UMass Lowell Computer Science 91.503 Graduate Analysis of Algorithms

Prof. Karen Daniels Spring, 2010

Lecture 3Tuesday, 2/9/10

Amortized Analysis

Overview

• Amortize: “To pay off a debt, usually by periodic payments” [Websters]

• Amortized Analysis:• “creative accounting” for operations• can show average cost of an operation is small (when averaged

over sequence of operations, not distribution of inputs) even though a single operation in the sequence is expensive

• result must hold for any sequence of these operations• no probability is involved (unlike average-case analysis)

• guarantee holds in worst-case • analysis method only; no effect on code operation

Overview (continued)

• 3 ways to determine amortized cost of an operation that is part of a sequence of operations:• Aggregate Method

• find upper bound T(n) on total cost of sequence of n operations• amortized cost = average cost per operation = T(n)/n• same for all the operations in the sequence

• Accounting Method• amortized cost can differ across operations• overcharge some operations early in sequence• store overcharge as “prepaid credit” on specific data structure objects

• Potential Method• amortized cost can differ across operations (as in accounting method)• overcharge some operations early in sequence (as in accounting method)• store overcharge as “potential energy” of data structure as a whole

• (unlike accounting method)

Aggregate Method:Stack Operations

• Aggregate Method• find upper bound T(n) on total cost of sequence of n operations• amortized cost = average cost per operation = T(n)/n• same for all the operations in the sequence

• Traditional Stack Operations• PUSH(S,x) pushes object x onto stack S• POP(S) pops top of stack S, returns popped object • O(1) time: consider cost as 1 for our discussion• Total actual cost of sequence of n PUSH/POP

operations = n• STACK-EMPTY(S) time in Q(1)

Aggregate Method:Stack Operations (continued)

• New Stack Operation• MULTIPOP(S,k) pops top k elements off stack S

• pops entire stack if it has < k items

• MULTIPOP actual cost for stack containing s items: • Use cost =1 for each POP• Cost = min(s,k)• Worst-case cost in O(s) in O(n)

MULTIPOP(S,k)

1 while not STACK-EMPTY(S) and k = 0

2 do POP(S)

3 k k - 1

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

Aggregate Method:Stack Operations (continued)

• Sequence of n PUSH, POP, MULTIPOP ops• initially empty stack • MULTIPOP worst-case O(n) O(n2) for sequence

• Aggregate method yields tighter upper bound• Sequence of n operations has O(n) worst-case cost

• Each item can be popped at most once for each push• # POP calls (including ones in MULTIPOP) <= #push calls

• <= n• Average cost of an operation = O(n)/n = O(1)

• = amortized cost of each operation• holds for PUSH, POP and MULTIPOP

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

Accounting Method

• Accounting Method• amortized cost can differ across operations• overcharge some operations early in sequence• store overcharge as “prepaid credit” on specific data structure

objects• Let be actual cost of ith operation• Let be amortized cost of ith operation (what we charge)

• Total amortized cost of sequence of operations must be upper bound on total actual cost of sequence

• Total credit in data structure =• must be nonnegative for all n

n

ii

n

ii cc

11

ˆ

n

ii

n

ii cc

11

ˆ

icic

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

Accounting Method:Stack Operations

• Paying for a sequence using amortized cost:• start with empty stack• PUSH of an item always precedes POP, MULTIPOP

• pay for PUSH & store 1 unit of credit • credit for each item pays for actual POP, MULTIPOP cost of that item• credit never “goes negative”

• total amortized cost of any sequence of n ops is in O(n)• amortized cost is upper bound on total actual cost

• PUSH 1 2• POP 1 0• MULTIPOP min(k,s) 0

Operation Actual Cost Assigned Amortized Cost

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

Potential Method

• Potential Method• amortized cost can differ across operations (as in accounting method)• overcharge some operations early in sequence (as in accounting method)• store overcharge as “potential energy” of data structure as a whole (unlike accounting method)

• Let ci be actual cost of ith operation• Let Di be data structure after applying ith operation• Let F(Di ) be potential associated with Di• Amortized cost of ith operation:

