Dynamic Set: Amortized Analysis

Tutorial 4

Dynamic Set Amortized Analysis

Computer Algorithms Design and Analysis

ReviewBinary tree

Complete binary treeFull binary tree

2-tree P115 Unlike common binary tree the base case is not an empty tree but a external node

Heap Binary search tree (BST)Balanced BST

Red-Black treehellip


In the classes we learned the following data structures

BSTHash Table

Closed address Linked listOpen address Rehashing

Dynamic SetUnion-FindwUnion-cFind


Amortized Analysis

Analyze the cost of a sequence of operationsThe total cost is averaged over all operations performedSome of the operations might be expensiveThe average cost is small

Average performance in the worst caseGives an upper bound


Amortized Analysis vs Average-case analysis

No probability is involvedGuarantee the average performance of each operation in the worst case

Amortized Analysis vs Adversary AnalysisAdversary gives worst case lower bound Amortized gives average case upper boundAdversary applies for all algorithms theoretically Amortized only applies in some special algorithms


Example Stack Operations

Consider the following two stack operationsPush push an object onto the stackPopAll pop all object from the stack

There are n operations what is the upper bound of total cost


Solution 1 Aggregate methodWe can prove the total cost of n operations in the worst cast is O(n)

The total cost of PopAll operations lt= the total cost of Push operationsThe total cost of Push lt=nThus the total cost of n operations lt=2n

So the average cost is O(n)n=O(1) in the worst case


Solution 2 Accounting methodActual cost

Push1 PopAll k where k is the number of objects in the stack

Accounting costPush 1PopAll -k

Amortized costPush 2PopAll 0

We can prove the total accounting cost is nonnegativeThus for any sequence of n operations the total amortized cost is an upper bound which is 2n


Solution 3 Potential methodDefine the potential function Φ on a stack to be the number of objects in the stackObviously it satisfies

Φ(D0)=0 Φ(Di)gt=0Amortized cost

If a Push operationPotential difference is Φ(Di)-Φ(Di-1)=(k+1)-k=1So the amortized cost of Push operation is

crsquoi=ci+Φ(Di)-Φ(Di-1)=1+1=2If a PopAll operation

Potential difference is Φ(Di)-Φ(Di-1)=0-k=-kSo the amortized cost of PopAll operation is

crsquoi=ci+Φ(Di)-Φ(Di-1)=k+(-k)=0So the total amortized cost is an upper bound which is 2n


Types of Amortized Analysis

Aggregate methodAccounting methodPotential method


(1) Aggregate Method

Consider a sequence of n operationsShow that for all n the n operations takes worse-case time T(n) in totalTherefore each operation is T(n)n (amortized cost) in the worst caseNote that this amortized cost applies to each operation even there are several types of operationsAggregate Method vs Accounting method and Potential method

The accounting method and potential method may assign different amortized cost to different type of operations


(2) Accounting MethodAssign different charges to different operationsSome operations are charged more or less than their actually costAmortized equation

amortized cost = actual cost + accounting costThe total accounting cost is the difference between the total amortized cost and the total actual costThe total accounting cost must be nonnegative at all times

AssumeEach operation i has an actual cost CiWe charge operation i a fictitious amortized cost CrsquoiWe must ensure that

Thus the total amortized costs provide an upper bound on the total actual costs


(3) Potential MethodIdea view the bank account as the potential energy of the data structureFramework

Start with an initial data structure D0 (n operations are going to perform on D0)Each operation i has an actual cost Ci Di is the data structure that results after applying the ith operation on data structure Di-1Operation i transforms Di-1 to DiDefine a potential function Φ DirarrR such that Φ(D0)=0 and Φ(Di) ge0 for all iThe amortized cost Crsquoi is defined as

Crsquoi= Ci+ Φ(Di)- Φ(Di-1)


The amortized cost bounded the actual cost

Choose an appropriate potential function is important


Example Dynamic TableGoal Make a table as small as possibleProblem What if we donrsquot know the proper size in advanceSolution Dynamic tables

