Amortized Analysis - Part 2 - Dynamic Tables. Objective: In this lecture, we shall explore Dynamic tables and its amortized analysis in detail.

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1 Idia Istitute of Iformatio Techology Desig ad Maufacturig, Kacheepuram Cheai , Idia A Autoomous Istitute uder MHRD, Govt of Idia COM 501 Advaced Data Structures ad Algorithms Istructor N.Sadagopa Scribe: Kuladeep Rejith.P Amortized Aalysis - Part - Dyamic Tables Objective: I this lecture, we shall explore Dyamic tables ad its amortized aalysis i detail. Dyamic Tables Programmig laguages such as C++ ad Java supports vector which is a dyamic array (table) that grows or shriks based o the cotext (applicatio). I this sectio, we shall aalyze dyamic tables i detail by cosiderig two operatios, amely, isert ad delete. Sice a applicatio may perform a sequece of isert ad delete operatios i some order, it is appropriate to ivestigate dyamic tables from the perspective of amortized aalysis. It is importat to come up with strategies for growig the table ad shrikig the table so that the uderlyig applicatio hadles requests (isert/delete) icely ad the applicatio does ot go for expasio or cotractio frequetly. Wheever the table overflows as the curret table size ca ot hadle ew requests, we grow the table by allocatig a ew, larger table ad free the space of the old table. Similarly if may objects are deleted from the table, it may be worthwhile to reallocate the table with a smaller size. Towards this ed, we adopt the followig strategy; The size of the table doubles as the table gets filled. i.e., the size expads from x x+1. Similarly, we shall discuss later a strategy for cotractio if too may deletios happe i a row. Isertio i a Dyamic Table Cosider a sequece of isertios. The worst-case time to execute oe isertio is Θ(). This is true because i th isertio triggers expasio. i.e., for i th isertio there is o free slot i the table ad the table must be expaded to accommodate the ew elemet. Time for copyig the cotets of old table to the ew table is θ() followed by isertio which icurs θ(1). Therefore, the worst-case time for isertios is Θ() = Θ( ). But this boud is ot tight because, expasio does ot occur so ofte i the course of operatios. Usig amortized aalysis, we see that the worst-case cost for isertios is oly Θ(). We assume that = k, where is the size of the table. Aggregate Aalysis Iitially the table is empty. i th operatio triggers a expasio oly whe i 1 is a power of. x x+1 (expasio cost is x ). Therefore, the total cost of isert operatios is, log AC = }{{} + i 1 per each isert i=1 }{{} expasio cost < < 3 = O() Amortized Cost = O() = O(1). Therefore, the cost of each isert is O(1) amortized.

2 Aalysis usig a potetial fuctio Amortized cost= Actual cost + chage i potetial. We shall work with the followig potetial fuctio; φ(t ) = Num(T ) Size(T ). N = Num(T ), the umber of elemets i table T. S = Size(T ), the size of the table T. Iitially, Num(T ) = 0 ad Size(T ) = 0. Therefore, φ(t ) = 0 Note: potetial fuctio must be defied i such a way that the potetial associated with data structure must be positive for all cofiguratios. i.e., φ(t ) 0 for all cofiguratios. Remark: 1. If Num(T ) = Size(T ) = φ(t ) = Num(T ). If Num(T ) = Size(T ) = φ(t ) = 0 3. Load Factor α = Num(T ) Size(T ). A measure of how much percetage of table is filled. Amortized cost for i th Isertio ito the Table AC i = c i + (φ i φ i 1 ) There are two cases possible here, either the i th isertio triggers a expasio or it does ot trigger a expasio. Case 1: i th isertio does ot trigger a expasio. S i 1 = S i N i = N i AC i = 1 + N i S i (N i 1) + S i 1 = 1 + N i S i N i + + S i = 3 Case : i th isertio triggers a expasio. Sice cotets of the old table must be copied to the ew table followed by isert, the actual cost is N i. S i 1 = Si N i 1 = N i 1 AC i = N i + N i S i (N i 1) + S i 1 = N i + N i S i N i + + Si = N i + Si + 1 But, N i = Si AC i = Si Si = 3 Thus, the amortized cost of isert is 3 = O(1). cost is O(1) = O(). Therefore, for a sequece of iserts, the amortized Remark: 1. Oce the table expads, the potetial associated with it is zero ad whe the table is full, the potetial is Num(T ). I all other cofiguratios, the potetial is betwee 0 ad Num(T ) ad thus, φ(t ) 0 always. Amortized cost for Deletio = 1 + N i S i (N i 1 S i 1 ) = 1 + (N i 1 1) S i N i 1 + S i Sice S i = S i 1, N i = N i 1 1 = 1

