Python Web Static Resource Management: Correctly Including CSS and JS Files in Flask
This article introduces methods to incorporate static resources like CSS and JS in Flask. Static resources, including CSS (styling), JS (interactivity), and images, should be placed in the `static` folder at the project root directory (automatically mapped to the `/static/` path by Flask). Template files are stored in the `templates` folder. The project structure should include `static` and `templates`. Static resources can be organized into subfolders by type (e.g., `css/`, `js/`). Within templates, use `url_for('static', filename='path')` to reference resources, for example: ```html <link rel="stylesheet" href="{{ url_for('static', filename='css/style.css') }}"> <script src="{{ url_for('static', filename='js/script.js') }}"></script> ``` Common issues: Path errors (such as incorrect filenames or missing subfolders) may result in 404 errors. Ensure the `static` folder exists and filenames are correct. Key points: Place static resources in `static`, use `url_for` for incorporation, and maintain a standardized structure to avoid issues.
Read MoreDjango ORM Practical Guide: CRUD Operations with SQLite Database
This article introduces the operation methods of Django framework combined with SQLite database, focusing on the use of ORM tools. Django ORM allows database operations through Python code, avoiding the need to write complex SQL statements. SQLite is lightweight and easy to use, making it suitable for development and learning. Preparatory work includes: creating a new Django project (`startproject`/`startapp`), configuring SQLite by default in `settings.py`, defining models (such as a `User` class), and executing `makemigrations` and `migrate` to generate database tables. The core is the CRUD operations of Django ORM: **Creation** can be done via `create()` or instantiation followed by `save()`; **Reading** uses `all()`, `filter()`, `get()` (supporting conditions and sorting); **Updating** uses `update()` for batch operations or querying first then modifying; **Deletion** uses `delete()` for batch operations or querying first then deleting. It is important to note the lazy execution of QuerySet, using `unique=True` to prevent duplicates, and exception handling for `get()`. Summary: By defining models, migrating table structures, and calling ORM methods, database operations can be completed. The code is concise and easy to maintain, making it suitable for quickly getting started with web development.
Read MoreFlask Form Handling: Complete Process from User Input to Data Display
This article introduces the complete process of implementing form handling using Flask and Flask-WTF, suitable for web development scenarios requiring user information collection. First, install the Flask and Flask-WTF extensions, then create form classes by inheriting the `FlaskForm` class, defining fields (e.g., username, password) and validation rules (required, length, email format, etc.). In Flask applications, view functions must handle GET (rendering the form) and POST (validating submitted data) requests. Use `form.validate_on_submit()` to check the request type and validate data. If validation fails, error messages are stored in `form.<field>.errors`, and templates display errors through loops. Templates must include `form.hidden_tag()` to enable CSRF protection and avoid form submission failures. Key details include: setting `SECRET_KEY` to ensure CSRF security, using redirects to prevent duplicate submissions, and encrypting stored data (e.g., passwords with bcrypt). The complete workflow is: user fills out form → frontend validation → backend validation → data processing → result display. Advanced features can extend to custom validators, multi-form handling, or file uploads. This article helps quickly master core skills of Flask form implementation from definition to data processing.
Read MoreRESTful API Getting Started: Developing a Simple GET Data Interface with Flask
This article introduces the concept of RESTful APIs and implementing a GET interface with Flask. RESTful APIs are HTTP-based front-end and back-end interaction architectures centered on resources, manipulating resources through GET/POST/PUT/DELETE operations, stateless, and returning JSON data. Flask is chosen for its lightweight and flexibility, making it suitable for beginner development. Flask can be installed via `pip install flask` (virtual environment is optional). Implementation involves two steps: first, define the root path `/` to return "Hello, Flask!", with the core code being the `@app.route('/')` decorator and the `return` statement with a string; second, implement the `/users` interface to return user list JSON data, requiring the import of `jsonify` and returning the converted list. After starting the application, access `http://localhost:5000/users` through a browser, curl, or Postman for testing. Core steps include: importing Flask, initializing the application, defining route functions, returning data, and starting the service. Future extensions can include more routes or database integration.
