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 MoreAdjacency 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 MoreState Transition in Dynamic Programming: The Process from Problem to State Transition Equation
Dynamic programming (DP) solves problems by breaking them down and storing intermediate results to avoid redundant calculations. It is applicable to scenarios with overlapping subproblems and optimal substructure. The core of DP lies in "state transition," which refers to the derivation relationship between states in different stages. Taking the staircase climbing problem as an example: define `dp[i]` as the number of ways to climb to the `i`-th step. The transition equation is `dp[i] = dp[i-1] + dp[i-2]`, with initial conditions `dp[0] = 1` (one way to be at the 0th step) and `dp[1] = 1` (one way to climb to the 1st step). In another extended example, the coin change problem, `dp[i]` represents the minimum number of coins needed to make `i` yuan. The transition equation is `dp[i] = min(dp[i-coin] + 1)` (where `coin` is a usable denomination), with initial conditions `dp[0] = 0` and the rest set to infinity. Beginners should master the steps of "defining the state → finding the transition relationship → writing the equation" and practice to become familiar with the state transition thinking. Essentially, dynamic programming is a "space-for-time" tradeoff, where the state transition equation serves as the bridge connecting intermediate results.
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 MoreMinimum Spanning Tree: A Classic Application of Greedy Algorithm, Introduction to Prim's Algorithm
This paper introduces spanning trees, Minimum Spanning Trees (MST), and the Prim algorithm. A spanning tree is an acyclic subgraph of a connected undirected graph that includes all vertices; an MST is the spanning tree with the minimum sum of edge weights, which is suitable for the greedy algorithm (selecting locally optimal choices at each step to achieve a globally optimal solution). The core steps of the Prim algorithm are: selecting a starting vertex, repeatedly choosing the edge with the smallest weight from the edges between the selected and unselected vertices, adding the corresponding vertex to the selected set, until all vertices are included in the set. The key is to use an adjacency matrix or adjacency list to record the graph structure. In the algorithm's pseudocode, the `key` array records the minimum edge weight, and the `parent` array records the parent node. The time complexity is O(n²) using an adjacency matrix, and can be optimized to O(m log n). The Prim algorithm is based on the greedy choice, and the cut property and cycle property ensure that the total weight is minimized. It is applied in scenarios requiring the minimum-cost connection of all nodes, such as network wiring and circuit design. In summary, MST is a classic application of the greedy algorithm, and Prim efficiently constructs the optimal spanning tree by incrementally expanding and selecting the smallest edge.
Read MoreSuffix Array: What is a Suffix Array? A Powerful Tool for Solving String Problems
Suffix array is an array that stores the starting positions of suffixes sorted lexicographically. A suffix is a substring starting from each position in the string to the end (e.g., for "banana", the suffixes are "banana", "anana", etc.). The lexicographical comparison rule is: compare the first different character by its size; if the characters are the same, compare subsequent characters in order; if one suffix is a prefix of the other, the shorter one is considered smaller. Taking "abrac" as an example, the sorted suffix starting positions array is [0, 3, 4, 1, 2] (e.g., the suffix starting at position 0 "abrac" is less than the one at position 3 "ac", and they are arranged in this order). The core value of suffix arrays lies in efficiently solving string problems: by leveraging the close relationship (long common prefix length) between adjacent suffixes after sorting, it can quickly handle tasks such as finding the longest repeated substring and verifying substring existence. For example, the LCP array can be used to find the longest repeated substring, or binary search can verify if a substring exists. In summary, the suffix array provides an efficient solution for string problems by sorting suffix starting positions and is a practical tool for string processing.
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 MoreQuick Sort: How to Choose the Pivot in Quick Sort? A Diagram of the Partition Process
Quick sort is based on the divide and conquer method, with the core being the selection of a pivot and partition. The choice of pivot affects efficiency: selecting the leftmost or rightmost element can lead to degradation on sorted arrays (O(n²)), while choosing the middle element results in slightly worse balance. The median-of-three method (median of the first, middle, and last elements) is most recommended as it avoids extreme cases. Partitioning is achieved by moving left and right pointers to place the pivot in its correct position, ensuring all elements to the left are smaller and all to the right are larger, followed by recursive sorting of the subarrays. With an average time complexity of O(n log n), quick sort is a highly efficient sorting algorithm commonly used in engineering.
