# Posts tagged as “math”

You have an inventory of different colored balls, and there is a customer that wants orders balls of any color.

The customer weirdly values the colored balls. Each colored ball’s value is the number of balls of that color you currently have in your inventory. For example, if you own 6 yellow balls, the customer would pay 6 for the first yellow ball. After the transaction, there are only 5 yellow balls left, so the next yellow ball is then valued at 5 (i.e., the value of the balls decreases as you sell more to the customer).

You are given an integer array, inventory, where inventory[i] represents the number of balls of the ith color that you initially own. You are also given an integer orders, which represents the total number of balls that the customer wants. You can sell the balls in any order.

Return the maximum total value that you can attain after selling orders colored balls. As the answer may be too large, return it modulo 109 + 7.

Example 1:

Input: inventory = [2,5], orders = 4
Output: 14
Explanation: Sell the 1st color 1 time (2) and the 2nd color 3 times (5 + 4 + 3).
The maximum total value is 2 + 5 + 4 + 3 = 14.


Example 2:

Input: inventory = [3,5], orders = 6
Output: 19
Explanation: Sell the 1st color 2 times (3 + 2) and the 2nd color 4 times (5 + 4 + 3 + 2).
The maximum total value is 3 + 2 + 5 + 4 + 3 + 2 = 19.


Example 3:

Input: inventory = [2,8,4,10,6], orders = 20
Output: 110


Example 4:

Input: inventory = [1000000000], orders = 1000000000
Output: 21
Explanation: Sell the 1st color 1000000000 times for a total value of 500000000500000000. 500000000500000000 modulo 109 + 7 = 21.


Constraints:

• 1 <= inventory.length <= 105
• 1 <= inventory[i] <= 109
• 1 <= orders <= min(sum(inventory[i]), 109)

## Solution: Greedy

1. Sort the colors by # of balls in descending order.
e.g. 3 7 5 1 => 7 5 3 1
2. Sell the color with largest number of balls until it has the same number of balls of next color
1. 7 5 3 1 => 6 5 3 1 => 5 5 3 1 # value = 7 + 6 = 13
2. 5 5 3 1 => 4 4 3 1 => 3 3 3 1 # value = 13 + (5 + 4) * 2 = 31
3. 3 3 3 1 => 2 2 2 1 => 1 1 1 1 # value = 31 + (3 + 2) * 3 = 46
4. 1 1 1 1 => 0 0 0 0 # value = 46 + 1 * 4 = 50
3. Need to handle the case if orders < total balls…

Time complexity: O(nlogn)
Space complexity: O(1)

## C++

Given an integer n, you must transform it into 0 using the following operations any number of times:

• Change the rightmost (0th) bit in the binary representation of n.
• Change the ith bit in the binary representation of n if the (i-1)th bit is set to 1 and the (i-2)th through 0th bits are set to 0.

Return the minimum number of operations to transform n into 0.

Example 1:

Input: n = 0
Output: 0


Example 2:

Input: n = 3
Output: 2
Explanation: The binary representation of 3 is "11".
"11" -> "01" with the 2nd operation since the 0th bit is 1.
"01" -> "00" with the 1st operation.


Example 3:

Input: n = 6
Output: 4
Explanation: The binary representation of 6 is "110".
"110" -> "010" with the 2nd operation since the 1st bit is 1 and 0th through 0th bits are 0.
"010" -> "011" with the 1st operation.
"011" -> "001" with the 2nd operation since the 0th bit is 1.
"001" -> "000" with the 1st operation.


Example 4:

Input: n = 9
Output: 14


Example 5:

Input: n = 333
Output: 393


Constraints:

• 0 <= n <= 109

## Solution 1: Graycode

Time complexity: O(logn)
Space complexity: O(1)

Ans is the order of n in graycode.

## C++

Given an array of positive integers arr, calculate the sum of all possible odd-length subarrays.

A subarray is a contiguous subsequence of the array.

Return the sum of all odd-length subarrays of arr.

Example 1:

Input: arr = [1,4,2,5,3]
Output: 58
Explanation: The odd-length subarrays of arr and their sums are:
[1] = 1
[4] = 4
[2] = 2
[5] = 5
[3] = 3
[1,4,2] = 7
[4,2,5] = 11
[2,5,3] = 10
[1,4,2,5,3] = 15
If we add all these together we get 1 + 4 + 2 + 5 + 3 + 7 + 11 + 10 + 15 = 58

Example 2:

Input: arr = [1,2]
Output: 3
Explanation: There are only 2 subarrays of odd length, [1] and [2]. Their sum is 3.

