Problem statement
Given an integer array nums, find three numbers whose product is maximum and return the maximum product.
Problem statement taken from: https://leetcode.com/problems/maximum-product-of-three-numbers
Example 1:
Input: nums = [1, 2, 3]
Output: 6
Example 2:
Input: nums = [1, 2, 3, 4]
Output: 24
Example 3:
Input: nums = [-1, -2, -3]
Output: -6
Constraints:
- 3 <= nums.length <= 10^4
- -1000 <= nums[i] <= 1000
Explanation
There are multiple ways we can solve this problem. Let's explore all the solutions from worst-case to best-case.
Brute force: 3 Nested loops
A simple solution is to check every triplet of the array using three nested loops.
A C++ snippet of this approach will look as below:
for (int i = 0; i < n - 2; i++)
for (int j = i + 1; j < n - 1; j++)
for (int k = j + 1; k < n; k++)
max_product = max(max_product, arr[i] * arr[j] * arr[k]);
As seen above, the time complexity is O(N^3) and, space complexity is O(1).
Using additional space
We can reduce the time complexity to O(N) using additional space.
- We construct four arrays leftMax[], rightMax[], leftMin[] and rightMin[] of same size as input array.
- We fill the above four arrays as below:
- leftMax[i] will contain maximum element on left of arr[i] excluding arr[i]. For index 0, left will contain -1.
- leftMin[i] will contain minimum element on left of arr[i] excluding arr[i]. For index 0, left will contain -1.
- rightMax[i] will contain maximum element on right of arr[i] excluding arr[i]. For index n - 1, right will contain -1.
- rightMin[i] will contain minimum element on right of arr[i] excluding arr[i]. For index n - 1, right will contain -1.
- For all array indexes i except first and last index, compute maximum of arr[i]*x*y where x can be leftMax[i] or leftMin[i] and y can be rightMax[i] or rightMin[i].
- Return the maximum from step 3.
A C++ snippet of this approach will look as below:
vector<int> leftMin(n, -1);
vector<int> rightMin(n, -1);
vector<int> leftMax(n, -1);
vector<int> rightMax(n, -1);
for (int i = 1; i < n; i++){
leftMax[i] = max_sum;
if (arr[i] > max_sum)
max_sum = arr[i];
leftMin[i] = min_sum;
if (arr[i] < min_sum)
min_sum = arr[i];
}
for (int j = n - 2; j >= 0; j--){
rightMax[j] = max_sum;
if (arr[j] > max_sum)
max_sum = arr[j];
rightMin[j] = min_sum;
if (arr[j] < min_sum)
min_sum = arr[j];
}
for (int i = 1; i < n - 1; i++){
int max1 = max(arr[i] * leftMax[i] * rightMax[i], arr[i] * leftMin[i] * rightMin[i]);
int max2 = max(arr[i] * leftMax[i] * rightMin[i], arr[i] * leftMin[i] * rightMax[i]);
max_product = max(max_product, max(max1, max2));
}
The space complexity of this approach is O(N).
Using sort
We can reduce the space complexity by sorting the array and consider the maximum between the product of the last three elements of the array and the product of the first two elements and the last element.
A C++ snippet of this approach will look as below:
sort(arr, arr + n);
return max(arr[0] * arr[1] * arr[n - 1], arr[n - 1] * arr[n - 2] * arr[n - 3]);
The time complexity of this approach is O(NlogN) and, space complexity is O(N).
Using five variables
The problem can be solved using five variables. Three variables will store the maximum values in an array. And the remaining two will store the minimum values present in the array.
Let's check the algorithm:
- set max1, max2 and max3 to INT_MIN
set min1, min2 to INT_MAX
- loop for i = 0; i < nums.size(); i++
- if nums[i] < min1
- set min2 = min1
- set min1 = nums[i]
- else if nums[i] < min2
- set min2 = nums[i]
- if nums[i] > max1
- set max3 = max2
- set max2 = max1
- set max1 = nums[i]
- else if nums[i] > max2
- set max3 = max2
- set max2 = nums[i]
- else if nums[i] > max3
- set max3 = nums[i]
- return max(min1 * min2 * max1, max1 * max2 * max3)
C++ solution
class Solution {
public:
int maximumProduct(vector<int>& nums) {
int max1 = INT_MIN, max2 = INT_MIN, max3 = INT_MIN;
int min1 = INT_MAX, min2 = INT_MAX;
for(int i = 0; i < nums.size(); i++){
if(nums[i] < min1){
min2 = min1;
min1 = nums[i];
} else if(nums[i] < min2){
min2 = nums[i];
}
if(nums[i] > max1){
max3 = max2;
max2 = max1;
max1 = nums[i];
} else if(nums[i] > max2){
max3 = max2;
max2 = nums[i];
} else if(nums[i] > max3){
max3 = nums[i];
}
}
return max(min1*min2*max1, max1*max2*max3);
}
};
Golang solution
const MAXINT = math.MaxInt32
const MININT = math.MinInt32
func maximumProduct(nums []int) int {
max1, max2, max3 := MININT, MININT, MININT
min1, min2 := MAXINT, MAXINT
for i := 0; i < len(nums); i++ {
if nums[i] < min1 {
min2 = min1
min1 = nums[i]
} else if nums[i] < min2 {
min2 = nums[i]
}
if nums[i] > max1 {
max3 = max2
max2 = max1
max1 = nums[i]
} else if nums[i] > max2 {
max3 = max2
max2 = nums[i]
} else if nums[i] > max3 {
max3 = nums[i]
}
}
return int(math.Max(float64(min1 *min2 * max1), float64(max1 * max2 * max3)))
}
Javascript solution
var maximumProduct = function(nums) {
let min1 = Number.POSITIVE_INFINITY, min2 = Number.POSITIVE_INFINITY;
let max1 = Number.NEGATIVE_INFINITY, max2 = Number.NEGATIVE_INFINITY, max3 = Number.NEGATIVE_INFINITY;
for(let i = 0; i < nums.length; i++) {
if( nums[i] < min1 ) {
min2 = min1;
min1 = nums[i];
} else if( nums[i] < min2 ) {
min2 = nums[i];
}
if( nums[i] > max1 ) {
max3 = max2;
max2 = max1;
max1 = nums[i];
} else if( nums[i] > max2 ) {
max3 = max2;
max2 = nums[i];
} else if( nums[i] > max3 ) {
max3 = nums[i];
}
}
return Math.max(min1 * min2 * max1, max1 * max2 * max3 );
};
Let's dry-run our algorithm to see how the solution works.