Read MoreMerge Sort: The Principle of Merge Sort and a Classic Application of Divide and Conquer Thought
Merge sort is based on the "divide and conquer" principle, with core steps including decomposition, recursion, and merging. It first recursively splits an array into subarrays of length 1, then merges adjacent ordered subarrays using a two-pointer technique (comparing element sizes and storing results in a temporary array). The complete process involves decomposing until the smallest subarrays are reached, then merging them layer by layer into an ordered array. The time complexity is stably O(n log n) (recursive depth is log n, and each layer requires traversing all elements during merging). The space complexity is O(n) due to the temporary array needed for storing merged results. As a stable sorting algorithm, the relative order of equal elements remains unchanged, making it suitable for large datasets or scenarios requiring stability. Its "decomposition-merge" logic intuitively embodies the divide and conquer concept, serving as a classic case for understanding recursion and simplifying complex problems.
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 MoreLinked List Reversal: Methods to Reverse a Singly Linked List, Implemented Recursively and Iteratively
A singly linked list consists of nodes with a data field and a pointer field (next), starting from a head node, with the tail node's next being None. Reversing a linked list is used in scenarios such as reverse output and palindrome judgment. **Iterative Method**: Traverse the list while maintaining three pointers: `prev` (initially None), `current` (the head node), and `next` (temporary storage). Steps: 1. Save `current.next` to `next`. 2. Reverse `current.next` to point to `prev`. 3. Move `prev` to `current` and `current` to `next`. 4. When `current` is None, return `prev` (the new head). Time complexity: O(n), Space complexity: O(1). Intuitive. **Recursive Method**: Recursively reverse sublists (terminates when the sublist is empty or has one node). After recursion, set `head.next.next = head` and `head.next = None`, then return the new head. Time complexity: O(n), Space complexity: O(n) (due to recursion stack). Concise code. **Comparison**: Iterative method avoids stack overflow risks; recursion relies on the call stack. Key points: - Iterative: Pay attention to pointer order. - Recursive: Clearly define the termination condition.
Read MoreHash Collisions: Why Do Hash Tables Collide? How to Resolve Them?
Hash tables map keys to array positions using hash functions, but when different keys map to the same position, a hash collision occurs. The core reasons are either the number of keys far exceeding the array capacity or an uneven hash function. The key to resolving collisions is to ensure conflicting keys "occupy distinct positions." Common methods include: 1. **Chaining (Zipper Method)**: The most widely used approach, where each array position is a linked list. Conflicting keys are appended sequentially to the corresponding linked list (e.g., keys 5, 1, and 9 colliding would form a list: 5→1→9). This method is simple to implement, has high space utilization, and allows efficient traversal during lookups. 2. **Open Addressing**: When a collision occurs, vacant positions are sought in subsequent slots. This includes linear probing (step size 1), quadratic probing (step size as a square), and double hashing (multiple hash functions). However, it may cause clustering and is more complex to implement. 3. **Public Overflow Area**: The main array stores non-colliding keys, while colliding keys are placed in an overflow area. Lookups require traversing both the main array and the overflow area, but space allocation is difficult. The choice of collision resolution method depends on the scenario. Chaining is widely adopted due to its efficiency and versatility. Understanding hash collisions and their solutions is crucial for optimizing hash table performance.
Read MoreBFS of Tree: Implementation Steps for Breadth-First Search and Level Order Traversal
BFS is a classic tree traversal method that accesses nodes in a "breadth-first" (level order) manner, with its core implementation relying on a queue (FIFO). The steps are as follows: initialize the queue by enqueueing the root node, then repeatedly dequeue the front node for access, enqueue its left and right children (in natural order) until the queue is empty. BFS is suitable for tree hierarchy problems, such as calculating tree height, determining a perfect binary tree, and finding the shortest root-to-leaf path. For the binary tree `1(2(4,5),3)`, the level order traversal sequence is 1→2→3→4→5. Key points: The queue ensures level order, the enqueue order of children (left→right), time complexity O(n) (where n is the number of nodes), and space complexity O(n) (worst-case scenario with n/2 nodes in the queue). Mastering BFS enables efficient solution of level-related problems and serves as a foundation for more complex algorithms.
Read MoreTree DFS: Depth-First Search, a Traversal Method from Root to Leaf
A tree consists of nodes and edges, where each node (except the root) has exactly one parent and can have multiple children. Depth-First Search (DFS) is a traversal method that "goes deep into one path until it ends, then backtracks." Tree DFS traversals include pre-order (root → left → right), in-order, and post-order, with pre-order most directly reflecting root-to-leaf paths. For recursive pre-order traversal: visit the current node → recursively traverse the left subtree → recursively traverse the right subtree. Using the example tree (root 1, left child 2, right child 3; 2 has left child 4 and right child 5), the traversal order is 1 → 2 → 4 → 5 → 3. For non-recursive implementation, a stack is used: initialize with the root, then repeatedly pop the top node, visit it, and push its right child first followed by its left child onto the stack. Root-to-leaf DFS traversal is applied to problems like path sum calculation and path output. Recursive implementation is intuitive, while non-recursive stack-based methods are suitable for large datasets. Mastering pre-order traversal is a core skill for tree structure manipulation.
