How can we find second maximum from array efficiently?

Question

Is it possible to find the second maximum number from an array of integers by traversing the array only once?

As an example, I have a array of five integers from whi

Answer 1

Your original code is okay, you just have to initialize the max and second_max variables. Use the first two elements in the array.

Answer 2

// Set the first two different numbers as the maximum and second maximum numbers

 int max = array[0];
 int i = 1;
//n is the amount of numbers

 while (array[i] == max && i < n) i++;
 int sec_max = array[i];
 if( max < sec_max ) {
    tmp = sec_max;
    sec_max = max;
    max = tmp;
 }

//find the second maximum number

 for( ; i < n; ++i ) {
   if( array[i] > max ) {
     sec_max = max;
     max = array[i];
   } else if( array[i] > sec_max && array[i] != max ) {
     sec_max = array[i];
   }
 }
 printf("The second maximum number is %d\n", sec_max);

Answer 3

Step 1. Decide on first two numbers.
Step 2. Loop through remaining numbers.
Step 3. Maintain latest maximum and second maximum.
Step 4. When updating second maximum, be aware that you are not making maximum and second maximum equal.

Tested for sorted input (ascending and descending), random input, input having duplicates, works fine.

#include <iostream>
#define MAX 50
int GetSecondMaximum(int* data, unsigned int size)
{
    int max, secmax;
    // Decide on first two numbers
    if (data[0] > data[1])
    {
        max = data[0];
        secmax = data[1];
    }
    else
    {
        secmax = data[0];
        max = data[1];
    }
    // Loop through remaining numbers
    for (unsigned int i = 2; i < size; ++i)
    {
        if (data[i] > max)
        {
            secmax = max;
            max = data[i];
        }
        else if (data[i] > secmax && data[i] != max/*removes duplicate problem*/)
            secmax = data[i];
    }
    return secmax;
}
int main()
{
    int data[MAX];
    // Fill with random integers
    for (unsigned int i = 0; i < MAX; ++i)
    {
        data[i] = rand() % MAX;
        std::cout << "[" << data[i] << "] "; // Display input
    }
    std::cout << std::endl << std::endl;
    // Find second maximum
    int nSecondMax = GetSecondMaximum(data, MAX);
    // Display output
    std::cout << "Second Maximum = " << nSecondMax << std::endl;
    // Wait for user input
    std::cin.get();
    return 0;
}

Answer 4

Quickselect is the way to go with this one. Pseudo code is available at that link so I shall just explain the overall algorithm:

QuickSelect for kth largest number:
    Select a pivot element
    Split array around pivot
    If (k < new pivot index)
       perform quickselect on left hand sub array
     else if (k > new pivot index)
       perform quickselect on right hand sub array (make sure to offset k by size of lefthand array + 1)
     else
       return pivot

This is quite obviously based on the good old quicksort algorithm.

Following this algorithm through, always selecting element zero as the pivot every time:

select 4th largest number:
1) array = {1, 3, 2, 7, 11, 0, -4}
partition with 1 as pivot
{0, -4, _1_, 3, 2, 7, 11}
4 > 2 (new pivot index) so...

2) Select 1st (4 - 3) largest number from right sub array
array = {3, 2, 7, 11}
partition with 3 as pivot
{2, _3_, 7, 11}
1 < 2 (new pivot index) so...

3) select 1st largest number from left sub array
array = {2}

4) Done, 4th largest number is 2

This will leave your array in an undefined order afterwards, it's up to you if that's a problem.

Answer 5

The upper bound should have be n+log2⁡n−2, but it bigger than O(n) in case of random selection algorithm, but in worst case it much smaller. The solution might be

1. build a tree like to find the MAX element with n - 1 comparisons

   max(N) / \ max(N/2) max(N/2)

2. remove the MAX and find the MAX again log2n - 1 comparison

PS. It uses additional memory, but it faster than random selection algorithm in worst case.

Answer 6

You need a second test:

 for(int i=0;i<5;i++){  
   if(arr[i]>max){  
     second_max=max;  
     max=arr[i];            
   }
   else if (arr[i] > second_max && arr[i] != max){
     second_max = arr[i];
   }
 }