Heap Sort: How to Implement Heap Sort and Detailed Explanation of Time Complexity
Heap sort is a sorting algorithm that utilizes "heaps" (a special type of complete binary tree), commonly using a max heap (where parent nodes are greater than or equal to their child nodes). The core idea is "build the heap first, then sort": first convert the array into a max heap (with the maximum value at the heap top), then repeatedly swap the heap top with the last element, adjust the remaining elements into a heap, and complete the sorting. Basic concepts of heaps: A complete binary tree structure where for an element at index i in the array, the left child is at 2i+1, the right child at 2i+2, and the parent is at (i-1)//2. In a max heap, parent nodes are greater than or equal to their children; in a min heap, parent nodes are less than or equal to their children. The implementation has two main steps: 1. Constructing the max heap: Starting from the last non-leaf node, use "heapify" (comparing parent and child nodes, swapping the maximum value, and recursively adjusting the subtree) to ensure the max heap property is maintained. 2. Sorting: Swap the heap top with the last unsorted element, reduce the heap size, and repeat the heapify process until sorting is complete. Time complexity: Building the heap takes O(n), and the sorting process takes O(n log n), resulting in an overall time complexity of O(n log n). Space complexity is O(1) (in-place sorting). It is an unstable sort and suitable for sorting large-scale data.
Read MoreImplementing the Radix Sort Algorithm with Python
Radix sort is a non-comparative integer sorting algorithm. Its core idea is to distribute elements into buckets and collect them by each digit (from the least significant to the most significant). The basic steps are as follows: first, determine the number of digits of the maximum number in the array. Then, from the least significant digit to the most significant digit, perform "bucket distribution" (10 buckets for digits 0-9) and "collection" operations for each digit. Elements with the same current digit are placed into the same bucket, and the array is updated by collecting them in bucket order until all digits are processed. In Python, this is implemented by looping through the digits, calculating the current digit to distribute into buckets, and then collecting. The time complexity is O(d×(n+k)) (where d is the number of digits of the maximum number, n is the array length, and k=10), and the space complexity is O(n+k). It is suitable for integer arrays with few digits. When handling negative numbers, they can first be converted to positive numbers for sorting and then their signs can be restored.
Read MoreImplementing the Shell Sort Algorithm with Python
Shell Sort is an improved version of Insertion Sort, which enhances efficiency by "coarsely sorting" and then "finely sorting" through grouping to reduce element intervals. The core involves selecting an initial increment (e.g., half the array length), dividing the array into multiple groups where elements within each group are spaced by the increment, and applying Insertion Sort to each group. The process then repeats with the increment halved until the increment reaches 1, completing the "fine sorting." Its key logic is reducing element movement through grouping: initially grouping with large intervals allows the array to become nearly sorted first, and gradually shrinking the increment ensures the final Insertion Sort phase finishes efficiently. The average time complexity is O(n log n), worst-case O(n²), with a space complexity of O(1). Shell Sort is suitable for arrays of moderate size with uneven element distribution and is an efficient in-place sorting algorithm.
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.
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