Each time we insert a new element into the table we check whether there is an empty unit If so the insertion needs only 1 operateIf the table overflows we ldquogrowrdquo it by allocating a new larger table (double size of the original one) Move all items from the old table into the new one insert the new element and then free the storage for the old table The total number of operations is 1+(the size of the old table)


Insert 1Insert 2Insert 3 Insert 4Insert 5Insert 6Insert 7

1 1

2

1

2

3

4

1

2

3

4

5

6

7

Overflow

Overflow

Overflow


Worst-case analysis

Consider a sequence of n insertions The worst-case time to execute one insertion is Θ(n) Therefore the worst-case time for n insertions is n Θ(n)= Θ(n2)

WRONG


Solution 1 aggregate methodLet Ci= the cost of the ith insertion

if i-1 is an exact power of 21 otherwisei

iC ⎧

= ⎨⎩

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

=1+2lg(i-1)


The cost of n insertion is

lg( 1)

1 0

2 2 3 ( )nn

ji

i j

C n n n n nminus⎢ ⎥⎣ ⎦

= =

= + le + = = Θsum sum


Solution 2 accounting methodActual costAccounting costAmortized cost Cirsquo=3we can prove Thus a upper bound of n insertion is

1

3 ( )n

ii

C n n=

= = Θsum

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

Store 1 2 2 2 2 2 2 2 2 2Crsquoi 2 3 3 3 3 3 3 3 3 3

Consume 0 -1 -2 0 -4 0 0 0 -8 0

lg( 1)1 2 if i-1 is an exact power of 21 otherwise

i

iCminus⎧ +

= ⎨⎩


i

iaminus⎧ minus

= ⎨⎩


Solution 3 Potential methodDefine a potential function We have

If i-1 is an exact power of 2

If i-1 is not an exact power of 2

So we have an upper bound

lg lg0( ) 2 2 (assume 2 =0)iiD i ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥Φ = minus

lg lg( 1)

lg lg( 1)

(2 2 ) (2( 1) 2 )

2 (2 2 ) 2 (2( 1) ( 1)) 3

i ii

i i

C i i i

i i i i

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

= + minus minus minus minus

= + minus minus = + minus minus minus minus =

lg lg( 1)

lg lg( 1)

1 (2 2 ) (2( 1) 2 )

1 2 (2 2 ) 3

i ii

i i

C i i minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥


= + minus minus =

1

3 ( )n

ii

C n n=

= = Θsum


i 1 2 3 4 5 6 7 8 9 10

table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1

Overhead 0 1 2 0 4 0 0 0 8 0

Φ(Di) 1 2 2 4 2 4 6 8 2 4

Crsquoi 2 3 3 3 3 3 3 3 3 3

ΔΦ 1 1 0 2 -2 2 2 2 -6 2

Tutorial 4 Dynamic Set Amortized Analysis

Amortized Analysis




(2) Accounting Method

(3) Potential Method


Example Dynamic Table

Worst-case analysis


ReviewBinary tree

Complete binary treeFull binary tree

2-tree P115 Unlike common binary tree the base case is not an empty tree but a external node

Heap Binary search tree (BST)Balanced BST

Red-Black treehellip



BSTHash Table




Amortized Analysis




















































1 1

2

1

2

3

4

1

2

3

4

5

6

7

Overflow

Overflow

Overflow


Worst-case analysis


WRONG




iC ⎧

= ⎨⎩

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

=1+2lg(i-1)



lg( 1)

1 0

2 2 3 ( )nn

ji

i j


= =




1

3 ( )n

ii

C n n=

= = Θsum

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

Store 1 2 2 2 2 2 2 2 2 2Crsquoi 2 3 3 3 3 3 3 3 3 3

Consume 0 -1 -2 0 -4 0 0 0 -8 0


i

iCminus⎧ +

= ⎨⎩


i

iaminus⎧ minus

= ⎨⎩







lg lg( 1)

lg lg( 1)

(2 2 ) (2( 1) 2 )

2 (2 2 ) 2 (2( 1) ( 1)) 3

i ii

i i

C i i i

i i i i

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥



lg lg( 1)

lg lg( 1)