Read MoreHash Functions: How Do Hash Functions Generate Hash Values? A Must-Know for Beginners
A hash function is a "translator" that converts input of arbitrary length into a fixed-length hash value, which serves as the "ID number" of the data. Its core characteristics include: fixed length (e.g., MD5 produces 32 hexadecimal characters), one-way irreversibility (original data cannot be derived from the hash value), near-uniqueness (extremely low collision probability), and the avalanche effect (minor input changes lead to drastic hash value changes). The generation process consists of three steps: input preprocessing into binary, segmented mathematical operations, and merging the results. Unlike encryption functions, hash functions are one-way and do not require a key, while encryption is reversible and requires a key. They have extensive applications: file verification (comparing hash values to prevent tampering), password storage (storing hash values for security), data indexing, and data distribution in distributed systems. As a data fingerprint, the key characteristics of hash functions make them indispensable in security and verification.
Read MoreInsertion Sort: How Does Insertion Sort Work? A Comparison with Bubble Sort
This article introduces the fundamental importance of sorting and focuses on two simple sorting algorithms: Insertion Sort and Bubble Sort. Insertion Sort works by gradually building a sorted sequence. Starting from the second element, each element is inserted into its correct position within the already sorted portion (similar to arranging playing cards). Its average time complexity is O(n²), with a best-case complexity of O(n) when the array is already sorted. It has a space complexity of O(1) and is stable, making it suitable for small-scale or nearly sorted data. Bubble Sort, on the other hand, compares adjacent elements and "bubbles" larger elements to the end (like bubbles rising to the surface), determining the position of the largest element in each pass. It also has an average time complexity of O(n²) and a space complexity of O(1), but it is stable and involves more element movements, making it less commonly used in practical applications. Both algorithms have an O(n²) complexity, with Insertion Sort being more efficient, especially when data is nearly sorted. Understanding these algorithms is foundational for learning more complex sorting techniques.
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 MoreAdjacency Matrix: Another Representation Method for Graphs and a Comparison of Advantages and Disadvantages
An adjacency matrix is a fundamental representation of a graph, essentially an n×n two-dimensional array where rows and columns correspond to the vertices of the graph, and element values indicate the existence or weight of edges between vertices. In an undirected graph, the value 1 represents the presence of an edge, and 0 represents its absence; in a weighted graph, the actual weight value is directly stored. Its advantages include: first, checking the existence of an edge takes only O(1) time, and calculating vertex degrees is efficient (for undirected graphs, it is the sum of a row, while for directed graphs, rows and columns correspond to out-degrees and in-degrees respectively); second, it is suitable for dense graphs (with edge counts close to n²), has high space utilization, and is simple to implement, making it easy for beginners to understand. Disadvantages include: a space complexity of O(n²), which wastes significant space for sparse graphs; traversing adjacent vertices requires O(n) time, making it less efficient than adjacency lists; and insufficient flexibility for dynamically adjusting the number of edges. In summary, the adjacency matrix trades space for time. It is suitable for dense graphs or scenarios requiring frequent edge queries or degree calculations, but unsuitable for sparse graphs or scenarios requiring frequent traversal of adjacent vertices. It serves as a foundational tool for understanding graph structures.
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 MoreDynamic Programming: An Introduction to Dynamic Programming and Efficient Solutions for the Fibonacci Sequence
The Fibonacci sequence is defined as f(0) = 0, f(1) = 1, and for n > 1, f(n) = f(n-1) + f(n-2). When calculated directly with recursion, the time complexity is O(2^n) due to excessive repeated computations, resulting in extremely low efficiency. Dynamic programming optimizes this by trading space for time: 1. Memoization recursion: Using a memoization array to store already computed results, each subproblem is solved only once, leading to both time and space complexities of O(n). 2. Iterative method: Using only two variables for iterative computation, with time complexity O(n) and space complexity O(1), which is the optimal solution. The core characteristics of dynamic programming are overlapping subproblems (subproblems reappearing) and optimal substructure (the current solution depends on the solutions of subproblems). Its essence is to avoid redundant calculations by storing subproblem results. The Fibonacci sequence is a classic introductory case, and mastering it can be generalized to similar problems such as climbing stairs.
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 MoreHeap: Structure and Applications, Introduction to Min-Heap and Max-Heap
A heap is a special type of complete binary tree, characterized by the size relationship between parent and child nodes (parent ≤ child for a min-heap, parent ≥ child for a max-heap). It efficiently retrieves extreme values (with the top element being the minimum or maximum), similar to a priority queue. The underlying structure is a complete binary tree, where each level is filled as much as possible, and the last level is filled from left to right. When stored in an array, the left child index is 2i+1, the right child index is 2i+2, and the parent index is (i-1)//2. Basic operations include insertion (appending to the end and then "bubbling up") and deletion (replacing the top element with the last element and then "bubbling down"), both with a time complexity of O(log n). Heaps are widely used in priority queues (e.g., task scheduling), finding the k-th largest element, and Huffman coding. They are a critical structure for efficiently handling extreme value problems.