3 Thus, the amortized cost of delete is 1 = O(1). Remark: I our earlier discussio, we focused o expasio whe a ew isert triggers expasio. However, we did ot focus o cotractio whe the table has a few elemets. I the ext sectio, we shall discuss a strategy for cotractio. Strategy for expasio ad cotractio Strategy for expasio is same as the strategy we discussed before. Expad whe the table is full ad create a ew table whose size is twice the size of the old table. The strategy for cotractio; if the table is half full ad i th operatio which is delete makes the table size go below size, the perform cotractio. Although, these strategies appear good, it triggers too may expasios ad cotractios as we ca see from the followig example. Cosider the followig sequece of operatios o the Table: I, I, I, I, I,...,I, D, D, I, I, D, D, I, I, D, D, I,... }{{}}{{} iserts operatios size + 1 operatio is isert, which triggers a expasio. Subsequetly, the table will cotai + 1 elemets. The subsequet two deletios brigs the table size to size 1 which itur triggers a cotractio. For the above example, we ca see that expasio ad cotractio happe alterately for 4 times. Let us aalyze the cost for the above sequece; The first operatios icurs a amortized cost of 3 O() = O(). The secod operatios icurs a amortized cost of O() (/) = O( ). The total cost of operatios is O( ) ad the amortized cost of each operatio is O(). Drawback of this strategy: -After a expasio, we do ot perform eough deletios to pay for a cotractio. -After a cotractio, we do ot perform eough isertios to pay for a expasio. Modified Strategy: We ca improve this strategy by allowig α(t ) to drop below 1. We cotiue to double the size whe a object is iserted ito a full table but, we cotract the table whe deletio causes α < 1 4. Therefore, 1 4 α(t ) 1. We ow defie a ew Potetial fuctio correspodig to the ew strategy, φ(t ) = Num(T ) Size(T ) for α 1 = Size(T ) Num(T ) for α < 1 Now, load factor α oscillates betwee 1 4 ad 1. Most importatly, the choice of potetial fuctio from the above two is also dictated by α. Suppose, i th operatio is a isert, the possible coditios are: Case 1: α i 1 1 α i 1 : Aalysis is idetical to that of table expasio i previous sectio AC i = 3 Case : α i 1 < 1 α i < 1 : = 1 + ( Si N i) ( Si 1 N i Si 1 + N i 1 = 0 this operatio comes for free, o eed to sped aythig from potetial. 3

4 Case 3: α i 1 < 1 α i 1 : = 1 + N i S i ( Si 1 = 1 + N i 1 + S i Si 1 + N i 1 ) = 3 + 3N i 1 3Si 1 < 3 + 3Si 1 3Si 1 < 3 If we charge 1 or per operatio, it works fie. Suppose, i th operatio is delete, the possible coditios are: Case 4: α i 1 1 α i 1 : = 1 + N i S i (N i 1 S i 1 ) = 1 + N i S i N i + S i = 1 Case 5: α i 1 1 α i < 1 : = 1 + ( Si N i) (N i 1 S i 1 ) N i N i 1 + S i = + 3Si 3N i 1 = + 3Si 3N i 3 = 1 Sice Ni S i < 1, N i < Si. α i 1 = 1 4 : This triggers a cotractio S i = Si 1, N i = Si 1 ad c i = N i + 1 = 1 + N i + ( Si N i) ( Si 1 = 1 + N i + Si N i S i + Si = 1 Case 6: α i α i < 1 : N i Si 1 + N i 1 Rearrage the terms; = 1 + N i 1 + Si N i Si = Therefore, the amortized cost of deletio is at most, which is O(1). Sice the amortized cost of each operatio is bouded above by a costat, for ay sequece of operatios cosistig of isert ad delete, o a Dyamic Table is O() ad O(1) amortized per operatio. 4

5 Ackowledgemets: Lecture cotets preseted i this module ad subsequet modules are based o the followig text books ad most importatly, author has greatly leart from lectures by algorithm expoets affiliated to IIT Madras/IMSc; Prof C. Padu Raga, Prof N.S.Narayaaswamy, Prof Vekatesh Rama, ad Prof Aurag Mittal. Author sicerely ackowledges all of them. Special thaks to Teachig Assistats Mr.Rejith.P ad Ms.Dhaalakshmi.S for their sicere ad dedicated effort ad makig this scribe possible. Author has beefited a lot by teachig this course to seior udergraduate studets ad juior udergraduate studets who have also cotributed to this scribe i may ways. Author sicerely thaks all of them. Refereces: 1. E.Horowitz, S.Sahi, S.Rajasekara, Fudametals of Computer Algorithms, Galgotia Publicatios.. T.H. Corme, C.E. Leiserso, R.L.Rivest, C.Stei, Itroductio to Algorithms, PHI. 3. Sara Baase, A.V.Gelder, Computer Algorithms, Pearso. 4. Eva Tardos ad Kleiberg, Algorithm Desig, Pearso. 5