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---
comments: true
---
# 4.2   Linked Lists
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Memory space is a shared resource among all programs. In a complex system environment, available memory can be dispersed throughout the memory space. We understand that the memory allocated for an array must be continuous. However, for very large arrays, finding a sufficiently large contiguous memory space might be challenging. This is where the flexible advantage of linked lists becomes evident.
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A "linked list" is a linear data structure in which each element is a node object, and the nodes are interconnected through "references". These references hold the memory addresses of subsequent nodes, enabling navigation from one node to the next.
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The design of linked lists allows for their nodes to be distributed across memory locations without requiring contiguous memory addresses.
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![Linked List Definition and Storage Method](linked_list.assets/linkedlist_definition.png){ class="animation-figure" }
<p align="center"> Figure 4-5 &nbsp; Linked List Definition and Storage Method </p>
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As shown in the figure, we see that the basic building block of a linked list is the "node" object. Each node comprises two key components: the node's "value" and a "reference" to the next node.
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- The first node in a linked list is the "head node", and the final one is the "tail node".
- The tail node points to "null", designated as `null` in Java, `nullptr` in C++, and `None` in Python.
- In languages that support pointers, like C, C++, Go, and Rust, this "reference" is typically implemented as a "pointer".
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As the code below illustrates, a `ListNode` in a linked list, besides holding a value, must also maintain an additional reference (or pointer). Therefore, **a linked list occupies more memory space than an array when storing the same quantity of data.**.
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=== "Python"
```python title=""
class ListNode:
"""Linked List Node Class"""
def __init__(self, val: int):
self.val: int = val # Node value
self.next: ListNode | None = None # Reference to the next node
```
=== "C++"
```cpp title=""
/* Linked List Node Structure */
struct ListNode {
int val; // Node value
ListNode *next; // Pointer to the next node
ListNode(int x) : val(x), next(nullptr) {} // Constructor
};
```
=== "Java"
```java title=""
/* Linked List Node Class */
class ListNode {
int val; // Node value
ListNode next; // Reference to the next node
ListNode(int x) { val = x; } // Constructor
}
```
=== "C#"
```csharp title=""
/* Linked List Node Class */
class ListNode(int x) { // Constructor
int val = x; // Node value
ListNode? next; // Reference to the next node
}
```
=== "Go"
```go title=""
/* Linked List Node Structure */
type ListNode struct {
Val int // Node value
Next *ListNode // Pointer to the next node
}
// NewListNode Constructor, creates a new linked list
func NewListNode(val int) *ListNode {
return &ListNode{
Val: val,
Next: nil,
}
}
```
=== "Swift"
```swift title=""
/* Linked List Node Class */
class ListNode {
var val: Int // Node value
var next: ListNode? // Reference to the next node
init(x: Int) { // Constructor
val = x
}
}
```
=== "JS"
```javascript title=""
/* Linked List Node Class */
class ListNode {
constructor(val, next) {
this.val = (val === undefined ? 0 : val); // Node value
this.next = (next === undefined ? null : next); // Reference to the next node
}
}
```
=== "TS"
```typescript title=""
/* Linked List Node Class */
class ListNode {
val: number;
next: ListNode | null;
constructor(val?: number, next?: ListNode | null) {
this.val = val === undefined ? 0 : val; // Node value
this.next = next === undefined ? null : next; // Reference to the next node
}
}
```
=== "Dart"
```dart title=""
/* 链表节点类 */
class ListNode {
int val; // Node value
ListNode? next; // Reference to the next node
ListNode(this.val, [this.next]); // Constructor
}
```
=== "Rust"
```rust title=""
use std::rc::Rc;
use std::cell::RefCell;
/* Linked List Node Class */
#[derive(Debug)]
struct ListNode {
val: i32, // Node value
next: Option<Rc<RefCell<ListNode>>>, // Pointer to the next node
}
```
=== "C"
```c title=""
/* Linked List Node Structure */
typedef struct ListNode {
int val; // Node value
struct ListNode *next; // Pointer to the next node
} ListNode;
/* Constructor */
ListNode *newListNode(int val) {
ListNode *node;
node = (ListNode *) malloc(sizeof(ListNode));
node->val = val;
node->next = NULL;
return node;
}
```
=== "Zig"
```zig title=""
// Linked List Node Class
pub fn ListNode(comptime T: type) type {
return struct {
const Self = @This();
val: T = 0, // Node value
next: ?*Self = null, // Pointer to the next node
// Constructor
pub fn init(self: *Self, x: i32) void {
self.val = x;
self.next = null;
}
};
}
```
## 4.2.1 &nbsp; Common Operations on Linked Lists
### 1. &nbsp; Initializing a Linked List
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Constructing a linked list is a two-step process: first, initializing each node object, and second, forming the reference links between the nodes. After initialization, we can traverse all nodes sequentially from the head node by following the `next` reference.