Read MoreJinja2 Template Engine: Dynamically Rendering Data in Web Pages with Flask (with Examples)
This article introduces the role of template engines in web development and the application of Jinja2 in Flask. Template engines solve the cumbersome problem of splicing backend data with frontend HTML, allowing developers to focus on separating data logic from page structure. Jinja2 is Flask's default template engine, with a concise syntax that supports variable substitution, conditional judgment, loops, filters, and other features. The basic process of using Jinja2 involves: first installing Flask, creating an application and defining routes, preparing backend data (such as user information and article lists), and rendering the template through render_template. Template files should be placed in the 'templates' folder, where data is embedded using {{ variables }}, conditional logic and loops are implemented with {% if %} and {% for %}, and filters are applied using the | operator to process data. Template inheritance, through base.html and child templates, promotes code reusability by reusing page structures. The core syntax of Jinja2 includes variable substitution, conditional judgment, loop traversal, and filters, while template inheritance further optimizes project structure. Mastering Jinja2 enables efficient implementation of dynamic page rendering and is a key tool for connecting data and interfaces in web development.
Read MoreEssential Python Web Tools: Installation and Dependency Management of Virtual Environment venv
Why are virtual environments needed? They resolve dependency conflicts between different projects (e.g., compatibility issues between Django 2.2 and 4.0), prevent pollution of the system Python environment, and facilitate shared dependencies in team collaboration. Python 3.3+ includes the built-in `venv` module, which is a lightweight tool for creating virtual environments without requiring additional installations. Steps to use: 1. **Create**: Navigate to the project directory and run `python -m venv venv` to generate an independent `venv` folder. 2. **Activate**: Commands vary by system: Use the appropriate path for activation in Windows (cmd/PowerShell) or Mac/Linux. A successful activation will show `(venv)` in the terminal. 3. **Verify**: Run `python --version` and `pip --version` to confirm the environment is active. 4. **Dependency Management**: Install packages with `pip install` while activated. After installation, export dependencies with `pip freeze > requirements.txt`. For a new environment or another project, install dependencies quickly using `pip install -r requirements.txt`. 5. **Deactivate and Delete**: Exit with `deactivate`, and delete the `venv` folder directly to remove the environment. `venv` effectively isolates project dependencies, ensuring safety and efficiency, making it an essential tool for Python development.
Read MoreHTTP Requests and Responses: Fundamental Core Concepts in Python Web Development
In Python web development, HTTP is the core for communication between the client (e.g., a browser) and the server, based on the "request-response" mechanism. The client generates an HTTP request, which includes methods (GET/POST, etc.), URLs, header information (e.g., User-Agent, Cookie), and a request body (POST data). After processing, the server returns an HTTP response, which contains a status code (e.g., 200 for success, 404 for not found), response headers (e.g., Content-Type), and a response body (web pages/data). The process is: client sends a request → server parses and processes it → returns a response → client renders the response (e.g., HTML to render a web page). Python frameworks (e.g., Flask) simplify this process. In the example, Flask uses `request` to access the request and `return` to directly send back the response content. Understanding this mechanism is fundamental to web development and lays the foundation for advanced studies such as HTTPS and Cookies.
Read MoreDjango from Scratch: Build a Simple Blog System with ORM and Template Engine in 3 Steps
This article introduces how to quickly build a blog system displaying article lists using Django, with a core understanding of ORM operations for data and template rendering for pages. Step 1: Environment preparation and project initialization. After installing Django, create the project `myblog` and the app `blog`. The project structure includes configuration directories, app directories, and command-line tools. Step 2: Define data models using ORM. Write a `Post` class (with fields: title, content, publication time) in `blog/models.py`, which is automatically mapped to a database table. Activate the model (configure `settings.py`) and execute migrations to generate the table. Step 3: Views and template rendering. Write a view function in `views.py` to retrieve article data and configure routing to distribute requests. Render the article list in the template `index.html` using Django template syntax, supporting loops and variable output. Running `python manage.py runserver` allows access to the blog. The core is to master Django's ORM model definition, view processing, and template rendering processes, with potential for subsequent feature expansion.
