Алгоритмы на Python 3. Лекция №6

Name: Алгоритмы на Python 3. Лекция №6
Uploaded: 2017-11-22T05:47:15.000Z
Duration: 2 h 38 min 35 s

Introduction to Algorithms and Arrays

Class Attendance and Setup

The speaker welcomes attendees, expressing satisfaction with the overall attendance despite some drop-off in the back rows.

A reminder is given about the importance of participation, especially for those who have not yet completed their assignments on arrays.

Understanding Python Lists vs. Arrays

The speaker explains that lists in Python are not fixed-size arrays; they differ significantly in memory allocation and management.

An example is provided where a static array has a maximum size (e.g., 10,000), but the actual number of elements can be zero initially.

Managing Array Elements

When adding an element to a static array, one must track the current count of stored elements using an additional variable.

The real number of elements in a list is managed internally by Python, which allows for dynamic resizing without manual tracking.

Dynamic Resizing and Memory Management

Adding Elements to Lists

Python provides built-in methods like append to add elements dynamically at the end of a list.

It’s emphasized that during contests or assignments, students should avoid using these dynamic features when instructed otherwise.

Retrieving Current Element Count

To find out how many elements are currently stored in a list, one can simply query its length without needing extra variables.

Removing Elements from Lists

Deleting Elements Efficiently

When removing an element from the end of a list, it’s important to decrease the count of stored items accordingly.

The method pop is introduced as a way to remove and return the last element from a list while automatically adjusting its size.

List Comprehension: Simplifying List Creation

Introduction to List Comprehension

The concept of list comprehension is briefly mentioned as an efficient way to create lists in Python.

While details on implementation are deferred for later discussion, it sets up anticipation for practical examples.

Understanding List Comprehension in Python

Introduction to List Comprehension

The speaker introduces the concept of list comprehension, demonstrating how to create a list using a concise syntax: x**2 for x in range(10).

Emphasizes that the variable x is temporary and only exists during the creation of the list, highlighting its ephemeral nature.

Efficiency of List Comprehension

Discusses how list comprehension can be more efficient than traditional loops because it allows for pre-computation and optimizations.

Notes that Python's for loop can iterate over existing lists without needing an index, making it easier to work with collections directly.

Understanding Array Operations

Explains why understanding static arrays is important; adding elements at the beginning requires shifting all subsequent elements, which is costly.

Mentions that while there are many advanced features in Python, beginners should focus on foundational concepts before exploring them.

Creating New Lists with Conditions

Introduces a task where a new array b needs to be created from an existing array a, squaring only even elements.

Describes initializing an empty list and using a loop to append squared values based on conditions (evenness).

Streamlining Code with List Comprehension

Suggests that the previous algorithm can be condensed into one line using list comprehension: [x**2 for x in a if x % 2 == 0].

Highlights that any arithmetic function can be used within this structure, emphasizing flexibility in defining temporary variables.

Advanced Filtering Techniques

Discusses filtering elements directly within list comprehension by adding conditions like checking if numbers are even.

Introduces the ternary operator as another tool for conditional expressions within comprehensions, enhancing code readability.

Conditional Expressions Explained

Explains how to use conditional expressions effectively within comprehensions, allowing different outputs based on input conditions.

Concludes with examples showing how these expressions improve clarity and maintainability of code.

Sorting Algorithms Overview

Introduction to Sorting Algorithms

The discussion begins with the introduction of sorting algorithms, emphasizing their gradual integration into daily life and the importance of understanding various sorting methods.

The task is set: given an array, the goal is to sort it. The speaker plans to explain several sorting techniques before demonstrating their implementations.

Quadratic Sorting Algorithms

The first category introduced is quadratic sorting algorithms, which require approximately n^2 operations for processing an array of length n .

Asymptotic analysis is discussed, focusing on memory usage and computational speed. It highlights that some algorithms may need more memory than initially anticipated.

A comparison is made between sorting operations and simpler array manipulations, noting that basic sorts often require more operations than simple shifts in arrays.

Insertion Sort Explained

The first specific algorithm covered is Insertion Sort (also known as Insert Sort), illustrated through a metaphor involving soldiers lining up by height.

An example with soldiers numbered 1 to 5 illustrates how insertion sort works by maintaining a sorted section of the array while inserting new elements in order.

Process of Insertion Sort

The process involves comparing a new soldier (element) with those already sorted and finding the correct position for insertion based on height (value).

If a soldier (element) arrives taller than those already lined up, they are placed at the end; if shorter, they are moved left until finding their place among taller soldiers.

Challenges in Implementation

Potential issues arise when trying to access elements outside the bounds of the array during comparisons. Care must be taken not to exceed index limits.

The algorithm continues until all elements are inserted correctly into their respective positions within the sorted section.

Selection Sort Introduction

Transitioning from Insertion Sort, Selection Sort is introduced as another method where elements are selected based on certain criteria for placement in order.

Sorting Algorithm Discussion Understanding Minimum Search and Sorting Techniques

The Concept of Finding the Minimum

The speaker discusses sorting elements, emphasizing that those already positioned must be considered while searching for the minimum.

