Elementary Data Structures Stacks, Queues, & Lists Amortized analysis Trees

Elementary Data Structures2 Abstract Data Types (ADTs) An abstract data type (ADT) is an abstraction of a data structure An ADT specifies: Data stored Operations on the data Error conditions associated with operations Example: ADT modeling a simple stock trading system The data stored are buy/sell orders The operations supported are  order buy(stock, shares, price)  order sell(stock, shares, price)  void cancel(order) Error conditions:  Buy/sell a nonexistent stock  Cancel a nonexistent order

Stacks

Elementary Data Structures4 Outline and Reading The Stack ADT (§2.1.1) Applications of Stacks (§2.1.1) Array-based implementation (§2.1.1) Growable array-based stack (§1.5)

Elementary Data Structures5 The Stack ADT The Stack ADT stores arbitrary objects Insertions and deletions follow the last-in first-out scheme Think of a spring-loaded plate dispenser Main stack operations: push(object): inserts an element object pop(): removes and returns the last inserted element Auxiliary stack operations: object top(): returns the last inserted element without removing it integer size(): returns the number of elements stored boolean isEmpty(): indicates whether no elements are stored

Elementary Data Structures6 Exceptions Attempting the execution of an operation of ADT may sometimes cause an error condition, called an exception Exceptions are said to be “thrown” by an operation that cannot be executed In the Stack ADT, operations pop and top cannot be performed if the stack is empty Attempting the execution of pop or top on an empty stack throws an EmptyStackException

Elementary Data Structures7 Applications of Stacks Direct applications Page-visited history in a Web browser Undo sequence in a text editor Chain of method calls in the Java Virtual Machine or C++ runtime environment Indirect applications Auxiliary data structure for algorithms Component of other data structures

Elementary Data Structures8 Method Stack in the JVM The Java Virtual Machine (JVM) keeps track of the chain of active methods with a stack When a method is called, the JVM pushes on the stack a frame containing Local variables and return value Program counter, keeping track of the statement being executed When a method ends, its frame is popped from the stack and control is passed to the method on top of the stack 1 main() { int i = 5; 3 foo(i); } foo(int j) { int k; k = j+1; 7 bar(k); …. } 9 bar(int m) { … } bar PC = 9 m = 6 foo PC = 7 j = 5 k = 6 main PC = 3 i = 5

Elementary Data Structures9 Array-based Stack A simple way of implementing the Stack ADT uses an array We add elements from left to right A variable keeps track of the index of the top element (size is t+1) S 012 t … Algorithm size() return t + 1 Algorithm pop() if isEmpty() then throw EmptyStackException else t  t  1 return S[t + 1]

Elementary Data Structures10 Array-based Stack (cont.) The array storing the stack elements may become full A push operation will then throw a FullStackException Limitation of the array- based implementation Not intrinsic to the Stack ADT S 012 t … Algorithm push(o) if t = S.length  1 then throw FullStackException else t  t + 1 S[t]  o

Elementary Data Structures11 Performance and Limitations Performance Let n be the number of elements in the stack The space used is O(n) Each operation runs in time O(1) Limitations The maximum size of the stack must be defined a priori and cannot be changed Trying to push a new element into a full stack causes an implementation-specific exception

Elementary Data Structures12 Growable Array-based Stack In a push operation, when the array is full, instead of throwing an exception, we can replace the array with a larger one How large should the new array be? incremental strategy: increase the size by a constant c doubling strategy: double the size Algorithm push(o) if t = S.length  1 then A  new array of size … for i  0 to t do A[i]  S[i] S  A t  t + 1 S[t]  o

Elementary Data Structures13 Comparison of the Strategies We compare the incremental strategy and the doubling strategy by analyzing the total time T(n) needed to perform a series of n push operations We assume that we start with an empty stack represented by an array of size 1 We call amortized time of a push operation the average time taken by a push over the series of operations, i.e., T(n)/n

Elementary Data Structures14 Incremental Strategy Analysis We replace the array k = n / c times The total time T(n) of a series of n push operations is proportional to n + c + 2c + 3c + 4c + … + kc = n + c( … + k) = n + ck(k + 1)/2 Since c is a constant, T(n) is O(n + k 2 ), i.e., O(n 2 ) The amortized time of a push operation is O(n)

Elementary Data Structures15 Doubling Strategy Analysis We replace the array k = log 2 n times The total time T(n) of a series of n push operations is proportional to n …+ 2 k = n  2 k + 1  1 = 2n  1 T(n) is O(n) The amortized time of a push operation is O(1) geometric series

