# Problem

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.

# Idea:

Find the top 3 numbers t1, t2, t3, and bottom 2 numbers, b1, b2.

If all numbers are positives,  answer must be t1 * t2 * t3.

Since the number can go negative, the answer must be either t1*t2*t3 or b1 * b2 * t1, if b1 and b2 are both negatives.

ex. nums: [5, 1, -6, 3, -1]

t1, t2, t3: 5, 3, 1

b1, b2: -6, -1

t1 * t2 * t3 = 15

t1 * b1 * b2 = 30

# Solution 1: Manual Tracking

Time complexity: O(n)

Space complexity: O(1)

# Solution 2: Sorting

Time complexity: O(nlogn)

Space complexity: O(1)

# Solution 3: Two Heaps (Priority Queues)

Time complexity: O(nlog3)

Space complexity: O(2 + 3)

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