Read MoreStep-by-Step Guide: Flask Routes and View Functions, Build Your First Web Page in 10 Minutes
Flask is a lightweight Python Web framework, simple and flexible, suitable for beginners, and supports extensibility as needed. Installation requires Python 3.6+, and can be done via `pip install flask`. To verify, use `flask --version`. The core of a basic application: Import the Flask class and instantiate an `app` object; define the root route with `@app.route('/')`, binding to the view function `home()`, which returns content (e.g., "Hello, Flask!"); `app.run()` starts the development server (default port 5000). For advanced support, dynamic routing is available, such as `/user/<username>`, where the view function receives parameters to implement personalized responses, supporting types like `int` and `float`. Core concepts: Routes bind URLs to functions, view functions handle requests and return content, and `app.run()` starts the service. Key techniques: `if __name__ == '__main__'` ensures the service starts when the script is run directly, and dynamic routing enhances page flexibility.
Read MoreFrom Insertion Sort to Quick Sort: A Beginner's Comparison of Sorting Algorithms
Sorting algorithms are methods to convert unordered data into ordered sequences. They are fundamental core algorithms in computer science, enabling optimization of subsequent operations such as searching and statistics. This article introduces four typical sorting algorithms: Insertion sort is similar to sorting playing cards, gradually building an ordered sequence. It has a time complexity of O(n²), space complexity of O(1), is stable, and is suitable for small-scale or nearly ordered data. Bubble sort involves comparing and swapping adjacent elements, allowing larger elements to "bubble up". It also has O(n²) time complexity, is stable but inefficient, and is only suitable for extremely small-scale data or educational purposes. Merge sort is based on the divide-and-conquer principle, decomposing arrays and merging ordered subarrays. It has O(n log n) time complexity, O(n) space complexity, is stable, and is suitable for large-scale data or scenarios requiring high stability. Quick sort combines divide-and-conquer with pivot partitioning. It has an average time complexity of O(n log n), O(log n) space complexity, is unstable, and is the most commonly used efficient algorithm in engineering, suitable for large-scale data. The article compares and summarizes the time complexity, space complexity, stability, and applicable scenarios of these algorithms. It suggests that beginners first understand the core ideas, learn from simple to complex cases gradually, and deepen their understanding through hands-on simulation.
Read More"Two-Dimensional" View of Sorting Algorithms: An Introduction to Time and Space Complexity
The two-dimensional complexity (time and space) of sorting algorithms is a core criterion for algorithm selection. In terms of time complexity, for small datasets (n ≤ 1000), simple quadratic-time algorithms like bubble sort, selection sort, and insertion sort (O(n²)) are suitable, while for large datasets (n > 10000), efficient linearithmic algorithms such as quicksort, mergesort, and heapsort (O(n log n)) are preferred. Regarding space complexity, heapsort and bubble sort are in-place with O(1) space, quicksort requires O(log n) auxiliary space, and mergesort needs O(n) space. A balance between time and space is essential: mergesort trades O(n) space for stable O(n log n) time, while quicksort uses O(log n) space to optimize efficiency. Beginners should first master simple algorithms before advancing to high-performance ones, selecting based on data size and space constraints to achieve "on-demand sorting."
Read MoreWhy is Bubble Sort Considered a "Beginner-Friendly" Sorting Algorithm?