Elementary Data Structures16 The accounting method determines the amortized running time with a system of credits and debits We view a computer as a coin-operated device requiring 1 cyber-dollar for a constant amount of computing. Accounting Method Analysis of the Doubling Strategy We set up a scheme for charging operations. This is known as an amortization scheme. The scheme must give us always enough money to pay for the actual cost of the operation. The total cost of the series of operations is no more than the total amount charged. (amortized time)  (total $ charged) / (# operations)

Elementary Data Structures17 Amortization Scheme for the Doubling Strategy Consider again the k phases, where each phase consisting of twice as many pushes as the one before. At the end of a phase we must have saved enough to pay for the array-growing push of the next phase. At the end of phase i we want to have saved i cyber-dollars, to pay for the array growth for the beginning of the next phase $$$$ $$$ $ $ $ We charge $3 for a push. The $2 saved for a regular push are “stored” in the second half of the array. Thus, we will have 2(i/2)=i cyber-dollars saved at then end of phase i. Therefore, each push runs in O(1) amortized time; n pushes run in O(n) time.

Queues

Elementary Data Structures19 Outline and Reading The Queue ADT (§2.1.2) Implementation with a circular array (§2.1.2) Growable array-based queue Queue interface in Java

Elementary Data Structures20 The Queue ADT The Queue ADT stores arbitrary objects Insertions and deletions follow the first-in first-out scheme Insertions are at the rear of the queue and removals are at the front of the queue Main queue operations: enqueue(object): inserts an element at the end of the queue object dequeue(): removes and returns the element at the front of the queue Auxiliary queue operations: object front(): returns the element at the front without removing it integer size(): returns the number of elements stored boolean isEmpty(): indicates whether no elements are stored Exceptions Attempting the execution of dequeue or front on an empty queue throws an EmptyQueueException

Elementary Data Structures21 Applications of Queues Direct applications Waiting lists, bureaucracy Access to shared resources (e.g., printer) Multiprogramming Indirect applications Auxiliary data structure for algorithms Component of other data structures

Elementary Data Structures22 Array-based Queue Use an array of size N in a circular fashion Two variables keep track of the front and rear f index of the front element r index immediately past the rear element Array location r is kept empty Q 012rf normal configuration Q 012fr wrapped-around configuration

Elementary Data Structures23 Queue Operations We use the modulo operator (remainder of division) Algorithm size() return (N  f + r) mod N Algorithm isEmpty() return (f  r) Q 012rf Q 012fr

Elementary Data Structures24 Queue Operations (cont.) Algorithm enqueue(o) if size() = N  1 then throw FullQueueException else Q[r]  o r  (r + 1) mod N Operation enqueue throws an exception if the array is full This exception is implementation- dependent Q 012rf Q 012fr

Elementary Data Structures25 Queue Operations (cont.) Operation dequeue throws an exception if the queue is empty This exception is specified in the queue ADT Algorithm dequeue() if isEmpty() then throw EmptyQueueException else o  Q[f] f  (f + 1) mod N return o Q 012rf Q 012fr

Elementary Data Structures26 Growable Array-based Queue In an enqueue operation, when the array is full, instead of throwing an exception, we can replace the array with a larger one Similar to what we did for an array-based stack The enqueue operation has amortized running time O(n) with the incremental strategy O(1) with the doubling strategy

Elementary Data Structures27 Queue Interface in Java Java interface corresponding to our Queue ADT Requires the definition of class EmptyQueueException No corresponding built-in Java class public interface Queue { public int size(); public boolean isEmpty(); public Object front() throws EmptyQueueException; public void enqueue(Object o); public Object dequeue() throws EmptyQueueException; }

Vectors

Elementary Data Structures29 Outline and Reading The Vector ADT (§2.2.1) Array-based implementation (§2.2.1)

Elementary Data Structures30 The Vector ADT The Vector ADT extends the notion of array by storing a sequence of arbitrary objects An element can be accessed, inserted or removed by specifying its rank (number of elements preceding it) An exception is thrown if an incorrect rank is specified (e.g., a negative rank) Main vector operations: object elemAtRank(integer r): returns the element at rank r without removing it object replaceAtRank(integer r, object o): replace the element at rank with o and return the old element insertAtRank(integer r, object o): insert a new element o to have rank r object removeAtRank(integer r): removes and returns the element at rank r Additional operations size() and isEmpty()

Elementary Data Structures31 Applications of Vectors Direct applications Sorted collection of objects (elementary database) Indirect applications Auxiliary data structure for algorithms Component of other data structures

Elementary Data Structures32 Array-based Vector Use an array V of size N A variable n keeps track of the size of the vector (number of elements stored) Operation elemAtRank ( r ) is implemented in O(1) time by returning V[r] V 012n r

