1 / 23

Lecture 3 Tuesday, 2/9/10 Amortized Analysis

UMass Lowell Computer Science 91.503 Graduate Analysis of Algorithms Prof. Karen Daniels Spring, 2010. Lecture 3 Tuesday, 2/9/10 Amortized Analysis. Overview. Amortize: “To pay off a debt, usually by periodic payments” [ Websters ] Amortized Analysis: “creative accounting” for operations

cael
Download Presentation

Lecture 3 Tuesday, 2/9/10 Amortized Analysis

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. UMass Lowell Computer Science 91.503GraduateAnalysis of AlgorithmsProf. Karen DanielsSpring, 2010 Lecture 3 Tuesday, 2/9/10 Amortized Analysis

  2. 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

  3. 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)

  4. Aggregate Method:Stack Operations see example • 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)

  5. MULTIPOP(S,k) 1 while not STACK-EMPTY(S) and k = 0 2 do POP(S) 3 k k - 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)

  6. 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

  7. Accounting Method stay out of debt! • 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 amortizedcost 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 nonnegativefor all n

  8. Actual Cost Operation Assigned Amortized Cost • PUSH 1 2 • POP 1 0 • MULTIPOP min(k,s) 0 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 see example

  9. terms telescope 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

  10. 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) see example

  11. 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

  12. 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) WHY? LOOSE “elementary” insertion Double only when table is already full.

  13. Dynamic Tables:Table Expansion (cont.) Amortized Analysis see example • Aggregate Method: • ci= i if i-1 is exact power of 2 • 1 otherwise • total cost of n inserts = count only elementary insertions 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

  14. Dynamic Tables:Table Expansion (cont.) Amortized Analysis • 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 cost of ith insert • Fi= potential after ith operation • Case 1: insert does not cause expansion • Case 2: insert causes expansion 3 functions: sizei, numi, Fi use these:

  15. Dynamic Tables:Table Expansion & Contraction count elementary insertions & deletions • Load factor bounds: • (double size when T is full) (halve size when T is ¼ full): • DELETE pseudocode analogous to INSERT same as INSERT Amortized Analysis • 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

  16. Dynamic Tables:Table Expansion & Contraction Amortized Analysis Potential Method

  17. Dynamic Tables:Table Expansion & Contraction Amortized Analysis Potential Method • 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 holds whether or not table expands no expansion no expansion see derivation in textbook

  18. Dynamic Tables:Table Expansion & Contraction Amortized Analysis Potential Method • Amortized cost of ith operation (continued) • Case 2: DELETE • Case 2a: ai-1 >= ½. • Case 2b: ai-1 < ½ and ai < ½ • Case 2c: ai-1 < ½ and ai < ½ textbook exercise no contraction contraction Conclusion: amortized cost of each operation is bounded above by a constant, so time for sequence of n operations is O(n).

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

  20. Example: Dynamic Closest Pair (continued) S S S source: “Fast hierarchical clustering and other applications of dynamic closest pairs,” David Eppstein, Journal of Experimental Algorithmics, Vol. 5, August 2000.

  21. Example: Dynamic Closest Pair (continued) Rules • 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 . We use log base 2. source: “Fast hierarchical clustering and other applications of dynamic closest pairs,” David Eppstein, Journal of Experimental Algorithmics, Vol. 5, August 2000.

  22. Example: Dynamic Closest Pair (continued) Rules: Operations 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 Gifor 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 Gifor new or merged subsets. • Merge subsets until : • Choose subsets Siand Sjto minimize size ratio |Sj|/ |Si|. See handout for example.

  23. Example: Dynamic Closest Pair (continued) Potential Function source: “Fast hierarchical clustering and other applications of dynamic closest pairs,” David Eppstein, Journal of Experimental Algorithmics, Vol. 5, August 2000. • Potential for a subset Si : Fi = n|Si|log|Si|. • Total potential F = n2logn - SFi. • Paper proves this Theorem: • HW#3 presents variation on these operations and asks you to determine the amortized cost per insertion and deletion. 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.

More Related