## 628. Maximum Product of Three Numbers

Given an integer array, find three numbers whose product is maximum and output the maximum product.

Example 1:

Input: [1,2,3]
Output: 6


Example 2:

Input: [1,2,3,4]
Output: 24


Note:

1. The length of the given array will be in range [3,104] and all elements are in the range [-1000, 1000].
2. Multiplication of any three numbers in the input won't exceed the range of 32-bit signed integer.

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## Solution

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#### Approach #1 Brute Force [Time Limit Exceeded]

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The simplest solution is to consider every triplet out of the given array and check their product and find out the maximum product out of them.

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Complexity Analysis

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Time complexity : . We need to consider every triplet from array of length .

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Space complexity : . Constant extra space is used.

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#### Approach #2 Using Sorting [Accepted]

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Algorithm

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Another solution could be to sort the given array(in ascending order) and find out the product of the last three numbers.

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But, we can note that this product will be maximum only if all the numbers in array are positive. But, in the given problem statement, negative elements could exist as well.

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Thus, it could also be possible that two negative numbers lying at the left extreme end could also contribute to lead to a larger product if the third number in the triplet being considered is the largest positive number in the array.

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Thus, either the product xx or xx will give the required result. Thus, we need to find the larger one from out of these values.

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Complexity Analysis

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Time complexity : . Sorting the array takes time.

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Space complexity : . Sorting takes O(logn) space.

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#### Approach #3 Single Scan [Accepted]

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Algorithm

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We need not necessarily sort the given array to find the maximum product. Instead, we can only find the required 2 smallest values( and ) and the three largest values() in the array, by iterating over the array only once.

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At the end, again we can find out the larger value out of xx and xx to find the required maximum product.

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Complexity Analysis

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Time complexity : . Only one iteration over the array of length is required.

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Space complexity : . Constant extra space is used.

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Analysis written by: @vinod23

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