W orst Case Ecien t Data Structures Gerth Stlting Bro dal - PowerPoint PPT Presentation

W orst Case E�cien t Data Structures Gerth St�lting Bro dal BRICS Departmen t of Computer Science Univ ersit y of Aarh us and Max�Planc k�Institut f � ur Informatik Saarbr uc � k en �

Ov erview � Priorit y queues Comparison based data structures � Lo w er b ounds for comparison based data structures � RAM data structures � P arallel data structures � � P artial p ersisten t data structures G� S� Bro dal� W orst case e�cien t data structures �

Priorit y Queues Main tain a set of elemen ts from a totally ordered univ erse �sa y n in tegers� under the op erations� � FindMin � Q � �� t � Q� e � Inser �� DeleteMin � Q � �� Delete � Q� e �� Meld � Q � � Q � � �� DecreaseKey � Q� �� e� e �� Assume the lo cation of is kno wn� e G� S� Bro dal� W orst case e�cien t data structures �

Priorit y Queues Man y priorit y queues are based on heap ordered trees� �� i�e� � eac h no de stores an elemen t and the elemen t stored �� at a no de is � the elemen t stored at the no de�s parren t� �� The priorit y queues �heaps� of Willi ams are �� based on one heap ordered binary tree� �� and can b e p er� Inser t DeleteMin �� formed in �log n � time� O Willi ams �� G� S� Bro dal� W orst case e�cien t data structures �

Linking Heap Ordered T rees More recen t data structures are �� based on linking heap ordered �� trees of the same size� �� Ex�� Binomial Queues are based on �log n � heap ordered trees� O �� Inser t �� Binomial queues supp ort and in amortized Theorem Inser t Meld constan t time and in amortized �log n � time� O DeleteMin V uillemin �� G� S� Bro dal� W orst case e�cien t data structures �

Constan t Time Meldable Priorit y Queues and can b e supp orted in w orst case Theorem Inser t Meld constan t time and in w orst case �log n � time� DeleteMin O � One heap ordered tree� �� 0 � One linking p er and Meld � Inser t �� 3 � �� or � sons of eac h rank �� 2 1 1 1 0 2 less than the paren t�s rank� �� one arbitrary rank ed son� 1 0 0 0 0 1 1 0 �� The ro ot has rank �� 0 0 0 1 0 �� 0 0 Bro dal �� G� S� Bro dal� W orst case e�cien t data structures �

Constan t Time DecreaseKey Fib onacci heaps supp ort in amortized Theorem DecreaseKey constan t time and in amortized �log n � time� DeleteMin O F redman� T arjan �� Relaxed heaps supp ort in w orst case Theorem DecreaseKey constan t time and in w orst case �log n � time� O DeleteMin Meld requires ��log n � time� Driscoll� Gab o w� Shrairman� T arjan �� and can b e supp orted in w orst Theorem DecreaseKey Meld case constan t time and in w orst case �log n � time� O DeleteMin Bro dal �� heap ordered trees� O � Relaxed heaps� � A n um b er of in v arian ts to solv e the tec hnical dep endencies � G� S� Bro dal� W orst case e�cien t data structures �

Comparison Based Priorit y Queues Driscoll Gab o w F redman Shrairman Williams V uillemin Bro dal T arjan T arjan Bro dal �� Heaps Binomial Fib onacci Relaxed Queues� Heaps� Heaps FindMin O�� O�� O�� O�� O�� O�� Inser t O�log n � O�� O�� O�� O�� O�� Meld O� n � O�� O�� O�� O�log n � O�� Delete�Min� O�log n � O�log n � O�log n � O�log n � O�log n � O�log n � DecreaseKey O�log n � O�log n � O�log n � O�� O�� O�� Amortized b ounds G� S� Bro dal� W orst case e�cien t data structures �

Comparison Based Priorit y Queues Lo w er Bounds Comparison based sorting requires �� n log n � Theorem comparisons� � or require ��log n � comparisons� Inser t DeleteMin If and mak e � t � comparisons� then Theorem Inser t Delete O n requires comparisons� FindMin � t � � O � a doubly link ed list is an optimal priorit y queue implemen tation� � If mak es o � n � comparisons� and � n � Theorem Meld FindMin O comparisons� �� then and require ��log n � � � Delete DeleteMin comparisons� Bro dal� Chaudh uri� Radhakrishnan �� G� S� Bro dal� W orst case e�cien t data structures �

RAM Priorit y Queues A unit cost Random Access Mac hine with w ord size � Mo del� w �� shifting� bit�wise b o olean op erations� W ord op erations� w In tegers in the range � �� Elemen ts� Op erations� � FindMin � Q � � t � Q� e � Inser � Delete � Q� e � � Pred � Q� e �� Pred � f �� g � �� The ab o v e op erations can b e p erformed in �log � time� Theorem O w v an Emde Boas �� O � an �log log n � priorit y queue for � log n � O w G� S� Bro dal� W orst case e�cien t data structures ��

