Adjacency List: An Efficient Graph Storage Method, What Makes It Better Than Adjacency Matrix?
This article introduces the basic concepts of graphs and two core storage methods: the adjacency matrix and the adjacency list. A graph consists of vertices (e.g., social network users) and edges (e.g., friendship relationships). The adjacency matrix is a 2D array where 0/1 indicates whether there is an edge between vertices. It requires O(n²) space with n being the number of vertices, and checking an edge takes O(1) time. However, it wastes significant space for sparse graphs (few edges). The adjacency list maintains a neighbor list for each vertex (e.g., a user’s friend list), with space complexity O(n + e) where e is the number of edges, as it only stores actual edges. Checking an edge requires traversing the neighbor list (O(degree(i)) time, with degree(i) being the number of neighbors of vertex i), but traversing neighbors is more efficient in practice. A comparison shows that the adjacency list significantly outperforms the adjacency matrix in both space and time efficiency for sparse graphs (most practical scenarios). It is the mainstream storage method for graph problems (e.g., shortest path algorithms), offering better space saving and faster traversal.
Read MorePath Compression in Union-Find: Optimizing Union-Find for Faster Lookups
Union-Find (Disjoint Set Union, DSU) is used to solve set merging and element membership problems, such as connectivity judgment. Its core operations are `find` (to locate the root node) and `union` (to merge sets). The basic version uses a `parent` array to record parent nodes, but long-chain structures lead to extremely low efficiency in the `find` operation. To optimize this, **path compression** is introduced: during the `find` process, all nodes along the path are directly pointed to the root node, flattening the tree structure and making the lookup efficiency nearly O(1). Path compression can be implemented recursively or iteratively, transforming long chains into "one-step" short paths. Combined with optimizations like rank-based merging, Union-Find efficiently handles large-scale set problems and has become a core tool for solving connectivity and membership judgment tasks.
Read MoreRed-Black Trees: A Type of Balanced Binary Tree, Understanding Its Rules Simply
A red-black tree is a self-balancing binary search tree that ensures balance through color marking and five rules, resulting in stable insertion, deletion, and search complexities of O(log n). The core rules are: nodes are either red or black; the root is black; empty leaves (NIL) are black; red nodes must have black children (to avoid consecutive red nodes); and the number of black nodes on any path from a node to its descendant NIL leaves (black height) is consistent. Rule 4 prevents consecutive red nodes, while Rule 5 ensures equal black heights, together limiting the tree height to O(log n). Newly inserted nodes are red, and adjustments (color changes or rotations) are performed if the parent is red. Widely used in Java TreeMap and Redis sorted sets, it enables efficient ordered operations through its balanced structure.
Read MoreTrie: 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 MoreBinary Search: Applicable Scenarios and Learning Guide for Beginners
This article introduces the binary search algorithm, whose core is to compare the middle element in an ordered array to gradually narrow down the search range and quickly locate the target. It is suitable for scenarios with ordered data, large data volumes, static (rarely modified) content, and the need for rapid search, such as dictionaries or configuration files. The search process uses left and right pointers to determine the middle value mid. Depending on the size of the target relative to the middle value, the pointers are adjusted: if the middle value equals the target, the search is successful; if the target is larger, left is moved right; if smaller, right is moved left, until the target is found or the range is invalid. The core code of the Python iterative implementation uses a loop with left <= right, calculates mid = (left + right) // 2, and handles boundaries to return -1 when the array is empty or the target does not exist. The time complexity is O(log n) (since the range is halved each time), and the space complexity is O(1) (using only constant variables). Key details include expanding the traversal when handling duplicate elements, directly judging single-element arrays, and returning -1 if the target is not found. The "divide and conquer" (reduction and governance) idea of binary search efficiently solves the problem of fast searching in ordered large datasets, making it an important tool in basic algorithms.