Read MoreGreedy Algorithm: What is the Greedy Algorithm? A Case Study on the Coin Change Problem
Greedy algorithm is an algorithm that makes the optimal choice (local optimum) at each step in the hope of achieving a global optimum. Its core is to satisfy the "greedy choice property"—that the local optimum at each step can lead to the global optimum. A classic application is the coin change problem: taking 25, 10, 5, and 1-cent coins as examples, to make 67 cents, we take 25×2 (50 cents), 10×1 (10 cents), 5×1 (5 cents), and 1×2 (2 cents) in descending order of denominations, totaling 6 coins, which is verified as optimal. However, its limitation is that if the problem does not satisfy the greedy choice property (e.g., with coin denominations [1, 3, 4] to make 6 cents), the greedy approach may fail (greedy would take 4+1+1=3 coins, while the optimal is 3+3=2 coins). Applicable scenarios include reasonable coin denominations (e.g., 25, 10, 5, 1) and activity scheduling (selecting the earliest-ending activities). In conclusion, the greedy algorithm is simple, intuitive, and efficient, but it only applies to problems that satisfy the greedy choice property and does not guarantee the global optimum for all problems.
Read MoreDivide and Conquer Algorithm: How Does the Divide and Conquer Idea Solve Problems? The Principle of Merge Sort
The core of the divide-and-conquer algorithm is "divide and conquer," which solves complex problems through three steps: divide (split into smaller subproblems), conquer (recursively solve subproblems), and combine (integrate results). It is suitable for scenarios with recursive structures. Taking array sum calculation as an example, the array is divided, the sum of subarrays is recursively computed, and the total sum is obtained through combination. Merge sort is a typical application: the array is first divided into individual elements (which are inherently ordered), and then the ordered subarrays are merged using the two-pointer technique. Its time complexity is O(n log n) and space complexity is O(n) (requiring a temporary array). Divide-and-conquer simplifies problems through recursion, and merge sort efficiently demonstrates its advantages. It serves as a foundation for understanding recursive and sorting algorithms.
Read MoreRecursion: What is Recursion? An Example with the Fibonacci Sequence, Explained for Beginners
This article explains the concept of recursion using everyday examples and classic cases. Recursion is a method that breaks down a large problem into smaller, similar subproblems until the subproblems are small enough to be solved directly (the termination condition), and then deduces the result of the large problem from the results of the small subproblems. The core lies in "decomposition" and "termination". Taking the Fibonacci sequence as an example, its recursive definition is: F(0) = 0, F(1) = 1, and for n > 1, F(n) = F(n-1) + F(n-2). To calculate F(5), we first need to compute F(4) and F(3), and so on, until we decompose down to F(0) or F(1) (the termination condition), then return the results layer by layer. The key points of recursion are having a clear termination condition (such as n=0, 1) and ensuring that each recursive call reduces the problem size; otherwise, it will lead to an infinite loop. The Python code implementation is concise: `def fibonacci(n): if n==0: return 0; elif n==1: return 1; else: return fibonacci(n-1)+fibonacci(n-2)`. Although recursive code is elegant, its efficiency is lower than the iterative method when calculating large numbers (e.g., F(100)), reflecting the idea of "retreating to advance" (though the original text's last sentence is incomplete, the translation captures the existing content).
Read MoreSearch Algorithms: Differences Between Sequential Search and Binary Search, and Which Is Faster?
The article introduces two basic search algorithms: sequential search and binary search, which are used to locate specific elements in data. Sequential search (linear search) works by comparing elements one by one. It does not require the data to be ordered, with a time complexity of O(n) (where n is the amount of data). Its advantage is simplicity, but its drawback is low efficiency, making it suitable for small data volumes or unordered data. Binary search (half-interval search) requires the data to be sorted. It narrows down the search range by half through comparison, with a time complexity of O(log n). It is highly efficient (e.g., only about 10 comparisons needed when n=1000), but it requires handling boundary conditions, and is suitable for large-sized ordered data. Comparison of the two: Sequential search does not require data ordering and is simple to implement but inefficient; binary search requires ordering and has higher complexity but is faster. The choice depends on data size and ordering: binary search for large ordered data and sequential search for small unordered data.
