[C/C++ Data Structure] In-depth exploration of algorithm complexity in data structures: from the perspective of C++ and mathematics

Directory title

• 1. Introduction
• • 1.1 Importance of Data Structures and Algorithms (Importance of Data Structures and Algorithms)
  • 1.2 Why Understanding Algorithm Complexity is Crucial
• 2. Basic Concepts
• • 2.1 Time Complexity and Space Complexity (Time Complexity and Space Complexity)
  • 2.2 Big O Notation (Big O Notation)
  • • 2.2.1 Common time complexity
• 3. Data Structures and Their Complexities
• • 3.1 Arrays
  • 3.2 Linked Lists
  • 3.3 Stacks and Queues
  • 3.4 Trees and Graphs
• 4. Calculating Depth
• • 4.1 Depth of Binary Trees
  • • 4.1.1 How to calculate depth
  • 4.2 Depth First Search in Graphs
  • • 4.2.1 Working principle of DFS
• 5. Implementation of Data Structures in C++ (Implementation of Data Structures in C++)
• • 5.1 Using the STL Library (Using the STL Library)
  • 5.2 Implementing Custom Data Structures
  • • 5.2.1 Binary Search Tree (Binary Search Tree)
  • 5.3 Diving into the Source Code
• 6. Advanced Mathematics and Algorithm Complexity
• • 6.1 Recursion and mathematical induction
  • 6.2 Graph theory and algorithms
  • 6.3 Depth of recursion and algorithm complexity
  • • 6.3.1 How recursion works
    • 6.3.2 Optimizing recursive algorithm
    • 6.3.3 Recursion and human thinking
• 7. Conclusion
• • 7.1 How to choose the appropriate data structure
  • 7.2 Suggestions for further study

1. Introduction

In the world of computer science, data structures and algorithms are two indispensable core concepts. They are the building blocks of computer programs and determine their efficiency and performance. But why are they so important? Why do we need to understand algorithm complexity? In this chapter, we will explore these issues in depth and understand their relationship to our daily lives from a deeper perspective.

1.1 Importance of Data Structures and Algorithms (Importance of Data Structures and Algorithms)

Data structures are the way data is stored and organized in computers, while algorithms are the steps and methods for solving specific problems. As stated in “Introduction to Algorithms”: “Algorithms are the core of computing and the soul of computer science.”

We deal with data every day, whether browsing information on social media or shopping on e-commerce websites. Behind every operation, there is an algorithm working silently to ensure fast storage, retrieval and processing of data.

1.2 Why Understanding Algorithm Complexity is Crucial (Why Understanding Algorithm Complexity is Crucial)

Algorithmic complexity describes the speed of algorithm execution or the storage space required, usually expressed in Big O notation. Understanding the complexity of an algorithm is crucial to writing efficient code. For example, an O(n^2) algorithm may be very slow when processing large amounts of data, while an O(n) algorithm may be faster.

But at a deeper level, algorithmic complexity is not just about speed or storage. It reflects how we view problems, how we solve them, and how we optimize solutions. As stated in Thinking, Fast and Slow: “The way we think determines our choices and decisions.”

In the following chapters, we’ll explore data structures and algorithmic complexity in more depth, illustrated with actual C++ code examples. I hope this article can provide you with a new perspective and help you better understand these core concepts.

Next, we will enter the core part of data structure and algorithm complexity. I hope you will continue to follow our pace and explore this exciting field in depth.

2. Basic Concepts

2.1 Time Complexity and Space Complexity (Time Complexity and Space Complexity)

In the world of exploring algorithms, we often hear “How efficient is this algorithm?” or “How much memory does this algorithm require?”. The answers behind these questions are time complexity and space complexity. Simply put, time complexity describes the time required to execute an algorithm, while space complexity describes the memory space required to execute the algorithm.

As Zhuangzi said in “Xiaoyaoyou”: “Those who have attained the Tao in the world have a generous and generous heart.” In the world of algorithms, we also pursue this kind of “magnanimity” and want our algorithms to be both fast and space-efficient. But often there is a trade-off between the two, where more space may need to be sacrificed for speed, and vice versa.

