Numpy Matrix Basics: Introduction to Multiplication, Transposition, and Inverse Matrix
This article introduces basic Numpy matrix operations, suitable for beginners to get started quickly. The core of Numpy is `ndarray`, created using `np.array`. Basic attributes include `shape` (number of rows and columns), `ndim` (dimension), and `dtype` (data type). Three core operations: 1. **Multiplication**: Distinguish between element-wise multiplication (`*`, requiring identical shapes) and matrix dot product (`np.dot`/`@`, where the number of columns of the first matrix equals the number of rows of the second matrix, resulting in a shape of `m×p`). 2. **Transposition**: Achieved using `.T` to swap rows and columns, suitable for adjusting shapes to fit operations. 3. **Inverse Matrix**: Exists only for square matrices with non-zero determinants, calculated using `np.linalg.inv`. Verification is done with `np.allclose` to check if it is the identity matrix. After mastering the basics, more complex operations can be advanced. Numpy requires more practice to improve proficiency.
Read MoreNumpy Random Number Generation: A Beginner's Guide to rand and randn
NumPy is the core library for scientific computing in Python. The `np.random` submodule provides random number generation functionality, with `rand` and `randn` being commonly used functions. These random numbers are pseudo-random, and fixing the seed allows for reproducible results. `np.random.rand(d0, …, dn)` generates random numbers from a **uniform distribution over [0, 1)**. The parameters specify the array shape (e.g., 1-dimensional, 2-dimensional, etc.), and all elements lie within [0, 1). It is suitable for scenarios requiring equal probability values (e.g., initializing weights). `np.random.randn(d0, …, dn)` generates random numbers from a **standard normal distribution** (mean 0, standard deviation 1). Elements are concentrated between -1 and 1, with a low probability of extreme values. To adjust the mean and standard deviation, the formula `μ + σ * randn` can be used. This is often applied to simulate natural data fluctuations (e.g., noise). Both functions accept shape parameters, with the former producing uniform distribution and the latter normal distribution. The results can be reproduced by fixing the seed using `np.random.seed(seed)`.
Read MoreNumpy for Beginners: Quick Reference for Common Functions arange and zeros
This article introduces two basic numerical array creation functions in Python Numpy: `arange` and `zeros`. `arange` is used to generate ordered arrays, similar to Python's built-in `range` but returns a Numpy array. Its syntax includes `start` (default 0), `stop` (required, exclusive), `step` (default 1), and `dtype`. Examples: The default parameters generate an array from 0 to 4; specifying `start=2, step=2` generates [2, 4, 6, 8] (note that `stop` is not included). When the step is a decimal, attention should be paid to floating-point precision. `zeros` is used to generate arrays filled with zeros, commonly for initialization. Its syntax parameters are `shape` (required, integer or tuple) and `dtype` (default float). Examples: `zeros(5)` generates a 1D array [0.0, 0.0, 0.0, 0.0, 0.0]; `zeros((2, 3))` generates a 2×3 2D array. Specifying `dtype=int` can produce arrays of integer zeros. Note that `shape` must be clearly specified, and a tuple should be passed for multi-dimensional arrays. Both are core tools for Numpy beginners. `arange` constructs ordered data,
Read MoreNumpy Broadcasting: A Core Technique to Simplify Array Operations
The Numpy broadcasting mechanism addresses element-wise operations for arrays of different shapes by automatically expanding smaller arrays to match the shape of larger arrays and aligning dimensions, eliminating the need for manual reshaping, thus saving memory and improving efficiency. Core rules: dimensions are matched from right to left, with each dimension size either being 1 or equal; smaller arrays are broadcasted to the merged shape of the larger array. For example, scalars (e.g., 10) can be broadcast to any array shape; when a 1D array (e.g., [10, 20, 30]) is broadcasted with a 2×3 2D array, the 1D array is repeated into 2 rows. When a 3D array (2×2×2) is broadcasted with a 2×2 2D array, the 2D array is expanded to 2×2×2. If dimensions are incompatible (e.g., 2×2 and 1×3), an error is raised. Practical applications include element-wise operations (e.g., adding a constant to an array) and matrix standardization, avoiding loops and simplifying code. Mastering broadcasting significantly enhances the efficiency and readability of Numpy array operations.
