设计一个栈来检索原始元素并返回 O(1) 时间和 O(1) 空间中的最小元素
我们的任务是设计一个数据结构 SpecialStack，它支持所有堆栈操作，如push() 、 pop() 、 isEmpty() 、isFull ()和一个额外的操作getMin( )，它应该从 SpecialStack 返回最小元素。 SpecialStack 的所有这些操作都必须以 O(1) 的时间复杂度来执行。要实现 SpecialStack，您应该只使用标准的 Stack 数据结构，而不能使用其他数据结构，如数组、列表等。
例子：
Consider the following Special-Stack
16 --> TOP
15
29
19
18
When getMin() is called it should return 15, which is the minimum
element in the current stack.
If we do pop two times on stack, the stack becomes
29 --> TOP
19
18
When getMin() is called, it should return 18 which is the minimum
in the current stack.这里讨论了一种使用 O(1) 时间和 O(1) 额外空间的方法。但是，在上一篇文章中，原始元素没有被恢复。在任何给定时间点仅返回最小元素。
在本文中，对之前的方法进行了修改，以便在pop()操作期间也可以检索原始元素。
方法：
考虑一个变量最小值，我们将最小元素存储在堆栈中。现在，如果我们从堆栈中弹出最小元素怎么办？我们如何将最小变量更新为下一个最小值？一种解决方案是按排序顺序维护另一个堆栈，以便最小的元素始终位于顶部。但是，这是一种 O(n) 空间方法。
为了在O(1)空间中实现这一点，我们需要一种方法来将元素的当前值和下一个最小值存储在同一节点中。这可以通过应用简单的数学来完成：
new_value = 2*current_value - minimum我们将这个 new_value 而不是 current_value 压入堆栈。从 new_value 检索 current_value 和下一个最小值：
current_value = (new_value + minimum)/2
minimum = new_value - 2*current当操作Push(x)完成时，我们遵循以下算法：
- If stack is empty
- insert x into the stack
- make minimum equal to x.
- If stack is not empty
- if x is less than minimum
- set temp equal to 2*x-minimum
- set minimum equal to x
- set x equal to temp
- insert x into stack
- if x is less than minimum
当操作Pop(x)完成时，我们遵循以下算法：
- If stack is not empty
- set x equal to topmost element
- if x is less than minimum
- set minimum equal to 2*minimum – x
- set x equal to (x+minimum)/2
- return x
当调用 getMin() 时，我们返回存储在变量minimum中的元素：
Java
// Java program to retrieve original elements of the
// from a Stack which returns the minimum element
// in O(1) time and O(1) space
class Stack {
Node top;
// Stores minimum element of the stack
int minimum;
// Function to push an element
void push(Node node)
{
int current = node.getData();
if (top == null) {
top = node;
minimum = current;
}
else {
if (current < minimum) {
node.setData(2 * current - minimum);
minimum = current;
}
node.setNext(top);
top = node;
}
}
// Retrieves topmost element
Node pop()
{
Node node = top;
if (node != null) {
int current = node.getData();
if (current < minimum) {
minimum = 2 * minimum - current;
node.setData((current + minimum) / 2);
}
top = top.getNext();
}
return node;
}
// Retrieves topmost element without
// popping it from the stack
Node peek()
{
Node node = null;
if (top != null) {
node = new Node();
int current = top.getData();
node.setData(current < minimum ? minimum : current);
}
return node;
}
// Function to print all elements in the stack
void printAll()
{
Node ptr = top;
int min = minimum;
if (ptr != null) { // if stack is not empty
while (true) {
int val = ptr.getData();
if (val < min) {
min = 2 * min - val;
val = (val + min) / 2;
}
System.out.print(val + " ");
ptr = ptr.getNext();
if (ptr == null)
break;
}
System.out.println();
}
else
System.out.println("Empty!");
}
// Returns minimum of Stack
int getMin()
{
return minimum;
}
boolean isEmpty()
{
return top == null;
}
}
// Node class which contains data
// and pointer to next node
class Node {
int data;
Node next;
Node()
{
data = -1;
next = null;
}
Node(int d)
{
data = d;
next = null;
}
void setData(int data)
{
this.data = data;
}
void setNext(Node next)
{
this.next = next;
}
Node getNext()
{
return next;
}
int getData()
{
return data;
}
}
// Driver Code
public class Main {
public static void main(String[] args)
{
// Create a new stack
Stack stack = new Stack();
Node node;
// Push the element into the stack
stack.push(new Node(5));
stack.push(new Node(3));
stack.push(new Node(4));
// Calls the method to print the stack
System.out.println("Elements in the stack are:");