Read MoreSorting Algorithms: An Introduction to Bubble Sort, Step-by-Step Explanation + Code Examples
Bubble Sort is one of the simplest sorting algorithms in computer science. Its core idea is to repeatedly compare adjacent elements and swap their positions, allowing larger elements to gradually "bubble" up to the end of the array. The basic steps are: the outer loop controls n-1 rounds of comparisons (each round determines the position of one large element), and the inner loop starts from the first element, comparing adjacent elements in sequence; if the previous element is larger and the next is smaller, they are swapped. An optimization is that if no swaps occur in a round, it indicates the array is already sorted, and the process can terminate early. In terms of time complexity, the worst-case scenario (completely reverse ordered) is O(n²), while the best case (already sorted) is O(n). The space complexity is O(1) (only constant extra space is required). This algorithm is simple to implement and easy to understand, making it suitable for sorting small-scale data and serving as a foundational entry point for sorting algorithms.
Read MoreHash Table: How Does a Hash Table Store Data? A Diagram of Collision Resolution Methods
A hash table is a key-value storage structure that maps keys to array bucket positions through a hash function, enabling O(1) efficient lookup, insertion, and deletion. Its underlying structure is an array, where keys are converted into array indices (bucket positions) via a hash function (e.g., "key % array length"), and corresponding values are directly stored at these indices. Collisions occur when different keys yield the same hash value (e.g., student IDs 12 and 22 both %10 to 2 when the array length is 10). Two classic collision resolution methods exist: 1. **Chaining**: Each bucket stores a linked list, with colliding elements appended to the tail of the list. This is simple to implement but requires additional space. 2. **Open Addressing**: Linear probing is a common variant, where the algorithm searches for the next empty bucket (e.g., h → h+1 → h+2 ... for a hash value h). This uses only array operations but may cause clustering. The core components of a hash table are the hash function and collision handling logic, making it a foundational topic in data structure learning.
Read MoreBinary Trees: Three Traversal Methods of Binary Trees, Recursive Implementation Made Super Simple
This article introduces three classic traversal methods of binary trees (pre-order, in-order, and post-order), implemented recursively, with the core being clarifying the position of root node access. Each node in a binary tree has at most left and right subtrees. Traversal refers to visiting nodes in a specific order. Recursion is key here, similar to "matryoshka dolls," where the function calls itself with a narrowed scope until empty nodes are encountered, terminating the recursion. The differences between the three traversal orders are: - Pre-order: Root → Left → Right; - In-order: Left → Root → Right; - Post-order: Left → Right → Root. Using an example tree (root 1 with left child 2 and right child 3; node 2 has left child 4 and right child 5), the traversal results are: - Pre-order: 1 2 4 5 3; - In-order: 4 2 5 1 3; - Post-order: 4 5 2 3 1. The core of recursive implementation lies in the termination condition (returning for empty nodes) and recursively traversing left and right subtrees in the traversal order. By clarifying the root position and recursive logic, the traversal process can be clearly understood.
Read MoreTree: What is a Tree Structure? Easily Understand with Real-Life Examples
This article uses a life analogy to explain the "tree" in data structures. The core is that a tree is similar to a tree in life: it has a root node (starting point), child/parent nodes (branches and their source), leaf nodes (no descendants), and subtrees (nodes and their descendants), with the characteristics of being non-linear, branching, and hierarchical. Unlike linear linked lists (single path), trees can have multiple branches (e.g., the root node can have multiple child nodes). Tree structures are ubiquitous in life: family relationships take elders as the root, corporate structures take the CEO as the root, and computer file systems take the disk as the root, all reflecting hierarchical branches. The core advantage of trees is their efficient handling of hierarchical branching problems, such as database indexing, navigation path planning, and game scene construction. Understanding tree structures allows one to master the thinking of handling branching problems. In life, families, companies, and file systems are typical applications of trees.
Read MoreQueue: How is the "First-In-First-Out" of Queues Implemented? A Simple Example to Illustrate
A queue is a data structure that follows the "First-In-First-Out" (FIFO) principle. It only allows insertion at the rear and deletion at the front. Key concepts include the front (earliest element) and the rear (latest element), with basic operations being Enqueue (insertion) and Dequeue (deletion). In array-based implementation, a queue requires a front pointer, a rear pointer, and a fixed-capacity array. The queue is empty when front == rear, and full when rear == max_size. During Enqueue, the rear pointer is moved forward to store the new element; during Dequeue, the front pointer is moved forward to retrieve the element. Example Demonstration: For a queue with capacity 5, initially front=0 and rear=0. After enqueuing 1, 2, 3, rear becomes 3, with the queue elements [1, 2, 3]. Dequeuing 1 makes front=1, and enqueuing 4 moves rear to 4. Enqueuing 5 results in a full queue. Dequeuing 2 (front=2) leaves the final queue as [3, 4, 5]. Applications include task scheduling, Breadth-First Search (BFS), printer queues, and network request handling, playing a critical role in data processing and task queuing scenarios.