Read MoreComprehensive Guide to Numpy Arrays: shape, Indexing, and Slicing
NumPy arrays are the foundation of Python data analysis, providing efficient multi-dimensional array objects with core operations including array creation, shape manipulation, indexing, and slicing. Creation methods: np.array() is commonly used to generate arrays from lists; zeros/ones create arrays filled with 0s/1s; arange generates sequences similar to Python's range. Shape is the dimension identifier of an array, viewed via .shape. The reshape() method adjusts dimensions (total elements must remain unchanged), with -1 indicating automatic dimension calculation. Indexing: 1D arrays behave like lists (0-based indexing with support for negative indices); 2D arrays use double indexing [i, j]. Slicing: Follows the syntax [start:end:step], with 1D/2D slicing producing subarrays. Slices return views by default (modifications affect the original array), requiring .copy() for independent copies. Mastering shape, indexing, and slicing is essential. Practical exercises are recommended to solidify these fundamental operations.
Read MoreGetting Started with Numpy from Scratch: From Array Creation to Basic Operations
NumPy is a core library for numerical computing in Python, providing high-performance multidimensional arrays and computational tools, suitable for scenarios such as data science and machine learning. Installation is done via `pip install numpy`, with the import typically abbreviated as `np`. Arrays can be created in various ways: from Python lists, using `np.zeros`/`ones` (arrays of all zeros/ones), `arange` (arithmetic sequences), `linspace` (uniformly distributed values), and `np.random` (random arrays). Array attributes include `shape` (dimensions), `ndim` (number of dimensions), `dtype` (data type), and `size` (total number of elements). Indexing and slicing are flexible: one-dimensional arrays behave like lists, while two-dimensional arrays use row and column indices, with support for boolean filtering (e.g., `arr[arr>3]`). Basic operations are efficient, including element-wise arithmetic (+, *, etc.), matrix multiplication (via `dot` or `@`), and the broadcasting mechanism (e.g., automatic expansion for array-scalar operations). Application examples include statistical analysis (using functions like `sum` and `mean`) and data filtering. Mastering these capabilities enables efficient numerical data processing and lays the foundation for advanced functionalities such as linear algebra.
Read MoreLearn Python OpenCV Easily: Drawing Basic Geometric Shapes
This article introduces methods to draw basic geometric shapes using OpenCV. The steps are as follows: First, install the opencv-python and numpy libraries. After importing these libraries, create a 500x500 black canvas. For drawing shapes: Lines are drawn using cv2.line, e.g., an anti-aliased red line from (50,50) to (450,450); Rectangles are drawn using cv2.rectangle, supporting both outlines (line width 3) and fill (line width -1), such as a green outlined rectangle and a blue filled rectangle; Circles are drawn using cv2.circle, supporting both outlines (line width 5) and fill (line width -1), such as a yellow outlined circle and a red filled circle; Polygons are drawn using cv2.polylines (for outlines) and cv2.fillPoly (for filling), with an example being a cyan triangular outline and a light red quadrilateral fill. Finally, display the image with cv2.imshow and wait for user input to close using cv2.waitKey. Key notes: Colors are in BGR format (e.g., red is (0,0,255)), line width -1 indicates filling, and the coordinate origin is at the top-left corner of the image.
Read MoreIntroduction to Python OpenCV: Denoising Methods in Image Preprocessing
In image preprocessing, denoising is a core step to eliminate noise (such as Gaussian, salt-and-pepper, Poisson noise) during acquisition/transmission and improve the accuracy of subsequent tasks. Python OpenCV provides multiple denoising methods: 1. **Mean Filtering**: A simple average of window pixels, fast but blurs edges. Suitable for Gaussian noise, implemented with `cv2.blur` (3×3 kernel). 2. **Median Filtering**: Replaces the center pixel with the median of window pixels. Effective against salt-and-pepper noise (0/255 specks), preserves edges well. Kernel size must be odd (e.g., 3×3), using `cv2.medianBlur`. 3. **Gaussian Filtering**: Weighted average using a Gaussian distribution kernel, balances denoising and edge preservation. Ideal for Gaussian noise, requires kernel size and standard deviation in `cv2.GaussianBlur`. 4. **Bilateral Filtering**: Combines spatial and color distance, excels at edge-preserving denoising with high computational cost. Suitable for high-precision scenarios (e.g., face images), implemented with `cv2.bilateralFilter`. **Selection Guidelines**: Gaussian noise → Gaussian filtering; salt-and-pepper noise → median filtering; mixed noise → Gaussian followed by median; high-frequency detail noise → bilateral filtering. Beginners are advised to start with Gaussian and median filters, adjusting based on... *(Note: The original text ends abruptly; the translation concludes at the logical cutoff point.)*
Read MorePython OpenCV Practical: Template Matching and Image Localization
This paper introduces an image localization method using Python OpenCV to implement template matching. The core of template matching is sliding a "template image" over a target image and calculating similarity to find the most matching region, which is suitable for simple scenarios (e.g., monitoring object localization). The steps include: preparing target and template images, converting them to grayscale to improve efficiency; using `matchTemplate` (e.g., the `TM_CCOEFF_NORMED` method) to calculate the similarity matrix; setting a threshold (e.g., 0.8) to filter high-similarity regions and using `np.where` to obtain their positions; finally, marking the matching results with rectangles and displaying/saving them. Note: Template matching is only applicable to scenarios where the target has no rotation or scaling; for complex scenarios, feature matching like ORB should be used instead. The matching method and threshold need to be adjusted according to actual conditions—too high a threshold may lead to missed detections, while too low may cause false positives. Through the practical example of "apple localization," this paper helps beginners master the basic process, making it suitable for quickly implementing simple image localization tasks.