stack.printAll();
// Print current minimum element if stack is
// not empty
System.out.println(stack.isEmpty() ? "\nEmpty Stack!" :
"\nMinimum: " + stack.getMin());
// Push new elements into the stack
stack.push(new Node(1));
stack.push(new Node(2));
// Printing the stack
System.out.println("\nStack after adding new elements:");
stack.printAll();
// Print current minimum element if stack is
// not empty
System.out.println(stack.isEmpty() ? "\nEmpty Stack!" :
"\nMinimum: " + stack.getMin());
// Pop elements from the stack
node = stack.pop();
System.out.println("\nElement Popped: "
+ (node == null ? "Empty!" : node.getData()));
node = stack.pop();
System.out.println("Element Popped: "
+ (node == null ? "Empty!" : node.getData()));
// Printing stack after popping elements
System.out.println("\nStack after removing top two elements:");
stack.printAll();
// Printing current Minimum element in the stack
System.out.println(stack.isEmpty() ? "\nEmpty Stack!" :
"\nMinimum: " + stack.getMin());
// Printing top element of the stack
node = stack.peek();
System.out.println("\nTop: " + (node == null ?
"\nEmpty!" : node.getData()));
}
}Python3
"""
Python program to retrieve original elements of the
from a Stack which returns the minimum element
in O(1) time and O(1) space
"""
# Class to make a Node
class Node:
# Constructor which assign argument to nade's value
def __init__(self, value):
self.value = value
self.next = None
# This method returns the string
# representation of the object.
def __str__(self):
return "Node({})".format(self.value)
# __repr__ is same as __str__
__repr__ = __str__
class Stack:
# Stack Constructor initialise
# top of stack and counter.
def __init__(self):
self.top = None
self.count = 0
self.minimum = None
# This method returns the string
# representation of the object (stack).
def __str__(self):
temp=self.top
m = self.minimum
out=[]
if temp is None:
print("Empty Stack")
else:
while not temp is None :
val = temp.value
if val < m:
m = (2 * m) -val
val = ( val + m ) / 2
out.append(str(int(val)))
temp=temp.next
out=' '.join(out)
return (out)
# __repr__ is same as __str__
__repr__=__str__
# This method is used to get minimum element of stack
def getMin(self):
if self.top is None:
return "Stack is empty"
else:
return self.minimum
# Method to check if Stack is Empty or not
def isEmpty(self):
# If top equals to None then stack is empty
if self.top == None:
return True
else:
# If top not equal to None then stack is empty
return False
# This method returns length of stack
def __len__(self):
self.count = 0
tempNode = self.top
while tempNode:
tempNode = tempNode.next
self.count+=1
return self.count
# This method returns top of stack
def peek(self):
if self.top is None:
print ("Stack is empty")
else:
if self.top.value < self.minimum:
return self.minimum
else:
return self.top.value
# This method is used to add node to stack
def push(self,value):
if self.top is None:
self.top = Node(value)
self.minimum = value
else:
new_node = Node(value)
if value < self.minimum:
temp = (2 * value) - self.minimum
new_node.value = temp
self.minimum = value
new_node.next = self.top
self.top = new_node
# This method is used to pop top of stack
def pop(self):
new_node = self.top
if self.top is None:
print( "Stack is empty")
else:
removedNode = new_node.value
if removedNode < self.minimum:
self.minimum = ( ( 2 * self.minimum ) - removedNode )
new_node.value = ( (removedNode + self.minimum) / 2 )
self.top = self.top.next
return int(new_node.value)
# Driver program to test above class
stack = Stack()
stack.push(5)
stack.push(3)
stack.push(4)
print("Elements in the stack are:")
print(stack)
print("Minimum: {}" .format( stack.getMin() ) )
