CS- Week 3

Algorithm is a set of instructions given in a specific sequence to solve a problem. Algorithms differ in speed and how much memory they occupy. In the programming process, most algorithms are based on data search (searching) and sorting (sorting). Let's get acquainted with data retrieval algorithms:

Linear search (Linear search)

Let us be given the following array:

[20, 500, 10, 5, 100, 1, 50]

When visualizing an array, it can be seen as seven red cabinets side by side like this:

We need to find 50 numbers from this array. The computer must check each locker to find the number 50. We call this process, that is, searching for a specific number, character or other element in an array "search".
We can pass our array to an algorithm and ask the algorithm to open the cupboards and determine if the number 50 is there. As a result, the algorithm will answer us “yes” or “no” (true or false).

We can construct an algorithm using the following instructions:

Chapdan o‘ngga har bir eshikni tekshirish:
    Agar 50 soni bor bo‘lsa:
        Ha deb qaytaramiz (return true)
Yo‘q deb qaytaramiz (return false)

The instructions above are human-readable pseudocode and are a simpler representation of the commands given to a computer.

We can implement the linear search algorithm in C using the following code:

#include <cs50.h>
#include <stdio.h>

int main(void)
{
    // Butun sonlardan iborat massiv berilgan
    int numbers[] = {20, 500, 10, 5, 100, 1, 50};

    // Kiritilgan sonni massivdan qidiramiz
    int n = get_int("Number: ");
    for (int i = 0; i < 7; i++)
    {
        if (numbers[i] == n)
        {
            printf("Topildi\n");
            return 0;
        }
    }
    printf("Topilmadi\n");
    return 1;
}

Here a linear search is performed using a for loop.
return 0 means that the program has finished successfully and the program is exited.
return 1 - indicates that an error occurred in the program.

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Binary search

Binary Search is another algorithm used to search for the number 50.
If the values ​​in the array are sorted in ascending order, we can give the pseudocode of binary search as follows:

Agar tekshiriladigan element qolmagan bo‘lsa:
    Yo‘q deb qaytaramiz (return false)
Agar massivning[o‘rta elementi] 50 soniga teng bo‘lsa:
    Ha deb qaytaramiz (return true)
Agar massivning[o‘rta elementi] > 50:
    Massivning chap yarmidan qidiramiz
Agar massivning[o‘rta elementi] < 50:
    Massivning o‘ng yarmidan qidiramiz

---

Big O notation

The Big O notation is used to analyze the time it takes to run the algorithm. Let's look at the following graph:

"Input data size" – x-axis; "Time for solution" – y-axis;
The efficiency of the algorithm is determined by the shape of its curve:
O(n²) is the worst performance time.
O(log n) is the fastest execution time.

The running time of the linear search algorithm is O(n), since in the worst case n steps may be required.
And the time it takes for the binary search algorithm to work is O(log n), because in the worst case, the number of steps decreases more and more.

There are two cases of interest to programmers:

• Worst case or upper bound (upper bound).
• Best case or lower bound (lower bound).

The symbol

Ω is used to denote the best case (lower bound) of the algorithm, for example Ω(n).

The symbol

TH indicates the case where the upper and lower bounds are the same, that is, the best and worst running times are the same.

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Sorting algorithms (Sorting)

Sorting is the process of changing an unordered list of values ​​into ordered ones.
When an array is sorted, it is much easier for a computer to search for a specific element in it. For example, binary search (binary search) works on a sorted array but not on an unsorted array.

There are many types of sorting algorithms. Let's consider one of them selection sort (selection sort). Let us be given an array like this:

The pseudocode of the selection method algorithm is as follows:

[20, 500, 10, 5, 100, 1, 50]

Steps analysis:

• Walking through array elements for the first time takes n - 1 steps.
• The second time n - 2 steps are required.
• Continuing this logic, the required steps can be expressed as:

Chapdan o‘ngga har bir eshikni tekshirish:
    Agar 50 soni bor bo‘lsa:
        Ha deb qaytaramiz (return true)
Yo‘q deb qaytaramiz (return false)

Simplifying this formula, we get: n(n-1)/2 or O(n²).
So, the algorithm of the selection method sorts in O(n²) order in the worst case. Even if all the values ​​are sorted, the number of steps does not change, so the best case is in O(n²) order.

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Bubble sort algorithm (Bubble sort)

Bubble sort is another sorting algorithm where we "promote" larger values ​​ by repeatedly permuting elements.

The pseudocode of the bubble sort algorithm is as follows:

#include <cs50.h>
#include <stdio.h>

int main(void)
{
    // Butun sonlardan iborat massiv berilgan
    int numbers[] = {20, 500, 10, 5, 100, 1, 50};

    // Kiritilgan sonni massivdan qidiramiz
    int n = get_int("Number: ");
    for (int i = 0; i < 7; i++)
    {
        if (numbers[i] == n)
        {
            printf("Topildi\n");
            return 0;
        }
    }
    printf("Topilmadi\n");
    return 1;
}

As we sort the array, we know that more of it will be sorted, so we only need to check the pairs that are not yet sorted.
Therefore, the bubble sort algorithm works in the worst case O(n²) if the array is not sorted, and in the best case O(n) if the array is already sorted.

We can see visually how the sorting algorithms work in this page.

This article uses CS50x 2024 source.

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