1 (2 2 ) (2( 1) 2 )

1 2 (2 2 ) 3

i ii

i i

C i i minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥


= + minus minus =

1

3 ( )n

ii

C n n=

= = Θsum


i 1 2 3 4 5 6 7 8 9 10

table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1

Overhead 0 1 2 0 4 0 0 0 8 0

Φ(Di) 1 2 2 4 2 4 6 8 2 4

Crsquoi 2 3 3 3 3 3 3 3 3 3

ΔΦ 1 1 0 2 -2 2 2 2 -6 2


Amortized Analysis








Worst-case analysis



BSTHash Table




Amortized Analysis




















































1 1

2

1

2

3

4

1

2

3

4

5

6

7

Overflow

Overflow

Overflow


Worst-case analysis


WRONG




iC ⎧

= ⎨⎩

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

=1+2lg(i-1)



lg( 1)

1 0

2 2 3 ( )nn

ji

i j


= =




1

3 ( )n

ii

C n n=

= = Θsum

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

Store 1 2 2 2 2 2 2 2 2 2Crsquoi 2 3 3 3 3 3 3 3 3 3

Consume 0 -1 -2 0 -4 0 0 0 -8 0


i

iCminus⎧ +

= ⎨⎩


i

iaminus⎧ minus

= ⎨⎩







lg lg( 1)

lg lg( 1)

(2 2 ) (2( 1) 2 )

2 (2 2 ) 2 (2( 1) ( 1)) 3

i ii

i i

C i i i

i i i i

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥



lg lg( 1)

lg lg( 1)

1 (2 2 ) (2( 1) 2 )

1 2 (2 2 ) 3

i ii

i i

C i i minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥


= + minus minus =

1

3 ( )n

ii

C n n=

= = Θsum


i 1 2 3 4 5 6 7 8 9 10

table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1

Overhead 0 1 2 0 4 0 0 0 8 0

Φ(Di) 1 2 2 4 2 4 6 8 2 4

Crsquoi 2 3 3 3 3 3 3 3 3 3

ΔΦ 1 1 0 2 -2 2 2 2 -6 2


Amortized Analysis








Worst-case analysis


Amortized Analysis




















































1 1

2

1

2

3

4

1

2

3

4

5

6

7

Overflow

Overflow

Overflow


Worst-case analysis


WRONG




iC ⎧

= ⎨⎩

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

=1+2lg(i-1)



lg( 1)

1 0

2 2 3 ( )nn

ji

i j


= =




1

3 ( )n

ii

C n n=

= = Θsum

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

Store 1 2 2 2 2 2 2 2 2 2Crsquoi 2 3 3 3 3 3 3 3 3 3

Consume 0 -1 -2 0 -4 0 0 0 -8 0


i

iCminus⎧ +

= ⎨⎩


i

iaminus⎧ minus

= ⎨⎩







lg lg( 1)

lg lg( 1)

(2 2 ) (2( 1) 2 )

2 (2 2 ) 2 (2( 1) ( 1)) 3

i ii

i i

C i i i

i i i i

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥



lg lg( 1)

lg lg( 1)

1 (2 2 ) (2( 1) 2 )

1 2 (2 2 ) 3

i ii

i i

C i i minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥


= + minus minus =

1

3 ( )n

ii

C n n=

= = Θsum


i 1 2 3 4 5 6 7 8 9 10

table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1

Overhead 0 1 2 0 4 0 0 0 8 0

Φ(Di) 1 2 2 4 2 4 6 8 2 4

Crsquoi 2 3 3 3 3 3 3 3 3 3

ΔΦ 1 1 0 2 -2 2 2 2 -6 2


Amortized Analysis








Worst-case analysis


















































1 1

2

1

2

3

4

1

2

3

4

5

6

7

Overflow

Overflow

Overflow


Worst-case analysis


WRONG




iC ⎧

= ⎨⎩

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

=1+2lg(i-1)



lg( 1)

