17.amortized analysis hsu, lih-hsing. computer theory lab. chapter 17p.2 the time required to...

17.Amortized analysis

Hsu, Lih-Hsing

Chapter 17 P.2

Computer Theory Lab.

The time required to perform a sequence of data structure operations in average over all the operations performed.

Average performance of each operation in the worst case.

Chapter 17 P.3


For all n, a sequence of n operations takes worst time T(n) in total. The amortize cost of each operation is

.n

nT )(

Chapter 17 P.4


Three common techniques

aggregate analysis accounting method potential method

Chapter 17 P.5


17.1 The aggregate analysis

Stack operation - PUSH(S, x) - POP(S) - MULTIPOP(S, k)

MULTIPOP(S, k) 1 while not STACK-EMPTY(S) and k 02 do POP(S)3 k k – 1

Chapter 17 P.6


Action of MULTIPOP on a stack S

top 23 17 6 39 10 47

initial

top 10 47

MULTIPOP(S,4)

MULTIPOP(S,7)

Chapter 17 P.7


Analysis a sequence of n PUSH, POP, and MULTIPOP operation on an initially empty stack.

O(n2) O(n) (better bound) The amortize cost of an operation is

.

O n

nO

( )( ) 1

Chapter 17 P.8


Incremental of a binary counter

Chapter 17 P.9


INCREMENT

INCREMENT(A)1 i 0

2 while i < length[A] and A[i] = 1

3 do A[i] 0

4 i i + 1

5 if i < length[A]

6 then A[i] 1

Chapter 17 P.10


Analysis:

O(n k) (k is the word length) Amortize Analysis:

)1()(

iscost amortize the

2

2

1

2 0

log

0

On

nO

n

nn

ii

n

ii

Chapter 17 P.11


17.2 The accounting method We assign different charges to different

operations, with some operations charged more or less than the actually cost. The amount we charge an operation is called its amortized cost.

Chapter 17 P.12


When an operation’s amortized cost exceeds its actually cost, the difference is assign to specific object in the data structure as credit. Credit can be used later on to help pay for operations whose amortized cost is less than their actual cost.

Chapter 17 P.13


If we want analysis with amortized costs to show that in the worst cast the average cost per operation is small, the total amortized cost of a sequence of operations must be an upper bound on the total actual cost of the sequence.

Moreover, as in aggregate analysis, this relationship must hold for all sequences of operations.

Chapter 17 P.14


If we denote the actual cost of the ith operation by ci and the amortized cost of the ith operation by , we require

for all sequence of n operations. The total credit stored in the data structure

is the difference between the total actual cost, or .

ic

n

ni

n

ii cc

11

ˆ

n

i i

n

i icc

11ˆ

Chapter 17 P.15


Stack operation

Amortize cost: O(1)

PUSH 1 PUSH 2

POP 1 POP 0

MULTIPOP min{k,s} MULTIPOP 0

Chapter 17 P.16


Incrementing a binary counter

Each time, there is exactly one 0 that is changed into 1.

The number of 1’s in the counter is never negative!

Amortized cost is at most 2 = O(1).

01 1 01 2

10 1 10 0

Chapter 17 P.17


17.3 The potential method

initial data structure D0 .

Di : the data structure of the result after applying the i-th

operation to the data structure Di1.

Ci : actual cost of the i-th operation.

Chapter 17 P.18


A p o t e n t i a l f u n c t i o n m a p s e a c h

d a t a s t r u c t u r e D i t o a r e a l n u m b e r

( )D i , w h i c h i s t h e p o t e n t i a l a s s o c i a t e d

w i t h d a t a s t r u c t u r e D i .

T h e a m o r t i z e d c o s t c i o f t h e i - t h

o p e r a t i o n w i t h r e s p e c t t o p o t e n t i a l

i s d e f i n e d b y )()(ˆ1

iiii

DDcc .

Chapter 17 P.19


If then . If then the potential incre

ases.

n

ini

n

iiii

n

ii

DDc

DDcc

10

11

1

)()(

))()((ˆ

( ) ( )D Di 0c ci

i

ni

i

n

1 1

( ) ( )D Di i 1

Chapter 17 P.20


Stack operations

the number of objects in the stack of the ith operation.

( )Di

( )D0 0( )Di 0

Chapter 17 P.21


PUSH

( ) ( ) ( )D D s si i 1 1 1

( ) ( )c c D Di i i i 1 1 1 2

Chapter 17 P.22


MULTIPOP

k k s' min{ , } ( ) ( ) 'D D ki i 1 ( ) ( ) ' 'c c D D k ki i i i 1 0

Chapter 17 P.23


POP

The amortized cost of each of these operations is O(1).

ci 0

Chapter 17 P.24


Incrementing a binary counter the number of 1’s in the counte

r after the ith operations = . = the ith INCREMENT operation rese

ts bits. Amortized cost = O(1)

( )Di bi

titi

1)(1

iiii

tbbDiiiiii

tbtbDD 1)1()()(111

2)1()1()()(ˆ1

iiiiiittDDcc

Chapter 17 P.25


Even if the counter does not start at zero:

