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@@ -16,13 +16,13 @@ There are three types of storage devices in computers: <u>hard disk</u>, <u>rand
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<div class="center-table" markdown>
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| | Hard Disk | Memory | Cache |
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| ---------- | -------------------------------------------------------------- | ------------------------------------------------------------------------ | ----------------------------------------------------------------------------------------------- |
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| Usage | Long-term storage of data, including OS, programs, files, etc. | Temporary storage of currently running programs and data being processed | Stores frequently accessed data and instructions, reducing the number of CPU accesses to memory |
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| Volatility | Data is not lost after power off | Data is lost after power off | Data is lost after power off |
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| Capacity | Larger, TB level | Smaller, GB level | Very small, MB level |
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| Speed | Slower, several hundred to thousands MB/s | Faster, several tens of GB/s | Very fast, several tens to hundreds of GB/s |
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| Price | Cheaper, a few cents to a few dollars / GB | More expensive, tens to hundreds of dollars / GB | Very expensive, priced with CPU |
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| | Hard Disk | Memory | Cache |
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| ----------- | -------------------------------------------------------------- | ------------------------------------------------------------------------ | ----------------------------------------------------------------------------------------------- |
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| Usage | Long-term storage of data, including OS, programs, files, etc. | Temporary storage of currently running programs and data being processed | Stores frequently accessed data and instructions, reducing the number of CPU accesses to memory |
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| Volatility | Data is not lost after power off | Data is lost after power off | Data is lost after power off |
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| Capacity | Larger, TB level | Smaller, GB level | Very small, MB level |
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| Speed | Slower, several hundred to thousands MB/s | Faster, several tens of GB/s | Very fast, several tens to hundreds of GB/s |
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| Price (USD) | Cheaper, a few cents / GB | More expensive, a few dollars / GB | Very expensive, priced with CPU |
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</div>
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@@ -517,7 +517,7 @@ Consider the following code, the term "worst-case" in worst-case space complexit
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/* Recursion O(n) */
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void recur(int n) {
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if (n == 1) return;
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return recur(n - 1);
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recur(n - 1);
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}
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```
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@@ -537,7 +537,7 @@ Consider the following code, the term "worst-case" in worst-case space complexit
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/* Recursion O(n) */
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void recur(int n) {
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if (n == 1) return;
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return recur(n - 1);
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recur(n - 1);
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}
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```
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@@ -666,7 +666,7 @@ Consider the following code, the term "worst-case" in worst-case space complexit
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/* Recursion O(n) */
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void recur(int n) {
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if (n == 1) return;
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return recur(n - 1);
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recur(n - 1);
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}
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```
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@@ -708,7 +708,7 @@ Consider the following code, the term "worst-case" in worst-case space complexit
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/* Recursion O(n) */
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void recur(int n) {
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if (n == 1) return;
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return recur(n - 1);
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recur(n - 1);
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}
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```
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@@ -35,7 +35,7 @@ Using the data from the preceding figure, we can follow the steps shown in the n
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2. Find the index of the root node 3 in the `inorder` sequence, and use this index to split `inorder` into `[ 9 | 3 ｜ 1 2 7 ]`.
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3. According to the split of the `inorder` sequence, it is straightforward to determine that the left and right subtrees contain 1 and 3 nodes, respectively, so we can split the `preorder` sequence into `[ 3 | 9 | 2 1 7 ]` accordingly.
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{ class="animation-figure" }
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{ class="animation-figure" }
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<p align="center"> Figure 12-6 Dividing the subtrees in pre-order and in-order traversals </p>
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@@ -7,13 +7,13 @@ icon: material/rocket-launch-outline
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A few years ago, I shared the "Sword for Offer" problem solutions on LeetCode, receiving encouragement and support from many readers. During interactions with readers, the most common question I encountered was "how to get started with algorithms." Gradually, I developed a keen interest in this question.
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Directly solving problems seems to be the most popular method — it's simple, direct, and effective. However, problem-solving is like playing a game of Minesweeper: those with strong self-study abilities can defuse the mines one by one, but those with insufficient basics might end up metaphorically bruised from explosions, retreating step by step in frustration. Going through textbooks is also common, but for those aiming for job applications, the energy spent on thesis writing, resume submissions, and preparation for written tests and interviews leaves little for tackling thick books, turning it into a daunting challenge.
