Introduction

CSCI 362: Data Structures

Prof. William Killian

Why study Data Structures and Algorithms?

• Course is required
• Write more efficient code (in terms of space, speed, power)
• Develop non-trivial software
• Become a better problem solver

Data Structures

What is a data structure?

• A particular way of organizing data in a computer so that it can be used efficiently Wikipedia

Examples?

• Primitive types: bool, char, short, int, etc.
• Composite types: array, struct/class, union
• Abstract Data types: stack, queue, list, set, map

Algorithms

What is an algorithm?

• A self-contained step-by-step set of operations to be performed (Wikipedia)

Examples?

• Sequential search, binary search
• Bubble sort, selection sort, merge sort
• Inserting an element into a Binary Search Tree

Data Structures and Algorithms

• Programs = Data Structures + Algorithms

• Data Structures
  • (efficient) storage and manipulation of (large) data sets
  • Arrays (both static and dynamic), linked lists, linked trees
  • Used to implement abstract data types e.g. set, list

• Algorithms
  • Operations on data
  • Insertion and removal of elements
  • Rearranging data (reverse, sort, merge)
  • Searching data

Complexity

Two types of complexity

• Speed
  • relate operations to input size
• Space
  • relate number of bytes to input size
  • example: space for linked list of ints vs array of ints

Big-O notation

• Machine-independent means for specifying efficiency (complexity)
• Concerned with *asymptotic behavior
• Generally count most significant statements/operations and related to input size ($N$) using a runtime function $T$
  • example: if $T(N) = 9N^2 + 43N + 7$ then the algorithm is $O(N^2)$

We can subdivide complexity into three cases:

• Best case
• Average case
• Worst case (default)

Example: linear search time complexity:

int linear_search (int A[], int N, int searchValue) {
    int i;
    for (i = 0; i < N; ++i) {
        if (A[i] == searchValue) {
            break;
        }
    }
    return i;
}

• What should we count? In the best case? Worst case? Average case?
• For worst case, $T(N) = ? = O(?)$

#include <iostream>
int a[10] = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 };

linear_search (a, 10, 0)

10

linear_search (a, 10, 1)

0

linear_search (a, 10, 5)

4

linear_search (a, 10, 10)

9

C++ Standard Library

Languages include data structures/ADTs/algorithms

• Java: Java Collections framework (ArrayList, TreeSet)
• C++: C++ Standard Library (commonly referred to as Standard Template Library {STL})
• C#: .NET class library

The C++ Standard Container Library provides:

Container classes

• array, vector, deque, list, forward_list
• set, multiset, map, multimap
• unordered_set, unordered_multiset, unordered_map, unordered_multimap
• stack, queue, priority_queue
• string, valarray, bitset (container-like)

Lots of algorithms

• accumulate
• copy
• sort
• lower_bound, upper_bound
• nth_element
• partition

Storage Containers (Vectors)

• Position-based
• Data structure: dynamic array

#include <vector>

std::vector<int> v { 7, 4, 9, 3, 1};

for (size_t i = 0; i < v.size(); ++i) {
    std::cout << v[i] << " ";
}

7 4 9 3 1 

v.push_back(2);
v

{ 7, 4, 9, 3, 1, 2 }

v.insert (v.begin(), 5);
v

{ 5, 7, 4, 9, 3, 1, 2 }

Resizing a vector

v = {7, 4, 9, 1, 3 }

{ 7, 4, 9, 1, 3 }

v.resize(8); // grow to 8 elements
v

{ 7, 4, 9, 1, 3, 0, 0, 0 }

v.resize(3); // shrink to 3 elements
v

{ 7, 4, 9 }

• Vectors allow efficient subscripting: $O(1)$
• Position-based indexing (why it's called a sequence container)
• Which vector operations are slow?

v.shrink_to_fit();
std::cout << "Size: " << v.size() << '\n';
std::cout << "Capacity: " << v.capacity() << '\n';
v

Size: 3
Capacity: 3

{ 7, 4, 9 }

v.insert(v.begin(), 10);
std::cout << "Size: " << v.size() << '\n';
std::cout << "Capacity: " << v.capacity() << '\n';
v

Size: 4
Capacity: 6

{ 10, 7, 4, 9 }

v.erase(v.begin());
std::cout << "Size: " << v.size() << '\n';
std::cout << "Capacity: " << v.capacity() << '\n';
v

Size: 3
Capacity: 6

{ 7, 4, 9 }

Storage Containers (Stacks and Queues)

a.k.a. Container Adapters

• Stack (std::stack)
  • Header <stack>
  • Data Structure: dynamic array
• Queue (std::queue)
  • Header <queue>
  • Data Structure: dynamic array
• Priority Queue (std::priority_queue)
  • Header <priority_queue>
  • Data Structure: dynamic array

Storage Containers (Maps, Sets)

std::set, std::map, std::multiset, std::multimap

• Headers: <set> and <map>
• Key-based data structure
• Data structure: (balanced) Binary Search Tree

Question: Maximum number of comparisons for search?