2.2 Big O Notation (Big O Notation)

Big O notation is a standard way of describing the complexity of an algorithm. It provides an upper bound on algorithm performance and helps us understand the worst-case execution time. For example, O(n) means that the execution time of the algorithm is linearly related to the size of the input data.

//Sample code: linear search
int linearSearch(int arr[], int n, int x) {<!-- -->
    for (int i = 0; i < n; i + + ) {<!-- -->
        if (arr[i] == x) {<!-- -->
            return i; // Find x and return its index
        }
    }
    return -1; // x is not found, return -1
}

In the above code, we can see that the time complexity of linear search is O(n) because in the worst case, we may need to check every element in the array.

As Mencius said in “Gongsun Chou”: “If you get the big one, you should do it; if you get the small one, you should also do it.” In the selection of algorithms, we should also choose the most suitable algorithm based on the actual situation, taking into account both its efficiency and its actual application scenarios.

2.2.1 Common time complexity

Complexity	Name	Example
O(1)	Constant time	Direct access to an element in the array
O(log n)	Logarithmic time	Binary search
O(n)	Linear time	Linear search
O(n log n)	Linear logarithmic time	Quick sort
O(n^2)	Squared time	Bubble sort

When choosing an algorithm, we should choose the most appropriate algorithm based on actual needs and the size of the data. For example, for small data sets, an O(n^2) algorithm may be faster than an O(n log n) algorithm, but for large data sets, the latter is significantly better.

3. Data Structures and Their Complexities

In computer science, a data structure is a way of storing and organizing data so that it can be accessed and modified efficiently. Different data structures are suitable for different problems, so choosing the right data structure is crucial to the efficiency of the algorithm.

3.1 Arrays

An array is a linear data structure used to store elements of the same type. Each element has an index, usually starting from 0. The main advantage of an array is that any of its elements can be accessed quickly via index. However, insertion and deletion operations may require moving a large number of elements.

Time complexity:

• Access:O(1)
• Insertion/deletion: O(n)

Space complexity: O(n)

//Array examples in C++
int arr[5] = {<!-- -->1, 2, 3, 4, 5};

3.2 Linked Lists

A linked list is a linear data structure consisting of a sequence of nodes, each node containing a data element and a pointer to the next node. Unlike an array, the elements of a linked list are not stored contiguously, which makes insertion and deletion operations more efficient.

Time complexity:

• Access: O(n)
• Insertion/deletion: O(1) (if node is known)

Space complexity: O(n)

//Singly linked list node example in C++
struct Node {<!-- -->
    int data;
    Node* next;
};

As said in “Human Weakness”: “People are usually more interested in things that are familiar to them.” This also explains why we prefer to use arrays instead of linked lists, because arrays are continuous in memory, while linked lists are not .

3.3 Stacks and Queues

The stack is a last-in-first-out (LIFO) data structure that only allows elements to be added or removed at the top. A queue is a first-in-first-out (FIFO) data structure that allows elements to be added at one end and deleted at the other end.

Time complexity:

• Access: O(n)
• Insertion/deletion: O(1)

Space complexity: O(n)

// Stack and queue examples in C++
#include <stack>
#include <queue>

std::stack<int> s;
std::queue<int> q;

3.4 Trees and Graphs

A tree is a hierarchical data structure used to represent data with a hierarchical structure. A graph is a data structure that can represent any binary relationship.

Time complexity:

• Access, insertion, deletion: O(log n) (for a balanced binary search tree)

Space complexity: O(n)

//Binary tree node example in C++
struct TreeNode {<!-- -->
    int data;
    TreeNode* left;
    TreeNode* right;
};

In our daily lives, we are often faced with choices. As said in “Existence and Time”: “Human existence is an existence of choice.” When choosing a data structure, we also face similar choices. Choosing the right data structure can make our algorithms more efficient, while the wrong choice can cause performance issues.