Read MoreUnion-Find: What is Union-Find? A Method to Solve "Friendship" Problems
Union-Find (Disjoint Set Union, DSU) is an efficient data structure for managing element groups, primarily solving the "Union" (merge groups) and "Find" (check if elements belong to the same group) problems. It is ideal for scenarios requiring quick determination of element membership in a set. At its core, it uses a parent array to maintain parent-child relationships, where each group is represented as a tree with the root node as the group identifier. Initially, each element forms its own group. Key optimizations include **path compression** (shortening the path during Find to make nodes directly point to the root) and **union by rank** (attaching smaller trees to larger trees to prevent the tree from degrading into a linked list), ensuring nearly constant time complexity for operations. The core methods `find` (finds the root and compresses the path) and `union` (merges two groups by attaching the root of the smaller tree to the root of the larger tree) enable efficient group management. Widely applied in network connectivity checks, family relationship queries, minimum spanning trees (via Kruskal's algorithm), and equivalence class problems, Union-Find is a concise and powerful tool for handling grouping scenarios.
Read MorePrefix Sum: How to Quickly Calculate Interval Sum Using Prefix Sum Array?
The prefix sum array is an auxiliary array used to quickly calculate the sum of intervals. It is defined as follows: for the original array A, the prefix sum array S has S[0] = 0, and for k ≥ 1, S[k] is the sum of elements from A[1] to A[k], i.e., S[k] = S[k-1] + A[k]. For example, if the original array A = [1, 2, 3, 4, 5], its prefix sum array S = [0, 1, 3, 6, 10, 15]. The core formula for calculating the sum of an interval is: the sum of elements from the i-th to the j-th element of the original array is S[j] - S[i-1]. For example, to calculate the sum of A[2] to A[4], we use S[4] - S[1] = 10 - 1 = 9, which gives the correct result. The advantages include: preprocessing the S array takes O(n) time, and each interval sum query only takes O(1) time, resulting in an overall complexity of O(n + q) (where q is the number of queries), which is much faster than the O(qn) complexity of direct calculation. It should be noted that index alignment (e.g., adjusting the formula if the original array starts from 0), interval validity, and the space-for-time tradeoff are important considerations. The prefix sum array is implemented through "pre-accumulation".
Read MoreBalanced Binary Trees: Why Balance Is Needed and A Simple Explanation of Rotation Operations
Binary Search Trees (BST) may degenerate into linked lists due to extreme insertions, causing operation complexity to rise to O(n). Balanced binary trees control balance through the **balance factor** (the height difference between the left and right subtrees of a node), requiring a balance factor of -1, 0, or 1. When unbalanced, **rotation operations** (LL right rotation, RR left rotation, LR left rotation followed by right rotation, RL right rotation followed by left rotation) are used to adjust the structure, keeping the tree height at a logarithmic level (log n) and ensuring that operations such as search, insertion, and deletion maintain a stable complexity of O(log n). Rotations essentially adjust the pivot point to restore the balanced structure of the tree.
Read MoreFigure: A Beginner's Guide to the Basic Concepts and Adjacency List Representation of Graphs
A graph consists of vertices (nodes) and edges (connections). Vertices are the basic units, and edges can be directed (digraph) or undirected. Weighted graphs have edges with weights (e.g., distances), while unweighted graphs only retain connection relationships. The adjacency list is an efficient representation method that solves the space waste problem of the adjacency matrix in sparse graphs (where the number of edges is much less than the square of the number of vertices). Its core is that each vertex stores a list of directly connected vertices. For an undirected graph, if vertex 0 is connected to 1, 2, and 3, its adjacency list is [1, 2, 3]. For a weighted graph, the adjacency list can store tuples of "neighbor + weight". The space complexity of an adjacency list is O(V + E) (where V is the number of vertices and E is the number of edges), making it suitable for sparse graphs. It facilitates traversal of neighboring vertices but requires traversing the adjacency list to check if an edge exists between two vertices. Mastering the adjacency list is fundamental for algorithms such as graph traversal and shortest path finding.