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=== "Python"
```python title="linked_list.py"
# Initialize linked list: 1 -> 3 -> 2 -> 5 -> 4
# Initialize each node
n0 = ListNode(1)
n1 = ListNode(3)
n2 = ListNode(2)
n3 = ListNode(5)
n4 = ListNode(4)
# Build references between nodes
n0.next = n1
n1.next = n2
n2.next = n3
n3.next = n4
```
=== "C++"
```cpp title="linked_list.cpp"
/* Initialize linked list: 1 -> 3 -> 2 -> 5 -> 4 */
// Initialize each node
ListNode* n0 = new ListNode(1);
ListNode* n1 = new ListNode(3);
ListNode* n2 = new ListNode(2);
ListNode* n3 = new ListNode(5);
ListNode* n4 = new ListNode(4);
// Build references between nodes
n0->next = n1;
n1->next = n2;
n2->next = n3;
n3->next = n4;
```
=== "Java"
```java title="linked_list.java"
/* Initialize linked list: 1 -> 3 -> 2 -> 5 -> 4 */
// Initialize each node
ListNode n0 = new ListNode(1);
ListNode n1 = new ListNode(3);
ListNode n2 = new ListNode(2);
ListNode n3 = new ListNode(5);
ListNode n4 = new ListNode(4);
// Build references between nodes
n0.next = n1;
n1.next = n2;
n2.next = n3;
n3.next = n4;
```
=== "C#"
```csharp title="linked_list.cs"
/* Initialize linked list: 1 -> 3 -> 2 -> 5 -> 4 */
// Initialize each node
ListNode n0 = new(1);
ListNode n1 = new(3);
ListNode n2 = new(2);
ListNode n3 = new(5);
ListNode n4 = new(4);
// Build references between nodes
n0.next = n1;
n1.next = n2;
n2.next = n3;
n3.next = n4;
```
=== "Go"
```go title="linked_list.go"
/* Initialize linked list: 1 -> 3 -> 2 -> 5 -> 4 */
// Initialize each node
n0 := NewListNode(1)
n1 := NewListNode(3)
n2 := NewListNode(2)
n3 := NewListNode(5)
n4 := NewListNode(4)
// Build references between nodes
n0.Next = n1
n1.Next = n2
n2.Next = n3
n3.Next = n4
```
=== "Swift"
```swift title="linked_list.swift"
/* Initialize linked list: 1 -> 3 -> 2 -> 5 -> 4 */
// Initialize each node
let n0 = ListNode(x: 1)
let n1 = ListNode(x: 3)
let n2 = ListNode(x: 2)
let n3 = ListNode(x: 5)
let n4 = ListNode(x: 4)
// Build references between nodes
n0.next = n1
n1.next = n2
n2.next = n3
n3.next = n4
```
=== "JS"
```javascript title="linked_list.js"
/* Initialize linked list: 1 -> 3 -> 2 -> 5 -> 4 */
// Initialize each node
const n0 = new ListNode(1);
const n1 = new ListNode(3);
const n2 = new ListNode(2);
const n3 = new ListNode(5);
const n4 = new ListNode(4);
// Build references between nodes
n0.next = n1;
n1.next = n2;
n2.next = n3;
n3.next = n4;
```
=== "TS"
```typescript title="linked_list.ts"
/* Initialize linked list: 1 -> 3 -> 2 -> 5 -> 4 */
// Initialize each node
const n0 = new ListNode(1);
const n1 = new ListNode(3);
const n2 = new ListNode(2);
const n3 = new ListNode(5);
const n4 = new ListNode(4);
// Build references between nodes
n0.next = n1;
n1.next = n2;
n2.next = n3;
n3.next = n4;
```
=== "Dart"
```dart title="linked_list.dart"
/* Initialize linked list: 1 -> 3 -> 2 -> 5 -> 4 */
// Initialize each node
ListNode n0 = ListNode(1);
ListNode n1 = ListNode(3);
ListNode n2 = ListNode(2);
ListNode n3 = ListNode(5);
ListNode n4 = ListNode(4);
// Build references between nodes
n0.next = n1;
n1.next = n2;
n2.next = n3;
n3.next = n4;
```
=== "Rust"
```rust title="linked_list.rs"
/* Initialize linked list: 1 -> 3 -> 2 -> 5 -> 4 */
// Initialize each node
let n0 = Rc::new(RefCell::new(ListNode { val: 1, next: None }));
let n1 = Rc::new(RefCell::new(ListNode { val: 3, next: None }));
let n2 = Rc::new(RefCell::new(ListNode { val: 2, next: None }));
let n3 = Rc::new(RefCell::new(ListNode { val: 5, next: None }));
let n4 = Rc::new(RefCell::new(ListNode { val: 4, next: None }));
// Build references between nodes
n0.borrow_mut().next = Some(n1.clone());
n1.borrow_mut().next = Some(n2.clone());
n2.borrow_mut().next = Some(n3.clone());
n3.borrow_mut().next = Some(n4.clone());
```
=== "C"
```c title="linked_list.c"
/* Initialize linked list: 1 -> 3 -> 2 -> 5 -> 4 */
// Initialize each node
ListNode* n0 = newListNode(1);
ListNode* n1 = newListNode(3);
ListNode* n2 = newListNode(2);
ListNode* n3 = newListNode(5);
ListNode* n4 = newListNode(4);
// Build references between nodes
n0->next = n1;
n1->next = n2;
n2->next = n3;
n3->next = n4;
```
=== "Zig"
```zig title="linked_list.zig"
// Initialize linked list
// Initialize each node
var n0 = inc.ListNode(i32){.val = 1};
var n1 = inc.ListNode(i32){.val = 3};
var n2 = inc.ListNode(i32){.val = 2};
var n3 = inc.ListNode(i32){.val = 5};
var n4 = inc.ListNode(i32){.val = 4};
// Build references between nodes
n0.next = &n1;
n1.next = &n2;
n2.next = &n3;
n3.next = &n4;
```
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The array as a whole is a variable, for instance, the array `nums` includes elements like `nums[0]`, `nums[1]`, and so on, whereas a linked list is made up of several distinct node objects. **We typically refer to a linked list by its head node**, for example, the linked list in the previous code snippet is referred to as `n0`.