Read MoreA Beginner's Guide to Python OpenCV Morphological Operations (Easy to Understand!)
Morphological operations are shape-based methods in image processing. Their core is to interact with images through a structuring element, altering the shape characteristics of objects. Primarily used for binary images, they implement functions such as denoising, connecting objects, and filling holes. Basic types include: Erosion (shrinking bright regions, expanding dark regions; denoising but edge contraction), Dilation (expanding bright regions, filling dark holes; connecting breaks), Opening (erosion followed by dilation; denoising while preserving shape), and Closing (dilation followed by erosion; hole filling and edge optimization). A structuring element is a small matrix defining the shape and size of operations. OpenCV supports rectangles, ellipses, crosses, etc., created via `cv2.getStructuringElement`. For code implementation, steps include reading the image, binarization, defining the structuring element, performing erosion, dilation, opening/closing operations, and displaying results. Advanced operations like morphological gradient, top hat, and black hat can also extract edges or noise. Summary: Morphology is a fundamental tool for denoising, object connection, and edge extraction. Beginners can start with opening/closing operations, adjusting structuring element size and shape to practice applications in different scenarios.
Read MoreIntroduction to Python OpenCV Filter Effects: Blur and Sharpen Image Processing
This article introduces the basic operations of blurring and sharpening in digital image processing, suitable for beginners to implement using Python+OpenCV. Blurring is used for denoising and smoothing, with common methods including: Mean filtering (simple averaging, fast denoising but blurs details), Gaussian filtering (weighted averaging, natural blurring, removes Gaussian noise), Median filtering (median substitution, anti-salt-and-pepper noise while preserving edges), and Bilateral filtering (edge-preserving blurring, used for portrait beauty). Sharpening enhances edge details, with methods such as: Laplacian operator (second-order derivative, general sharpening), simple pixel superposition (directly highlights edges), and Sobel operator (gradient calculation, enhances edges). The article summarizes the characteristics of these methods in a comparison table and provides exercise suggestions, serving as a foundational introduction to image processing.
Read MoreLearning Python OpenCV from Scratch: Real - time Capture and Display with Camera
This article introduces a method to achieve real - time camera capture and display using Python and OpenCV. The reasons for choosing OpenCV (Open Source Computer Vision Library) and Python (with concise syntax) are their ease of use and functional adaptability. The opencv - python interface for Python is easy to install. Installation steps: First, install Python 3.6 or higher, and then install the library through `pip install opencv - python` (numpy may need to be installed first if necessary). Core process: Open the camera (`cv2.VideoCapture(0)`), loop to read frames (`cap.read()`, which returns ret and frame), display the image (`cv2.imshow()`), press the 'q' key to exit, and release resources (`cap.release()` and `cv2.destroyAllWindows()`). Key code explanation: `cap.read()` checks the reading status, `cv2.waitKey(1)` waits for a key press (the 'q' key to exit), and ensures that resources are correctly released to avoid occupation. The article also mentions common problems (such as the camera not opening) and extended exercises (such as grayscale display, image flipping, etc.), laying a foundation for subsequent complex image processing.