Elementary Data Structures33 Insertion In operation insertAtRank ( r, o ), we need to make room for the new element by shifting forward the n  r elements V[r], …, V[n  1] In the worst case ( r  0 ), this takes O(n) time V 012n r V 012n r V 012n o r

Elementary Data Structures34 Deletion In operation removeAtRank ( r ), we need to fill the hole left by the removed element by shifting backward the n  r  1 elements V[r  1], …, V[n  1] In the worst case ( r  0 ), this takes O(n) time V 012n r V 012n o r V 012n r

Lists and Sequences

Elementary Data Structures36 Outline and Reading Singly linked list Position ADT and List ADT (§2.2.2) Doubly linked list (§ 2.2.2) Sequence ADT (§ 2.2.3) Implementations of the sequence ADT (§ 2.2.3) Iterators (2.2.3)

Elementary Data Structures37 Singly Linked List A singly linked list is a concrete data structure consisting of a sequence of nodes Each node stores element link to the next node next elem node ABCD 

Elementary Data Structures38 Stack with a Singly Linked List We can implement a stack with a singly linked list The top element is stored at the first node of the list The space used is O(n) and each operation of the Stack ADT takes O(1) time  t nodes elements

Elementary Data Structures39 Queue with a Singly Linked List We can implement a queue with a singly linked list The front element is stored at the first node The rear element is stored at the last node The space used is O(n) and each operation of the Queue ADT takes O(1) time f r  nodes elements

Elementary Data Structures40 Position ADT The Position ADT models the notion of place within a data structure where a single object is stored It gives a unified view of diverse ways of storing data, such as a cell of an array a node of a linked list Just one method: object element(): returns the element stored at the position

Elementary Data Structures41 List ADT The List ADT models a sequence of positions storing arbitrary objects It establishes a before/after relation between positions Generic methods: size(), isEmpty() Query methods: isFirst(p), isLast(p) Accessor methods: first(), last() before(p), after(p) Update methods: replaceElement(p, o), swapElements(p, q) insertBefore(p, o), insertAfter(p, o), insertFirst(o), insertLast(o) remove(p)

Elementary Data Structures42 Doubly Linked List A doubly linked list provides a natural implementation of the List ADT Nodes implement Position and store: element link to the previous node link to the next node Special trailer and header nodes prevnext elem trailer header nodes/positions elements node

Elementary Data Structures43 Insertion We visualize operation insertAfter(p, X), which returns position q ABXC ABC p ABC p X q pq

Elementary Data Structures44 Deletion We visualize remove(p), where p = last() ABCD p ABC D p ABC

Elementary Data Structures45 Performance In the implementation of the List ADT by means of a doubly linked list The space used by a list with n elements is O(n) The space used by each position of the list is O(1) All the operations of the List ADT run in O(1) time Operation element() of the Position ADT runs in O(1) time

Elementary Data Structures46 Sequence ADT The Sequence ADT is the union of the Vector and List ADTs Elements accessed by Rank, or Position Generic methods: size(), isEmpty() Vector-based methods: elemAtRank(r), replaceAtRank(r, o), insertAtRank(r, o), removeAtRank(r) List-based methods: first(), last(), before(p), after(p), replaceElement(p, o), swapElements(p, q), insertBefore(p, o), insertAfter(p, o), insertFirst(o), insertLast(o), remove(p) Bridge methods: atRank(r), rankOf(p)

Elementary Data Structures47 Applications of Sequences The Sequence ADT is a basic, general- purpose, data structure for storing an ordered collection of elements Direct applications: Generic replacement for stack, queue, vector, or list small database (e.g., address book) Indirect applications: Building block of more complex data structures

Elementary Data Structures48 Array-based Implementation We use a circular array storing positions A position object stores: Element Rank Indices f and l keep track of first and last positions 0123 positions elements S lf

Elementary Data Structures49 Sequence Implementations OperationArrayList size, isEmpty 11 atRank, rankOf, elemAtRank 1n first, last, before, after 11 replaceElement, swapElements 11 replaceAtRank 1n insertAtRank, removeAtRank nn insertFirst, insertLast 11 insertAfter, insertBefore n1 remove n1

Elementary Data Structures50 Iterators An iterator abstracts the process of scanning through a collection of elements Methods of the ObjectIterator ADT: object object() boolean hasNext() object nextObject() reset() Extends the concept of Position by adding a traversal capability Implementation with an array or singly linked list An iterator is typically associated with an another data structure We can augment the Stack, Queue, Vector, List and Sequence ADTs with method: ObjectIterator elements() Two notions of iterator: snapshot: freezes the contents of the data structure at a given time dynamic: follows changes to the data structure