How to find the Largest Difference in an Array

Question

Suppose I have an array of integers:

int[] A = { 10, 3, 6, 8, 9, 4, 3 };

My goal is to find the largest difference between A[Q] and A[P] such that Q > P.

For example, if P = 2 and Q = 3, then

diff = A[Q] - A[P]
diff = 8 - 6
diff = 2

If P = 1 and Q = 4

diff = A[Q] - A[P]
diff = 9 - 3
diff = 6

Since 6 is the largest number between all the difference, that is the answer.

My solution is as follows (in C#) but it is inefficient.

public int solution(int[] A) {

    int N = A.Length;
    if (N < 1) return 0;

    int difference;
    int largest = 0;

    for (int p = 0; p < N; p++)
    {
        for (int q = p + 1; q < N; q++)
        {
            difference = A[q] - A[p];
            if (difference > largest)
            {
                largest = difference;
            }
        }
    }

    return largest;
}

How can I improve this so it will run at O(N)? Thanks!

Simply getting the max and min wont work. Minuend (Q) should come after the Subtrahend (P).

This question is based on the "Max-profit" problem in codility (http://codility.com/train/). My solution only scored 66%. It requires O(N) for a score of 100%.

In light of your specification (`A[Q] - A[P] with Q > P`) 6 is indeed the correct answer. 7 would only be possible with `Q = 0` but that would violate `Q > P`. — Daniel Hilgarth, Sep 23 '13 at 10:19
I doubt that there is a better algorithm with this collection type. Is this a theoretical question? This is a prime discipline of a computer. — Tim Schmelter, Sep 23 '13 at 10:24

Daniel Hilgarth · Accepted Answer · 2016-02-26T08:09:57.107

20

The following code runs in O(n) and should conform to the specification (preliminary tests on codility were successful):

public int solution(int[] A)
{
    int N = A.Length;
    if (N < 1) return 0;

    int max = 0;
    int result = 0;

    for(int i = N-1; i >= 0; --i)
    {
        if(A[i] > max)
            max = A[i];

        var tmpResult = max - A[i];        
        if(tmpResult > result)
            result = tmpResult;
    }

    return result;
}

Update:
I submitted it as solution and it scores 100%.

Update 02/26/16:
The original task description on codility stated that "each element of array A is an integer within the range [0..1,000,000,000]."
If negative values would have been allowed as well, the code above wouldn't return the correct value. This could be fixed easily by changing the declaration of max to int max = int.MinValue;

edited Feb 26 '16 at 08:09

answered Sep 23 '13 at 10:59

Daniel Hilgarth

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@JoséHurtado: What do you mean with "doesn't work"? When the array is sorted descending, the biggest difference is 0, which the solution returns. – Daniel Hilgarth Sep 24 '13 at 07:46
@DanielHilgarth its ok, I forgot the Q > P restriction – JoseH Sep 24 '13 at 08:10
I have to agree with @JoseH . The restriction is Q > P, not Q >= P. For a strictly descending array, the answer should be negative, not 0. Both this algorithm and OP's O(n^2) algorithm are incorrect, unless OP meant Q >= P. – Jeff Irwin Feb 23 '16 at 19:01
@JeffIrwin: The answer is correct, at least according to the site that asked it. It was a codility training question and I submitted this code and it scored a hundred percent. The sort order of the array is irrelevant and has nothing to do with the question. Because the sort order has no relevance, the result can never be negative. – Daniel Hilgarth Feb 23 '16 at 19:05
This is will fail in this input int[] a = { 10,2,3,-1,4,-2,4,-9}; Expected 6 Output : 9 – Siva Kumar Feb 26 '16 at 07:41
@SivaKumar: Your input is invalid with regards to the original task description on codility. See [my other comment](http://stackoverflow.com/questions/18956740/how-to-find-the-largest-difference-in-an-array/18957562?noredirect=1#comment27997562_18956798). Unfortunately, the original task description is no longer online. – Daniel Hilgarth Feb 26 '16 at 08:06
@SivaKumar: However, if it were a valid input, the code could easily be fixed, by using `int.MinValue` as the initial value for `max`. – Daniel Hilgarth Feb 26 '16 at 08:06

score 4 · Answer 2 · answered Feb 23 '16 at 17:45

Here is the O(n) Java implementation

public static int largestDifference(int[] data) {
    int minElement=data[0], maxDifference=0;

    for (int i = 1; i < data.length; i++) {
        minElement = Math.min(minElement, data[i]);
        maxDifference = Math.max(maxDifference, data[i] - minElement);
    }
    return maxDifference;
}