Read MorePython OpenCV Image Scaling and Cropping: Essential Techniques for Beginners
This article introduces basic operations of image resizing and cropping in Python OpenCV, helping beginners master core techniques. **Image Resizing**: Use the `cv2.resize()` function, supporting two target size specification methods: scaling by ratio (controlled via `fx`/`fy`, e.g., `fx=0.5` to halve the size) or directly specifying width and height (e.g., `(200, 200)`). Recommended interpolation methods: `INTER_AREA` for shrinking and `INTER_LINEAR` for enlarging to avoid distortion. In examples, pay attention to correct image path and window operations (`waitKey` and `destroyAllWindows`). **Image Cropping**: Essentially involves NumPy array slicing with the format `img[y_start:y_end, x_start:x_end]`, ensuring coordinates do not exceed bounds (`y_end` ≤ height, `x_end` ≤ width). Examples include fixed-region cropping and center-region cropping (calculating center offsets `(w-target_w)//2` and `(h-target_h)//2` before slicing). **Summary**: Resizing requires attention to path and interpolation methods, while cropping must focus on coordinate ranges. These two operations are often used together (e.g., cropping first then resizing) and are fundamental in image preprocessing.
Read MoreStep-by-Step Guide to Image Contour Detection with Python OpenCV
This article introduces a method for image contour recognition using Python OpenCV. First, the OpenCV and NumPy libraries need to be installed. Image contours are the boundary lines of objects, used to locate target objects (such as faces, circles). The core steps include: preprocessing (grayscale conversion + binarization to simplify the image), edge detection (Canny algorithm to determine boundaries through thresholds), contour extraction (obtaining coordinates via findContours), and filtering and drawing (filtering by area and other criteria and visualizing). In practice, taking "shapes.jpg" as an example, the process is demonstrated: reading the image → grayscale conversion + binarization → Canny edge detection → findContours to extract contours → filtering the largest contour by area and drawing it. Common issues like incomplete contours can be addressed by adjusting Canny thresholds, and excess contours can be resolved through area filtering. It can also be extended to recognize objects using shape features such as circularity. In summary, contour recognition is a foundation in computer vision. Beginners can start with simple images and optimize results through parameter adjustments.
Read MoreEasy Guide: Python OpenCV Edge Detection Fundamentals
This article introduces the concept of image edge detection, its implementation in Python with OpenCV, and core algorithms. Edge detection identifies regions with significant changes in pixel intensity (e.g., object contours), a foundational technique in computer vision with applications in facial recognition, autonomous driving, etc. For environment setup, install Python and OpenCV (`pip install opencv-python`). The core workflow has three steps: image preprocessing (grayscale conversion, noise reduction), edge detection algorithms, and result visualization. The Canny edge detection algorithm (proposed by John Canny in 1986) is emphasized with the following steps: 1) Grayscale conversion (reduces computational complexity); 2) Gaussian blur (noise reduction, 5×5 kernel size is common); 3) Gradient calculation (using Sobel operators); 4) Non-maximum suppression (refines edges); 5) Double thresholding (low threshold 50-150, high threshold 150-200; threshold values affect edge sensitivity). Python code example: read image → grayscale conversion → blur → Canny detection → display results. Other algorithms include Sobel (gradient calculation) and Laplacian (second-order derivative), which require prior blur for noise reduction. Practical tips: prioritize blurring, adjust thresholds; common issues: image read failure (check file path).
Read MoreFrom Beginner to Practical: A Detailed Explanation of Python OpenCV Color Space Conversion
This article introduces the concept of image color spaces and the conversion applications in Python using OpenCV. Common color spaces include RGB (for display, with red/green/blue channels), BGR (OpenCV default, in blue/green/red order), and HSV (hue H, saturation S, value V, suitable for color segmentation). The conversion reasons are that different spaces serve different purposes (RGB for display, HSV for color recognition, BGR as OpenCV's native format). The core tool is `cv2.cvtColor()`, with the syntax `cv2.cvtColor(img, cv2.COLOR_originalSpace2targetSpace)`, e.g., `cv2.COLOR_BGR2HSV`. In practice, taking red object detection as an example: read the image → convert to HSV → define the red HSV range (H values in 0-10 and 160-179 intervals) → extract via mask. It can also be extended to real-time detection with a camera. Key points: master the conversion function, note the difference between BGR and RGB, and adjust HSV ranges according to light conditions.
Read MorePython OpenCV Tutorial: Master Image Binarization in 5 Minutes
Image binarization is a process that classifies pixels into black and white categories based on a threshold, simplifying images for easier analysis, and is commonly used in scenarios such as text recognition. The core implementation relies on the `cv2.threshold()` function, which requires inputting a grayscale image, a threshold value, a maximum value, and a type, returning the actual threshold and the binarized image. Common threshold types include: `THRESH_BINARY` (pixels above the threshold turn white), `THRESH_BINARY_INV` (the opposite), and `THRESH_OTSU` (automatically calculates the optimal threshold). For threshold selection: manual selection is suitable for images with uniform brightness, Otsu's method is ideal for high-contrast scenarios, and adaptive thresholds are used for uneven lighting. The key steps are: reading the image and converting it to grayscale → selecting the threshold type → performing binarization → displaying the result. Mastering binarization supports tasks such as edge detection and object segmentation.