Read MoreStack: What Does "Last-In-First-Out" Mean? Principle Diagram
This article uses "stacking plates" as an example to explain the core concepts of the data structure "stack". A stack is a linear list where insertions and deletions can only be performed from one end (the top), with the other end being the bottom. Its core feature is "Last-In-First-Out" (LIFO) — the last element added is the first to be removed. Basic operations of a stack include: push (adding an element to the top), pop (removing and returning the top element), top (viewing the top element), and empty (checking if the stack is empty). For example, when stacking plates, new plates are placed on top (push), and the top plate must be taken first (pop), which aligns with LIFO. Stacks are widely applied in life and programming: bracket matching (using the stack to record left brackets, popping to match right brackets), function call stacks (functions called later return first), and browser back functionality (successively popping recently visited webpages). Understanding the "LIFO" feature of stacks helps solve problems like recursion and dynamic programming, making it a foundational tool in data structures.
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 MoreArrays: Why Are They the Cornerstone of Data Structures? A Must-Learn for Beginners
This article introduces the core position of arrays as a basic data structure. An array is a sequence of elements of the same type, enabling random access through indices (starting from 0). It features simplicity, intuitive design, continuous storage, and efficient index-based access. As a foundational structure, arrays underpin complex data structures like stacks, queues, and hash tables (e.g., stacks use arrays for Last-In-First-Out behavior, while queues utilize circular arrays for First-In-First-Out operations). They also form the basis of multi-dimensional arrays (e.g., matrices). Arrays support fundamental operations such as traversal, search, and sorting, with a random access time complexity of O(1), significantly outperforming linked lists' O(n). However, arrays have limitations: fixed size (static arrays) and inefficient insertion/deletion (requiring element shifting). In summary, arrays serve as the "key to entry" in data structures, and mastering them lays the foundation for learning complex structures and algorithms.
Read MoreWhat is Time Complexity O(n)? A Must-Learn Efficiency Concept for Data Structure Beginners
The article explains the necessity of time complexity: due to differences in hardware and compilers, direct timing is impractical, and an abstract description of the algorithm's efficiency trend is required. The core is linear time complexity O(n), which indicates that the number of operations is proportional to the input size n (such as the length of an array). Scenarios like traversing an array to find a target or printing all elements require n operations. Big O notation ignores constants and lower-order terms, focusing only on the growth trend (e.g., O(2n+5) is still O(n)). Comparing common complexities (O(1) constant, O(n) linear, O(n²) quadratic, O(log n) logarithmic), O(n) is more efficient than O(n²) but less efficient than O(1) or O(log n). Understanding O(n) is fundamental to algorithms, helping optimize code and avoid timeout issues caused by "brute-force solutions."
Read MoreHow to Handle Hash Table Collisions? Understand Hash Resolution Methods in Data Structures Simply
Hash tables map keys to array slots via hash functions, and hash collisions occur when different keys map to the same slot. The core solution is to find new positions for colliding elements, with the main methods as follows: 1. **Chaining (Separate Chaining)**: Each slot stores a linked list, where colliding elements are appended to the list. For example, keys 1, 6, and 11 hashing to the same slot form a linked list [1, 6, 11]. **Advantages**: Simple implementation, suitable for dynamic data. **Disadvantages**: Extra space for linked lists; worst-case O(n) lookup. 2. **Open Addressing**: When collisions occur, empty slots are probed. Includes linear probing (i+1, wrapping around) and quadratic probing (i+1², etc.). For example, key 6 hashing to slot 1 (collision) probes slot 2; key 11 probes slot 3. **Advantages**: High space utilization. **Disadvantages**: Linear probing causes primary clustering; quadratic probing requires a larger array. Other methods: Double hashing (multiple hash functions) and common overflow area (primary table + overflow table), suitable for low-collision scenarios. Selection depends on the use case: Chaining (e.g., Java HashMap) suits dynamic, large datasets; Open addressing is better for fixed-size arrays with few collisions.
Read MoreHow to Learn Tree Traversal? Easily Understand Preorder, Inorder, and Postorder Traversals
Trees are fundamental data structures, and traversal is the process of visiting all nodes. This article focuses on explaining the preorder, inorder, and postorder traversals of binary trees, with their core difference lying in the timing of visiting the root node. - **Preorder Traversal (Root → Left → Right)**: Visit the root first, then recursively traverse the left subtree, and finally the right subtree. Example: 1→2→4→5→3→6→7. - **Inorder Traversal (Left → Root → Right)**: Recursively traverse the left subtree first, then visit the root, and finally the right subtree. Example: 4→2→5→1→6→3→7. - **Postorder Traversal (Left → Right → Root)**: Recursively traverse the left subtree first, then the right subtree, and finally visit the root. Example: 4→5→2→6→7→3→1. Memory mnemonic: Preorder has the root first, inorder has the root in the middle, postorder has the root last. In applications, preorder is used for tree copying, inorder sorts binary search trees, and postorder is used for node deletion. Traversal essentially embodies the recursive thought; mastering the order and scenarios enables proficiency.
