CS294-1 On-line Computation & Net w ork Algorithms Spring - - PDF document

SLIDE 1 CS294-1 On-line Computation & Net w

• rk

Algorithms Spring 1997 Lecture 2: Jan uary 23 L e ctur er: Y air Bartal Scrib e: Keith V anderve en This lecture in tro duces amortized analysis tec hniques and demonstrates their use in analyzing the cost

• f

t w

• simple

data structures problems. 2.1 Amorti ze d Anlysis Amortized analysis [T arjan85 , CLR90] is

• rginated

in the study

• f

dynamic data structures. Eac h

• p

eration is regarded as starting with some initial amoun t

• f

\money ." If the

• p

eration costs less than this amoun t

• f

money , the excess is stored as credit to w ard pa ying the costs

• f
• p

erations whic h cost more than the initial amoun t

• f

money . In Data Structures: Supp

• se

y

• u

carry

• ut

a sequence

• f
• p

erations

• 1

;

• 2

; :::;

• n

. Then w

• rst-case

cost = max i c(o i ): total cost = X i c(o i ): amortized cost = total cost n

• ~

c F

• r

example, in dynamic hashing the w

• rst-case

searc h cost is (log n) whereas the amortized cost is O (1). 2.2 Amorti zatri

• n

T ec hniques Through Examples 2.2.1 The Binary Coun ter Initially , all the bits are set to 0. T

• incremen

t the coun ter b y 1,

• ne

go es righ t to left ipping bits un til a bit is ipp ed from to 1. The cost

• f

the ith up date, c i , is the n um b er

• f

bits whic h are ipp ed. w

• rst-case

cost = k : 2-1

SLIDE 2 2-2 Lecture 2: Jan uary 23

• b

b b b b

k-1 k-2 k-3 1

Figure 2.1: k-bit binary coun ter total cost = c(n) = n X i c i = O (nk ): amortized cost ~ c = c(n) n : Claim 2.1 ~ c

• 2.

Pro

• f:

3 tec hniques to pro v e this:

• The

Engineer's view

• The

Bank er's view

• The

Ph ysicist's view 2.2.1.1 The Engineer's view Idea: brute force calculation! Observ e: b ips ev ery up date b 1 ips ev ery 2 up dates . . . b i ips ev ery 2 i up dates c(n) = blog nc X i=0 b n 2 i c < 2n: ~ c < 2:

SLIDE 3 Lecture 2: Jan uary 23 2-3 2.2.1.2 The Bank er's View: (The Credit-Debit Metho d) Idea: Eac h

• p

eration comes in with ~ c $. During execution, if the cost

• f

an

• p

eration is less than ~ c, store the excess ( ~ c

• cost)

as credit. If the cost

• f

an

• p

eration is greater than ~ c , withdra w previously stored credit to complete pa ymen t

• f

(cost ( ~ c). Theorem 2.2 If system has non-ne gative cr e dit at al l times, then ~ c is a b

• und
• n

the amotize d c

• st.

Bac k to the Binary Coun ter problem, w e set ~ c = 2 and asso ciate credits with bit p

• sitions.

Case 1: W e ip a bit in to 1, pa y 1 unit and store 1 unit at that bit's p

• sition.

Case 2: W e ip a bit in to 0, withdra w 1 unit from credit at that bit's p

• sition.
• 1

b b b b b

k-1 k-2 k-3 1

1

Figure 2.2: after rst up date: 1 credit

• n

bit

• 1

1

b b b b b

k-1 k-2 k-3 1

Figure 2.3: after 2nd up date: 1 credit

• n

bit 1 (paid to ip bit with credit for that bit) It can b e sho wn b y an induction argumen t that ev ery time a bit i is ipp ed to 1, it will ha v e a credit

• f

1 dep

• sited

at its p

• sition.

This is b ecause all

• f

the bits to its righ t m ust ha v e b een 1 after the previous up date, hence they had credit to pa y for their ips, and w e got t w

• units
• f

credit for the up date,

• ne

to ip the bit i and the

• ther

to store as credit at i's p

• sition.

2.2.1.3 The Ph ysicist's View: (The P

• ten

tial F unction Metho d) Idea: Store pre-paid credit as a "p

• ten

tial energy"

• f

the en tire system. Use p

• ten

tial to pa y for

• p

erations whic h cost more energy than they bring in.

SLIDE 4 2-4 Lecture 2: Jan uary 23 F

• rmally

, let

• i

denote the p

• ten

tial at time i. Clearly , ~ c = c i + , where

• =
• i
• i1

, th us assume ~ c i = b, the total amortized cost is b

• n

= n X i=0 ~ c i = n X i=1 (c i +

• i
• i1

) = n X i=1 c i +

• n
• Theorem

2.3 If

• n
• (usual

ly

• =

0), then ~ c (n) = c(n) n

• b.

Imp

• rtan

t: W an t

• i
• for

all i

• n.

T

• pro

v e ~ c i b

• unded,

w e can sho w that c i

• ~

c i

• .

The p

• ten

tial metho d is applied to the Binary Coun ter problem as follo ws. Let

• i

= n um b er

• f

bits set to 1 at time i. Let

• =

0. Then 8i; i

• 0.

Consider the ith up date. There exists a j suc h that bit j is ipp ed in to a 1 and all bits less than j are ipp ed to 0. Therefore,

• i
• i1

= 1

• j:

W e w an t to sho w ~ c = 2. c i = j + 1 = 2

• (1
• j

) = ~ c i

• 2.2.2

A Queue F rom Tw

• Stac

ks W e ma y use t w

• stac

ks A and B and need to implemen t a queue. The Queue

• p

erations include: insert item x (I (x)) and delete item x (D (x)). An insert places a new item at the tail

• f

the queue, and a delete remo v es an item from the head

• f

the queue. W e are giv en a sequence

• f

insert and delete

• p

erations. Consider the follo wing algorithm. Newly inserted items are pushed in to stac k A. When a delete

• ccurs

w e rst c hec k if there are items in stac k B. If so w e p

• p

the item at the top

• f

stac k B. Otherwise all items in stac k A are p

• pp

ed

• the

stac k and all except the last

• ne

are pushed in to stac k B. It is easy to c hec k the correctness

• f

this algorithm. W e pro v e the follo wing:

SLIDE 5 Lecture 2: Jan uary 23 2-5 Claim 2.4 The 2-Stacks algorithm has amortize d c

• st

at most 3. Pro

• f:

W e use the Credit-Debit metho d. Let eac h insert

• p

eration come with 3 units and eac h delete

• p

eration come with 1 unit. Then inserting an item in to stac k A tak es 1 unit, and p

• pping

the item and inserting it in to stac k 2 tak es 2 units. Deleting the item from stac k 2 then tak es 1 unit. Then the cost (whic h is the total n um b er

• f

pushes and p

• ps)
• b

eys total cost

• 3
• #
• f

Inserts + #

• f

Deletes

• 3n;

where n is the

• v

erall n um b er

• f
• p

erations. References [CLR90] T.H. Cormen, C.E. Leiserson, and R.L. Riv est. Intr

• duction

to A lgorithms. The MIT Press, 1990. [T arjan85] R.E. T arjan. Amortized Computational Complexit y . SIAM Journal

• n

A lgebr aic and Discr ete Metho ds, 6(2):306-318, 1985.