4. Calculating Depth

4.1 Depth of Binary Trees

Binary tree is one of the most commonly used data structures in computer science. It consists of nodes and edges, each node has up to two child nodes. The depth of a binary tree, also called the height, is the number of edges on the longest path from the root node to the furthest leaf node.

4.1.1 How to calculate depth

To calculate the depth of a binary tree, we can use a recursive method. The basic idea is: the depth of a binary tree is the maximum value of the depth of its left and right subtrees plus 1.

int depth(TreeNode* root) {<!-- -->
    if (root == nullptr) return 0;
    int leftDepth = depth(root->left);
    int rightDepth = depth(root->right);
    return max(leftDepth, rightDepth) + 1;
}

This code first checks if the root node is empty. If empty, the depth is 0. Otherwise, it recursively calculates the depth of the left and right subtrees and returns the largest value plus 1.

As said in “The Illusion of Free Will”: “The depth of our thinking is closely related to our understanding of the external world.” This is similar to how we understand and calculate the depth of a binary tree.

4.2 Depth First Search in Graphs

A graph is made up of nodes (called vertices) and the lines (called edges) that connect these nodes. Depth-first search (DFS) is an algorithm for traversing or searching graphs and trees. The idea is to start from the starting point and search as deep as possible until the target is found or a point that has already been visited is encountered.

4.2.1 Working principle of DFS

DFS uses a stack to store vertices to be accessed. Starting from the starting vertex, it is pushed onto the stack and marked as visited. Then, move to the next unvisited adjacent vertex, push it onto the stack and mark it as visited. Repeat this process until the stack is empty.

void DFS(int v, bool visited[], vector<int> adj[]) {<!-- -->
    visited[v] = true;
    for (int i = 0; i < adj[v].size(); i + + ) {<!-- -->
        if (!visited[adj[v][i]]) {<!-- -->
            DFS(adj[v][i], visited, adj);
        }
    }
}

This code uses recursion to implement DFS. It first marks the current vertex as visited and then calls DFS recursively on all unvisited adjacent vertices.

As it is said in Being and Time: “The depth of our existence is closely related to the way we connect to the external world.” This is deeply connected to how we explore every corner of the graph.

When exploring data structures and algorithms in depth, we must understand not only how they work, but also the philosophy and thinking behind them. In this way, we can better apply them and solve practical problems.

5. Implementation of Data Structures in C++ (Implementation of Data Structures in C++)

5.1 Using the STL Library (Using the STL Library)

The C++ Standard Template Library (STL) is the core part of the C++ library and provides several template classes to handle common data structures and algorithms. The design goal of the STL library is to make data structures and algorithms independent of data types, thereby improving code reusability.

For example, std::vector is a template class that can be used to handle dynamic arrays of any data type. Here is a simple example showing how to use std::vector to store and access integers:

#include <iostream>
#include <vector>

int main() {<!-- -->
    std::vector<int> vec = {<!-- -->1, 2, 3, 4, 5};
    for(int num : vec) {<!-- -->
        std::cout << num << " ";
    }
    return 0;
}

In this code, we first include the necessary header files, and then define an integer type std::vector. With a range for loop, we can easily access and print each element in vec.

As said in “C++ Primer”: “The design philosophy of the STL library is to separate data and operations, which can reduce the coupling between data structures and algorithms and improve the maintainability of the code.”

5.2 Implementing Custom Data Structures

In addition to using the data structures provided by STL, we sometimes need to implement custom data structures based on specific needs. For example, we might need a data structure that can store large amounts of data and support fast lookups.

5.2.1 Binary Search Tree (Binary Search Tree)

Binary Search Tree (BST) is a self-balancing binary tree in which each node has a key and a value. All nodes in the left subtree of the BST have keys that are less than the key of their root node, while all nodes in the right subtree have keys that are greater than the key of their root node.

The following is a simple BST implementation:

template <typename Key, typename Value>
class BSTNode {<!-- -->
public:
    Key key;
    Value value;
    BSTNode* left;
    BSTNode* right;

    BSTNode(Key k, Value v) : key(k), value(v), left(nullptr), right(nullptr) {<!-- -->}
};

template <typename Key, typename Value>
class BinarySearchTree {<!-- -->
private:
    BSTNode<Key, Value>* root;

public:
    // ... (other functions and methods)
};

In this example, we first define a template class BSTNode, and then define a BinarySearchTree class to manage these nodes.