RAM Priorit y Queues v an Emde Boas Thorup� Andersson Bro dal �� FindMin �� O O O O O p Inser t O �log w � O �log log n � O � log n � O � f � n �� O �log log n � p Delete �log � �log log n � � log n � � f � n �� log log n � O w O O O O p log log n n Pred �log � � log n � � � � � O w O O O � n � log log f n p log log � n � log n � f � n �Amortized b ounds G� S� Bro dal� W orst case e�cien t data structures ��

Outline of RAM Priorit y Queue � � � n � lev els f van Emde Boas � f � n � � elemen ts �t in to a w ord � log n lev els � n � f �� f � n �� P ac k ed searc h trees of degree � with bu�ers of dela y ed Inser t and op erations� supp orting and in w orst Delete Inser t Delete case � f � n �� time� O � Tw o lev el data structure �v an Emde Boas and pac k ed�� P ac k ed searc h trees� Andersson �� Bu�er trees for external memory � Arge �� List merging in �� w ords� Alb ers� Hagerup �� O � Standard deamortization tec hniques� Bro dal �� G� S� Bro dal� W orst case e�cien t data structures ��

Adopting P arallelism to Priorit y Queues Question� Is it p ossible to obtain comparison based priorit y queues supp orting op erations in o �log n � time b y using a non�constan t n um b er of pro cessors � Answ er� Y es� � n � pro cessors can supp ort and O Inser t DeleteMin in constan t time� P P P P P P � � � � � � �� F olklore G� S� Bro dal� W orst case e�cien t data structures ��

Outline of P arallel Priorit y Queue P P P P � � � � �� i � Pro cessor P main tains �� or � trees of size � � i � P arallel linking and unlinking of trees� �log n � pro cessors can supp ort t � and Theorem O Inser Meld in constan t time� An extension of the data structure DeleteMin supp orts and in constan t time to o� Delete DecreaseKey Bro dal �� G� S� Bro dal� W orst case e�cien t data structures ��

W orst Case Ecien t Data Structures Gerth Stlting Bro dal - PowerPoint PPT Presentation

W orst Case Ecien t Data Structures Gerth Stlting Bro dal BRICS Departmen t of Computer Science Univ ersit y of Aarh us and MaxPlanc kInstitut f ur Informatik Saarbr uc k en Ov erview

Exams Gerth Stlting Brodal Algorithms and Data Structures Retreat, Sandbjerg, Denmark, March 3,

Data Structures and Models of Computation Gerth Stlting Brodal Inauguration talk, Department

Beyond assertion: setup and teardown UN IT TES TIN G F OR DATA S CIEN CE IN P YTH ON Dibya

Mastering assert statements UN IT TES TIN G F OR DATA S CIEN CE IN P YTH ON Dibya Chakravorty

Cache-Oblivious Algorithms and Data Structures Gerth Stlting Brodal University of Aarhus

Skewed Binary Search Trees 6 2 10 1 3 7 12 4 8 11 13 5 9 14 15 Gerth Stlting

Bu Busines ess a and S Sci cien en,fic W fic Wri,ng g I N S T R U C TO R :

Cache Oblivious Sorting Gerth Stlting Brodal University of Aarhus Algorithms and Data

Massive Data Algorithmics Gerth Stlting Brodal Aarhus University Forskningsdag for

Worst-Case Ef fi cient External-Memory Priority Queues Gerth Stlting Brodal

Why unit test? UN IT TES TIN G F OR DATA S CIEN CE IN P YTH ON Dibya Chakravorty Test

Comparison Based Dictionaries: Fault Tolerance versus I/O Efficiency Gerth Stlting Brodal

Optimal Finger Search Trees in the Pointer Machine Gerth Stlting Brodal BRICS University of

Data Structures 1 / 27 Built-in Data Structures Values can be collected in data structures:

Exercises: Special Case Search Structures Design super-fast search structures for the following

The Case for R244 Learned Index Structures Michael Chi Ian Tang Kraska, T., Beutel, A., Chi, E.

Python and Data Structures (continued) Tyler Moore CSE 3353, SMU, Dallas, TX February 5, 2013

MA/CSSE 474 Theory of Computation More Math Review Many of today's ICQ questions involve

Fra ss es Theorem A class of finite structures is the age of a countable homogeneous

Topological spaces of monadic MV-algebras R. Grigolia (Tbilisi), A. Di Nola and G. Lenzi (Salerno)

Preference Aggregation with Restricted Ballot Languages: Sincerity and Strategy-Proofness Ulle

total order broadcast VL Networked Embedded Systems Markus Kammerstetter (e0226196) overview

Theory of Computation Course note based on Computability, Complexity, and Languages: Fundamentals

Induction in Isabelle: Lecture 14 1 Notation In lecture 14 we introduced some recursive