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### 2. &nbsp; Inserting a Node
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Inserting a node into a linked list is very easy. As shown in the figure, let's assume we aim to insert a new node `P` between two adjacent nodes `n0` and `n1`. **This can be achieved by simply modifying two node references (pointers)**, with a time complexity of $O(1)$.
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By comparison, inserting an element into an array has a time complexity of $O(n)$, which becomes less efficient when dealing with large data volumes.
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![Linked List Node Insertion Example](linked_list.assets/linkedlist_insert_node.png){ class="animation-figure" }
<p align="center"> Figure 4-6 &nbsp; Linked List Node Insertion Example </p>
=== "Python"
```python title="linked_list.py"
def insert(n0: ListNode, P: ListNode):
"""在链表的节点 n0 之后插入节点 P"""
n1 = n0.next
P.next = n1
n0.next = P
```
=== "C++"
```cpp title="linked_list.cpp"
/* 在链表的节点 n0 之后插入节点 P */
void insert(ListNode *n0, ListNode *P) {
ListNode *n1 = n0->next;
P->next = n1;
n0->next = P;
}
```
=== "Java"
```java title="linked_list.java"
/* 在链表的节点 n0 之后插入节点 P */
void insert(ListNode n0, ListNode P) {
ListNode n1 = n0.next;
P.next = n1;
n0.next = P;
}
```
=== "C#"
```csharp title="linked_list.cs"
/* 在链表的节点 n0 之后插入节点 P */
void Insert(ListNode n0, ListNode P) {
ListNode? n1 = n0.next;
P.next = n1;
n0.next = P;
}
```
=== "Go"
```go title="linked_list.go"
/* 在链表的节点 n0 之后插入节点 P */
func insertNode(n0 *ListNode, P *ListNode) {
n1 := n0.Next
P.Next = n1
n0.Next = P
}
```
=== "Swift"
```swift title="linked_list.swift"
/* 在链表的节点 n0 之后插入节点 P */
func insert(n0: ListNode, P: ListNode) {
let n1 = n0.next
P.next = n1
n0.next = P
}
```
=== "JS"
```javascript title="linked_list.js"
/* 在链表的节点 n0 之后插入节点 P */
function insert(n0, P) {
const n1 = n0.next;
P.next = n1;
n0.next = P;
}
```
=== "TS"
```typescript title="linked_list.ts"
/* 在链表的节点 n0 之后插入节点 P */
function insert(n0: ListNode, P: ListNode): void {
const n1 = n0.next;
P.next = n1;
n0.next = P;
}
```
=== "Dart"
```dart title="linked_list.dart"
/* 在链表的节点 n0 之后插入节点 P */
void insert(ListNode n0, ListNode P) {
ListNode? n1 = n0.next;
P.next = n1;
n0.next = P;
}
```
=== "Rust"
```rust title="linked_list.rs"
/* 在链表的节点 n0 之后插入节点 P */
#[allow(non_snake_case)]
pub fn insert<T>(n0: &Rc<RefCell<ListNode<T>>>, P: Rc<RefCell<ListNode<T>>>) {
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let n1 = n0.borrow_mut().next.take();
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P.borrow_mut().next = n1;
n0.borrow_mut().next = Some(P);
}
```
=== "C"
```c title="linked_list.c"
/* 在链表的节点 n0 之后插入节点 P */
void insert(ListNode *n0, ListNode *P) {
ListNode *n1 = n0->next;
P->next = n1;
n0->next = P;
}
```
=== "Zig"
```zig title="linked_list.zig"
// 在链表的节点 n0 之后插入节点 P
fn insert(n0: ?*inc.ListNode(i32), P: ?*inc.ListNode(i32)) void {
var n1 = n0.?.next;
P.?.next = n1;
n0.?.next = P;
}
```
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??? pythontutor "Code Visualization"
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<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20insert%28n0%3A%20ListNode,%20P%3A%20ListNode%29%3A%0A%20%20%20%20%22%22%22%E5%9C%A8%E9%93%BE%E8%A1%A8%E7%9A%84%E8%8A%82%E7%82%B9%20n0%20%E4%B9%8B%E5%90%8E%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%20P%22%22%22%0A%20%20%20%20n1%20%3D%20n0.next%0A%20%20%20%20P.next%20%3D%20n1%0A%20%20%20%20n0.next%20%3D%20P%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%93%BE%E8%A1%A8%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%90%84%E4%B8%AA%E8%8A%82%E7%82%B9%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%0A%20%20%20%20p%20%3D%20ListNode%280%29%0A%20%20%20%20insert%28n0,%20p%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=39&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20insert%28n0%3A%20ListNode,%20P%3A%20ListNode%29%3A%0A%20%20%20%20%22%22%22%E5%9C%A8%E9%93%BE%E8%A1%A8%E7%9A%84%E8%8A%82%E7%82%B9%20n0%20%E4%B9%8B%E5%90%8E%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%20P%22%22%22%0A%20%20%20%20n1%20%3D%20n0.next%0A%20%20%20%20P.next%20%3D%20n1%0A%20%20%20%20n0.next%20%3D%20P%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%93%BE%E8%A1%A8%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%90%84%E4%B8%AA%E8%8A%82%E7%82%B9%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20%E6%8F%92%E5%85%A5%E8%8A%82%E7%82%B9%0A%20%20%20%20p%20%3D%20ListNode%280%29%0A%20%20%20%20insert%28n0,%20p%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=39&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
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### 3. &nbsp; Deleting a Node
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As shown in the figure, deleting a node from a linked list is also very easy, **involving only the modification of a single node's reference (pointer)**.
