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- 0.2 How to Read
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- Chapter 1. Introduction to Algorithms
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- 中文
+ 简体中文
+
+
+
+
+
+ 繁體中文
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- 0.1 About This Book
+ 0.1 About this book
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- 0.2 How to Read
+ 0.2 How to read
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- Chapter 1. Introduction to Algorithms
+ Chapter 1. Introduction to algorithms
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+ 1.1 Algorithms are everywhere
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+ 1.2 What is an algorithm
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- Chapter 2. Complexity Analysis
+ Chapter 2. Complexity analysis
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+ 2.1 Algorithm efficiency assessment
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+ 2.2 Iteration and recursion
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- 2.3 Time Complexity
+ 2.3 Time complexity
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- 2.4 Space Complexity
+ 2.4 Space complexity
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- Chapter 3. Data Structures
+ Chapter 3. Data structures
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- 3.1 Classification of Data Structures
+ 3.1 Classification of data structures
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+ 3.2 Fundamental data types
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+ 3.3 Number encoding *
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+ 3.4 Character encoding *
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- Chapter 4. Array and Linked List
+ Chapter 4. Array and linked list
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- 4.1.1 Common Operations on Arrays
+ 4.1.1 Common operations on arrays
-
+
- 1. Initializing Arrays
+ 1. Initializing arrays
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- 2. Accessing Elements
+ 2. Accessing elements
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- 3. Inserting Elements
+ 3. Inserting elements
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- 4. Deleting Elements
+ 4. Deleting elements
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- 5. Traversing Arrays
+ 5. Traversing arrays
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- 6. Finding Elements
+ 6. Finding elements
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- 7. Expanding Arrays
+ 7. Expanding arrays
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+ 4.1.2 Advantages and limitations of arrays
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- 4.1.3 Typical Applications of Arrays
+ 4.1.3 Typical applications of arrays
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- 4.2 Linked List
+ 4.2 Linked list
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- 4.4 Memory and Cache
+ 4.4 Memory and cache
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- Chapter 5. Stack and Queue
+ Chapter 5. Stack and queue
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@@ -1368,7 +1374,7 @@
- 5.3 Double-ended Queue
+ 5.3 Double-ended queue
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+ Chapter 6. Hash table
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+ 6.1 Hash table
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+ 6.2 Hash collision
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- 6.3 Hash Algorithm
+ 6.3 Hash algorithm
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+ 4.1.1 Common operations on arrays
-
+
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+ 1. Initializing arrays
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+ 2. Accessing elements
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+ 3. Inserting elements
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+ 6. Finding elements
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- 7. Expanding Arrays
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-4.1 Arrays
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An "array" is a linear data structure that operates as a lineup of similar items, stored together in a computer's memory in contiguous spaces. It's like a sequence that maintains organized storage. Each item in this lineup has its unique 'spot' known as an "index". Please refer to the Figure 4-1 to observe how arrays work and grasp these key terms.
-
- Figure 4-1 Array Definition and Storage Method
+
+ Figure 4-1 Array definition and storage method
-4.1.1 Common Operations on Arrays
-1. Initializing Arrays
+4.1.1 Common operations on arrays
+1. Initializing arrays
Arrays can be initialized in two ways depending on the needs: either without initial values or with specified initial values. When initial values are not specified, most programming languages will set the array elements to \(0\) :
2. Accessing Elements
+2. Accessing elements
Elements in an array are stored in contiguous memory spaces, making it simpler to compute each element's memory address. The formula shown in the Figure below aids in determining an element's memory address, utilizing the array's memory address (specifically, the first element's address) and the element's index. This computation streamlines direct access to the desired element.
-
- Figure 4-2 Memory Address Calculation for Array Elements
+
+ Figure 4-2 Memory address calculation for array elements
As observed in the above illustration, array indexing conventionally begins at \(0\) . While this might appear counterintuitive, considering counting usually starts at \(1\) , within the address calculation formula, an index is essentially an offset from the memory address . For the first element's address, this offset is \(0\) , validating its index as \(0\) .
Accessing elements in an array is highly efficient, allowing us to randomly access any element in \(O(1)\) time.
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-3. Inserting Elements
+3. Inserting elements
Array elements are tightly packed in memory, with no space available to accommodate additional data between them. Illustrated in Figure below, inserting an element in the middle of an array requires shifting all subsequent elements back by one position to create room for the new element.
-
- Figure 4-3 Array Element Insertion Example
+
+ Figure 4-3 Array element insertion example
It's important to note that due to the fixed length of an array, inserting an element will unavoidably result in the loss of the last element in the array. Solutions to address this issue will be explored in the "List" chapter.
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4. Deleting Elements
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4. Deleting elements
Similarly, as depicted in the Figure 4-4 , to delete an element at index \(i\) , all elements following index \(i\) must be moved forward by one position.
-
-
Figure 4-4 Array Element Deletion Example
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+
Figure 4-4 Array element deletion example
Please note that after deletion, the former last element becomes "meaningless," hence requiring no specific modification.
