Trie: How Does a Trie Store and Look Up Words? A Practical Example
A trie (prefix tree) is a data structure for handling string prefix problems. Its core is to save space and improve search efficiency by utilizing common prefixes. Each node contains a character, up to 26 child nodes (assuming lowercase letters), and an isEnd flag (indicating whether the node is the end of a word). When inserting, start from the root node and process each character one by one. If there is no corresponding child node, create a new one. After processing all characters, mark the end node's isEnd as true. For search, also start from the root and match each character one by one, then check the isEnd flag to confirm existence. In examples, "app" and "apple" share the prefix "app", while "banana" and "bat" share "ba", demonstrating the space advantage. Its strengths include more space-efficient storage (sharing prefixes), fast search (time complexity O(n), where n is the word length), and support for prefix queries.
Read MoreApplications of Stacks and Queues: Parentheses Matching Problem, Super Simple with Stacks
### Parentheses Matching Problem: The "Ultra-Simple" Application of Stacks This article introduces a method to solve the parentheses matching problem using stacks (with the Last-In-First-Out property). The problem requires determining if a string composed of `()`, `[]`, and `{}` is valid, meaning left parentheses and right parentheses correspond one-to-one and in the correct order. The "Last-In-First-Out" property of stacks is well-suited for this problem: left parentheses are pushed onto the stack for temporary storage, and right parentheses must match the most recently pushed left parenthesis. The specific steps are as follows: initialize a stack; when traversing the string, directly push left parentheses onto the stack; for right parentheses, check if the top element of the stack matches (using a dictionary to map right parentheses to their corresponding left parentheses). If they match, pop the top element; otherwise, the string is invalid. After traversal, if the stack is empty, the string is valid; otherwise, it is invalid. Key details include: distinguishing parenthesis types (using a dictionary for mapping), immediately returning invalid if the stack is empty when encountering a right parenthesis, and ensuring the stack is empty at the end as a necessary condition for validity. Through the logic of pushing left parentheses, checking right parentheses, and popping on match, this method efficiently determines the validity of any parenthesis string.
Read MoreBinary Search Trees: How to Implement Efficient Search Using Binary Search Trees?
A Binary Search Tree (BST) is an efficient data structure designed to solve the problem of "quickly locating targets" in daily data retrieval. It is a special type of binary tree where each node satisfies the following condition: all values in the left subtree are less than the current node's value, and all values in the right subtree are greater than the current node's value. The efficiency of BST stems from its "left smaller, right larger" rule. When searching, starting from the root node, we compare the target value with the current node's value at each step. If the target is smaller, we recursively search the left subtree; if it is larger, we recursively search the right subtree. This process eliminates half of the nodes with each comparison, resulting in a stable time complexity of O(log n), which outperforms unordered arrays (O(n)) and binary search on sorted arrays (which has low insertion efficiency). The core of the search process is "comparison - narrowing the range": starting from the root node, if the target value equals the current node's value, the target is found. If it is smaller, we move to the left subtree; if it is larger, we move to the right subtree, repeating this recursively. This can be implemented using either recursion or iteration. For example, the recursive method compares values level by level starting from the root, while the iterative method uses a loop to narrow down the search range. It is important to note that if a BST becomes unbalanced (e.g., degenerating into a linked list), its efficiency degrades to O(n). However, balanced trees such as Red-Black Trees and AVL Trees can maintain a stable O(log n) time complexity. BST achieves efficient ordered search by "navigating" to narrow down the range step by step.
Read MoreImplementing Common Sorting Algorithms in Python
Thank you very much for sharing the implementations of these sorting algorithms. To provide a more comprehensive and understandable version, I will briefly explain each sorting algorithm and include complete code snippets. Additionally, I will add necessary import statements and comments within each function to enhance code readability. ### 1. Bubble Sort Bubble Sort is a simple sorting method that repeatedly traverses the list to be sorted, comparing two elements at a time and swapping them if their order is incorrect. After multiple traversals, the largest element "bubbles up" to the end. ```python def bubble_sort(arr): n = len(arr) for i in range(n): # Last i elements are already in place swapped = False for j in range(0, n-i-1): if arr[j] > arr[j+1]: arr[j], arr[j+1] = arr[j+1], arr[j] swapped = True # If no swaps occurred, the array is sorted if not swapped: break return arr ```
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