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Directly solving problems seems to be the most popular method — it's simple, direct, and effective. However, problem-solving is like playing Minesweeper: those with strong self-study skills can navigate the pitfalls one by one, while those lacking a solid foundation may find themselves repeatedly stumbling and retreating in frustration. Reading through textbooks is also a common practice, but for job seekers, writing graduation thesis, submitting resumes, preparing for written tests and interviews have already consumed most of their energy, and reading thick books often becomes a daunting challenge.
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If you're facing similar troubles, then this book is lucky to have found you. This book is my answer to the question. While it may not be the best solution, it is at least a positive attempt. This book may not directly land you an offer, but it will guide you through the "knowledge map" in data structures and algorithms, help you understand the shapes, sizes, and locations of different "mines," and enable you to master various "demining methods." With these skills, I believe you can solve problems and read literature more comfortably, gradually building a knowledge system.
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If you're facing similar troubles, then this book is lucky to have found you. This book is my answer to the question. While it may not be the best solution, it is at least a positive attempt. Although this book is not enough to get you an offer directly, it will guide you to explore the "knowledge map" of data structures and algorithms, help you understand the shapes, sizes, and locations of different "mines", and enable you to master various "mine clearance methods". With these skills, I believe you can solve problems and read literature more comfortably, gradually building a knowledge system.
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I deeply agree with Professor Feynman's statement: "Knowledge isn't free. You have to pay attention." In this sense, this book is not entirely "free." To not disappoint the precious "attention" you pay for this book, I will do my best, dedicating my utmost "attention" to this book.
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I deeply agree with Professor Feynman's statement: "Knowledge isn't free. You have to pay attention." In this sense, this book is not entirely "free." In order to live up to your precious "attention" for this book, I will do my best and devote my greatest "attention" to write this book.
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Knowing my limitations, although the content of this book has been refined over time, there are surely many errors remaining. I sincerely request critiques and corrections from all teachers and students.
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Aware of my limitations, I recognize that despite the content of this book being refined over time, errors surely remain. I sincerely welcome critiques and corrections from both teachers and students.
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{ class="cover-image" }
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@@ -4,63 +4,63 @@ comments: true
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# 1.1 Algorithms are everywhere
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When we hear the word "algorithm," we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which can be seen everywhere in our daily lives.
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When we hear the term "algorithm," we naturally think of mathematics. However, many algorithms do not involve complex mathematics but rely more on basic logic, which can be seen everywhere in our daily lives.
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Before formally discussing algorithms, there's an interesting fact worth sharing: **you have already unconsciously learned many algorithms and have become accustomed to applying them in your daily life**. Here, I will give a few specific examples to prove this point.
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Before we start discussing about algorithms officially, there's an interesting fact worth sharing: **you've learned many algorithms unconsciously and are used to applying them in your daily life**. Here, I will give a few specific examples to prove this point.
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**Example 1: Looking Up a Dictionary**. In an English dictionary, words are listed alphabetically. Suppose we're searching for a word that starts with the letter $r$. This is typically done in the following way:
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**Example 1: Looking Up a Dictionary**. In an English dictionary, words are listed alphabetically. Assuming we're searching for a word that starts with the letter $r$, this is typically done in the following way:
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1. Open the dictionary to about halfway and check the first letter on the page, let's say the letter is $m$.
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2. Since $r$ comes after $m$ in the alphabet, we can ignore the first half of the dictionary and focus on the latter half.
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1. Open the dictionary to about halfway and check the first vocabulary of the page, let's say the letter starts with $m$.
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2. Since $r$ comes after $m$ in the alphabet, the first half can be ignored and the search space is narrowed down to the second half.
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3. Repeat steps `1.` and `2.` until you find the page where the word starts with $r$.
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=== "<1>"
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{ class="animation-figure" }
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{ class="animation-figure" }
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=== "<2>"
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{ class="animation-figure" }
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{ class="animation-figure" }
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=== "<3>"
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{ class="animation-figure" }
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{ class="animation-figure" }
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=== "<4>"
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{ class="animation-figure" }
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{ class="animation-figure" }
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=== "<5>"
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{ class="animation-figure" }
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{ class="animation-figure" }
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<p align="center"> Figure 1-1 Process of Looking Up a Dictionary </p>
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<p align="center"> Figure 1-1 Process of looking up a dictionary </p>
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This essential skill for elementary students, looking up a dictionary, is actually the famous "Binary Search" algorithm. From a data structure perspective, we can consider the dictionary as a sorted "array"; from an algorithmic perspective, the series of actions taken to look up a word in the dictionary can be viewed as "Binary Search."