Bubble Sort is known as a "beginner-friendly" sorting algorithm due to its intuitive logic, simple code, and clear examples, which help beginners quickly grasp the core idea of sorting. **Definition**: It repeatedly compares adjacent elements and gradually "bubbles" larger elements to the end of the array, ultimately achieving order. The core is a "compare-swap" loop: the outer loop controls the number of rounds (at most n-1 rounds), and the inner loop traverses adjacent elements, comparing and swapping them. If no swaps occur in a round, the process terminates early. **Reasons for being beginner-friendly**: 1. **Intuitive Logic**: Similar to "adjusting a queue," it intuitively understands pairwise swaps and gradual ordering. 2. **Simple Code**: Implemented with nested loops, with a clear structure (outer loop for rounds, inner loop for comparison/swaps, optimized with early termination). 3. **Clear Examples**: The sorting process of [5, 3, 8, 4, 2] (where the largest number "bubbles up" to the end in each round) is easy to follow with step-by-step understanding. 4. **Understanding the Essence**: Helps grasp core sorting elements such as "order," "swaps," and "termination conditions." Despite its O(n²) time complexity and low efficiency, as a sorting启蒙 tool, it allows beginners to "first learn to walk" and lays the foundation for subsequent algorithms like Quick Sort. (Note: "启蒙" was translated as "enlightenment" here to maintain the technical educational context; "启蒙 tool" effectively conveys the purpose of foundational learning.) **Corrected translation (more precise term for 启蒙)**: Despite its O(n²) time complexity and low efficiency, as a **foundational sorting tool**, it enables beginners to "first learn to walk" and lays the groundwork for subsequent algorithms like Quick Sort. **Final translation**: Bubble Sort is known as a "beginner-friendly" sorting algorithm due to its intuitive logic, simple code, and clear examples, which help beginners quickly grasp the core idea of sorting. **Definition**: It repeatedly compares adjacent elements and gradually "bubbles" larger elements to the end of the array, ultimately achieving order. The core is a "compare-swap" loop: the outer loop controls the number of rounds (at most n-1 rounds), and the inner loop traverses adjacent elements, comparing and swapping them. If no swaps occur in a round, the process terminates early. **Reasons for being beginner-friendly**: 1. **Intuitive Logic**: Similar to "adjusting a queue," it intuitively understands pairwise swaps and gradual ordering. 2. **Simple Code**: Implemented with nested loops, with a clear structure (outer loop for rounds, inner loop for comparison/swaps, optimized with early termination). 3. **Clear Examples**: The sorting process of [5, 3, 8, 4, 2] (where the largest number "bubbles up" to the end in each round) is easy to follow with step-by-step understanding. 4. **Understanding the Essence**: Helps grasp core sorting elements such as "order," "swaps," and "termination conditions." Despite its O(n²) time complexity and low efficiency, as a **foundational sorting tool**, it enables beginners to "first learn to walk" and lays the groundwork for subsequent algorithms like Quick Sort.
Read More"Memory Consumption of Sorting Algorithms: An Introduction to Space Complexity"
The space complexity (memory consumption) of sorting algorithms is a critical consideration, especially in scenarios with limited memory. Space complexity describes the relationship between the additional storage space used by the algorithm during execution and the data size \( n \), denoted as \( O(1) \), \( O(n) \), or \( O(\log n) \). Key space characteristics of major sorting algorithms: - **In-place sorting** (Bubble Sort, Selection Sort, Insertion Sort, Heap Sort): Require no additional arrays, with space complexity \( O(1) \); - **Merge Sort**: Requires temporary arrays during the merge step of divide-and-conquer, with space complexity \( O(n) \); - **Quick Sort**: Uses recursive partitioning, with average space complexity \( O(\log n) \). Selection strategy: Prioritize \( O(1) \) algorithms when memory is limited; choose Merge Sort for stable sorting with sufficient memory; pursue average performance (e.g., standard library sorting) with Quick Sort. Understanding space characteristics enables balancing time and space to write efficient code.
Read MoreInspiration from Poker Sorting: A Life Analogy and Implementation of Insertion Sort
This article introduces the insertion sort algorithm. Its core idea is to gradually build an ordered sequence: the first element is defaulted as sorted, and starting from the second element, each element (the element to be inserted) is inserted into the correct position in the previously sorted sequence (where larger elements need to be moved to make space). Taking the array `[5, 3, 8, 4, 2]` as an example, the process of comparing and moving elements from back to front is demonstrated. The key steps are: traversing the array, temporarily storing the element to be inserted, moving the sorted elements, and inserting at the correct position. Algorithm characteristics: Advantages include simplicity and intuitiveness, in-place sorting (space complexity O(1)), stability, and suitability for small-scale or nearly ordered data; disadvantages include the worst-case time complexity of O(n²) and a relatively large number of move operations. In summary, insertion sort is a foundation for understanding sorting algorithms. It is explained through a life analogy (e.g., sorting playing cards) to aid comprehension and is applicable to simple scenarios or sorting small-scale data.
Read MoreBubble, Selection, Insertion Sort: Who is the Entry-Level 'Sorting King'?