001

011

22

)()(ˆ

bbnbb

DDcc

nn

n

i

n

n

ii

n

ii

17.4 Dynamic tables

Chapter 17 P.27


17.4.1 Table expansion

TABLE-INSERT TABLE-DELETE (discuss later) load-factor

(load factor)

( ) ( ( ) )T T 12

( )[ ]

[ ]T

num T

size T

Chapter 17 P.28


TABLE_INSERT TABLE_INSERT(T, x)1 if size[T] = 02 then allocate table[T] with 1 slot3 size[T] 14 if num[T] = size[T]5 then allocate new-table with 2 size[T] slots6 insert all items in table[T] in new-table7 free table[T]8 table[T] new-table9 size[T] 2 size[T] 10 insert x into table[T]11 num[T] num[T] + 1

Chapter 17 P.29


Aggregate method:

amortized cost = 3

otherwise1

2 ofpower exact an is 1 if i-ici

n

i

n

j

j

i nnnnc1

lg

1322

Chapter 17 P.30


Accounting method: each item pays for 3 elementary

insertions; 1. inserting itself in the current table, 2. moving itself when the table is

expanded, and 3. moving another item that has

already been moved once when the table is expanded.

Chapter 17 P.31


Potential method:

(not expansion)

][][2)( TsizeTnumT size sizei i 1

3

))1(2()2(1

)2()2(1

ˆ

11

1

iiii

iiii

iiii

sizenumsizenum

sizenumsizenum

cc

Chapter 17 P.32


Example:

size size numi i i 1 16 13,

3

)16122()16132(1

)16122()16132(1

ˆ 1

iiii cc

Chapter 17 P.33


(expansion)

size size numi i i/ 2 11

3

)1(2

))1()1(2(

))22(2(

)2()2(

ˆ

11

1

ii

ii

iii

iiiii

iiii

numnum

numnum

numnumnum

sizenumsizenumnum

cc

Chapter 17 P.34


Example:

Amortized cost = 3

size size numi i i/ 2 1 161

3

16217

)16162())2172(172(17

)16162()32172(17

ˆ 1

iiii cc

Chapter 17 P.35


17.4.2 Table expansion and contraction

To implement a TABLE-DELETE operation, it is desirable to contract the table when the load factor of the table becomes too small, so that the waste space is not exorbitant.

Chapter 17 P.36


Goal:

The load factor of the dynamic table is bounded below by a constant.

The amortized cost of a table operation is bounded above by a constant.

Set load factor12

Chapter 17 P.37


The first operations are inserted. The second operations, we perform I, D, D, I, I, D, D, …

Total cost of these n operations is . Hence the amortized cost is .

Set load factor (as TABLE_DELETE) (after the contraction, the load factor

become )

n

2n

2

( )n2

( )n14

1

2

Chapter 17 P.38


2

1)(][

2

][2

1)(][][2

)(TifTnum

Tsize

TifTsizeTnumT

Chapter 17 P.39


Initial

0

1

0

0

0

0

0

0

size

num

Chapter 17 P.40


TABLE-INSERT

if , same as before.

if if

i 112

i 112

i 12

( ) ( )

( ) ( ( ))

c c

sizenum

sizenum

sizenum

sizenum

i i i i

ii

ii

ii

ii

1

111

2 2

12 2

1

0

Chapter 17 P.41


Example:


( ) ( )c ci i i i

1 116

26

16

25

0

Chapter 17 P.42


If i 12

3

32

3

2

3

32

33

32

33

))(2

())1(2(1

)2

()2(1

ˆ

11

111

11

11

11

11

1

ii

iii

ii

ii

ii

ii

ii

iiii

sizesize

sizesize

sizenum

numsize

sizenum

numsize

sizenum

cc

Chapter 17 P.43


Example:


( ) ( )

c ci i i i

1

1 2 8 1616

27

3

Amortized cost of Table insert is O(1).Amortized cost of Table insert is O(1).

Chapter 17 P.44


TABLE-DELETE

If does not cause a contraction

(i.e.,

i 11

2isize sizei i 1

( ) ( )

( ) ( ( ))

c c

sizenum

sizenum

sizenum

sizenum

i i i i

ii

ii

ii

ii

1

111

2 2

12 2

1

2

Chapter 17 P.45


Example:


( ) ( )c ci i i i

1 116

26

16

27

2

Chapter 17 P.46


causes a contraction i

c num

size sizenum

i i

i ii

1

2 411

( )actual cost

Chapter 17 P.47


1

))1()22((

))1(()1(

)2

()2

()1(

ˆ

1

1

1

ii

iii

i

i

i

i

i

iiii

numnum

numnumnum

numsize

numsize

num

cc

Chapter 17 P.48


Example:

size sizenumi i

i2 41 41

( ) ( ) ( )

c ci i i i

1

3 18

23

16

24

1

Chapter 17 P.49


if (Exercise 18.4.3) Amortized cost O(1).

i 11

2

17.amortized analysis hsu, lih-hsing. computer theory lab. chapter 17p.2 the time required to...

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