Abstract Data Types (ADTs)

• Use ADT to specify a type
• Abstract Data Types (ADTs)
  • collection of values (e.g. ints, Persons, Lists)
  • set of operations on the data with pre- and post- conditions
  • Note: what's missing?
• Typically implement ADTs with a class
• A class encapsulates two things:
  • data (data members {Java: instance variables})
  • operations (member functions {Java: methods})

ADT Operations Description

• operationName -- action statement specifies:
  • input parameters
  • type of operations on elemenets of data structure
  • output parameter(s)
• Preconditions:
  • necessary conditions that must apply to
    1. input parameters of operations
    2. current state of object
  • which then allows for successful execution of the operation
• Postconditions:
  • changes in data structure caused by operation

Abstract Data Type Example: time24

Suppose we have a time24 ADT that has a duration method

duration$(t)$

• time $t$ is input parameter of type time24
• measures length of time from $this$ time to time $t$
• returns result as time24 value
• Precondition: $this \le t$
• Postcondition: object is unaffected

struct time24 {
    int hour;
    int minute;
    
    time24 duration(time24 other) const {
        // assert ((other.minute + 60 * other.hour) >= (minute + 60 * hour))
        if (minute > other.minute) {
            --other.hour;
            other.minute += 60;
        }
        other.hour -= hour;
        other.minute -= minute;
        return other;
    }
};

#include <string>
#include "nlohmann/json.hpp"

using namespace std::literals::string_literals;
auto mime_bundle_repr(time24 const& n) {
    auto bundle = nlohmann::json::object();
    bundle["text/html"] = "<dl><dt><strong>Hour</strong></dt><dd>"s + std::to_string(n.hour) + "</dd><strong>Minute</strong><dd>"s + std::to_string(n.minute) + "</dd></dl>"s;
    return bundle;
}

time24 now {14, 45};
time24 later {16, 25};
now.duration (later)

Hour

1

Minute

40

now = {14, 45};
later = {12, 15};
now.duration (later)

Hour

-3

Minute

30

struct time24 {
    int hour;
    int minute;

    time24 duration(time24 other) const {
        // assert ((other.minute + 60 * other.hour) >= (minute + 60 * hour))
        if (minute > other.minute) {
            --other.hour;
            other.minute += 60;
        }
        other.hour -= hour;
        other.minute -= minute;
        return other;
    }
};

Classes (Implementing ADTs)

class MyClass
{
public:
    // public member functions and (maybe) data members
    // Form an interface -- Choose wisely!
private:
    // implementation details go here!
    // - private data members
    // - private helper functions
};

• public section visible to clients
• private section visible to member functions only (and friends)

class C
{
public:
    // methods with access to 'field'
private:
    int field;
};

C c;
c.field = 3;

input_line_18:10:3: error: 'field' is a private member of 'C'
c.field = 3;
  ^
input_line_18:6:9: note: declared private here
    int field;
        ^

Interpreter Error: 

• Private section contains implementation
  • Data values of the object
  • Utility (helper) functions that support the class
  • Only accessible to member functions (and friends)
• Key object-oriented programming concept: Data Abstraction
  • Separation of implementation from interface

• In C++, place class definition (interface) in a header file
  • Header files usually end in .h or .hpp (e.g. MyClass.h)
• In C++, place class implementation in a source file
  • Header files usually end in .cpp (e.g. MyClass.cpp)
• Clients of MyClass include MyClass.h

#include "MyClass.h"

Example: Rectangle class

• Default constructor
  • default arguments
  • member initializers
• Const member functions
  • compiler contract to not modify this
• Method definitions are inlined
  • see (future) examples on website for non-inlined versions
  • uses the special scope resolution operator ::

class Rectangle
{
public:  
  // This is the constructor. It's a special function
  // that allows us to initialize a new instance of
  // a Rectangle with a specified length and width
  Rectangle (double length = 0.0, double width = 0.0)

    // below is a member initializer list. For each
    // member variable of our class, we should always
    // initialize them according to some reasonable
    // default
  : m_length (length)
  , m_width (width)
  {
    // in this case, our constructor "body" is empty
  }

// area is a const member function (denote the trailing const)
  // it offers a read-only view of the current class and calculates
  // the area of the rectangle
  double
  area() const
  {
    // inside we can access (read) any member variable
    return m_length * m_width;
  }

  double
  perimeter() const
  {
    return 2 * (m_length + m_width);
  }

double
  getLength() const
  {
    return m_length;
  }

  double
  getWidth() const
  {
    return m_width;
  }

  // mutator method which sets a new length and width
  double
  setSides (double length, double width) {
    m_length = length;
    m_width = width;
  }

// member variables (usually) go at the end of the class
  // definition. They should probably be marked as "private"
  // most of the time
private:
  double m_length;
  double m_width;
};

APIs

Application Programming Interface

• ADTs + syntax
• Function prototypes for all accessible functions
  • constructors are usually listed separately
• Description of methods, sometimes with pre- and post-conditions
• Complexities (sometimes, not always required/necessary)
• NOTE: does not expose implementation details

RandomNumber

Constructors

RandomNumber (unsigned int seed = 0);

• Sets the seed for the random number generator
• Postconditions: With the default value 0, the system clock initializes the seed; otherwise, the user provides the seed for the generator

Operations

double frandom();

• Return a real number $x \mid 0.0 \le x < 1.0$

int random();

• Return a 32-bit random integer $m \mid 0 \le m < 2^31 - 1$

int random(unsigned int n);

• Return a 32-bit random integer $m \mid 0 \le m < n$

Use

RandomNumber rndA, rndB (7);
for (int i = 0; i < 5; ++i)
{
    // 0 <= item < 40
    int item = rndA.random (40);
    // 0.0 <= x < 1.0
    double x = rndB.frandom ();
    std::cout << item << "  " << x << '\n';
}

std::string

• portion of string API follows
• Strings are mutable
• Some useful member functions:
  • find_first_of
  • find_last_of
  • find
  • substr
  • insert
  • erase
  • string::npos (static)
• Free operator+() to concatenate

C++ Documentation

http://cppreference.com/w/cpp