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 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 Insertion Sort Algorithm in C++
Insertion sort is a simple and intuitive sorting algorithm whose core idea is to insert elements one by one into their appropriate positions in a sorted subarray (similar to sorting playing cards). The basic approach is: starting from the second element, take the current element, compare it with the previously sorted elements. If a previous element is larger, shift it backward until the insertion position is found. Insert the current element there and continue processing the next element. When implementing, an outer loop iterates through the elements, and an inner loop uses a temporary variable to save the current element. By comparing and shifting the previous elements to make space, the current element is finally inserted. The time complexity is O(n²) in the worst case and O(n) in the best case, with a space complexity of O(1). It is suitable for small-scale data or nearly sorted data. Its advantages include stability and simplicity, making it a foundation for understanding more complex sorting algorithms.
Read MoreImplementing the Insertion Sort Algorithm in Java
Insertion sort is a simple and intuitive sorting algorithm. Its core idea is to insert unsorted elements one by one into their correct positions in the sorted part, similar to organizing playing cards. It is suitable for small-scale data and has a simple implementation. Basic idea: Starting from the second element, mark the current element as the "element to be inserted". Compare it with the elements in the sorted part from back to front. If the sorted element is larger, shift it backward until the insertion position is found. Repeat this process until all elements are processed. In Java implementation, the element to be inserted needs to be saved, and the insertion is completed by looping through comparisons and shifting elements backward. The time complexity of the algorithm is: best O(n) (when already sorted), worst and average O(n²); space complexity O(1) (in-place sorting); stable sort, suitable for small-scale data or nearly sorted data. Its core lies in "gradual insertion", with simple implementation. Its stability and in-place nature make it perform well in small-scale sorting.
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