1 0

2 2 3 ( )nn

ji

i j


= =




1

3 ( )n

ii

C n n=

= = Θsum

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

Store 1 2 2 2 2 2 2 2 2 2Crsquoi 2 3 3 3 3 3 3 3 3 3

Consume 0 -1 -2 0 -4 0 0 0 -8 0


i

iCminus⎧ +

= ⎨⎩


i

iaminus⎧ minus

= ⎨⎩







lg lg( 1)

lg lg( 1)

(2 2 ) (2( 1) 2 )

2 (2 2 ) 2 (2( 1) ( 1)) 3

i ii

i i

C i i i

i i i i

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥



lg lg( 1)

lg lg( 1)

1 (2 2 ) (2( 1) 2 )

1 2 (2 2 ) 3

i ii

i i

C i i minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥


= + minus minus =

1

3 ( )n

ii

C n n=

= = Θsum


i 1 2 3 4 5 6 7 8 9 10

table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1

Overhead 0 1 2 0 4 0 0 0 8 0

Φ(Di) 1 2 2 4 2 4 6 8 2 4

Crsquoi 2 3 3 3 3 3 3 3 3 3

ΔΦ 1 1 0 2 -2 2 2 2 -6 2


Amortized Analysis








Worst-case analysis


Worst-case analysis


WRONG




iC ⎧

= ⎨⎩

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

=1+2lg(i-1)



lg( 1)

1 0

2 2 3 ( )nn

ji

i j


= =




1

3 ( )n

ii

C n n=

= = Θsum

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

Store 1 2 2 2 2 2 2 2 2 2Crsquoi 2 3 3 3 3 3 3 3 3 3

Consume 0 -1 -2 0 -4 0 0 0 -8 0


i

iCminus⎧ +

= ⎨⎩


i

iaminus⎧ minus

= ⎨⎩







lg lg( 1)

lg lg( 1)

(2 2 ) (2( 1) 2 )

2 (2 2 ) 2 (2( 1) ( 1)) 3

i ii

i i

C i i i

i i i i

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥



lg lg( 1)

lg lg( 1)

1 (2 2 ) (2( 1) 2 )

1 2 (2 2 ) 3

i ii

i i

C i i minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥


= + minus minus =

1

3 ( )n

ii

C n n=

= = Θsum


i 1 2 3 4 5 6 7 8 9 10

table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1

Overhead 0 1 2 0 4 0 0 0 8 0

Φ(Di) 1 2 2 4 2 4 6 8 2 4

Crsquoi 2 3 3 3 3 3 3 3 3 3

ΔΦ 1 1 0 2 -2 2 2 2 -6 2


Amortized Analysis








Worst-case analysis




iC ⎧

= ⎨⎩

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

=1+2lg(i-1)



lg( 1)

1 0

2 2 3 ( )nn

ji

i j


= =




1

3 ( )n

ii

C n n=

= = Θsum

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

Store 1 2 2 2 2 2 2 2 2 2Crsquoi 2 3 3 3 3 3 3 3 3 3

Consume 0 -1 -2 0 -4 0 0 0 -8 0


i

iCminus⎧ +

= ⎨⎩


i

iaminus⎧ minus

= ⎨⎩







lg lg( 1)

lg lg( 1)

(2 2 ) (2( 1) 2 )

2 (2 2 ) 2 (2( 1) ( 1)) 3

i ii

i i

C i i i

i i i i

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥



lg lg( 1)

lg lg( 1)

1 (2 2 ) (2( 1) 2 )

1 2 (2 2 ) 3

i ii

i i

C i i minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥


= + minus minus =

1

3 ( )n

ii

C n n=

= = Θsum


i 1 2 3 4 5 6 7 8 9 10

table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1

Overhead 0 1 2 0 4 0 0 0 8 0

Φ(Di) 1 2 2 4 2 4 6 8 2 4

Crsquoi 2 3 3 3 3 3 3 3 3 3

ΔΦ 1 1 0 2 -2 2 2 2 -6 2


Amortized Analysis








Worst-case analysis



lg( 1)