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It's important to note that even though node `P` continues to point to `n1` after being deleted, it becomes inaccessible during linked list traversal. This effectively means that `P` is no longer a part of the linked list.
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![Linked List Node Deletion](linked_list.assets/linkedlist_remove_node.png){ class="animation-figure" }
<p align="center"> Figure 4-7 &nbsp; Linked List Node Deletion </p>
=== "Python"
```python title="linked_list.py"
def remove(n0: ListNode):
"""删除链表的节点 n0 之后的首个节点"""
if not n0.next:
return
# n0 -> P -> n1
P = n0.next
n1 = P.next
n0.next = n1
```
=== "C++"
```cpp title="linked_list.cpp"
/* 删除链表的节点 n0 之后的首个节点 */
void remove(ListNode *n0) {
if (n0->next == nullptr)
return;
// n0 -> P -> n1
ListNode *P = n0->next;
ListNode *n1 = P->next;
n0->next = n1;
// 释放内存
delete P;
}
```
=== "Java"
```java title="linked_list.java"
/* 删除链表的节点 n0 之后的首个节点 */
void remove(ListNode n0) {
if (n0.next == null)
return;
// n0 -> P -> n1
ListNode P = n0.next;
ListNode n1 = P.next;
n0.next = n1;
}
```
=== "C#"
```csharp title="linked_list.cs"
/* 删除链表的节点 n0 之后的首个节点 */
void Remove(ListNode n0) {
if (n0.next == null)
return;
// n0 -> P -> n1
ListNode P = n0.next;
ListNode? n1 = P.next;
n0.next = n1;
}
```
=== "Go"
```go title="linked_list.go"
/* 删除链表的节点 n0 之后的首个节点 */
func removeItem(n0 *ListNode) {
if n0.Next == nil {
return
}
// n0 -> P -> n1
P := n0.Next
n1 := P.Next
n0.Next = n1
}
```
=== "Swift"
```swift title="linked_list.swift"
/* 删除链表的节点 n0 之后的首个节点 */
func remove(n0: ListNode) {
if n0.next == nil {
return
}
// n0 -> P -> n1
let P = n0.next
let n1 = P?.next
n0.next = n1
P?.next = nil
}
```
=== "JS"
```javascript title="linked_list.js"
/* 删除链表的节点 n0 之后的首个节点 */
function remove(n0) {
if (!n0.next) return;
// n0 -> P -> n1
const P = n0.next;
const n1 = P.next;
n0.next = n1;
}
```
=== "TS"
```typescript title="linked_list.ts"
/* 删除链表的节点 n0 之后的首个节点 */
function remove(n0: ListNode): void {
if (!n0.next) {
return;
}
// n0 -> P -> n1
const P = n0.next;
const n1 = P.next;
n0.next = n1;
}
```
=== "Dart"
```dart title="linked_list.dart"
/* 删除链表的节点 n0 之后的首个节点 */
void remove(ListNode n0) {
if (n0.next == null) return;
// n0 -> P -> n1
ListNode P = n0.next!;
ListNode? n1 = P.next;
n0.next = n1;
}
```
=== "Rust"
```rust title="linked_list.rs"
/* 删除链表的节点 n0 之后的首个节点 */
#[allow(non_snake_case)]
pub fn remove<T>(n0: &Rc<RefCell<ListNode<T>>>) {
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if n0.borrow().next.is_none() {
return;
};
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// n0 -> P -> n1
let P = n0.borrow_mut().next.take();
if let Some(node) = P {
let n1 = node.borrow_mut().next.take();
n0.borrow_mut().next = n1;
}
}
```
=== "C"
```c title="linked_list.c"
/* 删除链表的节点 n0 之后的首个节点 */
// 注意：stdio.h 占用了 remove 关键词
void removeItem(ListNode *n0) {
if (!n0->next)
return;
// n0 -> P -> n1
ListNode *P = n0->next;
ListNode *n1 = P->next;
n0->next = n1;
// 释放内存
free(P);
}
```
=== "Zig"
```zig title="linked_list.zig"
// 删除链表的节点 n0 之后的首个节点
fn remove(n0: ?*inc.ListNode(i32)) void {
if (n0.?.next == null) return;
// n0 -> P -> n1
var P = n0.?.next;
var n1 = P.?.next;
n0.?.next = n1;
}
```
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??? pythontutor "Code Visualization"
