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This study examines the computational time required to solve the Tower of Hanoi problem using a recursive approach, with detailed measurements of various operations and overhead deduction. The analysis shows the complexity of solving the problem for different numbers of disks and highlights the substantial time involved in large-scale scenarios.
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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. • 1E+9 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 = 5.171/1E+9 s • To move a single disk w/ necessary bookkeeping: T[1] = 14.157/1E+9 s • To compute auxiliary tower and pass over n=1 case: Taux = 2.742/1E+9 s
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 T[n] be the time required to solve problem for n disks then : T[n] = Tf + Taux + 2T[n-1] + T[1] = C + 2 T[n-1] (letting C= Tf + Taux + T[1] ) = C + 2 (C +2 T[n-2]) = C + 2 (C + 2 (C + 2 T[n-3]) = C + 2C + 2^2C + …+ 2^(k-1)C + 2^k T[n-k] (In general) = C (2^k – 1) + 2^k T[n – k] = C (2^(n-1) – 1) + 2^(n-1)*T[1] By n – 1 k • Substituting the times found earlier : T[64] = 2.036E+11 + 1.306E+11 = 3.342E+11 s or 10,590 years