This article introduces the significance of sorting and three basic sorting algorithms. Sorting is a fundamental operation that rearranges data according to rules to achieve a more ordered state, and it is a prerequisite for understanding complex algorithms. The core ideas and characteristics of the three algorithms are as follows: Bubble Sort repeatedly "bubbles" the largest number to the end through multiple passes, with an intuitive logic but frequent swaps, having a time complexity of O(n²). Selection Sort selects the smallest number in each round and inserts it, with fewer swaps but instability, also with O(n²) complexity. Insertion Sort is similar to arranging playing cards, suitable for small-scale or nearly ordered data, and its complexity is close to O(n). Although these three algorithms are simple, they form the foundation of more complex sorts (such as Heap Sort and Merge Sort). For beginners, it is more important to grasp the core ideas of "selecting the smallest, inserting appropriately, and bubbling the largest," and to understand the thinking of "gradually building an ordered sequence," rather than getting caught up in efficiency. This is the key to understanding the essence of sorting.
Read MoreHow Sorting Algorithms Implement Ascending/Descending Order? A Guide for Data "Obedience"
This article introduces methods for sorting algorithms to implement data in ascending/descending order, with the core being to make data "obey" through algorithmic rules. The significance of sorting: arranging messy data in ascending order (from small to large) or descending order (from large to small), such as organizing playing cards. Three basic algorithm implementation rules: 1. **Bubble Sort**: When ascending, large numbers "bubble" and move backward (swap if front > back); when descending, small numbers "sink" and move backward (swap if front < back), like bubbles rising/falling. 2. **Selection Sort**: When ascending, select the smallest number in each round and place it on the left; when descending, select the largest number and place it on the left, similar to selecting class monitors to take their positions in order. 3. **Insertion Sort**: When ascending, insert the new number into the correct position in the already sorted part (from left to right, increasing); similarly for descending (from left to right, decreasing), like inserting playing cards into a sorted sequence. Core logic: Adjust the comparison rule (> or <) to determine the data movement direction. Ascending/descending order essentially involves "making data move according to the rules". It is recommended to first master basic algorithms and manually simulate data movement to understand the logic.
Read MoreThe "Fairness" of Sorting: What is Stability? A Case Study of Insertion Sort
The "stability" of sorting refers to whether the relative order of equal elements remains unchanged after sorting; if it does, the sort is stable. Insertion sort achieves stability through its unique insertion logic. The core of insertion sort is to insert unordered elements one by one into the ordered portion. When inserting equal elements, no swap occurs; instead, the element is directly inserted after the equal elements. For example, in the array [3, 1, 3, 2], when processing the second 3, since it is equal to the 3 at the end of the ordered portion, it is directly inserted after it. The final sorted result is [1, 2, 3, 3], and the relative order of the original two 3s remains unchanged. The key to stability lies in preserving the original order of equal elements. This is crucial in practical scenarios such as grade ranking and order processing, where fair sorting according to the original order is required. Due to its logic of "no swap for equal elements, insert later", insertion sort naturally guarantees stability, ensuring that duplicate elements always remain in their original order and reflecting the "fairness" of the sort.
Read MoreSelection Sort: One of the Simplest Sorting Algorithms with the Least Swaps Implementation Method
Selection Sort is a fundamental sorting algorithm in computer science. Due to its simple logic and the minimum number of swaps, it is the first choice for beginners to get started. The core idea of Selection Sort is to divide the array into two parts: sorted and unsorted. In each iteration, the smallest element in the unsorted part is found and swapped with the first element of the unsorted part, gradually expanding the sorted part until the entire array is sorted. For example, for the array [64, 25, 12, 22, 11], through multiple rounds of finding the minimum element and swapping (such as swapping 11 to the first position in the first round and 12 to the second position in the second round), an ascending order can be achieved. Selection Sort involves the minimum number of swaps (at most n-1 swaps). Its time complexity is O(n²) (for all best, worst, and average cases), while the space complexity is only O(1). Its advantages include simplicity, low swap cost, and minimal space requirements. However, its drawbacks are low time efficiency and being an unstable sort (the relative order of equal elements may change). It is suitable for sorting small-scale data or scenarios where swap count is sensitive (e.g., embedded systems). Mastering Selection Sort helps in understanding the core ideas of sorting and laying a foundation for learning more complex algorithms.