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<div style="height: 549px; width: 100%;"><iframe class="pythontutor-iframe" src="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20remove%28n0%3A%20ListNode%29%3A%0A%20%20%20%20%22%22%22%E5%88%A0%E9%99%A4%E9%93%BE%E8%A1%A8%E7%9A%84%E8%8A%82%E7%82%B9%20n0%20%E4%B9%8B%E5%90%8E%E7%9A%84%E9%A6%96%E4%B8%AA%E8%8A%82%E7%82%B9%22%22%22%0A%20%20%20%20if%20not%20n0.next%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20n0%20-%3E%20P%20-%3E%20n1%0A%20%20%20%20P%20%3D%20n0.next%0A%20%20%20%20n1%20%3D%20P.next%0A%20%20%20%20n0.next%20%3D%20n1%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%93%BE%E8%A1%A8%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%90%84%E4%B8%AA%E8%8A%82%E7%82%B9%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%0A%20%20%20%20remove%28n0%29&codeDivHeight=472&codeDivWidth=350&cumulative=false&curInstr=34&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false"> </iframe></div>
<div style="margin-top: 5px;"><a href="https://pythontutor.com/iframe-embed.html#code=class%20ListNode%3A%0A%20%20%20%20%22%22%22%E9%93%BE%E8%A1%A8%E8%8A%82%E7%82%B9%E7%B1%BB%22%22%22%0A%20%20%20%20def%20__init__%28self,%20val%3A%20int%29%3A%0A%20%20%20%20%20%20%20%20self.val%3A%20int%20%3D%20val%20%20%23%20%E8%8A%82%E7%82%B9%E5%80%BC%0A%20%20%20%20%20%20%20%20self.next%3A%20ListNode%20%7C%20None%20%3D%20None%20%20%23%20%E5%90%8E%E7%BB%A7%E8%8A%82%E7%82%B9%E5%BC%95%E7%94%A8%0A%0Adef%20remove%28n0%3A%20ListNode%29%3A%0A%20%20%20%20%22%22%22%E5%88%A0%E9%99%A4%E9%93%BE%E8%A1%A8%E7%9A%84%E8%8A%82%E7%82%B9%20n0%20%E4%B9%8B%E5%90%8E%E7%9A%84%E9%A6%96%E4%B8%AA%E8%8A%82%E7%82%B9%22%22%22%0A%20%20%20%20if%20not%20n0.next%3A%0A%20%20%20%20%20%20%20%20return%0A%20%20%20%20%23%20n0%20-%3E%20P%20-%3E%20n1%0A%20%20%20%20P%20%3D%20n0.next%0A%20%20%20%20n1%20%3D%20P.next%0A%20%20%20%20n0.next%20%3D%20n1%0A%0A%22%22%22Driver%20Code%22%22%22%0Aif%20__name__%20%3D%3D%20%22__main__%22%3A%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E9%93%BE%E8%A1%A8%0A%20%20%20%20%23%20%E5%88%9D%E5%A7%8B%E5%8C%96%E5%90%84%E4%B8%AA%E8%8A%82%E7%82%B9%0A%20%20%20%20n0%20%3D%20ListNode%281%29%0A%20%20%20%20n1%20%3D%20ListNode%283%29%0A%20%20%20%20n2%20%3D%20ListNode%282%29%0A%20%20%20%20n3%20%3D%20ListNode%285%29%0A%20%20%20%20n4%20%3D%20ListNode%284%29%0A%20%20%20%20%23%20%E6%9E%84%E5%BB%BA%E8%8A%82%E7%82%B9%E4%B9%8B%E9%97%B4%E7%9A%84%E5%BC%95%E7%94%A8%0A%20%20%20%20n0.next%20%3D%20n1%0A%20%20%20%20n1.next%20%3D%20n2%0A%20%20%20%20n2.next%20%3D%20n3%0A%20%20%20%20n3.next%20%3D%20n4%0A%0A%20%20%20%20%23%20%E5%88%A0%E9%99%A4%E8%8A%82%E7%82%B9%0A%20%20%20%20remove%28n0%29&codeDivHeight=800&codeDivWidth=600&cumulative=false&curInstr=34&heapPrimitives=nevernest&origin=opt-frontend.js&py=311&rawInputLstJSON=%5B%5D&textReferences=false" target="_blank" rel="noopener noreferrer">Full Screen ></a></div>
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### 4. &nbsp; Accessing Nodes
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**Accessing nodes in a linked list is less efficient**. As previously mentioned, any element in an array can be accessed in $O(1)$ time. In contrast, with a linked list, the program involves starting from the head node and sequentially traversing through the nodes until the desired node is found. In other words, to access the $i$-th node in a linked list, the program must iterate through $i - 1$ nodes, resulting in a time complexity of $O(n)$.