King King · Answer 3 · 2013-09-23T10:49:56.777

1

After some attempts, I end up with this:

int iMax = N - 1;
int min = int.MaxValue, max = int.MinValue;
for (int i = 0; i < iMax; i++) {
    if (min > A[i]) min = A[i];                                     
    if (max < A[N - i - 1]){
      iMax = N - i - 1;
      max = A[iMax];
    }        
 }
 int largestDiff = max - min;

NOTE: I have just tested it with some cases. Please if you find any case in which it doesn't work, let me know in the comment. I'll try to improve it or remove the answer. Thanks!

edited Sep 23 '13 at 10:49

answered Sep 23 '13 at 10:18

King King

61,710
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@DanielHilgarth it's a shame for me without reading it carefully, could you review my updated code. Thank you! – King King Sep 23 '13 at 10:50
@IlyaIvanov I don't get well what you mean, could you make it clearer? or provide some input of `A` for me to test. thanks – King King Sep 23 '13 at 10:58
@wudzik: According to the test on codility that is not a valid input: *"each element of array A is an integer within the range [0..1,000,000,000]."* – Daniel Hilgarth Sep 23 '13 at 11:02
nice solution, I didn't catch `i < iMax` part at first review. Sorry. – Ilya Ivanov Sep 23 '13 at 11:02
@DanielHilgarth didin't knew :) – Kamil Budziewski Sep 23 '13 at 11:04

score 0 · Answer 4 · answered Sep 30 '13 at 13:05

  int FirstIndex = -1;
            int SecondIndex = -1;
            int diff = 0;

            for (int i = A.Length-1; i >=0; i--)
            {
                int FirstNo = A[i];
                int tempDiff = 0;
                for (int j = 0; j <i ; j++)
                {
                    int SecondNo = A[j];
                    tempDiff = FirstNo - SecondNo;
                    if (tempDiff > diff)
                    {
                        diff = tempDiff;
                        FirstIndex = i;
                        SecondIndex = j;
                    }
                }
            }

            MessageBox.Show("Diff: " + diff + "   FirstIndex: " + (FirstIndex+1) + "   SecondIndex: " + (SecondIndex+1));

score 0 · Answer 5 · answered Oct 23 '18 at 09:17

PHP Solution

<?php
$a = [0,5,0,5,0];
$max_diff = -1;
$min_value = $a[0];
for($i = 0;$i<count($a)-1;$i++){
    if($a[$i+1] > $a[$i]){
        $diff = $a[$i+1] - $min_value;
        if($diff > $max_diff){
            $max_diff = $diff;
        }
    } else {
        $min_value = $a[$i+1];
    }
}
echo $max_diff;
?>

score 0 · Answer 6 · answered Sep 28 '20 at 21:27

We can do it in a much simpler way by calculating biggest and smallest element of the array. I know that you're also looking for time complexity. But for anyone looking to understand and solve this problem in a simple and easy to understand way, then here is my code:

#include<stdio.h>  
  
#define N 6  
  
int main()  
{  
    int num[N], i, big, small, pos = 0;  
  
    printf("Enter %d integer numbers\n", N);  
    for(i = 0; i < N; i++)  
        scanf("%d", &num[i]);  
  
    big = small = num[0];  
  
    for(i = 1; i < N; i++)  
    {  
        if(num[i] > big)  
        {  
            big = num[i];  
            pos = i;  
        }  
    }  
  
    for(i = 1; i < pos; i++)  
    {  
        if(num[i] < small)  
            small = num[i];  
    }  
  
    printf("The largest difference is %d, ", (big - small));  
    printf("and its between %d and %d.\n", big, small);  
  
    return 0;  
}

Output:

Enter 6 integer numbers

7

9

5

6

13

2

The largest difference is 8, and its between 13 and 5.