Human understanding of tree structures is often similar to our understanding of family, society, and nature. As stated in “A Brief History of Humankind”: “Our brains evolved to understand and parse tree structures, which is why we can easily understand and use data structures like BST.”

5.3 Diving into the Source Code

To gain a deeper understanding of how the STL library works, we can view its source code. For example, the implementation of std::vector is based on a dynamic array, which has three pointers inside: _M_start, _M_finish and _M_end_of_storage, respectively points to the beginning, end and storage end position of the array.

This design allows std::vector to dynamically adjust its size at runtime without the need to reallocate memory each time. This is also why the average time complexity of std::vector‘s push_back operation is O(1).

As stated in “Modern Operating Systems”: “Dynamic memory management is a core issue in computer science, which involves how to allocate and reclaim memory effectively and efficiently.”

6. Advanced Mathematics and Algorithm Complexity

When exploring the deep structure of algorithmic complexity, we are not just looking at how efficiently an algorithm performs. At a deeper level, we are exploring the nature of mathematics and how the human mind works. As said in “The Beauty of Mathematics”: “Mathematics is the crystallization of human thinking.”

6.1 Recursion and mathematical induction

Recursion is an algorithm design strategy in which an algorithm calls itself to solve part of a problem. The idea of recursion is closely related to mathematical induction. Mathematical induction is a proof technique used to prove that each of a series of propositions is true.

Consider a simple recursive function that computes the factorial of a number. The definition of the factorial function is based on recursion, because the factorial of n is the factorial of n times (n-1).

int factorial(int n) {<!-- -->
    if (n == 0) return 1; // basic case
    return n * factorial(n - 1); // Recursive call
}

Here we can see how recursion is similar to mathematical induction. We have a base case (n = 0) and then we have a recursive step where we assume the function is valid for n-1 and use this assumption to calculate the value of n.

As said in “The Story of Philosophy”: “Real knowledge is not just knowing the external properties of things, but knowing their essence.”

6.2 Graph theory and algorithms

Graph theory is a branch of mathematics that studies the properties and applications of graphs. In algorithms, graphs are used as data structures to represent relationships between objects.

Consider a simple graph algorithm: BFS (Breadth First Search). This is an algorithm that traverses a graph starting from a specified starting vertex and visiting all reachable vertices.

void BFS(Graph G, Node start) {<!-- -->
    queue<Node> q;
    q.push(start);
    while (!q.empty()) {<!-- -->
        Node current = q.front();
        q.pop();
        // Process the current node
        for (Node neighbor : G.neighbors(current)) {<!-- -->
            if (!visited[neighbor]) {<!-- -->
                visited[neighbor] = true;
                q.push(neighbor);
            }
        }
    }
}

Here we can see how queues can be used to help us traverse the graph. The beauty of this approach is its simplicity and efficiency.

As said in “Thinking, Fast and Slow”: “Intuition is the crystallization of knowledge and experience.”

When exploring algorithmic complexity, we’re not just learning how to solve problems more efficiently. We are still learning how to better understand the world, and our own way of thinking. This is a journey with both depth and breadth, as it is said in “Being and Time”: “Human existence is not only the existence of matter, it is also the existence of time.”

6.3 Recursion depth and algorithm complexity

Recursion is a powerful programming technique, but it also poses some challenges, especially related to algorithm complexity. To understand this better, we first need to dive into how recursion works.

6.3.1 How recursion works

When a function calls itself, a new function call is placed on the call stack. This means that if the recursive call goes too deep, we may run out of stack space and cause a stack overflow.

Consider a simple recursive function that computes the Fibonacci sequence:

int fibonacci(int n) {<!-- -->
    if (n <= 1) return n;
    return fibonacci(n - 1) + fibonacci(n - 2);
}

The time complexity of this function is O(2^n) because it generates two new recursive calls for every n. This is a very inefficient implementation because it recalculates the same value many times.