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=== "Python"
```python title="linked_list.py"
def access(head: ListNode, index: int) -> ListNode | None:
"""访问链表中索引为 index 的节点"""
for _ in range(index):
if not head:
return None
head = head.next
return head
```
=== "C++"
```cpp title="linked_list.cpp"
/* 访问链表中索引为 index 的节点 */
ListNode *access(ListNode *head, int index) {
for (int i = 0; i < index; i++) {
if (head == nullptr)
return nullptr;
head = head->next;
}
return head;
}
```
=== "Java"
```java title="linked_list.java"
/* 访问链表中索引为 index 的节点 */
ListNode access(ListNode head, int index) {
for (int i = 0; i < index; i++) {
if (head == null)
return null;
head = head.next;
}
return head;
}
```
=== "C#"
```csharp title="linked_list.cs"
/* 访问链表中索引为 index 的节点 */
ListNode? Access(ListNode? head, int index) {
for (int i = 0; i < index; i++) {
if (head == null)
return null;
head = head.next;
}
return head;
}
```
=== "Go"
```go title="linked_list.go"
/* 访问链表中索引为 index 的节点 */
func access(head *ListNode, index int) *ListNode {
for i := 0; i < index; i++ {
if head == nil {
return nil
}
head = head.Next
}
return head
}
```
=== "Swift"
```swift title="linked_list.swift"
/* 访问链表中索引为 index 的节点 */
func access(head: ListNode, index: Int) -> ListNode? {
var head: ListNode? = head
for _ in 0 ..< index {
if head == nil {
return nil
}
head = head?.next
}
return head
}
```
=== "JS"
```javascript title="linked_list.js"
/* 访问链表中索引为 index 的节点 */
function access(head, index) {
for (let i = 0; i < index; i++) {
if (!head) {
return null;
}
head = head.next;
}
return head;
}
```
=== "TS"
```typescript title="linked_list.ts"
/* 访问链表中索引为 index 的节点 */
function access(head: ListNode | null, index: number): ListNode | null {
for (let i = 0; i < index; i++) {
if (!head) {
return null;
}
head = head.next;
}
return head;
}
```
=== "Dart"
```dart title="linked_list.dart"
/* 访问链表中索引为 index 的节点 */
ListNode? access(ListNode? head, int index) {
for (var i = 0; i < index; i++) {
if (head == null) return null;
head = head.next;
}
return head;
}
```
=== "Rust"
```rust title="linked_list.rs"
/* 访问链表中索引为 index 的节点 */
pub fn access<T>(head: Rc<RefCell<ListNode<T>>>, index: i32) -> Rc<RefCell<ListNode<T>>> {
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if index <= 0 {
return head;
};
if let Some(node) = &head.borrow().next {
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return access(node.clone(), index - 1);
}
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return head;
}
```
=== "C"
```c title="linked_list.c"
/* 访问链表中索引为 index 的节点 */
ListNode *access(ListNode *head, int index) {
for (int i = 0; i < index; i++) {
if (head == NULL)
return NULL;
head = head->next;
}
return head;
}
```
=== "Zig"
```zig title="linked_list.zig"
// 访问链表中索引为 index 的节点
fn access(node: ?*inc.ListNode(i32), index: i32) ?*inc.ListNode(i32) {
var head = node;
var i: i32 = 0;
while (i < index) : (i += 1) {
head = head.?.next;
if (head == null) return null;
}
return head;
}
```
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??? pythontutor "Code Visualization"
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### 5. &nbsp; Finding Nodes
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Traverse the linked list to locate a node whose value matches `target`, and then output the index of that node within the linked list. This procedure is also an example of linear search. The corresponding code is provided below:
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=== "Python"
```python title="linked_list.py"
def find(head: ListNode, target: int) -> int:
"""在链表中查找值为 target 的首个节点"""
index = 0
while head:
if head.val == target:
return index
head = head.next
index += 1
return -1
```
=== "C++"
```cpp title="linked_list.cpp"
/* 在链表中查找值为 target 的首个节点 */
int find(ListNode *head, int target) {
int index = 0;
while (head != nullptr) {
if (head->val == target)
return index;
head = head->next;
index++;
}
return -1;
}
```
=== "Java"
```java title="linked_list.java"
/* 在链表中查找值为 target 的首个节点 */
int find(ListNode head, int target) {
int index = 0;
while (head != null) {
if (head.val == target)
return index;
head = head.next;
index++;
}
return -1;
}
```
=== "C#"
```csharp title="linked_list.cs"
/* 在链表中查找值为 target 的首个节点 */
int Find(ListNode? head, int target) {
int index = 0;
while (head != null) {
if (head.val == target)
return index;
head = head.next;
index++;
}