Read MoreLearning Python OpenCV from Scratch: A Step-by-Step Guide to Reading and Displaying Images
This article introduces basic operations of Python OpenCV, including installation, image reading, and displaying. OpenCV is an open-source computer vision library. It can be installed via `pip install opencv-python` (or accelerated by domestic mirror sources). To verify, import the library and print the version number. For reading images, use `cv2.imread()`, specifying the path and parameters (color, grayscale, or original image), and check if the return value is `None` to confirm success. To display images, use `cv2.imshow()`, which should be accompanied by `cv2.waitKey(0)` to wait for a key press and `cv2.destroyAllWindows()` to close windows. Common issues: OpenCV reads images in BGR channels by default; use `cv2.cvtColor()` to convert to RGB to avoid color abnormalities. Path errors may cause reading failure; use absolute paths or confirm the image format. The core steps are installation, reading, and displaying, and hands-on practice can quickly master these operations.
Read MoreImplementing Radix Sort Algorithm in C++
Radix sort is a non-comparison integer sorting algorithm that uses the least significant digit first (LSD) approach, sorting numbers digit by digit (units, tens, etc.) without comparing element sizes. Its core idea is to process each digit using a stable counting sort, ensuring that the result of lower-digit sorting remains ordered during higher-digit sorting. Implementation steps: 1. Identify the maximum number in the array to determine the highest number of digits to process; 2. From the lowest digit to the highest, process each digit using counting sort: count the frequency of the current digit, compute positions, place elements stably from back to front, and finally copy back to the original array. In the C++ code, the `countingSort` helper function implements digit-wise sorting (counting frequencies, calculating positions, and stable placement), while the `radixSort` main function loops through each digit. The time complexity is O(d×(n+k)) (where d is the maximum number of digits, n is the array length, and k=10), making it suitable for scenarios with a large range of integers. The core lies in leveraging the stability of counting sort to ensure that the results of lower-digit sorting are not disrupted during higher-digit sorting. Test results show that the sorted array is ordered, verifying the algorithm's effectiveness.
Read MoreImplementing Bucket Sort Algorithm in C++
Bucket sort is a non-comparison sorting algorithm that sorts elements by distributing them into multiple "buckets", sorting each bucket individually, and then merging the sorted buckets. The core is to reasonably partition the buckets so that each bucket contains a small number of elements, thereby reducing the sorting cost. Taking floating-point numbers in the range [0,1) as an example, the algorithm steps are as follows: 1. Create n empty buckets (where n is the length of the array); 2. Assign each element x to the corresponding bucket using the bucket index calculated as the integer part of x * n; 3. Sort each bucket using std::sort; 4. Merge all elements from the buckets. In the C++ implementation, the `bucketSort` function creates n buckets using a vector of vectors of doubles, distributes elements into the buckets through traversal, sorts each bucket, and then merges the results. Testing verifies the correctness of the algorithm. Complexity analysis: The average time complexity is O(n) (when elements are uniformly distributed), and the worst-case time complexity is O(n log n) (when all elements are placed in the same bucket). The space complexity is O(n). It is suitable for numerical data with uniformly distributed values and a clear range; performance degrades when data distribution is uneven. This algorithm is efficient when the data distribution is reasonable, especially suitable for sorting interval data in statistical analysis.
Read MoreImplementing the Counting Sort Algorithm in C++
**Counting Sort** is a non-comparison sorting algorithm. Its core idea is to construct a sorted array by counting the occurrences of elements, making it suitable for scenarios where the range of integers is not large (e.g., student scores, ages). **Basic Idea**: Taking the array `[4, 2, 2, 8, 3, 3, 1]` as an example, the steps are: 1. Determine the maximum value (8) and create a count array `count` to statistics the occurrences of each element (e.g., `count[2] = 2`); 2. Insert elements into the result array in the order of the count array to obtain the sorted result `[1, 2, 2, 3, 3, 4, 8]`. **Implementation Key Points**: In C++ code, first find the maximum value, count the occurrences, construct the result array, and copy it back to the original array. Key steps include initializing the count array, counting occurrences, and filling the result array according to the counts. **Complexity**: Time complexity is O(n + k) (where n is the array length and k is the data range), and space complexity is O(k). **Applicable Scenarios**: Non-negative integers with a small range, requiring efficient sorting; negative numbers can be handled by offset conversion (e.g., adding the minimum value). Counting Sort achieves linear-time sorting through the "counting-construction" logic and is ideal for processing small-range integers.