Read MoreThe 'Speed Code' of Sorting Algorithms: Time Complexity and Bubble Sort
This article focuses on the "speed code" of sorting algorithms, with a core emphasis on time complexity and bubble sort. Time complexity is measured using the Big O notation, with common types including O(n) (linear, growing proportionally with data size), O(n²) (quadratic, extremely inefficient for large datasets), and O(log n) (logarithmic, extremely fast). It serves as a key indicator for judging the efficiency of algorithms. Bubble sort, a fundamental algorithm, works by comparing and swapping adjacent elements to "float" smaller elements upward and "sink" larger elements downward. Using the array [5, 3, 8, 4, 2] as an example, it repeatedly traverses and compares adjacent elements until the array is sorted. Its time complexity is O(n²), with a space complexity of O(1) (in-place sorting). Its advantages include simplicity, intuitive logic, while its main drawback is low efficiency, making it extremely time-consuming for large datasets. In summary, despite its slowness (O(n²)), bubble sort is a valuable introductory tool that helps understand sorting principles and time complexity, laying the foundation for learning more efficient algorithms like quicksort (optimized to O(n log n)).
Read MoreLearning Insertion Sort: Principles and Code, Just Like Organizing Your Desktop
This article analogizes "sorting the desktop" to insertion sort, with the core idea being inserting elements one by one into their correct positions within the already sorted portion. The basic steps are: initializing the first element as sorted, starting from the second element, comparing it backward with the sorted portion to find the insertion position, shifting elements, and then inserting the current element. Taking the array `[5,3,8,2,4]` as an example: initially sorted as `[5]`, inserting 3 (after shifting 5) results in `[3,5]`; inserting 8 (directly appending) gives `[3,5,8]`; inserting 2 (shifting 8, 5, and 3 sequentially, then inserting at the beginning) yields `[2,3,5,8]`; inserting 4 (shifting 8 and 5, then inserting after 3) completes the sorting. The Python code implementation uses a double loop: the outer loop iterates over elements to be inserted, and the inner loop compares backward and shifts elements. It has a time complexity of O(n²), space complexity of O(1), and is suitable for small-scale data or nearly sorted data. It is an in-place sorting algorithm with no additional space required. This sorting analogy intuitively reflects the essence of "inserting one by one" and aids in understanding the sorting logic.
Read MoreLearning Bubble Sort from Scratch: Step-by-Step Teaching and Code Implementation
### Summary of Bubble Sort Sorting is the process of rearranging unordered data according to a set of rules. Bubble Sort is a fundamental sorting algorithm whose core principle is to gradually "bubble up" larger elements to the end of the array through adjacent element comparisons and swaps. **Core Idea**: In each iteration, start from the beginning of the array and compare adjacent elements sequentially. If a preceding element is larger than the following one, swap them. After each iteration, the largest element "sinks" to its correct position at the end, reducing the length of the unsorted portion by 1. Repeat until all elements are sorted. **Steps**: The outer loop controls the number of iterations (n-1 times, where n is the array length). The inner loop compares and swaps adjacent elements in each iteration. An optimization is to terminate early if no swaps occur in a round. **Complexity**: Time complexity is O(n²) in the worst case (completely reversed order) and O(n) in the best case (already sorted). Space complexity is O(1) (in-place sorting). **Characteristics and Applications**: It is simple to implement but inefficient (O(n²)). Suitable for small datasets or scenarios with low efficiency requirements (e.g., teaching demonstrations). By embodying the comparison-swap paradigm, Bubble Sort lays the foundation for understanding more complex sorting algorithms.