Input: nums = [-6, 5, 1, 2, 3, -4, 9]
Step 1: int max1 = INT_MIN, max2 = INT_MIN, max3 = INT_MIN;
int min1 = INT_MAX, min2 = INT_MAX;
Step 2: loop for int i = 0; i < nums.size()
i < nums.size()
0 < 7
true
if nums[i] < min1
nums[0] < INT_MAX
-6 < INT_MAX
true
- min2 = min1
= INT_MAX
- min1 = nums[i]
= nums[0]
= -6
if nums[i] > max1
nums[0] > INT_MIN
-6 > INT_MIN
true
- max3 = max2
= INT_MIN
- max2 = max1
= INT_MIN
- max1 = nums[i]
= nums[0]
= -6
i++
i = 1
Step 3: loop for int i = 0; i < nums.size()
i < nums.size()
1 < 7
true
if nums[i] < min1
nums[1] < INT_MAX
1 < -6
false
else if nums[i] < min2
5 < INT_MAX
true
- min2 = nums[i]
= 5
if nums[i] > max1
nums[1] > -6
5 > -6
true
- max3 = max2
= INT_MIN
- max2 = max1
= -6
- max1 = nums[i]
= nums[1]
= 5
i++
i = 2
Step 4: loop for int i = 0; i < nums.size()
i < nums.size()
2 < 7
true
if nums[i] < min1
nums[2] < -6
1 < -6
false
else if nums[i] < min2
1 < 5
true
- min2 = nums[2]
= 1
if nums[i] > max1
nums[2] > 5
1 > 5
false
else if nums[i] > max2
nums[2] > -6
1 > -6
true
- max3 = max2
= -6
- max2 = nums[i]
= nums[2]
= 1
i++
i = 3
Step 5: loop for int i = 0; i < nums.size()
i < nums.size()
3 < 7
true
if nums[i] < min1
nums[3] < -6
2 < -6
false
else if nums[i] < min2
2 < 1
false
if nums[i] > max1
nums[3] > 5
2 > 5
false
else if nums[i] > max2
nums[3] > 1
2 > 1
true
- max3 = max2
= 1
- max2 = nums[i]
= nums[3]
= 2
i++
i = 4
Step 6: loop for int i = 0; i < nums.size()
i < nums.size()
4 < 7
true
if nums[i] < min1
nums[4] < -6
3 < -6
false
else if nums[i] < min2
3 < 1
false
if nums[i] > max1
nums[4] > 5
3 > 5
false
else if nums[i] > max2
nums[4] > 2
3 > 2
true
- max3 = max2
= 2
- max2 = nums[i]
= nums[4]
= 3
i++
i = 5
Step 7: loop for int i = 0; i < nums.size()
i < nums.size()
5 < 7
true
if nums[i] < min1
nums[5] < -6
-4 < -6
false
else if nums[i] < min2
-4 < 1
true
- min2 = nums[i]
= -4
if nums[i] > max1
nums[5] > 5
-4 > 5
false
else if nums[i] > max2
nums[5] > 3
-4 > 3
false
else if nums[i] > max3
nums[5] > 2
-4 > 2
false
i++
i = 6
Step 8: loop for int i = 0; i < nums.size()
i < nums.size()
6 < 7
true
if nums[i] < min1
nums[6] < -6
9 < -6
false
else if nums[i] < min2
9 < -4
false
if nums[i] > max1
nums[6] > 5
9 > 5
true
- max3 = max2
= 3
- max2 = max1
= 5
- max1 = nums[i]
= nums[6]
= 9
i++
i = 7
Step 9: loop for int i = 0; i < nums.size()
i < nums.size()
7 < 7
false
Step 10: return max(min1 * min2 * max1, max1 * max2 * max3)
max(-6 * -4 * 9, 9 * 5 * 3)
max(216, 135)
= 216
So we return the answer as 216.
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