stack.push(1)
stack.push(2)
print("Stack after adding new elements:")
print(stack)
print("Minimum:{}" .format( stack.getMin() ) )
print("Element Popped: {}".format(stack.pop()))
print("Element Popped: {}".format(stack.pop()))
print("Stack after removing top two elements: ")
print(stack)
print("Minimum: {}" .format( stack.getMin() ) )
print("Top: {}".format( stack.peek() ) )
# This code is contributed by BlinkiiC#
// C# program to retrieve original elements of the
// from a Stack which returns the minimum element
// in O(1) time and O(1) space
using System;
public class Stack
{
Node top;
// Stores minimum element of the stack
public int minimum;
// Function to push an element
public void push(Node node)
{
int current = node.getData();
if (top == null)
{
top = node;
minimum = current;
}
else
{
if (current < minimum)
{
node.setData(2 * current - minimum);
minimum = current;
}
node.setNext(top);
top = node;
}
}
// Retrieves topmost element
public Node pop()
{
Node node = top;
if (node != null)
{
int current = node.getData();
if (current < minimum)
{
minimum = 2 * minimum - current;
node.setData((current + minimum) / 2);
}
top = top.getNext();
}
return node;
}
// Retrieves topmost element without
// popping it from the stack
public Node peek()
{
Node node = null;
if (top != null)
{
node = new Node();
int current = top.getData();
node.setData(current < minimum ? minimum : current);
}
return node;
}
// Function to print all elements in the stack
public void printAll()
{
Node ptr = top;
int min = minimum;
if (ptr != null)
{
// if stack is not empty
while (true)
{
int val = ptr.getData();
if (val < min)
{
min = 2 * min - val;
val = (val + min) / 2;
}
Console.Write(val + " ");
ptr = ptr.getNext();
if (ptr == null)
break;
}
Console.WriteLine();
}
else
Console.WriteLine("Empty!");
}
// Returns minimum of Stack
public int getMin()
{
return minimum;
}
public bool isEmpty()
{
return top == null;
}
}
// Node class which contains data
// and pointer to next node
public class Node
{
public int data;
public Node next;
public Node()
{
data = -1;
next = null;
}
public Node(int d)
{
data = d;
next = null;
}
public void setData(int data)
{
this.data = data;
}
public void setNext(Node next)
{
this.next = next;
}
public Node getNext()
{
return next;
}
public int getData()
{
return data;
}
}
// Driver Code
public class MainClass
{
public static void Main(String[] args)
{
// Create a new stack
Stack stack = new Stack();
Node node;
// Push the element into the stack
stack.push(new Node(5));
stack.push(new Node(3));
stack.push(new Node(4));
// Calls the method to print the stack
Console.WriteLine("Elements in the stack are:");
stack.printAll();
// Print current minimum element if stack is
// not empty
Console.WriteLine(stack.isEmpty() ? "\nEmpty Stack!" :
"\nMinimum: " + stack.getMin());
// Push new elements into the stack
stack.push(new Node(1));
stack.push(new Node(2));
// Printing the stack
Console.WriteLine("\nStack after adding new elements:");
stack.printAll();
// Print current minimum element if stack is
// not empty
Console.WriteLine(stack.isEmpty() ? "\nEmpty Stack!" :
"\nMinimum: " + stack.getMin());
// Pop elements from the stack
node = stack.pop();
if(node == null)
Console.WriteLine("\nElement Popped: "+"Empty!");
else
Console.WriteLine("\nElement Popped: "+node.getData());
node = stack.pop();
if(node == null)
Console.WriteLine("Element Popped: "+"Empty!");
else
Console.WriteLine("Element Popped: "+node.getData());
// Printing stack after popping elements
Console.WriteLine("\nStack after removing top two elements:");
stack.printAll();
// Printing current Minimum element in the stack