Source: C Program To Find Largest Difference Between Two Elements of Array

score 0 · Answer 7 · answered Dec 02 '20 at 23:19

C++ solution for MaxProfit of codility test task giving 100/100 https://app.codility.com/programmers/lessons/9-maximum_slice_problem/max_profit/

int Max(vector<int> &A)
{
    if (A.size() == 1 || A.size() == 0)
        return 0;
    int min_price = A[0];
    int max_val = 0;
    for (int i = 1; i < A.size(); i++)
    {
        max_val = std::max(max_val, A[i] - min_price);
        min_price = std::min(min_price, A[i]);
    }
    return max_val;
}

score 0 · Answer 8 · answered Mar 06 '22 at 17:30

My 100% JavaScript solution with O(N) time complexity:

function solution(A) {

    // each element of array A is an integer within the range [0..200,000]
    let min  = 200000;
    // The function should return 0 if it was impossible to gain any profit.
    let maxDiff = 0;

    for (const a of A) {
        min = Math.min(min, a);
        // find the maximum positive difference (profit) between current global minimum and current value of a
        maxDiff = Math.max(maxDiff, a - min);
    }

    return maxDiff;

}

score 0 · Answer 9 · answered Apr 07 '22 at 00:00

0

function solution(A) {
  var n = A.length;
  var min = Infinity, max = -Infinity, maxNet=0;
  // find smallest and largest in the array following each other 
  for(let i = 0; i < n; i++){
      if(A[i]<min) { // if you are updating the min you cannot consider the old max
          min = A[i];
          max = -Infinity;
      } else if(A[i]> max){
          max = A[i];
      }
      if(max!=-Infinity && max-min>maxNet) maxNet = max-min;
  }
  return maxNet;
}

answered Apr 07 '22 at 00:00

Muhammed kerim

31
3

The question is about finding the *Largest Difference* in the given array. your code snippet will get the *Largest* number – Mahmoud Samy Apr 12 '22 at 08:37

score -1 · Answer 10 · answered Jun 06 '14 at 15:01

PHP solution for MaxProfit of codility test task giving 100/100 found at http://www.rationalplanet.com/php-related/maxprofit-demo-task-at-codility-com.html

function solution($A) {
    $cnt = count($A);
    if($cnt == 1 || $cnt == 0){
        return 0;
    }

    $max_so_far = 0;
    $max_ending_here = 0;
    $min_price = $A[0];

    for($i = 1; $i < $cnt; $i++){
        $max_ending_here = max(0, $A[$i] - $min_price);
        $min_price = min($min_price, $A[$i]);
        $max_so_far = max($max_ending_here, $max_so_far);
    }

    return $max_so_far;
}

score -1 · Answer 11 · answered May 26 '15 at 03:13

100% score JavaScript solution.

function solution(A) {
  if (A.length < 2)
    return 0;

  // Init min price and max profit
  var minPrice = A[0];
  var maxProfit = 0;

  for (var i = 1; i < A.length; i++) {
    var profit = A[i] - minPrice;
    maxProfit = Math.max(maxProfit, profit);
    minPrice = Math.min(minPrice, A[i]);
  }
  return maxProfit;
}

score -1 · Answer 12 · edited Jun 20 '20 at 09:12

Python solution

def max_diff_two(arr):
    #keep tab of current diff and min value
    min_value = arr[0]

    #begin with something
    maximum = arr[1] - arr[0]

    new_min = min_value

    for i,value in enumerate(arr):
        if i == 0:
            continue

        if value < min_value and value < new_min:
            new_min = value

        current_maximum = value - min_value
        new_maximum = value - new_min

        if new_maximum > current_maximum:
            if new_maximum > maximum:
                maximum = new_maximum
                min = new_min
        else:
            if current_maximum > maximum:
                maximum = current_maximum

    return  maximum

You've seen that `c#`-tag over there? – cramopy Feb 23 '16 at 17:57 — cramopy, Feb 23 '16 at 17:57

score -1 · Answer 13 · answered May 26 '16 at 13:26

-1

100% for Javascript solution using a more elegant functional approach.

function solution(A) {
    var result = A.reverse().reduce(function (prev, val) {
        var max = (val > prev.max) ? val : prev.max
        var diff = (max - val > prev.diff) ? max - val : prev.diff
        return {max: max, diff: diff}
    }, {max: 0, diff: 0})
    return result.diff
}

answered May 26 '16 at 13:26

stukennedy

1,088
1
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You could (and should) do it with LINQ. – AlexMelw Feb 18 '18 at 18:45

How to find the Largest Difference in an Array

13 Answers13

PHP Solution

Python solution