1 0

2 2 3 ( )nn

ji

i j


= =




1

3 ( )n

ii

C n n=

= = Θsum

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

Store 1 2 2 2 2 2 2 2 2 2Crsquoi 2 3 3 3 3 3 3 3 3 3

Consume 0 -1 -2 0 -4 0 0 0 -8 0


i

iCminus⎧ +

= ⎨⎩


i

iaminus⎧ minus

= ⎨⎩







lg lg( 1)

lg lg( 1)

(2 2 ) (2( 1) 2 )

2 (2 2 ) 2 (2( 1) ( 1)) 3

i ii

i i

C i i i

i i i i

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥



lg lg( 1)

lg lg( 1)

1 (2 2 ) (2( 1) 2 )

1 2 (2 2 ) 3

i ii

i i

C i i minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥


= + minus minus =

1

3 ( )n

ii

C n n=

= = Θsum


i 1 2 3 4 5 6 7 8 9 10

table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1

Overhead 0 1 2 0 4 0 0 0 8 0

Φ(Di) 1 2 2 4 2 4 6 8 2 4

Crsquoi 2 3 3 3 3 3 3 3 3 3

ΔΦ 1 1 0 2 -2 2 2 2 -6 2


Amortized Analysis








Worst-case analysis



1

3 ( )n

ii

C n n=

= = Θsum

i 1 2 3 4 5 6 7 8 9 10table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1Overhead 0 1 2 0 4 0 0 0 8 0

Store 1 2 2 2 2 2 2 2 2 2Crsquoi 2 3 3 3 3 3 3 3 3 3

Consume 0 -1 -2 0 -4 0 0 0 -8 0


i

iCminus⎧ +

= ⎨⎩


i

iaminus⎧ minus

= ⎨⎩







lg lg( 1)

lg lg( 1)

(2 2 ) (2( 1) 2 )

2 (2 2 ) 2 (2( 1) ( 1)) 3

i ii

i i

C i i i

i i i i

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥



lg lg( 1)

lg lg( 1)

1 (2 2 ) (2( 1) 2 )

1 2 (2 2 ) 3

i ii

i i

C i i minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥


= + minus minus =

1

3 ( )n

ii

C n n=

= = Θsum


i 1 2 3 4 5 6 7 8 9 10

table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1

Overhead 0 1 2 0 4 0 0 0 8 0

Φ(Di) 1 2 2 4 2 4 6 8 2 4

Crsquoi 2 3 3 3 3 3 3 3 3 3

ΔΦ 1 1 0 2 -2 2 2 2 -6 2


Amortized Analysis








Worst-case analysis







lg lg( 1)

lg lg( 1)

(2 2 ) (2( 1) 2 )

2 (2 2 ) 2 (2( 1) ( 1)) 3

i ii

i i

C i i i

i i i i

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥



lg lg( 1)

lg lg( 1)

1 (2 2 ) (2( 1) 2 )

1 2 (2 2 ) 3

i ii

i i

C i i minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

minus⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥


= + minus minus =

1

3 ( )n

ii

C n n=

= = Θsum


i 1 2 3 4 5 6 7 8 9 10

table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1

Overhead 0 1 2 0 4 0 0 0 8 0

Φ(Di) 1 2 2 4 2 4 6 8 2 4

Crsquoi 2 3 3 3 3 3 3 3 3 3

ΔΦ 1 1 0 2 -2 2 2 2 -6 2


Amortized Analysis








Worst-case analysis


i 1 2 3 4 5 6 7 8 9 10

table size 1 2 4 4 8 8 8 8 16 16

Ci 1 2 3 1 5 1 1 1 9 1

Overhead 0 1 2 0 4 0 0 0 8 0

Φ(Di) 1 2 2 4 2 4 6 8 2 4

Crsquoi 2 3 3 3 3 3 3 3 3 3

ΔΦ 1 1 0 2 -2 2 2 2 -6 2


Amortized Analysis








Worst-case analysis

Dynamic Set: Amortized Analysis

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Transcript of Dynamic Set: Amortized Analysis