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Looking up a dictionary, an essential skill for elementary school students is actually the famous "Binary Search" algorithm. From a data structure perspective, we can consider the dictionary as a sorted "array"; from an algorithmic perspective, the series of actions taken to look up a word in the dictionary can be viewed as the algorithm "Binary Search."
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**Example 2: Organizing Playing Cards**. When playing cards, we need to arrange the cards in our hand in ascending order, as shown in the following process.
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**Example 2: Organizing Card Deck**. When playing cards, we need to arrange the cards in our hands in ascending order, as shown in the following process.
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1. Divide the playing cards into "ordered" and "unordered" sections, assuming initially the leftmost card is already in order.
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2. Take out a card from the unordered section and insert it into the correct position in the ordered section; after this, the leftmost two cards are in order.
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3. Continue to repeat step `2.` until all cards are in order.
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3. Repeat step `2` until all cards are in order.
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{ class="animation-figure" }
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{ class="animation-figure" }
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<p align="center"> Figure 1-2 Playing cards sorting process </p>
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<p align="center"> Figure 1-2 Process of sorting a deck of cards </p>
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The above method of organizing playing cards is essentially the "Insertion Sort" algorithm, which is very efficient for small datasets. Many programming languages' sorting functions include the insertion sort.
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The above method of organizing playing cards is practically the "Insertion Sort" algorithm, which is very efficient for small datasets. Many programming languages' sorting functions include the insertion sort.
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**Example 3: Making Change**. Suppose we buy goods worth $69$ yuan at a supermarket and give the cashier $100$ yuan, then the cashier needs to give us $31$ yuan in change. They would naturally complete the thought process as shown in Figure 1-3.
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**Example 3: Making Change**. Assume making a purchase of $69$ at a supermarket. If you give the cashier $100$, they will need to provide you with $31$ in change. This process can be clearly understood as illustrated in Figure 1-3.
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1. The options are currencies smaller than $31$, including $1$, $5$, $10$, and $20$.
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1. The options are currencies valued below $31$, including $1$, $5$, $10$, and $20$.
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2. Take out the largest $20$ from the options, leaving $31 - 20 = 11$.
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3. Take out the largest $10$ from the remaining options, leaving $11 - 10 = 1$.
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4. Take out the largest $1$ from the remaining options, leaving $1 - 1 = 0$.
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5. Complete the change-making, with the solution being $20 + 10 + 1 = 31$.
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5. Complete change-making, the solution is $20 + 10 + 1 = 31$.
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{ class="animation-figure" }
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{ class="animation-figure" }
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<p align="center"> Figure 1-3 Change making process </p>
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<p align="center"> Figure 1-3 Process of making change </p>
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In the above steps, we make the best choice at each step (using the largest denomination possible), ultimately resulting in a feasible change-making plan. From the perspective of data structures and algorithms, this method is essentially a "Greedy" algorithm.
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In the steps described, we choose the best option at each stage by utilizing the largest denomination available, which leads to an effective change-making strategy. From a data structures and algorithms perspective, this approach is known as a "Greedy" algorithm.
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From cooking a meal to interstellar travel, almost all problem-solving involves algorithms. The advent of computers allows us to store data structures in memory and write code to call the CPU and GPU to execute algorithms. In this way, we can transfer real-life problems to computers, solving various complex issues more efficiently.
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From cooking a meal to interstellar travel, almost all problem-solving involves algorithms. The advent of computers allows us to store data structures in memory and write code to call the CPU and GPU to execute algorithms. In this way, we can transfer real-life problems to computers and solve various complex issues in a more efficient way.
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!!! tip
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If concepts such as data structures, algorithms, arrays, and binary search still seem somewhat obscure, I encourage you to continue reading. This book will gently guide you into the realm of understanding data structures and algorithms.
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If you are still confused about concepts like data structures, algorithms, arrays, and binary searches, I encourage you to keep reading. This book will gently guide you into the realm of understanding data structures and algorithms.
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@@ -8,6 +8,19 @@ comments: true
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- The principle of looking up a word in a dictionary is consistent with the binary search algorithm. The binary search algorithm embodies the important algorithmic concept of divide and conquer.
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- The process of organizing playing cards is very similar to the insertion sort algorithm. The insertion sort algorithm is suitable for sorting small datasets.
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- The steps of making change in currency essentially follow the greedy algorithm, where each step involves making the best possible choice at the moment.
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- An algorithm is a set of instructions or steps used to solve a specific problem within a finite amount of time, while a data structure is the way data is organized and stored in a computer.
