Computational Time Required to Solve the Tower of Hanoi

1
Computational Time Required to Solve the Tower
of Hanoi

• To solve the Tower of Hanoi problem, a recursive
  approach was taken. To move n disk from one tower
  to another we first move (n-1) disk to an
  auxiliary tower, then move the bottom disk to its
  final destination, and then move the (n-1) disk
  to their final destination.
• The time required for various operations was
  measure and any overhead not associated with the
  function was subtracted out.
• 1E9 Operations were performed on a Pentium(4)
  running at 2400.153 MHz. The times required for a
  single operation are as follows
• To call the function (w/o performing any
  operations)
• Tf 5.171/1E9 s
• To move a single disk w/ necessary bookkeeping
• T1 14.157/1E9 s
• To compute auxiliary tower and pass over n1
  case
• Taux 2.742/1E9 s

2
Computational Time Required to Solve the Tower
of Hanoi

• Now a recursive relationship can be written and
  solved for the time as a function of n
• Let Tn be the time required to solve problem
  for n disks then
• Tn Tf Taux 2Tn-1 T1
• C 2 Tn-1 (letting C Tf Taux T1 )
• C 2 (C 2 Tn-2)
• C 2 (C 2 (C 2 Tn-3)
• C 2C 22C 2(k-1)C 2k Tn-k (In
  general)
• C (2k 1) 2k Tn k
• C (2(n-1) 1) 2(n-1)T1 By n 1 ? k
• Substituting the times found earlier
• T64 2.036E11 1.306E11 3.342E11 s or
  10,590 years