CS 221 Analysis of Algorithms Data Structures Vectors, Lists

Portions of the following slides are from  Goodrich and Tamassia, Algorithm Design: Foundations, Analysis and Internet Examples, 2002  and  Material provided by the publisher’s John Wiley & Son, Inc.) companion website for this book

Data Structures  We have seen data structures Stacks – LIFO Queues – FIFO These were very efficient where  we need back most recently stored data – Stack  we need to retrieve data in the order that is was stored How efficient?

Data Structures  Stacks and Queue are not so efficient When? We need to handle data somewhere other than the end of the structure

Vectors  Suppose we have a linear sequence of data What does linear sequence means? Call sequence S S contains n elements Rank = number of elements before a given element in the sequence First element in S has rank = 0 Last element in S has rank = n-1

Vectors  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, rank > n)  Can be implemented using array construct

Vectors  Main vector operations: elemAtRank(integer r): returns the element at rank r without removing it 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 removeAtRank(integer r): removes and returns the element

Vectors  Auxiliary vector operations: Size(): how many elements are in S isEmpty(): return boolean for whether vector has no elements

Vectors - applications  How would you use this type of structure?

Vectors – array implementation  Use an array V of size N  A variable n keeps track of the size of the vector (number of elements stored) V 012n r from: Goodrich and Tamassia, 2002

Vectors – array implementation  elementAtRank(r) run-time? O(1) V 012n r from: Goodrich and Tamassia, 2002

Vectors – array implementation  replaceAtRank(r) run-time? O(1) V 012n r from: Goodrich and Tamassia, 2002

Vectors – array implementation  insertAtRank(r,o) keep in mind- we need to make room for the new element by shifting forward the n  r elements V[r], …, V[n  1] Worst case? run-time? O(n) from: Goodrich and Tamassia, 2002 V 012n r V 012n r V 012n o r

Vectors – array implementation  removeAtRank(r,o) We need to shift every element after removed element to fill hole.. V[r]=V[r+1], V[r+1]=V[r+2],… Worst case? run-time? O(n) from: Goodrich and Tamassia, 2002 V 012n r V 012n o r V 012n r

Vectors - performance OperationT(n) size()O(1) isEmpty()O(1) elementAtRank()O(1) replaceAtRank()O(1) insertAtRank()O(n) removeAtRank()O(n)

Lists  Vectors (or the algorithms to use them)are pretty efficient data structures  …but not so efficient if we have to reorganize them (insertAtRank, removeAtRank)  What if we want to access data with respect to its position in structure, particularly with respect to other elements?

Lists  Consider the concept of an element in a structure as a node  Node contains value of element location of its neighbor node - link next elem node

Lists  As a single linked list we maintain sequence if we have an element we know next element ABCD …

Lists  How about a Stack (LIFO) as a list  Queue (FIFO)as a list ABCD …

Double LinkedLists  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

Position Abstract Data Type  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

List Abstract Data Type  It establishes a before/after relation between positions  Generic methods:  size(), isEmpty()  Query methods:  isFirst(p), isLast(p)

List Abstract Data Type 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)

Element Insertion  insertAfter(p, X), which returns position q ABXC ABC ABC p X q pq

Element Deletion  remove(p), where p = last() ABCD p ABC D p ABC

List Performance  Consider a double linked list  Space?  run-time insertAfter(p) remove(p)  What about growth?