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The Towers of Hanoi Problem in Graph Form. Finding bounds for the number of moves. Presented by: Shibo Fang 9 July 2010 Mentor: Dr. Ernst Leiss. Background. Towers of Hanoi: a classic problem Three poles to move disks on Goal is to move all disks to third pole

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the towers of hanoi problem in graph form

The Towers of Hanoi Problem in Graph Form

Finding bounds for the number of moves

  • Presented by:
  • Shibo Fang
  • 9 July 2010
  • Mentor:
  • Dr. Ernst Leiss
background
Background
  • Towers of Hanoi: a classic problem
    • Three poles to move disks on
    • Goal is to move all disks to third pole
    • Certain rules apply in movement of disks
  • Can be represented as a direct graph
    • S = start node, A = auxiliary node (between S and D nodes), D = destination node
    • Two edges between every two nodes
solvable and finite graphs
Solvable and Finite Graphs
  • A graph is solvable when:
    • There exist vertices S, D, and A
    • There exist paths from S to A, from A to D, and from D to S
  • Solvable means any number of disks can be moved from the S to the D node
  • Any graph that is not solvable is a finite graph
hanoi graphs
Hanoi Graphs
  • Original Hanoi problem with three nodes requires 2d – 1 moves (d = number of disks)
  • Modified Hanoi problem (there are no edges between S and D) requires 3d – 1 moves
special hanoi graphs
Special Hanoi Graphs
  • First special Hanoi graph (k graph)
    • Has k+3 nodes, S0 through Sk, A, D
    • Requires k*d + 3d – 1 moves
special hanoi graphs cont
Special Hanoi Graphs (Cont.)
  • Second special Hanoi graph (cycle graph)
    • An edge between each two nodes
    • First edge from S node to first A node
    • Last edge from D node to S node
findings
Findings
  • Needed to find algorithm and formula for number of moves for cycle graph
    • Found for d ≤ n – 1 (n = number of nodes)
    • Need to find for d ≥ n
  • K graph thought to be upper bound
    • Requires most moves out of all Hanoi graphs
  • Cycle graph worse than K graph
    • When d ≤ n – 1, small d and large n
questions
Questions

Thank you. Any questions?