Example 3:

Input: arr = [10,11,12]
Output: 66


Constraints:

• 1 <= arr.length <= 100
• 1 <= arr[i] <= 1000

## Solution 0: Brute Force

Enumerate all odd length subarrys: O(n^2), each take O(n) to compute the sum.

Total time complexity: O(n^3)
Space complexity: O(1)

## Solution 1: Running Prefix Sum

Reduce the time complexity to O(n^2)

## Solution 2: Math

Count how many times arr[i] can be in of an odd length subarray
we chose the start, which can be 0, 1, 2, … i, i + 1 choices
we chose the end, which can be i, i + 1, … n – 1, n – i choices
Among those 1/2 are odd length.
So there will be upper((i + 1) * (n – i) / 2) odd length subarrays contain arr[i]

ans = sum(((i + 1) * (n – i) + 1) / 2 * arr[i] for in range(n))

Time complexity: O(n)
Space complexity: O(1)

## C++

Given a binary string s (a string consisting only of ‘0’s and ‘1’s), we can split s into 3 non-empty strings s1, s2, s3 (s1+ s2+ s3 = s).

Return the number of ways s can be split such that the number of characters ‘1’ is the same in s1, s2, and s3.

Since the answer may be too large, return it modulo 10^9 + 7.

Example 1:

Input: s = "10101"
Output: 4
Explanation: There are four ways to split s in 3 parts where each part contain the same number of letters '1'.
"1|010|1"
"1|01|01"
"10|10|1"
"10|1|01"


Example 2:

Input: s = "1001"
Output: 0


Example 3:

Input: s = "0000"
Output: 3
Explanation: There are three ways to split s in 3 parts.
"0|0|00"
"0|00|0"
"00|0|0"


Example 4:

Input: s = "100100010100110"
Output: 12


Constraints:

• s[i] == '0' or s[i] == '1'
• 3 <= s.length <= 10^5

## Solution: Counting

Count how many ones in the binary string as T, if not a factor of 3, then there is no answer.

Count how many positions that have prefix sum of T/3 as l, and how many positions that have prefix sum of T/3*2 as r.

Ans = l * r

But we need to special handle the all zero cases, which equals to C(n-2, 2) = (n – 1) * (n – 2) / 2

Time complexity: O(n)
Space complexity: O(1)

## Python3

One pass: Space complexity: O(n)

## Python3

Given an array nums that represents a permutation of integers from 1 to n. We are going to construct a binary search tree (BST) by inserting the elements of nums in order into an initially empty BST. Find the number of different ways to reorder nums so that the constructed BST is identical to that formed from the original array nums.

For example, given nums = [2,1,3], we will have 2 as the root, 1 as a left child, and 3 as a right child. The array [2,3,1] also yields the same BST but [3,2,1] yields a different BST.

Return the number of ways to reorder nums such that the BST formed is identical to the original BST formed from nums.

Since the answer may be very large, return it modulo 10^9 + 7.

Example 1:

Input: nums = [2,1,3]
Output: 1
Explanation: We can reorder nums to be [2,3,1] which will yield the same BST. There are no other ways to reorder nums which will yield the same BST.


Example 2:

Input: nums = [3,4,5,1,2]
Output: 5
Explanation: The following 5 arrays will yield the same BST:
[3,1,2,4,5]
[3,1,4,2,5]
[3,1,4,5,2]
[3,4,1,2,5]
[3,4,1,5,2]


Example 3:

Input: nums = [1,2,3]
Output: 0
Explanation: There are no other orderings of nums that will yield the same BST.


Example 4:

Input: nums = [3,1,2,5,4,6]
Output: 19


Example 5:

Input: nums = [9,4,2,1,3,6,5,7,8,14,11,10,12,13,16,15,17,18]
Output: 216212978
Explanation: The number of ways to reorder nums to get the same BST is 3216212999. Taking this number modulo 10^9 + 7 gives 216212978.


Constraints:

• 1 <= nums.length <= 1000
• 1 <= nums[i] <= nums.length
• All integers in nums are distinct.

## Solution: Recursion + Combinatorics

For a given root (first element of the array), we can split the array into left children (nums[i] < nums[0]) and right children (nums[i] > nums[0]). Assuming there are l nodes for the left and r nodes for the right. We have C(l + r, l) different ways to insert l elements into a (l + r) sized array. Within node l / r nodes, we have ways(left) / ways(right) different ways to re-arrange those nodes. So the total # of ways is:
C(l + r, l) * ways(l) * ways(r)
Don’t forget to minus one for the final answer.

Time complexity: O(n^2)
Space complexity: O(n^2)

## python3

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