A reminder is given about how to organize the search for a minimum or maximum value, where incoming values are compared against a current reference point.

The need to retain the current minimum in the array without discarding it is highlighted; instead, positions will be swapped.

Swapping Elements and Maintaining Order

The process of finding a new minimum involves swapping positions with an element that is smaller, demonstrating how order can change during sorting.

An invariant concept is introduced: each "officer" maintains a section of the array that remains sorted as they progress through their task.

Expanding Sorted Sections

As elements are sorted, the size of the ordered section increases. Initially empty, this section grows with each successful swap.

It’s crucial not to revisit already sorted sections when searching for new minima; thus, searches begin from where previous sorts left off.

Efficient Searching Strategies

Emphasis on avoiding unnecessary comparisons by only looking among unsorted elements helps streamline sorting processes.

Each time a new position is declared vacant for placing a minimum found in subsequent iterations reinforces efficient organization.

Finalizing Sorted Positions

After several passes through the data set, portions become increasingly sorted until only one element remains unprocessed.

The last element does not require sorting since it automatically falls into place once all others are arranged correctly.

Bubble Sort Introduction Exploring Basic Sorting Algorithms

Overview of Bubble Sort Mechanics

Transitioning to bubble sort methodology introduces another approach where adjacent elements are compared and swapped if out of order.

Characteristics of Bubble Sort Process

In this method, each pass through the list requires checking pairs from start to finish repeatedly until no swaps are needed.

This structured overview captures key insights from discussions on sorting algorithms and their mechanics while providing timestamps for easy reference.

Sorting Algorithm Insights

Understanding the Sorting Process

The speaker discusses a scenario where individuals are incorrectly positioned and need to swap places, emphasizing the importance of correct positioning in sorting.

It is noted that previous sorting methods had clear invariants, allowing for predictable outcomes after a series of sorting actions, which required n - 1 operations for an array of size n .

The method described involves moving elements based on their order; if they are not correctly placed, they must be swapped. This highlights the dynamic nature of the sorting process.

Bubble Sort Mechanism

The term "bubble sort" is introduced, illustrating how the largest element rises to its correct position at the end of the array through successive comparisons and swaps.

After one complete pass through the array, it becomes evident that at least one element is sorted (the largest), indicating progress in achieving a fully sorted array.

Efficiency Considerations

If an element is found to be in order during comparisons, no action is taken; however, if elements are out of order, they are swapped. This leads to fewer necessary comparisons as more elements become sorted.

The invariant established here indicates that once an element reaches its final position at the end of the array, it does not need further comparison.

Comparison Count Analysis

A detailed analysis reveals that with four elements requiring three comparisons (not four due to pairwise checking), efficiency improves as fewer comparisons are needed with each pass.

As sorting progresses and more elements reach their final positions, subsequent passes require even fewer comparisons.

Summation and Complexity

The total number of comparisons across multiple passes forms an arithmetic progression: 4 (first pass), 3 (second), 2 (third), and 1 (fourth). This totals ten comparisons overall.

The formula for this arithmetic series suggests a complexity proportional to n^2 , where n represents the number of elements being sorted.

Final Thoughts on Sorting Algorithms

While bubble sort has a worst-case time complexity of O(n^2) , other algorithms like insertion sort exhibit similar behavior under certain conditions but can perform better depending on data characteristics.

Observations indicate that optimizations can lead to faster execution times when no swaps occur during a pass. Thus, understanding these nuances helps refine algorithm performance assessments.

Sorting Algorithms Implementation and Testing

Introduction to Sorting Functions

The speaker introduces the implementation of sorting functions, emphasizing the importance of testing. They express a desire to start testing immediately after defining functions.

Function Definitions

The first function defined is insert_sort, which will sort a list but initially does not implement any logic. It will take a list as input without specifying the number of elements.

The insert_sort function is intended to perform insertion sort on an array or list.

Additional Sorting Functions

Two more sorting functions are mentioned: choice_sort (selection sort) and bubble_sort. Both are placeholders at this stage, promising functionality without actual implementation yet.

Testing Framework Setup

The speaker plans to pass sorting algorithms as parameters for testing purposes, allowing for easy comparison between different sorting methods using the same dataset.

Test Function Development

A test function named test_sort is introduced, which will accept an array and a sorting algorithm as parameters. This function aims to execute the sorting and compare results with expected outcomes.

Initial Test Case Creation

The speaker begins creating test cases, starting with an empty string for output formatting. They discuss how they will structure their test cases using specific arrays.

Array Comparison Logic

To compare two sorted arrays element-wise, the speaker notes that if any element differs between them, it indicates a failure in sorting. This process can be time-consuming.

Python's Built-in Comparison Features

The speaker mentions Python's built-in capabilities for comparing lists directly but acknowledges that while it's easy to write (a == assorted), implementing it efficiently can be complex.

Streamlining Output Logic

There’s discussion about optimizing print statements based on whether arrays match or not, suggesting a ternary operator approach for cleaner code execution.

