How to efficiently implement k Queues in a single array?

We have discussed efficient implementation of k stack in an array. In this post, same for queue is discussed. Following is the detailed problem statement.


Create a data structure kQueues that represents k queues. Implementation of kQueues should use only one array, i.e., k queues should use the same array for storing elements. Following functions must be supported by kQueues.

enqueue(int x, int qn) –> adds x to queue number ‘qn’ where qn is from 0 to k-1
dequeue(int qn) –> deletes an element from queue number ‘qn’ where qn is from 0 to k-1

Method 1 (Divide the array in slots of size n/k)
A simple way to implement k queues is to divide the array in k slots of size n/k each, and fix the slots for different queues, i.e., use arr[0] to arr[n/k-1] for the first queue, and arr[n/k] to arr[2n/k-1] for queue2 where arr[] is the array to be used to implement two queues and size of array be n.

The problem with this method is an inefficient use of array space. An enqueue operation may result in overflow even if there is space available in arr[]. For example, consider k as 2 and array size n as 6. Let we enqueue 3 elements to first and do not enqueue anything to the second queue. When we enqueue the 4th element to the first queue, there will be overflow even if we have space for 3 more elements in the array.



Method 2 (A space efficient implementation)
The idea is similar to the stack post, here we need to use three extra arrays. In stack post, we needed two extra arrays, one more array is required because in queues, enqueue() and dequeue() operations are done at different ends.

Following are the three extra arrays are used:
1) front[]: This is of size k and stores indexes of front elements in all queues.
2) rear[]: This is of size k and stores indexes of rear elements in all queues.
2) next[]: This is of size n and stores indexes of next item for all items in array arr[].

Here arr[] is the actual array that stores k stacks.

Together with k queues, a stack of free slots in arr[] is also maintained. The top of this stack is stored in a variable ‘free’.

All entries in front[] are initialized as -1 to indicate that all queues are empty. All entries next[i] are initialized as i+1 because all slots are free initially and pointing to the next slot. Top of the free stack, ‘free’ is initialized as 0.

Following is C++ implementation of the above idea.

C++

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// A C++ program to demonstrate implementation of k queues in a single
// array in time and space efficient way
#include<iostream>
#include<climits>
using namespace std;
  
// A C++ class to represent k queues in a single array of size n
class kQueues
{   
    // Array of size n to store actual content to be stored in queue
    int *arr;
    // Array of size k to store indexes of front elements of the queue  
    int *front;   
    // Array of size k to store indexes of rear elements of queue
    int *rear; 
    // Array of size n to store next entry in all queues            
    int *next;  
    int n, k;
    int free; // To store the beginning index of the free list
public:
    //constructor to create k queue in an array of size n
    kQueues(int k, int n);
  
    // A utility function to check if there is space available
    bool isFull()   {  return (free == -1);  }
  
    // To enqueue an item in queue number 'qn' where qn is from 0 to k-1
    void enqueue(int item, int qn);
  
    // To dequeue an from queue number 'qn' where qn is from 0 to k-1
    int dequeue(int qn);
  
    // To check whether queue number 'qn' is empty or not
    bool isEmpty(int qn)  {  return (front[qn] == -1); }
};
  
// Constructor to create k queues in an array of size n
kQueues::kQueues(int k1, int n1)
{
    // Initialize n and k, and allocate memory for all arrays
    k = k1, n = n1;
    arr = new int[n];
    front = new int[k];
    rear = new int[k];
    next = new int[n];
  
    // Initialize all queues as empty
    for (int i = 0; i < k; i++)
        front[i] = -1;
  
    // Initialize all spaces as free
    free = 0;
    for (int i=0; i<n-1; i++)
        next[i] = i+1;
    next[n-1] = -1;  // -1 is used to indicate end of free list
}
  
// To enqueue an item in queue number 'qn' where qn is from 0 to k-1
void kQueues::enqueue(int item, int qn)
{
    // Overflow check
    if (isFull())
    {
        cout << "\nQueue Overflow\n";
        return;
    }
  
    int i = free;      // Store index of first free slot
  
    // Update index of free slot to index of next slot in free list
    free = next[i];
  
    if (isEmpty(qn))
        front[qn] = i;
    else
        next[rear[qn]] = i;
  
    next[i] = -1;
  
    // Update next of rear and then rear for queue number 'qn'
    rear[qn] = i;
  
    // Put the item in array
    arr[i] = item;
}
  
// To dequeue an from queue number 'qn' where qn is from 0 to k-1
int kQueues::dequeue(int qn)
{
    // Underflow checkSAS
    if (isEmpty(qn))
    {
         cout << "\nQueue Underflow\n";
         return INT_MAX;
    }
  
    // Find index of front item in queue number 'qn'
    int i = front[qn];
  
    front[qn] = next[i];  // Change top to store next of previous top
  
    // Attach the previous front to the beginning of free list
    next[i] = free;
    free = i;
  
