Recurrence Applications

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# Recurrence Applications ## Tower of Hanoi Number of moves $T_n = 2T_{n-1} + 1$, $T_1 = 1$. Solution: $T_n = 2^n - 1$.

Number of moves

T_n = 2T_{n-1} + 1

T_1 = 1

. Solution:

T_n = 2^n - 1

. For

n = 64

disks:

2^{64} - 1 \approx 1.84 \times 10^{19}

moves — about 585 billion years!

T(n) = aT(n/b) + f(n)

Master Theorem:

If $f(n) = O(n^{\log_b a - \epsilon})$ , then $T(n) = \Theta(n^{\log_b a})$
If $f(n) = \Theta(n^{\log_b a})$ , then $T(n) = \Theta(n^{\log_b a} \log n)$
If $f(n) = \Omega(n^{\log_b a + \epsilon})$ and $af(n/b) \leq cf(n)$ , then $T(n) = \Theta(f(n))$ Example: Merge sort: $T(n) = 2T(n/2) + n$ . $n^{\log_2 2} = n = f(n)$ , so $T(n) = \Theta(n \log n)$ . Join Discord PreviousRecurrence RelationsNextGenerating Functions