return -1;
}
```
=== "Go"
```go title="linked_list.go"
/* 在链表中查找值为 target 的首个节点 */
func findNode(head *ListNode, target int) int {
index := 0
for head != nil {
if head.Val == target {
return index
}
head = head.Next
index++
}
return -1
}
```
=== "Swift"
```swift title="linked_list.swift"
/* 在链表中查找值为 target 的首个节点 */
func find(head: ListNode, target: Int) -> Int {
var head: ListNode? = head
var index = 0
while head != nil {
if head?.val == target {
return index
}
head = head?.next
index += 1
}
return -1
}
```
=== "JS"
```javascript title="linked_list.js"
/* 在链表中查找值为 target 的首个节点 */
function find(head, target) {
let index = 0;
while (head !== null) {
if (head.val === target) {
return index;
}
head = head.next;
index += 1;
}
return -1;
}
```
=== "TS"
```typescript title="linked_list.ts"
/* 在链表中查找值为 target 的首个节点 */
function find(head: ListNode | null, target: number): number {
let index = 0;
while (head !== null) {
if (head.val === target) {
return index;
}
head = head.next;
index += 1;
}
return -1;
}
```
=== "Dart"
```dart title="linked_list.dart"
/* 在链表中查找值为 target 的首个节点 */
int find(ListNode? head, int target) {
int index = 0;
while (head != null) {
if (head.val == target) {
return index;
}
head = head.next;
index++;
}
return -1;
}
```
=== "Rust"
```rust title="linked_list.rs"
/* 在链表中查找值为 target 的首个节点 */
pub fn find<T: PartialEq>(head: Rc<RefCell<ListNode<T>>>, target: T, index: i32) -> i32 {
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if head.borrow().val == target {
return index;
};
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if let Some(node) = &head.borrow_mut().next {
return find(node.clone(), target, index + 1);
}
return -1;
}
```
=== "C"
```c title="linked_list.c"
/* 在链表中查找值为 target 的首个节点 */
int find(ListNode *head, int target) {
int index = 0;
while (head) {
if (head->val == target)
return index;
head = head->next;
index++;
}
return -1;
}
```
=== "Zig"
```zig title="linked_list.zig"
// 在链表中查找值为 target 的首个节点
fn find(node: ?*inc.ListNode(i32), target: i32) i32 {
var head = node;
var index: i32 = 0;
while (head != null) {
if (head.?.val == target) return index;
head = head.?.next;
index += 1;
}
return -1;
}
```
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??? pythontutor "Code Visualization"
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## 4.2.2 &nbsp; Arrays vs. Linked Lists
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The Table 4-1 summarizes the characteristics of arrays and linked lists, and it also compares their efficiencies in various operations. Because they utilize opposing storage strategies, their respective properties and operational efficiencies exhibit distinct contrasts.
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<p align="center"> Table 4-1 &nbsp; Efficiency Comparison of Arrays and Linked Lists </p>
<div class="center-table" markdown>
| | Arrays | Linked Lists |
| ------------------ | ------------------------------------------------ | ----------------------- |
| Storage | Contiguous Memory Space | Dispersed Memory Space |
| Capacity Expansion | Fixed Length | Flexible Expansion |
| Memory Efficiency | Less Memory per Element, Potential Space Wastage | More Memory per Element |
| Accessing Elements | $O(1)$ | $O(n)$ |
| Adding Elements | $O(n)$ | $O(1)$ |
| Deleting Elements | $O(n)$ | $O(1)$ |
</div>
## 4.2.3 &nbsp; Common Types of Linked Lists
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As shown in the figure, there are three common types of linked lists.
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- **Singly Linked List**: This is the standard linked list described earlier. Nodes in a singly linked list include a value and a reference to the next node. The first node is known as the head node, and the last node, which points to null (`None`), is the tail node.
- **Circular Linked List**: This is formed when the tail node of a singly linked list points back to the head node, creating a loop. In a circular linked list, any node can function as the head node.
- **Doubly Linked List**: In contrast to a singly linked list, a doubly linked list maintains references in two directions. Each node contains references (pointer) to both its successor (the next node) and predecessor (the previous node). Although doubly linked lists offer more flexibility for traversing in either direction, they also consume more memory space.