Finalizing Tests and Operations

As tests are finalized, the speaker reflects on creating various test cases that share similarities but differ in content. They also touch upon operations like concatenation in Python lists.

This structured overview captures key insights from the transcript regarding sorting algorithms' implementation and testing processes while providing timestamps for further reference.

Creating Lists with Arithmetic Progression

Understanding List Creation

The speaker discusses creating a list based on an arithmetic progression using the range function, emphasizing that it is not a traditional list but rather a generator for arithmetic sequences.

A new list is being created from an arithmetic sequence, highlighting the convenience of reusing variable names without concern for previous assignments due to garbage collection in Python.

Concatenating Lists

The speaker explains concatenating two lists: one ranging from 10 to 20 (exclusive) and another from 0 to 10 (exclusive), demonstrating how elements can be rearranged easily.

An example of generating a continuous arithmetic progression up to 20 is provided, indicating the ease of creating sorted lists even with duplicate elements.

Testing Sorting Algorithms

The speaker transitions into testing sorting algorithms, mentioning that they will run tests on three different sorting methods while keeping track of their documentation strings for clarity.

Emphasis is placed on accessing documentation strings within test cases to understand which sorting algorithm is currently being tested.

Implementing Insertion Sort

The implementation of insertion sort begins, where the sorted portion of the array expands as new elements are added. The top element being inserted starts at index 1 and moves through the array.

A temporary variable holds the index value during insertion, allowing for comparisons and shifts leftward until the correct position for insertion is found.

Moving Elements During Insertion

As elements are shifted left during insertion, comparisons ensure that larger values move rightward while maintaining order. This process continues until all elements are correctly positioned.

The mechanics of shifting involve swapping values between indices as necessary while tracking positions accurately throughout the operation.

Understanding Logical Operations in Python

Importance of Logical Operators

The discussion emphasizes that logical operations should only occur under specific conditions, particularly when the first condition is true before evaluating subsequent conditions.

Lazy Evaluation in Python

It is highlighted that the logical AND operator (and) in Python employs lazy evaluation, meaning it does not evaluate the second operand if the first one is false.

Control Flow and Array Boundaries

The speaker explains how lazy evaluation prevents out-of-bounds errors by stopping execution when a condition fails, thus avoiding unnecessary checks on array indices.

Error Handling in Different Languages

A comparison is made between Python and languages like C/C++, where accessing out-of-bounds elements can lead to severe issues. Python's built-in controls help mitigate these risks.

Testing for Algorithm Reliability

The importance of writing test functions to ensure algorithm correctness is discussed, asserting that preemptive testing guarantees reliability in code execution.

Sorting Algorithms: Selection and Insertion Sort

Implementing Selection Sort

The speaker outlines how selection sort operates using two nested loops to identify positions for inserting minimum elements from an unsorted section of an array.

Range Considerations in Sorting

When implementing sorting algorithms, it's crucial to iterate through indices starting from position + 1 to avoid comparing already sorted elements.

Swapping Elements During Sorting

An analogy involving a "far-sighted" character illustrates how elements are swapped based on their values during sorting processes.

Testing Selection Sort Functionality

After implementing selection sort, a test confirms its functionality with successful results indicating no errors occurred during execution.

Insertion Sort Mechanics

Similarities with Selection Sort

Insertion sort shares similarities with selection sort but differs in approach; it counts passes rather than directly swapping positions as done previously.

Pass Count Optimization

The speaker notes that n - 1 passes are sufficient for sorting all elements correctly without needing to reach the end of the list unnecessarily.

Element Comparison Logic

During insertion sort, comparisons are made between adjacent elements; if they are out of order, they are swapped accordingly until all elements are sorted properly.

Final Thoughts on Sorting Algorithms

Complexity Analysis

All three discussed sorting algorithms (selection sort and insertion sort included here) exhibit quadratic time complexity (O(n²)), which may not be efficient for large datasets but serves educational purposes well.

Emphasis on Practical Programming Techniques

The speaker stresses the importance of practical programming methods that yield reliable results through rigorous testing rather than superficial validation after coding completion.

Testing Drives Development Methodology

Overview of Testing Drives Development

The speaker discusses the methodology of testing drives development, emphasizing the importance of writing strict tests that account for various edge cases where all elements are equal.

Algorithmic Approach to Sorting

Introduction of two algorithms: Counting Sort and its efficiency in sorting large datasets quickly, requiring O(n) time complexity while using memory proportional to the number of distinct elements.

Practical Example of Counting Sort

A practical example is provided with a set of numbers (e.g., 1235, 2760), illustrating how traditional sorting methods would require numerous operations compared to counting sort's efficiency.

One-Pass Algorithms

The concept of one-pass algorithms is introduced, highlighting their ability to find maximum values or count occurrences without storing all numbers in an array. This approach simplifies processing by reducing memory usage.

Frequency Analysis in Counting Sort

The speaker explains frequency analysis as part of counting sort, likening it to Sherlock Holmes' method for deciphering letter frequencies. This technique aids in understanding data distribution and enhances sorting efficiency.