  • int w adj.beg()
  • if (adj.end())
  • finished true
  • else
  • pStack.push(v)
  • graph.remove(Edge(v, w))
  • v w
  • return v

17
Linear-time Euler path
  • template ltclass Graphgt
  • void EPathltGraphgtshow()
  • int curV v
  • if (!found)
  • return
  • while(tour(curV) curV !pStack.empty())
  • curV sStack.pop()
  • cout ltlt "-" ltlt curV
  • cout ltlt endl
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