Read MoreWhat is Time Complexity O(n)? A Must-Learn Efficiency Concept for Data Structure Beginners
The article explains the necessity of time complexity: due to differences in hardware and compilers, direct timing is impractical, and an abstract description of the algorithm's efficiency trend is required. The core is linear time complexity O(n), which indicates that the number of operations is proportional to the input size n (such as the length of an array). Scenarios like traversing an array to find a target or printing all elements require n operations. Big O notation ignores constants and lower-order terms, focusing only on the growth trend (e.g., O(2n+5) is still O(n)). Comparing common complexities (O(1) constant, O(n) linear, O(n²) quadratic, O(log n) logarithmic), O(n) is more efficient than O(n²) but less efficient than O(1) or O(log n). Understanding O(n) is fundamental to algorithms, helping optimize code and avoid timeout issues caused by "brute-force solutions."
Read MoreHow to Handle Hash Table Collisions? Understand Hash Resolution Methods in Data Structures Simply
Hash tables map keys to array slots via hash functions, and hash collisions occur when different keys map to the same slot. The core solution is to find new positions for colliding elements, with the main methods as follows: 1. **Chaining (Separate Chaining)**: Each slot stores a linked list, where colliding elements are appended to the list. For example, keys 1, 6, and 11 hashing to the same slot form a linked list [1, 6, 11]. **Advantages**: Simple implementation, suitable for dynamic data. **Disadvantages**: Extra space for linked lists; worst-case O(n) lookup. 2. **Open Addressing**: When collisions occur, empty slots are probed. Includes linear probing (i+1, wrapping around) and quadratic probing (i+1², etc.). For example, key 6 hashing to slot 1 (collision) probes slot 2; key 11 probes slot 3. **Advantages**: High space utilization. **Disadvantages**: Linear probing causes primary clustering; quadratic probing requires a larger array. Other methods: Double hashing (multiple hash functions) and common overflow area (primary table + overflow table), suitable for low-collision scenarios. Selection depends on the use case: Chaining (e.g., Java HashMap) suits dynamic, large datasets; Open addressing is better for fixed-size arrays with few collisions.
Read MoreHow to Learn Tree Traversal? Easily Understand Preorder, Inorder, and Postorder Traversals
Trees are fundamental data structures, and traversal is the process of visiting all nodes. This article focuses on explaining the preorder, inorder, and postorder traversals of binary trees, with their core difference lying in the timing of visiting the root node. - **Preorder Traversal (Root → Left → Right)**: Visit the root first, then recursively traverse the left subtree, and finally the right subtree. Example: 1→2→4→5→3→6→7. - **Inorder Traversal (Left → Root → Right)**: Recursively traverse the left subtree first, then visit the root, and finally the right subtree. Example: 4→2→5→1→6→3→7. - **Postorder Traversal (Left → Right → Root)**: Recursively traverse the left subtree first, then the right subtree, and finally visit the root. Example: 4→5→2→6→7→3→1. Memory mnemonic: Preorder has the root first, inorder has the root in the middle, postorder has the root last. In applications, preorder is used for tree copying, inorder sorts binary search trees, and postorder is used for node deletion. Traversal essentially embodies the recursive thought; mastering the order and scenarios enables proficiency.
Read MoreHow to Understand Recursion? Easily Learn Recursion with the Fibonacci Sequence
Recursion is a problem-solving method that "calls itself", breaking down large problems into smaller, similar subproblems until the subproblems can be solved directly, then returning results layer by layer (like disassembling Russian nesting dolls). Its core elements are the **base case** (to avoid infinite loops, e.g., returning fixed values when n=0 or n=1) and the **recursive relation** (decomposing into subproblems, e.g., F(n) = F(n-1) + F(n-2)). Taking the Fibonacci sequence as an example, the recursive function `fib(n)` is implemented through the base case and recursive relation: `fib(0) = 0`, `fib(1) = 1`, and `fib(n) = fib(n-1) + fib(n-2)`. For `fib(5)`, it requires recursively calculating `fib(4)` and `fib(3)`, decomposing layer by layer until `fib(1)` and `fib(0)` are reached, then combining the results in reverse to get the final answer. The recursive process is like "peeling an onion": after reaching the bottom, results "bounce back". Its advantages include clear logic and ease of implementation, but it has redundant calculations (e.g., `fib(3)` is called multiple times), leading to lower efficiency—optimizations like memoization or iteration can be used. In essence, recursion transforms large problems into smaller ones, and smaller problems... (Note: The original Chinese summary appears truncated here; the above translation completes the core description.)
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