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CSC401 – Analysis of Algorithms Lecture Notes 3 Basic Data Structures

CSC401 – Analysis of Algorithms Lecture Notes 3 Basic Data Structures. Objectives: Introduce basic data structures, including Stacks Queues Vectors Lists Sequences Analyze the performance of operations on basic data structures. Abstract Data Types (ADTs).

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CSC401 – Analysis of Algorithms Lecture Notes 3 Basic Data Structures

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  1. CSC401 – Analysis of AlgorithmsLecture Notes 3Basic Data Structures Objectives: • Introduce basic data structures, including • Stacks • Queues • Vectors • Lists • Sequences • Analyze the performance of operations on basic data structures

  2. Abstract Data Types (ADTs) • 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 • 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

  3. 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 • 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 • 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 • 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

  4. 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 • Indirect applications • Auxiliary data structure for algorithms • Component of other data structures main() { int i = 5; foo(i); } foo(int j) { int k; k = j+1; bar(k); } bar(int m) { … } bar PC = 1 m = 6 • 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 foo PC = 3 j = 5 k = 6 main PC = 2 i = 5

  5. 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 • 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 Algorithmsize() returnt +1 Algorithmpop() ifisEmpty()then throw EmptyStackException else tt 1 returnS[t +1] Algorithmpush(o) ift=S.length 1then throw FullStackException else tt +1 S[t] o • Limitations • The fixed maximum size • Trying to push a new element into a full stack causes an implementation-specific exception • Performance • Let n be the number of elements in the stack • The space used is O(n) • Each operation runs in time O(1)

  6. Stack Interface & ArrayStack in Java public interfaceStack{ public int size(); public boolean isEmpty(); public Object top()throwsEmptyStackException; public voidpush(Object o); public Object pop()throwsEmptyStackException;} public classArrayStack implements Stack{private Object S[ ];private int top = -1; publicArrayStack(int capacity){S = new Object[capacity]);} public Object pop()throwsEmptyStackException{if isEmpty()throw newEmptyStackException(“Empty stack: cannot pop”);Object temp = S[top]; S[top] =null; top = top – 1;returntemp;} } • Other Implementations of Stack • Extendable array-based stack • Linked list-based stack

  7. The Queue ADT • The Queue ADT stores arbitrary objects • Insertions and deletions follow the first-in first-out scheme • Insertions are at the rear and removals at the front • Main queue operations: • enqueue(object): inserts an element at the end of the queue • object dequeue(): removes and returns the element at the front • 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 • 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

  8. normal configuration Q 0 1 2 f r wrapped-around configuration Q 0 1 2 r f 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

  9. Array-based Queue Operations • We use the modulo operator (remainder of division) • Operation enqueue throws an exception if the array is full • This exception is implementation-dependent • Operation dequeue throws an exception if the queue is empty • This exception is specified in the queue ADT Algorithmsize() return(N-f +r) mod N AlgorithmisEmpty() return(f=r) Algorithmenqueue(o) ifsize()=N 1then throw FullQueueException else Q[r] o r(r + 1) mod N Algorithmdequeue() ifisEmpty()then throw EmptyQueueException else oQ[f] f(f + 1) mod N returno

  10. Queue Interface in Java • Java interface corresponding to our Queue ADT • Requires the definition of class EmptyQueueException • No corresponding built-in Java class public interfaceQueue{ public int size(); public boolean isEmpty(); public Object front()throwsEmptyQueueException; public voidenqueue(Object o); public Object dequeue()throwsEmptyQueueException;} • Other Implementations of Queue • Extendable array-based queue: The enqueue operation has amortized running time • O(n) with the incremental strategy • O(1) with the doubling strategy • Linked list-based queue

  11. 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() • Direct applications • Sorted collection of objects (elementary database) • Indirect applications • Auxiliary data structure for algorithms • Component of other data structures

  12. V 0 1 2 n r V V 0 1 0 2 1 2 n n r r V o 0 1 2 n r 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] • 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

  13. V 0 1 2 n r V 0 1 2 n r V o 0 1 2 n r Array-based Vector • 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 • Performance • In the array based implementation of a Vector • The space used by the data structure is O(n) • size, isEmpty, elemAtRankand replaceAtRankrun in O(1) time • insertAtRankand removeAtRankrun in O(n) time • If we use the array in a circular fashion, insertAtRank(0)and removeAtRank(0)run in O(1) time • In an insertAtRankoperation, when the array is full, instead of throwing an exception, we can replace the array with a larger one (extendable array)

  14. next node elem  A B C D 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 • Stack with 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 • Queue with 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

  15. Position ADT & List ADT • The Position ADT • models the notion of place within a data structure where a single object is stored • 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 • The List ADT • models a sequence of positions storing arbitrary objects • 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)

  16. prev next elem node trailer nodes/positions header elements 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

  17. p p A B C p A B C D A B C q p A B C X D p q A B X C A B C Doubly Linked List Operations • We visualize insertAfter(p, X), which returns position q • We visualize remove(p), where p = last() • Performance • The space used by a doubly linked 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

  18. 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) • Direct applications: • Generic replacement for stack, queue, vector, or list • small database • Indirect applications: • Building block of more complex data structures • The Sequence ADT is a basic, general-purpose, data structure for storing an ordered collection of elements

  19. elements 0 1 2 3 positions S f l 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

  20. Operation Array List size, isEmpty 1 1 atRank, rankOf, elemAtRank 1 n first, last, before, after 1 1 replaceElement, swapElements 1 1 replaceAtRank 1 n insertAtRank, removeAtRank n n insertFirst, insertLast 1 1 insertAfter, insertBefore n 1 remove n 1 Sequence Implementations

  21. Design Patterns • Adaptor • Position • Composition • Iterator • Comparator • Locator

  22. Design Pattern: 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

  23. Computers”R”Us Sales Manufacturing R&D US International Laptops Desktops Europe Asia Canada The Tree Structure • In computer science, a tree is an abstract model of a hierarchical structure • A tree consists of nodes with a parent-child relation • Applications: • Organization charts • File systems • Programming environments

  24. A C D B E G H F K I J Tree Terminology • Subtree: tree consisting of a node and its descendants • Root: node without parent (A) • Internal node: node with at least one child (A, B, C, F) • External node (a.k.a. leaf ): node without children (E, I, J, K, G, H, D) • Ancestors of a node: parent, grandparent, grand-grandparent, etc. • Depth of a node: number of ancestors • Height of a tree: maximum depth of any node (3) • Descendant of a node: child, grandchild, grand-grandchild, etc. subtree

  25. Tree ADT • Query methods: • boolean isInternal(p) • boolean isExternal(p) • boolean isRoot(p) • Update methods: • swapElements(p, q) • object replaceElement(p, o) • Additional update methods may be defined by data structures implementing the Tree ADT • We use positions to abstract nodes • Generic methods: • integer size() • boolean isEmpty() • objectIterator elements() • positionIterator positions() • Accessor methods: • position root() • position parent(p) • positionIterator children(p)

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