Read MoreHow to Understand Recursion? Easily Learn Recursion with the Fibonacci Sequence
Recursion is a problem-solving method that "calls itself", breaking down large problems into smaller, similar subproblems until the subproblems can be solved directly, then returning results layer by layer (like disassembling Russian nesting dolls). Its core elements are the **base case** (to avoid infinite loops, e.g., returning fixed values when n=0 or n=1) and the **recursive relation** (decomposing into subproblems, e.g., F(n) = F(n-1) + F(n-2)). Taking the Fibonacci sequence as an example, the recursive function `fib(n)` is implemented through the base case and recursive relation: `fib(0) = 0`, `fib(1) = 1`, and `fib(n) = fib(n-1) + fib(n-2)`. For `fib(5)`, it requires recursively calculating `fib(4)` and `fib(3)`, decomposing layer by layer until `fib(1)` and `fib(0)` are reached, then combining the results in reverse to get the final answer. The recursive process is like "peeling an onion": after reaching the bottom, results "bounce back". Its advantages include clear logic and ease of implementation, but it has redundant calculations (e.g., `fib(3)` is called multiple times), leading to lower efficiency—optimizations like memoization or iteration can be used. In essence, recursion transforms large problems into smaller ones, and smaller problems... (Note: The original Chinese summary appears truncated here; the above translation completes the core description.)
Read MoreWhat is a Heap? A Detailed Explanation of Basic Operations on Heaps in Data Structures
A heap is a special structure based on a complete binary tree, stored in an array, and satisfies the properties of a max heap (parent node value ≥ child node) or a min heap (parent node value ≤ child node). It can efficiently retrieve the maximum or minimum value and is widely used in algorithms. The array indices map parent-child relationships: left child is at 2i+1, right child at 2i+2, and parent is at (i-1)//2. A max heap has the largest root (e.g., [9,5,7,3,6,2,4]), while a min heap has the smallest root (e.g., [3,6,5,9,7,2,4]). Core operations include insertion (appending new element to the end and adjusting upward to satisfy heap property), deletion (swapping root with last element and adjusting downward), heap construction (adjusting from the last non-leaf node downward), and retrieving the root (directly accessing the root node). It is applied in priority queues, heap sort, and Top K problems. The efficient structure and operations of heaps are crucial for understanding algorithms, and beginners can start with array simulation to master them.
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 MoreBubble Sort: The Simplest Sorting Algorithm, Easy to Learn in 3 Minutes
Bubble Sort is a basic sorting algorithm that simulates the "bubble rising" process to gradually "bubble up" the largest element to the end of the array for sorting. The core idea is to repeatedly compare adjacent elements, swapping them if the preceding element is larger than the following one. After each round of traversal, the largest element is placed in its correct position. If no swaps occur in a round, the algorithm can terminate early. Taking the array [5, 3, 8, 4, 2] as an example: In the first round, adjacent elements are compared, and the largest number 8 "bubbles up" to the end, resulting in the array [3, 5, 4, 2, 8]. In the second round, the first four elements are compared, and the second largest number 5 moves to the second-to-last position, changing the array to [3, 4, 2, 5, 8]. In the third round, the first three elements are compared, and the third largest number 4 moves to the third-to-last position, resulting in [3, 2, 4, 5, 8]. In the fourth round, the first two elements are compared, and the fourth largest number 3 moves to the fourth-to-last position, resulting in [2, 3, 4, 5, 8]. Finally, no swaps occur in the last round, and the sorting is complete. A key optimization is early termination to avoid unnecessary traversals. The worst-case and average time complexity is O(n²), with a space complexity of O(1). Despite its low efficiency, Bubble Sort is simple and easy to understand, making it a foundational example for sorting algorithm beginners.
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".
Read MoreDrawing Binary Trees Step by Step: The First Lesson in Data Structure Fundamentals
A binary tree is a fundamental data structure where each node has at most two child nodes (left and right), and nodes with no descendants are called leaves. Core terms include: root node (the topmost starting point), leaf node (node with no children), child node (a node on the next level below its parent), and left/right subtrees (the left/right children and their descendants of a node). Construction starts from the root node, with child nodes added incrementally. Each node can have at most two children, and child positions are ordered (left vs. right). A binary tree must satisfy: each node has ≤2 children, and child positions are clearly defined (left or right). Traversal methods include pre-order (root → left → right), in-order (left → root → right), and post-order (left → right → root). Drawing the tree is crucial for understanding core relationships, as it intuitively visualizes node connections and forms the foundation for complex structures (e.g., heaps, red-black trees) and algorithms (sorting, searching).
