Algorithm Design and Analysis Course Review notes 2-- recursive relationship

Algorithm Design and Analysis Course Review notes 2-- recursive relationship

Recursive relationship

Recurrence relation describes the function of a natural number
for some n> 0 is a function of the value, a value smaller than n by the function represented

Why should analyze recursive relationship?
Many algorithms, in particular, recursive algorithm, the time cost function can be used to describe a recursive relationship

Solution

• Replacement Act (Substitution)
• Recursive tree (Recursion Tree)
• Iterative method (Iteration)
• Master mode (Master Theorem)

Replacement

1. I guess (experience, replacing the argument, by loosening tight)
2. Verify conjecture for n = $n_0$Correctness
3. For the guess verification n> $n_0$Correctness

example：T(n) = 2T( $\ lfloor$n/2 $\rfloor$ ) + n-
guess: T (n) = Θ (
is assumed to when n≥2 $\ lfloor$n/2 $\rfloor$ established,
namely T ( $\ lfloor$n/2 $\rfloor$ )≤c $\ lfloor$n/2 $\rfloor$lg( $\ lfloor$n/2 $\rfloor$)

T(n)= 2T( $\ lfloor$n/2 $\rfloor$ ) + n
≤2(c $\ lfloor$n/2 $\rfloor$lg( $\ lfloor$n/2 $\rfloor$ )) + n-
≤cnlg (n-/ 2) + n-
= cnlgn-n-cnlg2 +
= cnlgn-CN + n-
≤cnlgn for c> 1

examples:
T(n) = 2T(n/2) +Θ(n) → T(n) = Θ(n lg n)
T(n) = 2T( $\ lfloor$n/2 $\rfloor$ ) + n → T (n) = Θ (n lg n)
T (n) = 2T ( $\ lfloor$n/2 $\rfloor$ + 17) + n → T (n)

Recursive tree

Iterative method

1. Expansion
2. Algebraic operations
3. Summing

example：
$T(n)=\left\{ \begin{aligned} c　ｎ＝１\\ aT(\frac nb)+cn　ｎ＞１ \end{aligned} \right.$

$T(n)=\left\{ \begin{aligned} Θ(n)　ａ＜ｂ\\ Θ(nlog_bn) 　ａ＝ｂ\\ Θ(n^{log_ba})　ａ＞ｂ\\ \end{aligned} \right.$

Main way

example:
$T (n) = 9T (n / 3) + n$
a=9
b=3
f(n)=n

$n ^ {} log_ba$= $n^{log_39}$ = Θ (N²)f
(N) = o ( $n ^ {log_39-e}$ ) n-=
[epsilon] =. 1, a first satisfies the formula

Therefore, T (n) = Θ ( $n ^ {} log_ba$)
T(n)=Θ( $n^2$)

Key:

• Replacement Act (Substitution)
• Recursive tree (Recursion Tree)
• Iterative method (Iteration)
• Master mode (Master Theorem)

Reference: classroom teachers Professor Khoo Teck red courseware