ECE250: Algorithms and Data Structures

Elementary Data Structures

Materials from CLRS: Chapter 10.1, 10.2

Ladan Tahvildari, PEng, SMIEEE Professor

Software Technologies Applied Research (STAR) Group

Dept. of Elect. & Comp. Eng.

University of Waterloo

Acknowledgements

v The following resources have been used to prepare materials for this course: Ø  MIT OpenCourseWare Ø  Introduction To Algorithms (CLRS Book) Ø  Data Structures and Algorithm Analysis in C++ (M. Wiess) Ø  Data Structures and Algorithms in C++ (M. Goodrich)

v Thanks to many people for pointing out mistakes, providing suggestions, or helping to improve the quality of this course over the last ten years: Ø  http://www.stargroup.uwaterloo.ca/~ece250/acknowledgment/

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Definition

v Data Type Ø  The set of allowed values for a variable

v Data Structure Ø  Systematic way of organizing and accessing data

v Principle of Abstraction Ø  Focus on what not how when solving a problem

v Abstract Data Type Ø  Set of elements together with a well-defined set of operations

on these elements

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Abstract Data Types (ADTs)

v  Allow to break work into pieces that can be worked on independently without compromising correctness

v  Serve as specifications of requirements for the building blocks of solutions to algorithmic problems

v  Encapsulate data structures and algorithms that implement them

v  Provide a language to talk on a higher level of abstraction

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Dictionary

v An element has a key part and a data part

v Dictionary ADT – a dynamic set with methods: Ø Search(S, k) – an access operation that returns a

pointer x to an element where x.key = k

Ø  Insert(S, x) – a manipulation operation that adds the element pointed to by x to S

Ø Delete(S, x) – a manipulation operation that removes the element pointed to by x from S

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Dictionary (cont.)

v Dictionary ADT – a dynamic set with methods: Ø  Minimum (S) – an access operation that returns a pointer to

the element of S with the smallest key

Ø  Maximum (S) – an access operation that returns a pointer to the element of S with the largest key

Ø  Successor (S, x) – an access operation that returns a pointer to the next larger element in S for a given x, or NIL if x is the maximum element

Ø  Predecessor (S, x) – an access operation that returns a pointer to the next smaller element in S for a given x, or NIL if x is the minimum element

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Dictionaries v Dictionaries store elements so that they can be

located quickly using keys v A dictionary may hold bank accounts

Ø each account is an object that is identified by an account number

Ø each account stores a wealth of additional information

Ø an application wishing to operate on an account would have to provide the account number as a search key

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Dictionaries (cont.)

v Different data structures to realize dictionaries

v  Arrays v  Binary Trees

v  Linked List v  AVL Trees

v  Queues v  B-Trees v  Stacks v  Heaps v  Hash Tables

Data Storage for ADT

v Data storage can be classified as either: –  Contiguous storage –  Node-based storage

v Examples:

–  Contiguous storage is the array –  Node-based storage is the linked list

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Contiguous Storage

v An array stores n objects in a single contiguous space of memory

v Unfortunately, if more memory is required, a request for new memory usually requires copying all information into the new memory –  In general, you cannot request for

the operating system to allocate to you the next n memory locations

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Node-Based Storage

v Node-based storage such as a linked list associates two pieces of data with each item being stored: – The object itself, and – A reference to the next item

•  In C++ that reference is the address of the next node

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v A data structure in which the objects are arranged in a linear order with dynamic allocation

v Singly linked list – nodes (data, pointer) connected in a chain by links

Linked List ADT

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ADT Operation

Time Complexity

isEmpty()

O(1)

first(), last(), insertFirst(), insertLast()

O(1)

after(p), insertAfter()

O(1)

before(p), insertBefore()

O(n)

replace(p,e)

O(1)

remove(d)

O(n)

Listed List – Running Time

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Doubly Linked Lists

v To implement all of the linked list operations in constant time we use doubly linked lists

v A node of a doubly linked list has a next and a prev link

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Stack ADT

v A stack is a container of objects that are inserted and removed according to the last-in-first-out (LIFO) principle.

Ø  Objects can be inserted at any time, but only the last (the most-recently inserted) object can be removed.

v Inserting an item is known as “pushing” onto the stack. “Popping” off the stack is synonymous with removing an item.

§  Push (S:Stack, x:element) - Inserts o onto top of S §  Pop (S:Stack) - Removes the top object of stack S

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Stack - An Array Implementation

v Create a stack using an array by specifying a maximum size N for our stack.

v The stack consists of an N-element array S and an integer variable t, the index of the top element in array S.

v Array indices start at 0, so we initialize t to -1

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Stack – Pseudo Code

Algorithm push(o) if size()==N then return Error t=t+1 S[t]=o

Algorithm pop() if isEmpty() then return Error S[t]=null t=t-1

Algorithm size() return t+1

Algorithm isEmpty() return (t<0)

Algorithm top() if isEmpty() then return Error return S[t]

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Stack - Running Time

v The array implementation is simple and efficient (methods performed in O(1)).

v There is an upper bound, N, on the size of the stack. The arbitrary value N may be too small for a given application, or a waste of memory.

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Queue ADT v A queue differs from a stack in that its insertion and removal

routines follows the first-in-first-out (FIFO) principle.

v Elements may be inserted at any time, but only the element which has been in the queue the longest may be removed.

v Elements are inserted at the rear (enqueued) and removed from the front (dequeued)

Front Rear Queue

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Queue - An Array Implementation

v Create a queue using an array in a circular fashion

v A maximum size N is specified

v The queue consists of an N-element array Q and two integer variables:

Ø  f, index of the front element (head – for dequeue) Ø  r, index of the element after the rear one (tail – for enqueue)

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Queue – Pseudo Code

Algorithm size() return (N-f+r) mod N

Algorithm isEmpty() return size()=0

Algorithm front() if isEmpty() then return Error return Q[f]

Algorithm dequeue() if isEmpty() then return Error Q[f]=null f=(f+1) mod N

Algorithm enqueue(o) if size = N - 1 then return Error Q[r]=o r=(r+1) mod N

Methods performed in O(1)

Stack and Queue Example

v  Consider we have one stack (LIFO principle) data structure and one circular queue (FIFO principle) data structure; both in array implementations. Draw the content of the stack and the circular queue for “each step” in the following given order:

add(A), add(B), add(C), remove, add(D), add(E), remove, add(F), and add(G)

v  Assume an initial size of 5 for the array in each data structure. v  Remember to show “the top of the stack” and “the front and the back

of the circular queue” in each step. v  In circular queue, assume we sacrifice one storage space to make a

difference between FULL and EMPTY.

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