Console.WriteLine(stack.isEmpty() ? "\nEmpty Stack!" :
"\nMinimum: " + stack.getMin());
// Printing top element of the stack
node = stack.peek();
if(node == null)
Console.WriteLine("\nTop: "+"\nEmpty!");
else
Console.WriteLine("\nTop: "+node.getData());
}
}
// This code is contributed by Rajput-JiC++
// Cpp program to retrieve original elements of the
// from a Stack which returns the minimum element
// in O(1) time and O(1) space
#include
using namespace std;
// Node class which contains data
// and pointer to next node
class Node {
public:
int data;
Node* next;
Node()
{
data = -1;
next = NULL;
}
Node(int d)
{
data = d;
next = NULL;
}
void setData(int data)
{
this->data = data;
}
void setNext(Node* next)
{
this->next = next;
}
Node* getNext()
{
return next;
}
int getData()
{
return data;
}
};
class Stack{
public:
Node* top;
// Stores minimum element of the stack
int minimum;
// Function to push an element
void push(Node* node)
{
int current=node->getData();
if(top == NULL)
{
top=node;
minimum=current;
}
else{
if(current < minimum)
{
node->setData(2*current - minimum);
minimum = current;
}
node->setNext(top);
top=node;
}
}
// Retrieves topmost element
Node* pop()
{
Node* node = top;
if(node!=NULL)
{
int current=node->getData();
if(currentsetData((current + minimum) / 2);
}
top = top->getNext();
}
return node;
}
// Retrieves topmost element without
// popping it from the stack
Node* peek()
{
Node* node = NULL;
if (top != NULL) {
node = new Node();
int current = top->getData();
node->setData(current < minimum ? minimum : current);
}
return node;
}
// Function to print all elements in the stack
void printAll()
{
Node* ptr = top;
int min = minimum;
if (ptr != NULL) { // if stack is not empty
while (true) {
int val = ptr->getData();
if (val < min) {
min = 2 * min - val;
val = (val + min) / 2;
}
cout << val << " ";
ptr = ptr->getNext();
if (ptr == NULL)
break;
}
cout << '\n';
}
else
cout << "Empty!\n";
}
// Returns minimum of Stack
int getMin()
{
return minimum;
}
bool isEmpty()
{
return top == NULL;
}
};
int main()
{
Stack* stack = new Stack();
Node* node;
stack->push(new Node(5));
stack->push(new Node(3));
stack->push(new Node(4));
// Calls the method to print the stack
cout << "Elements in the stack are:\n";
stack->printAll();
// Print current minimum element if stack is
// not empty
if(stack->isEmpty())
cout << "Empty Stack!\n";
else
cout << "Minimum: " << stack->getMin() << '\n';
// Push new elements into the stack
stack->push(new Node(1));
stack->push(new Node(2));
// Printing the stack
cout << "Stack after adding new elements:\n";
stack->printAll();
// Print current minimum element if stack is
// not empty
if(stack->isEmpty())
cout << "Empty Stack!\n";
else
cout << "Minimum: " << stack->getMin() << '\n';
// Pop elements from the stack
node = stack->pop();
cout << "Element Popped: ";
if(node == NULL)
cout << "Empty!\n";
else
cout << node->getData() << '\n';
node = stack->pop();
cout << "Element Popped: ";
if(node == NULL)
cout << "Empty!\n";
else
cout << node->getData() << '\n';
// Printing stack after popping elements
cout << "Stack after removing top two elements:\n";
stack->printAll();
// Printing current Minimum element in the stack
if(stack->isEmpty())
cout << "Empty Stack!\n";
else
cout << "Minimum: " << stack->getMin() << '\n';
// Printing top element of the stack
node = stack->peek();
cout << "Top: " ;
if(node == NULL)
cout << "Empty!\n";
else
cout << node->getData() << '\n';
return 0;
} 输出：
Elements in the stack are:
4 3 5
Minimum: 3
Stack after adding new elements:
2 1 4 3 5
Minimum: 1
Element Popped: 2
Element Popped: 1
Stack after removing top two elements:
4 3 5
Minimum: 3
Top: 4