Read MoreImplementing the Merge Sort Algorithm in C++
Merge sort is based on the divide-and-conquer principle, with the core being "divide-merge": first recursively split the array into individual elements (where subarrays are ordered), then merge two ordered subarrays into a larger ordered array. **Divide process**: Recursively split the array from the middle until each subarray contains only one element. **Merge process**: Compare elements from two ordered subarrays, take the smaller value and place it in the result array sequentially, then handle the remaining elements. The C++ implementation includes two core functions: `mergeSort` for recursively dividing the array, and `merge` for merging two ordered subarrays. The time complexity is O(n log n), and the space complexity is O(n) (due to the need for a temporary array). Merge sort is stable and efficient, making it suitable for sorting large-scale data. In the example, the array `[5,3,8,6,2,7,1,4]` is sorted into the ordered array `[1,2,3,4,5,6,7,8]` through division and merging, verifying the algorithm's correctness.
Read MoreImplementing the Heap Sort Algorithm in C++
Heap sort is an efficient sorting algorithm based on the heap data structure, with a time complexity of O(n log n) and a space complexity of O(1), making it suitable for large-scale data. A heap is a special complete binary tree, divided into max heaps (parent ≥ children) and min heaps, with max heaps commonly used in sorting. It is stored in an array where the parent of index i is (i-1)/2, and the left and right children are 2i+1 and 2i+2, respectively. The core steps are: 1. Constructing the initial max heap (adjusting from the last non-leaf node upwards); 2. Sorting (swapping the top element with the end of the unsorted part, adjusting the heap, and repeating until completion). The C++ implementation includes swap, max_heapify (iteratively adjusting the subtree to form a max heap), and heap_sort (constructing the heap and performing sorting) functions. The main function tests array sorting, and the output result is correct.
Read MoreImplementing the Selection Sort Algorithm in C++
Selection sort is a simple and intuitive sorting algorithm. Its core idea is to repeatedly select the smallest (or largest) element from the unsorted elements and place it at the end of the sorted sequence until all elements are sorted. The basic steps are as follows: the outer loop controls the current starting position of the unsorted elements; the inner loop finds the minimum value among the remaining elements; the swap operation moves the minimum value to the current starting position; this process repeats until all elements are sorted. Taking the array {64, 25, 12, 22, 11} as an example, the process is demonstrated: when i=0, the minimum value 11 is found and swapped to the first position; when i=1, 12 is found and swapped to the second position; when i=2, 22 is found and swapped to the third position; no swap is needed when i=3, and the array is finally sorted. The C++ code is implemented with two nested loops: the outer loop controls the position i, the inner loop finds the index minIndex of the minimum value, and swaps arr[i] with arr[minIndex]. The time complexity is O(n²) and the space complexity is O(1). Selection sort is easy to implement and requires no additional space. It is suitable for sorting small-scale data and serves as a foundational example for understanding sorting algorithms.
Read MoreImplementing the Shell Sort Algorithm in C++
Shell Sort is an improved version of Insertion Sort, also known as "diminishing increment sort". It efficiently sorts arrays by performing insertion sorts on grouped subsequences and gradually reducing the increment. The basic idea is: select an initial increment `gap` (e.g., half the array length), group elements with intervals of `gap` (forming subsequences), perform insertion sort on each group; repeat by reducing `gap` (usually halving it) until `gap=1` to complete the overall sorting. Core principle: Larger `gap` reduces the number of moves by grouping, while smaller `gap` leaves the array partially sorted, significantly lowering the total number of moves in the final insertion sort. For instance, take the array `[12, 34, 54, 2, 3]` – after initial `gap=2` grouping and sorting, the array becomes more ordered, and then `gap=1` completes the final sort. The code implements Shell Sort with three nested loops: the outer loop controls the `gap`, the middle loop iterates through each group, and the inner loop shifts elements. The average time complexity is `O(n^1.3)` (dependent on the increment), with the worst-case `O(n²)`, and a space complexity of `O(1)`. It is unstable. By optimizing insertion sort through grouping, Shell Sort is suitable for larger arrays. Its core logic lies in "grouping → sorting → reducing increment → final sorting".
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