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- An algorithm is a set of step-by-step instructions for solving a specific problem within a finite time, while a data structure defines how data is organized and stored in a computer.
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- Data structures and algorithms are closely linked. Data structures are the foundation of algorithms, and algorithms are the stage to utilize the functions of data structures.
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- We can liken data structures and algorithms to building blocks. The blocks represent data, the shape and connection method of the blocks represent data structures, and the steps of assembling the blocks correspond to algorithms.
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- We can compare data structures and algorithms to assembling building blocks. The blocks represent data, the shape and connection method of the blocks represent data structures, and the steps of assembling the blocks correspond to algorithms.
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### 1. Q & A
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**Q**：As a programmer, I’ve rarely needed to implement algorithms manually in my daily work. Most commonly used algorithms are already built into programming languages and libraries, ready to use. Does this suggest that the problems we encounter in our work haven’t yet reached the level of complexity that demands custom algorithm design?
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If specific work skills are like the "moves" in martial arts, then fundamental subjects are more like "internal strength".
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I believe the significance of learning algorithms (and other fundamental subjects) isn’t necessarily to implement them from scratch at work, but to enable more professional decision-making and problem-solving based on a solid understanding of the concepts. This, in turn, raises the overall quality of our work. For example, every programming language provides a built-in sorting function:
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- If we have not learned data structures and algorithms, then given any data, we might just give it to this sorting function. It runs smoothly, has good performance, and seems to have no problems.
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- However, if we’ve studied algorithms, we understand that the time complexity of a built-in sorting function is typically $O(n \log n)$. Moreover, if the data consists of integers with a fixed number of digits (such as student IDs), we can apply a more efficient approach like radix sort, reducing the time complexity to O(nk) , where k is the number of digits. When handling large volumes of data, the time saved can turn into significant value — lowering costs, improving user experience, and enhancing system performance.
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In engineering, many problems are difficult to solve optimally; most are addressed with ‘near-optimal’ solutions. The difficulty of a problem depends not only on its inherent complexity but also on the knowledge and experience of the person tackling it. The deeper one’s expertise and experience, the more thorough the analysis, and the more elegantly the problem can be solved.
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@@ -23,14 +23,14 @@ A <u>data structure</u> is a way of organizing and storing data in a computer, w
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**Designing data structures is a balancing act, often requiring trade-offs**. If you want to improve in one aspect, you often need to compromise in another. Here are two examples:
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- Compared to arrays, linked lists offer more convenience in data addition and deletion but sacrifice data access speed.
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- Graphs, compared to linked lists, provide richer logical information but require more memory space.
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- Compared with linked lists, graphs provide richer logical information but require more memory space.
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## 1.2.3 Relationship between data structures and algorithms
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As shown in Figure 1-4, data structures and algorithms are highly related and closely integrated, specifically in the following three aspects:
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- Data structures are the foundation of algorithms. They provide structured data storage and methods for manipulating data for algorithms.
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- Algorithms are the stage where data structures come into play. The data structure alone only stores data information; it is through the application of algorithms that specific problems can be solved.
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- Algorithms inject vitality into data structures. The data structure alone only stores data information; it is through the application of algorithms that specific problems can be solved.
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- Algorithms can often be implemented based on different data structures, but their execution efficiency can vary greatly. Choosing the right data structure is key.
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{ class="animation-figure" }
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@@ -62,4 +62,4 @@ It's worth noting that data structures and algorithms are independent of program
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!!! tip "Conventional Abbreviation"
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In real-life discussions, we often refer to "Data Structures and Algorithms" simply as "Algorithms". For example, the well-known LeetCode algorithm problems actually test both data structure and algorithm knowledge.
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In real-life discussions, we often refer to "Data Structures and Algorithms" simply as "Algorithms". For example, the well-known LeetCode algorithm questions actually test knowledge of both data structures and algorithms.
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+1
-1
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<!-- contributors -->
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<div style="margin: 2em auto;">
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<h3>Contributors</h3>
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<p>This book has been refined by the efforts of over 180 contributors. We sincerely thank them for their invaluable time and contributions!</p>
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<p>This book has been refined by the efforts of over 200 contributors. We sincerely thank them for their invaluable time and contributions!</p>
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<a href="https://github.com/krahets/hello-algo/graphs/contributors">
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<img src="https://contrib.rocks/image?repo=krahets/hello-algo&max=300&columns=12" alt="Contributors" style="width: 100%; max-width: 38.5em;">
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</a>
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