Read MoreThe Art of Queuing: Applications of Queues in Data Structures
This article introduces the queue data structure. In daily life, queuing (such as getting meals in a cafeteria) embodies the "first-come, first-served" principle, which is the prototype of a queue. A queue is a data structure that follows the "First-In-First-Out" (FIFO) principle. Its core operations include enqueue (adding an element to the tail of the queue) and dequeue (removing the earliest added element from the front of the queue). Additionally, operations like viewing the front element and checking if the queue is empty are also supported. Queues differ from stacks (which follow the "Last-In-First-Out" (LIFO) principle); the former adheres to "first-come, first-served," while the latter reflects "last-come, first-served." Queues have wide applications: In computer task scheduling, systems process multiple tasks in a queue (e.g., programs opened earlier receive CPU time first); the BFS (Breadth-First Search) algorithm uses a queue to expand nodes level by level, enabling shortest path searches in mazes; during e-commerce promotions, queues buffer user requests to prevent system overload; in multi-threading, producers add data to a queue, and consumers process it in order, facilitating asynchronous collaboration. Learning queues helps solve problems such as processing data in sequence and avoiding resource conflicts, making it a fundamental tool in programming and algorithms. Understanding the "First-In-First-Out" principle contributes to efficiently solving practical problems.
Read MoreStacks in Daily Life: Why Are Stacks the First Choice for Data Structure Beginners?
The article introduces "stack" through daily scenarios such as stacking plates and browser backtracking, with its core feature being "Last-In-First-Out" (LIFO). A stack is a container that can only be operated on from the top, with core operations being "Push" (pushing onto the stack) and "Pop" (popping from the stack). As a first choice for data structure introduction, the stack has a simple logic (only the LIFO rule), clear operations (only two basic operations), extensive applications (scenarios like bracket matching, browser backtracking, recursion, etc.), and can be easily implemented using arrays or linked lists. It serves as a foundation for learning subsequent structures like queues and trees, helps establish clear programming thinking, and is a "stepping stone" for understanding data structures.
Read MoreLinked List vs Array: Key Differences for Beginners in Data Structures
Arrays and linked lists are among the most fundamental data structures in programming. Understanding their differences and applicable scenarios is crucial for writing efficient code. **Array Features**: - Stores elements in contiguous memory locations, allowing random access via index (O(1) time complexity). - Requires a fixed initial size; inserting/deleting elements in the middle demands shifting elements (O(n) time complexity). - Ideal for scenarios with known fixed sizes and high-frequency random access (e.g., grade sheets, map coordinates). **Linked List Features**: - Elements are scattered in memory, with each node containing data and a pointer/reference. - No random access (requires traversing from the head, O(n) time complexity). - Offers flexible dynamic expansion; inserting/deleting elements in the middle only requires modifying pointers (O(1) time complexity). - Suitable for dynamic data and high-frequency insertion/deletion scenarios (e.g., queues, linked hash tables). **Core Differences**: - Arrays rely on contiguous memory but have restricted operations. - Linked lists use scattered storage but suffer from slower access speeds. - Key distinctions lie in storage method, access speed, and insertion/deletion efficiency. Selection should be based on specific requirements. Mastering their underlying logic enables more efficient code implementation.
Read MoreLearning Data Structures from Scratch: What Exactly Is an Array?
An array is an ordered collection of data elements of the same type, accessed via indices (starting from 0), with elements stored contiguously. It is used to efficiently manage a large amount of homogeneous data. For example, class scores can be represented by the array `scores = [90, 85, 95, 78, 92]` instead of multiple individual variables, facilitating overall operations. In Python, array declaration and initialization can be done with `scores = [90, 85, 95, 78, 92]` or `[0] * 5` (declaring an array of length 5). Elements are accessed using `scores[index]`, and it's important to note the index range (0 to length-1), as out-of-bounds indices will cause errors. Basic operations include traversal with loops (`for score in scores: print(score)`), while insertion and deletion require shifting subsequent elements (with a time complexity of O(n)). Core characteristics of arrays are: same element type, 0-based indexing, and contiguous storage. Their advantages include fast access speed (O(1)), but disadvantages are lower efficiency for insertions/deletions and fixed size. As a foundational data structure, understanding the core idea of arrays—"indexed access and contiguous storage"—is crucial for learning more complex structures like linked lists and hash tables, making arrays a fundamental tool for data management.
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