• Total amortized cost of n operations:

• Require: so total amortized cost is upper bound on total actual cost

• Since n might not be known in advance, guarantee “payment in advance” by requiring

)()(ˆ 1FF iiii DDcc

)()( 0DDn FF

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

)()())()((ˆ 01

111

DDcDDcc n

n

iiii

n

ii

n

ii FFFF

terms telescope

)()( 0DDi FF

Potential Method: Stack Operations

• Potential function value = number of items in stack• guarantees nonnegative potential after ith operation• Amortized operation costs (assuming stack has s items)

• PUSH:• potential difference= • amortized cost =

• MULTIPOP(S,k) pops k’=min(k,s) items off stack• potential difference= • amortized cost =

• POP amortized cost also = 0• Amortized cost O(1) total amortized cost of

sequence of n operations in O(n)

1)1()()( 1 FF ssDD ii

211)()(ˆ 1 FF iiii DDcc

')()( 1 kDD ii FF

0'')()(ˆ 1 FF kkDDcc iiii

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

Dynamic Tables:Overview

• Dynamic Table T:• array of slots

• Ignore implementation choices: stack, heap, hash table...• if too full, increase size & copy entries to T’• if too empty, decrease size & copy entries to T’

• Analyze dynamic table insert and delete • Actual expansion or contraction cost is large• Show amortized cost of insert or delete in O(1)

• Load factor a(T) = num[T]/size[T]• empty table: a(T) = 1 (convention guaranteeing load factor can be

lower bounded by a useful constant)• full table: a(T) = 1

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

Dynamic Tables:Table (Expansion Only)

• Load factor bounds (double size when T is full):• Sequence of n inserts on initially empty table

• Worst-case cost of insert is in O(n)• Worst-case cost of sequence of n inserts is in O(n2)

1)(21

Ta

LOOSE

“elementary” insertion

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

Double only when table is already full.

WHY?

Dynamic Tables:Table Expansion (cont.)

• Aggregate Method:• ci= i if i-1 is exact power of 2 1 otherwise• total cost of n inserts =

Accounting Method: • charge cost = 3 for each element inserted• intuition for 3

• each item pays for 3 elementary insertions• inserting itself into current table• expansion: moving itself• expansion: moving another item that has already been moved

Amortized Analysis

nnnnc

n

j

jn

ii 322

lg

01

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

count only elementary insertions

Dynamic Tables:Table Expansion (cont.)

• Potential Method:• Value of potential function F(T)

• 0 right after expansion (then becomes 2)• builds to table size by time table is full• always nonnegative, so sum of amortized costs of

n inserts is upper bound on sum of actual costs

Amortized Analysis][][2)( TsizeTnumT F

3)2()2(1ˆ 111 FF iiiiiiii sizenumsizenumcc

• Amortized cost of ith insert • Fi = potential after ith operation • Case 1: insert does not cause expansion

• Case 2: insert causes expansion 3)2()2(ˆ 111 FF iiiiiiiii sizenumsizenumnumcc

11 ii sizenum 11 ii numnum 12 ii sizesizeuse these:

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

3 functions: sizei, numi, Fi

Dynamic Tables:Table Expansion & Contraction

• Load factor bounds:• (double size when T is full) (halve size when T is ¼ full):

• DELETE pseudocode analogous to INSERT

1)(41

Ta

• Potential Method:• Value of potential function F(T)

• = 0 for empty table• 0 right after expansion or contraction• builds as a(T) increases to 1 or decreases to ¼• always nonnegative, so sum of amortized costs of

n inserts is upper bound on sum of actual costs• = 0 when a(T)=1/2• = num[T] when a(T)=1• = num[T] when a(T)=1/4

Amortized Analysis2/1)(][][2 )( F TifTsizeTnumT a2/1)(][2/][ TifTnumTsize a

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

count elementary insertions & deletions

same as INSERT

Dynamic Tables:Table Expansion & Contraction

Amortized Analysis Potential Method

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

Dynamic Tables:Table Expansion & Contraction

Amortized Analysis

• Analyze cost of sequence of n inserts and/or deletes• Amortized cost of ith operation