Read MoreLinked List: Difference Between Singly Linked List and Doubly Linked List, Easy for Beginners to Understand
This article uses the example of storing a list of game players to illustrate how linked lists solve the problem of node movement required when deleting intermediate elements from an array. A linked list is a linear structure composed of nodes, where each node contains a data field and a pointer field. It is stored in non - contiguous memory, and only pointers need to be modified during insertion and deletion operations. A singly linked list is the simplest form. Each node only contains a next pointer, allowing for one - way traversal (from head to tail). When inserting or deleting elements, it is necessary to first find the predecessor node and then modify the pointer. It saves memory and is suitable for one - way scenarios (such as queues). A doubly linked list has an additional prev pointer in each node, supporting two - way traversal. During insertion and deletion, operations can be directly performed through the prev and next pointers without needing to find the predecessor node. However, it consumes slightly more memory and is suitable for two - way operations (such as browser history and address books). Comparison of singly and doubly linked lists: The singly linked list has a simple structure and saves memory, while the doubly linked list is fully functional but slightly more memory - intensive. The choice should be based on the requirements: use a singly linked list for one - way operations and a doubly linked list for two - way operations or frequent operations.
Read MoreImplementing the Selection Sort Algorithm in Java
Selection sort is a simple and intuitive sorting algorithm. Its core idea is to repeatedly select the smallest (or largest) element from the unsorted portion and place it at the end of the sorted portion until the entire array is sorted. The basic approach involves an outer loop to determine the end position of the sorted portion, and an inner loop to find the minimum value in the unsorted portion, followed by swapping this minimum value with the element at the current position of the outer loop. In Java implementation, the `selectionSort` method is implemented with two nested loops: the outer loop iterates through the array (with `i` ranging from 0 to `n-2`), and the inner loop (with `j` ranging from `i+1` to `n-1`) finds the index `minIndex` of the minimum value in the unsorted portion. Finally, the element at position `i` is swapped with the element at `minIndex`. Taking the array `{64, 25, 12, 22, 11}` as an example, the sorted array `[11, 12, 22, 25, 64]` is gradually constructed through each round of swaps. The time complexity is O(n²), making it suitable for small-scale data. This algorithm has a simple logic and easy-to-implement code, serving as a typical example for understanding the basic sorting concepts.
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 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 MoreBinary Search: How Much Faster Than Linear Search? Search Techniques in Data Structures
This article introduces search algorithms in computers, focusing on linear search and binary search. Linear search (sequential search) is a basic method that checks data one by one from the beginning to the end. It has a time complexity of O(n) and is suitable for scenarios with small data volumes or unordered data. In the worst case, it needs to traverse all data. Binary search, on the other hand, requires an ordered array. Its core is to eliminate half of the data each time, with a time complexity of O(log n). When the data volume is large, it is far more efficient than linear search (e.g., for n=1 million, binary search only needs 20 times, while linear search needs 1 million times). The two have different applicable scenarios: binary search is suitable for ordered, large - data - volume, and frequently searched scenarios; linear search is suitable for unordered, small - data - volume, or dynamically changing data. In summary, binary search significantly improves efficiency through "half - by - half elimination" and is an efficient choice for large - volume ordered data. Linear search is more flexible in scenarios with small data volumes or unordered data.
Read MoreHow Does a Hash Table Store Data? Hash Functions Simplify Lookup
The article uses the analogy of searching for a book to introduce the problem: sequential search of data (such as arrays) is inefficient, while a hash table is an efficient storage tool. The core of a hash table is the hash function, which maps data to "buckets" (array positions) to enable fast access and storage. The hash function converts data into a hash value (bucket number), for example, taking the last two digits of a student ID to get the hash value. When storing, the hash value is first calculated to locate the bucket. If multiple data items have the same hash value (collision), it can be resolved using the linked list method (chaining within the bucket) or open addressing (finding the next empty bucket). When searching, the hash value is directly calculated to locate the bucket, and then the search is performed within the bucket, eliminating the need to traverse all data, resulting in extremely fast speed. Hash tables are widely used (such as in address books and caches), with their core being the use of a hash function to transform the search from "scanning through everything" to "direct access".
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