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=== "Python"
```python title=""
class ListNode:
"""Bidirectional linked list node class""""
def __init__(self, val: int):
self.val: int = val # Node value
self.next: ListNode | None = None # Reference to the successor node
self.prev: ListNode | None = None # Reference to a predecessor node
```
=== "C++"
```cpp title=""
/* Bidirectional linked list node structure */
struct ListNode {
int val; // Node value
ListNode *next; // Pointer to the successor node
ListNode *prev; // Pointer to the predecessor node
ListNode(int x) : val(x), next(nullptr), prev(nullptr) {} // Constructor
};
```
=== "Java"
```java title=""
/* Bidirectional linked list node class */
class ListNode {
int val; // Node value
ListNode next; // Reference to the next node
ListNode prev; // Reference to the predecessor node
ListNode(int x) { val = x; } // Constructor
}
```
=== "C#"
```csharp title=""
/* Bidirectional linked list node class */
class ListNode(int x) { // Constructor
int val = x; // Node value
ListNode next; // Reference to the next node
ListNode prev; // Reference to the predecessor node
}
```
=== "Go"
```go title=""
/* Bidirectional linked list node structure */
type DoublyListNode struct {
Val int // Node value
Next *DoublyListNode // Pointer to the successor node
Prev *DoublyListNode // Pointer to the predecessor node
}
// NewDoublyListNode initialization
func NewDoublyListNode(val int) *DoublyListNode {
return &DoublyListNode{
Val: val,
Next: nil,
Prev: nil,
}
}
```
=== "Swift"
```swift title=""
/* Bidirectional linked list node class */
class ListNode {
var val: Int // Node value
var next: ListNode? // Reference to the next node
var prev: ListNode? // Reference to the predecessor node
init(x: Int) { // Constructor
val = x
}
}
```
=== "JS"
```javascript title=""
/* Bidirectional linked list node class */
class ListNode {
constructor(val, next, prev) {
this.val = val === undefined ? 0 : val; // Node value
this.next = next === undefined ? null : next; // Reference to the successor node
this.prev = prev === undefined ? null : prev; // Reference to the predecessor node
}
}
```
=== "TS"
```typescript title=""
/* Bidirectional linked list node class */
class ListNode {
val: number;
next: ListNode | null;
prev: ListNode | null;
constructor(val?: number, next?: ListNode | null, prev?: ListNode | null) {
this.val = val === undefined ? 0 : val; // Node value
this.next = next === undefined ? null : next; // Reference to the successor node
this.prev = prev === undefined ? null : prev; // Reference to the predecessor node
}
}
```
=== "Dart"
```dart title=""
/* Bidirectional linked list node class */
class ListNode {
int val; // Node value
ListNode next; // Reference to the next node
ListNode prev; // Reference to the predecessor node
ListNode(this.val, [this.next, this.prev]); // Constructor
}
```
=== "Rust"
```rust title=""
use std::rc::Rc;
use std::cell::RefCell;
/* Bidirectional linked list node type */
#[derive(Debug)]
struct ListNode {
val: i32, // Node value
next: Option<Rc<RefCell<ListNode>>>, // Pointer to successor node
prev: Option<Rc<RefCell<ListNode>>>, // Pointer to predecessor node
}
/* Constructors */
impl ListNode {
fn new(val: i32) -> Self {
ListNode {
val,
next: None,
prev: None,
}
}
}
```
=== "C"
```c title=""
/* Bidirectional linked list node structure */
typedef struct ListNode {
int val; // Node value
struct ListNode *next; // Pointer to the successor node
struct ListNode *prev; // Pointer to the predecessor node
} ListNode;
/* Constructors */
ListNode *newListNode(int val) {
ListNode *node, *next;
node = (ListNode *) malloc(sizeof(ListNode));
node->val = val;
node->next = NULL;
node->prev = NULL;
return node;
}
```
=== "Zig"
```zig title=""
// Bidirectional linked list node class
pub fn ListNode(comptime T: type) type {
return struct {
const Self = @This();
val: T = 0, // Node value
next: ?*Self = null, // Pointer to the successor node
prev: ?*Self = null, // Pointer to the predecessor node
// Constructor
pub fn init(self: *Self, x: i32) void {
self.val = x;
self.next = null;
self.prev = null;
}
};
}
```
![Common Types of Linked Lists](linked_list.assets/linkedlist_common_types.png){ class="animation-figure" }
<p align="center"> Figure 4-8 &nbsp; Common Types of Linked Lists </p>
## 4.2.4 &nbsp; Typical Applications of Linked Lists
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Singly linked lists are frequently utilized in implementing stacks, queues, hash tables, and graphs.
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- **Stacks and Queues**: In singly linked lists, if insertions and deletions occur at the same end, it behaves like a stack (last-in-first-out). Conversely, if insertions are at one end and deletions at the other, it functions like a queue (first-in-first-out).
- **Hash Tables**: Linked lists are used in chaining, a popular method for resolving hash collisions. Here, all collided elements are grouped into a linked list.
- **Graphs**: Adjacency lists, a standard method for graph representation, associate each graph vertex with a linked list. This list contains elements that represent vertices connected to the corresponding vertex.
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Doubly linked lists are ideal for scenarios requiring rapid access to preceding and succeeding elements.
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- **Advanced Data Structures**: In structures like red-black trees and B-trees, accessing a node's parent is essential. This is achieved by incorporating a reference to the parent node in each node, akin to a doubly linked list.
- **Browser History**: In web browsers, doubly linked lists facilitate navigating the history of visited pages when users click forward or back.
- **LRU Algorithm**: Doubly linked lists are apt for Least Recently Used (LRU) cache eviction algorithms, enabling swift identification of the least recently used data and facilitating fast node addition and removal.
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Circular linked lists are ideal for applications that require periodic operations, such as resource scheduling in operating systems.
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- **Round-Robin Scheduling Algorithm**: In operating systems, the round-robin scheduling algorithm is a common CPU scheduling method, requiring cycling through a group of processes. Each process is assigned a time slice, and upon expiration, the CPU rotates to the next process. This cyclical operation can be efficiently realized using a circular linked list, allowing for a fair and time-shared system among all processes.
- **Data Buffers**: Circular linked lists are also used in data buffers, like in audio and video players, where the data stream is divided into multiple buffer blocks arranged in a circular fashion for seamless playback.
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