    // Return the previous front item
    return arr[i];
}
  
/* Driver program to test kStacks class */
int main()
{
    // Let us create 3 queue in an array of size 10
    int k = 3, n = 10;
    kQueues ks(k, n);
  
    // Let us put some items in queue number 2
    ks.enqueue(15, 2);
    ks.enqueue(45, 2);
  
    // Let us put some items in queue number 1
    ks.enqueue(17, 1);
    ks.enqueue(49, 1);
    ks.enqueue(39, 1);
  
    // Let us put some items in queue number 0
    ks.enqueue(11, 0);
    ks.enqueue(9, 0);
    ks.enqueue(7, 0);
  
    cout << "Dequeued element from queue 2 is " << ks.dequeue(2) << endl;
    cout << "Dequeued element from queue 1 is " << ks.dequeue(1) << endl;
    cout << "Dequeued element from queue 0 is " << ks.dequeue(0) << endl;
  
    return 0;
}

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Java

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// A Java program to demonstrate implementation of k queues in a single
// array in time and space efficient way
public class KQueues {
  
    int k;
    int n;
    int[] arr;
    int[] front;
    int[] rear;
    int[] next;
    int free;
      
    KQueues(int k, int n){
          
        // Initialize n and k, and allocate memory for all arrays 
        this.k = k;
        this.n = n;
        this.arr = new int[n];
        this.front = new int[k];
        this.rear = new int[k];
        this.next = new int[n];
          
        // Initialize all queues as empty 
        for(int i= 0; i< k; i++) {
            front[i] = rear[i] = -1;
        }
          
        // Initialize all spaces as free 
        free = 0;
        for(int i= 0; i< n-1; i++) {
            next[i] = i+1;
        }
        next[n-1] = -1;
          
          
    }
      
    public static void main(String[] args) 
    
        // Let us create 3 queue in an array of size 10 
        int k = 3, n = 10
        KQueues ks=  new KQueues(k, n); 
         
          
        // Let us put some items in queue number 2 
        ks.enqueue(15, 2); 
        ks.enqueue(45, 2); 
        
        // Let us put some items in queue number 1 
        ks.enqueue(17, 1); 
        ks.enqueue(49, 1); 
        ks.enqueue(39, 1); 
        
        // Let us put some items in queue number 0 
        ks.enqueue(11, 0); 
        ks.enqueue(9, 0); 
        ks.enqueue(7, 0); 
          
        System.out.println("Dequeued element from queue 2 is "
                                ks.dequeue(2)); 
        System.out.println("Dequeued element from queue 1 is "
                                ks.dequeue(1)); 
        System.out.println("Dequeued element from queue 0 is "
                                ks.dequeue(0) ); 
        
    
      
    // To check whether queue number 'i' is empty or not 
    private boolean isEmpty(int i) {
        return front[i] == -1;
    }
      
    // To dequeue an from queue number 'i' where i is from 0 to k-1 
    private boolean isFull(int i) {
        return free == -1;
    }
  
    // To enqueue an item in queue number 'j' where j is from 0 to k-1 
    private void enqueue(int item, int j) {
        if(isFull(j)) {
            System.out.println("queue overflow");
            return;
        }
          
        int nextFree = next[free];
          
        if(isEmpty(j)) {
            rear[j] = front[j] = free;
        }else {
            // Update next of rear and then rear for queue number 'j' 
            next[rear[j]] = free;
            rear[j] = free;
        }
        next[free] = -1;
          
        // Put the item in array
        arr[free] = item;
          
        // Update index of free slot to index of next slot in free list 
        free = nextFree;
    }
      
    // To dequeue an from queue number 'i' where i is from 0 to k-1 
    private int dequeue(int i) {
        // Underflow checkSAS 
        if(isEmpty(i)) {
            System.out.println("Stack underflow");
            return Integer.MIN_VALUE;
        }
          
        // Find index of front item in queue number 'i' 
        int frontIndex = front[i];
  
        // Change top to store next of previous top
        front[i] = next[frontIndex];
          
        // Attach the previous front to the beginning of free list 
        next[frontIndex] = free; 
        free = frontIndex;
          
        return arr[frontIndex];
    }
      
}

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Output:

Dequeued element from queue 2 is 15
Dequeued element from queue 1 is 17
Dequeued element from queue 0 is 11

Time complexities of enqueue() and dequeue() is O(1).

The best part of the above implementation is, if there is a slot available in the queue, then an item can be enqueued in any of the queues, i.e., no wastage of space. This method requires some extra space. Space may not be an issue because queue items are typically large, for example, queues of employees, students, etc where every item is of hundreds of bytes. For such large queues, the extra space used is comparatively very less as we use three integer arrays as extra space.

This article is contributed by Sachin. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above



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Improved By : siddhartha33, bejuzb, viping74



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