• Case 1: INSERT • Case 1a: ai-1 >= ½. By previous insert analysis:

• Case 1b: ai-1 < ½ and ai < ½

• Case 1c: ai-1 < ½ and ai >= ½

3))2/(()2(1ˆ 111 FF iiiiiiii numsizesizenumcc

0))2/(())2/((1ˆ 111 FF iiiiiiii numsizenumsizecc

Potential Method

3ˆ icholds whether or not table expands

no expansion

no expansion

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

2/1)(][][2)( F TifTsizeTnumT a2/1)(][2/][ TifTnumTsize a

see derivation in textbook

Dynamic Tables:Table Expansion & Contraction

Amortized Analysis• Amortized cost of ith operation (continued)

Potential Method

• Case 2: DELETE • Case 2a: ai-1 >= ½. • Case 2b: ai-1 < ½ and ai < ½

• Case 2c: ai-1 < ½ and ai < ½

no contraction

contraction

2)2/()2/(1ˆ 111 FF iiiiiiii numsizenumsizecc

1)2/()2/()1(ˆ 111 FF iiiiiiiii numsizenumsizenumcc

Conclusion: amortized cost of each operation is bounded above by a constant, so time for sequence of n operations is O(n).

source: 91.503 textbook source: 91.503 textbook CormenCormen et al. et al.

textbook exercise

Example: Dynamic Closest Pair

source: “Fast hierarchical clustering and other applications of dynamic closest pairs,” David Eppstein, Journal of Experimental Algorithmics, Vol. 5, August 2000.

S S S

Example: Dynamic Closest Pair (continued)

source: “Fast hierarchical clustering and other applications of dynamic closest pairs,” David Eppstein, Journal of Experimental Algorithmics, Vol. 5, August 2000.

S S S

Example: Dynamic Closest Pair (continued) Rules

nk log

source: “Fast hierarchical clustering and other applications of dynamic closest pairs,” David Eppstein, Journal of Experimental Algorithmics, Vol. 5, August 2000.

• Partition dynamic set S into subsets.• Each subset Si has an associated digraph Gi consisting of a set of

disjoint, directed paths.• Total number of edges in all graphs remains linear

• Combine and rebuild if number of edges reaches 2n.• Closest pair is always in some Gi.• Initially all points are in single set S1.• Operations:

• Create Gi for a subset Si.• Insert a point.• Delete a point.• Merge subsets until .

nk log

We use log base 2.

Example: Dynamic Closest Pair (continued) Rules: Operations

nk log

source: “Fast hierarchical clustering and other applications of dynamic closest pairs,” David Eppstein, Journal of Experimental Algorithmics, Vol. 5, August 2000.• Create Gi for a subset Si:

• Select starting point (we choose leftmost (or higher one in case of a tie))• Iteratively extend the path P, selecting next vertex as:

• Case 1: nearest neighbor in S \ P if last point on path belongs to Si• Case 2: nearest neighbor in Si \ P if last point on path belongs to S \ Si

• Insert a point x:• Create new subset Sk+1={x}.• Merge subsets if necessary. • Create Gi for new or merged subsets.

• Delete a point x:• Create new subset Sk+1= all points y such that (y,x) is a directed edge in some Gi.• Remove x and adjacent edges from all Gi. (We also remove y from its subset.)• Merge subsets if necessary. • Create Gi for new or merged subsets.

• Merge subsets until :• Choose subsets Si and Sj to minimize size ratio |Sj|/ |Si|.

See handout for example.

Example: Dynamic Closest Pair (continued) Potential Function

• Potential for a subset Si : Fi = n|Si|log|Si|.• Total potential F = n2logn - S Fi.• Paper proves this Theorem:

• HW#3 presents variation on these operations and asks you to determine the amortized cost per insertion and deletion.

source: “Fast hierarchical clustering and other applications of dynamic closest pairs,” David Eppstein, Journal of Experimental Algorithmics, Vol. 5, August 2000.

Theorem: The data structure maintains the closest pair in S in O(n) space, amortized time O(nlogn) per insertion, and amortized time O(nlog2n) per deletion.