Amortized Analysis
主講人:陳建源
研究室 :法401
Email: [email protected]
日期:99/12/18
Outline
Analysis of algorithms
Aggregate Analysis
Accounting method
Potential function
Analysis of algorithms

Measure the goodness of algorithms






Measure the difficulty of problems




efficiency?
asymptotic notations: O(n2)
worst case
average case
amortized
NP-complete
undecidable
lower bound
Is the algorithm optimal?
Amortized Analysis
Average performance of each operation
in the worst-case a sequence of n
operations takes worst-case Time T(N).
Amortized Analysis
Aggregate Analysis-Stack operation
PUSH(S,X)
Multipop(S,k)k個元素，or empty
worst-case Multipop => n
o(n2)/n=o(n)
not true
每個元素最多pop一次，n個operation
最多push n個元素，即最多有n個被pop。
∴2n/n=2
Amortized Analysis
Aggregate Analysis
Incrementing a cost binary counter
0 0 0
0
0 0 1
1
0 1 0
1+2=3
0 1 1
3+1=4
1 0 0
4+3=7
logn 

i 0
increment worst-case k

1
n
 2i   n  2i  2n
i 0
Total T(nk)
Amortized Analysis
Accounting method
 Differing charge to different operation [Note: 原來是一
律相同cost]
 actual cost +credit=amortized cost
令amortized cast
actual cast
ĉ
c
若credit不為負
 ĉ   c   credit
 0
  ĉ   c   c   ĉ

為upper bound
Amortized Analysis
Accounting method
Stack operation：
actual
cost
PUSH
1
PUSH
2
POP
1
POP
0
MULTIPOP
0
MULTIPOP min(k,s)
取幾個  
stack
amortized cost
Amortized Analysis
Accounting method
PUSH：即一個cost為實際cost
另一個為credit預備將來POP之用
當stack有3元素，表示credit為3，而stack之元素個
數0
N
N
i 1
i 1
  c   ĉ  2N
Amortized Analysis
Accounting method-Increasing a binary counter
000
001
1
2
010
actual cost

amortized cost
0 1
1
0 1
2
1 0
1
1 0
0
當set為1時，有 1 cost為實際成本，
另一個預備將來轉回0之用
Amortized Analysis
Accounting method-Increasing a binary counter
Amortized cost of an increment
=2
(頂多一次set為1)
credit為binary 1’s個數
N
N
i 1
i 1
∴
 c   ĉ  2N
0
Amortized Analysis
Potential function
建一 data structure D
D0

D i 1
Di


Di
i operation
Dn
建一potential function
 ：from data structure
to real number  Di 
Amortized Analysis
Potential function
針對第i個operation
ĉ i  ci   (D i )   (D i 1 )
N
N
i 1
i 1
  ĉ i   ci   (D N )   (D 0 )
if  (D N )   (D 0 )  0
N
N
i 1
i 1
  c i   ĉ i
推出
 ( Di )   ( D 0 )
Amortized Analysis
Potential function
如何選potential function
 初值最小
 將實際的cost消去
Amortized Analysis
Potential function-Stack operate
 : the number of objects in the stack
且 (D i )  0
 (D 0 )  0

empty stack
 PUSH
ĉi  ci   (D i )   (D i 1 )
 11  2
 POP
ĉi  ci   (D i )   (D i 1 )
 1  (1)  0
 MultiPOP ĉi  k  (k )  0
Amortized Analysis
Potential function-Incrementing a binary counter
 ：the number of 1’s in the counter
bi
第 i 個 operator 之1’s個數
ti
reset ti bits 1 0
第 i 個 operation actual cost ti+1
 (Di )   (Di 1 )  (bi 1  t i  1)  bi 1
 1 ti
當bi>0，表示bi-1-reset+1
Amortized Analysis
Potential function-Incrementing a binary counter
ĉ i  c i   (D i )   (D i 1 )
 t i 1  (1  t i )
2
 (D 0 ) ,  (D i )  0
N
N
i 1
i 1
  c i   ĉ i  2 N