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In summary, the insertion and deletion operations in arrays present the following disadvantages:
-High Time Complexity : Both insertion and deletion in an array have an average time complexity of \(O(n)\) , where \(n\) is the length of the array.Loss of Elements : Due to the fixed length of arrays, elements that exceed the array's capacity are lost during insertion.Waste of Memory : Initializing a longer array and utilizing only the front part results in "meaningless" end elements during insertion, leading to some wasted memory space.High time complexity : Both insertion and deletion in an array have an average time complexity of \(O(n)\) , where \(n\) is the length of the array.Loss of elements : Due to the fixed length of arrays, elements that exceed the array's capacity are lost during insertion.Waste of memory : Initializing a longer array and utilizing only the front part results in "meaningless" end elements during insertion, leading to some wasted memory space.
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5. Traversing Arrays
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5. Traversing arrays
In most programming languages, we can traverse an array either by using indices or by directly iterating over each element:
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6. Finding Elements
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6. Finding elements
Locating a specific element within an array involves iterating through the array, checking each element to determine if it matches the desired value.
Because arrays are linear data structures, this operation is commonly referred to as "linear search."
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7. Expanding Arrays
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7. Expanding arrays
In complex system environments, ensuring the availability of memory space after an array for safe capacity extension becomes challenging. Consequently, in most programming languages, the length of an array is immutable .
To expand an array, it's necessary to create a larger array and then copy the elements from the original array. This operation has a time complexity of \(O(n)\) and can be time-consuming for large arrays. The code are as follows:
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-
4.1.2 Advantages and Limitations of Arrays
+
4.1.2 Advantages and limitations of arrays
Arrays are stored in contiguous memory spaces and consist of elements of the same type. This approach provides substantial prior information that systems can leverage to optimize the efficiency of data structure operations.
-High Space Efficiency : Arrays allocate a contiguous block of memory for data, eliminating the need for additional structural overhead.Support for Random Access : Arrays allow \(O(1)\) time access to any element.Cache Locality : When accessing array elements, the computer not only loads them but also caches the surrounding data, utilizing high-speed cache to enchance subsequent operation speeds.High space efficiency : Arrays allocate a contiguous block of memory for data, eliminating the need for additional structural overhead.Support for random access : Arrays allow \(O(1)\) time access to any element.Cache locality : When accessing array elements, the computer not only loads them but also caches the surrounding data, utilizing high-speed cache to enchance subsequent operation speeds.
However, continuous space storage is a double-edged sword, with the following limitations:
-Low Efficiency in Insertion and Deletion : As arrays accumulate many elements, inserting or deleting elements requires shifting a large number of elements.Fixed Length : The length of an array is fixed after initialization. Expanding an array requires copying all data to a new array, incurring significant costs.Space Wastage : If the allocated array size exceeds the what is necessary, the extra space is wasted.Low efficiency in insertion and deletion : As arrays accumulate many elements, inserting or deleting elements requires shifting a large number of elements.Fixed length : The length of an array is fixed after initialization. Expanding an array requires copying all data to a new array, incurring significant costs.Space wastage : If the allocated array size exceeds the what is necessary, the extra space is wasted.
-
4.1.3 Typical Applications of Arrays
+
4.1.3 Typical applications of arrays
Arrays are fundamental and widely used data structures. They find frequent application in various algorithms and serve in the implementation of complex data structures.
-Random Access : Arrays are ideal for storing data when random sampling is required. By generating a random sequence based on indices, we can achieve random sampling efficiently.Sorting and Searching : Arrays are the most commonly used data structure for sorting and searching algorithms. Techniques like quick sort, merge sort, binary search, etc., are primarily operate on arrays.Lookup Tables : Arrays serve as efficient lookup tables for quick element or relationship retrieval. For instance, mapping characters to ASCII codes becomes seamless by using the ASCII code values as indices and storing corresponding elements in the array.Machine Learning : Within the domain of neural networks, arrays play a pivotal role in executing crucial linear algebra operations involving vectors, matrices, and tensors. Arrays serve as the primary and most extensively used data structure in neural network programming.Data Structure Implementation : Arrays serve as the building blocks for implementing various data structures like stacks, queues, hash tables, heaps, graphs, etc. For instance, the adjacency matrix representation of a graph is essentially a two-dimensional array.Random access : Arrays are ideal for storing data when random sampling is required. By generating a random sequence based on indices, we can achieve random sampling efficiently.Sorting and searching : Arrays are the most commonly used data structure for sorting and searching algorithms. Techniques like quick sort, merge sort, binary search, etc., are primarily operate on arrays.Lookup tables : Arrays serve as efficient lookup tables for quick element or relationship retrieval. For instance, mapping characters to ASCII codes becomes seamless by using the ASCII code values as indices and storing corresponding elements in the array.Machine learning : Within the domain of neural networks, arrays play a pivotal role in executing crucial linear algebra operations involving vectors, matrices, and tensors. Arrays serve as the primary and most extensively used data structure in neural network programming.Data structure implementation : Arrays serve as the building blocks for implementing various data structures like stacks, queues, hash tables, heaps, graphs, etc. For instance, the adjacency matrix representation of a graph is essentially a two-dimensional array.
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