As said in “A Brief History of Humankind”: “Repetition is the mother of learning, but it can also be the source of waste.”

6.3.2 Optimizing recursive algorithm

To optimize recursive algorithms, we can use a technique called “memoization”. This means we store already calculated values to avoid double calculations.

Considering the Fibonacci function above, we can use an array to store the calculated values:

int memo[1000];

int fibonacci(int n) {<!-- -->
    if (n <= 1) return n;
    if (memo[n] != -1) return memo[n];
    return memo[n] = fibonacci(n - 1) + fibonacci(n - 2);
}

The time complexity of this optimized version is O(n) since we only calculate each value once.

As said in “Zen and the Art of Motorcycle Maintenance”: “The best way is often the simplest way.”

6.3.3 Recursion and human thinking

Recursion is more than just a programming technique. It reflects how we think about and solve problems. When we are faced with a complex problem, we often try to break it down into smaller, more manageable parts. This is very similar to the idea of recursion.

As said in “Paradox”: “By simplifying complexity, we can understand it more deeply”.

When exploring recursion and algorithmic complexity, we’re not just learning programming skills. We are also learning how to better understand problems and how to solve them more effectively. This is a journey of both depth and breadth, as said in “The Joy of Thinking”: “True wisdom is not just knowing the answers, but knowing how to ask questions.”

7. Conclusion

7.1 How to choose the appropriate data structure

In our daily lives, choosing the right tool is crucial to getting the job done. Likewise, in programming, choosing the right data structure is crucial. Just as Zhuangzi said in “Zhuangzi Xiaoyaoyou”: “The fish is what I want, and the bear’s paw is what I want. You can’t have both. One would give up the fish and get the bear’s paw.” This means that in When faced with a choice, we must weigh the pros and cons and choose the option that best suits the situation.

// For example, when we need to frequently insert and delete elements in the middle, a Linked List may be a better choice.
struct Node {<!-- -->
    int data;
    Node* next;
    // Constructor
    Node(int d) : data(d), next(nullptr) {<!-- -->}
};

// When we need to access elements quickly and randomly, an array (Array) may be more suitable.
int arr[10];
arr[5] = 20; // Quickly access the 6th element of the array

7.2 Suggestions for further study

Learning is a never-ending process. As Zhuangzi said in “Zhuangzi·Xiaoyaoyou”: “Heaven and earth live together with me, and all things are one with me.” This tells us that knowledge is infinite, we and knowledge are one, and we should continue to explore and study.

For those readers who wish to delve deeper into data structures and algorithms, I recommend:

1. Practice: Writing code and actually running it is the best way to learn. With practice, you can better understand how data structures and algorithms work.
2. Reading: Read classic computer science books, such as “Introduction to Algorithms”. These books provide you with in-depth theoretical knowledge and practical experience.
3. Challenge yourself: Participate in online coding challenges and competitions like LeetCode or HackerRank. This will not only improve your programming skills, but also help you better apply this knowledge in real work.

On the road of exploring knowledge, we should always maintain curiosity and exploration spirit, constantly challenge ourselves and surpass ourselves.

In our programming learning journey, understanding is an important step for us to move to a higher level. However, mastering new skills and ideas always requires time and persistence. From a psychological point of view, learning is often accompanied by constant trial and error and adjustment, which is like our brain gradually optimizing its “algorithm” for solving problems.

That’s why when we encounter mistakes, we should view them as opportunities to learn and improve, not just as annoyances. By understanding and solving these problems, we can not only fix the current code, but also improve our programming skills and prevent making the same mistakes in future projects.

I encourage everyone to actively participate and continuously improve their programming skills. Whether you are a beginner or an experienced developer, I hope my blog will be helpful on your learning journey. If you find this article useful, you may wish to click to bookmark it, or leave your comments to share your insights and experiences. You are also welcome to make suggestions and questions about the content of my blog. Every like, comment, share and attention is the greatest support for me and the motivation for me to continue sharing and creating.

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