# Posts published in “Math”

A decimal number is called deci-binary if each of its digits is either 0 or 1 without any leading zeros. For example, 101 and 1100 are deci-binary, while 112 and 3001 are not.

Given a string n that represents a positive decimal integer, return the minimum number of positive deci-binary numbers needed so that they sum up to n.

Example 1:

Input: n = "32"
Output: 3
Explanation: 10 + 11 + 11 = 32


Example 2:

Input: n = "82734"
Output: 8


Example 3:

Input: n = "27346209830709182346"
Output: 9


Constraints:

• 1 <= n.length <= 105
• n consists of only digits.
• n does not contain any leading zeros and represents a positive integer.

## Solution: Return the max digit

Proof: For a given string, we find the maximum number m, we create m binary strings.
for each one, check each digit, if it’s greater than 0, we mark 1 at that position and decrease the digit by 1.

e.g. 21534
max is 5, we need five binary strings.
1. 11111: 21534 -> 10423
2. 10111: 10423 -> 00312
3: 00111: 00312 -> 00201
4: 00101: 00201 -> 00100
5: 00100: 00100 -> 00000

We can ignore the leading zeros.

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

## C++

You are given an integer array nums sorted in non-decreasing order.

Build and return an integer array result with the same length as nums such that result[i] is equal to the summation of absolute differences between nums[i] and all the other elements in the array.

In other words, result[i] is equal to sum(|nums[i]-nums[j]|) where 0 <= j < nums.length and j != i (0-indexed).

Example 1:

Input: nums = [2,3,5]
Output: [4,3,5]
Explanation: Assuming the arrays are 0-indexed, then
result[0] = |2-2| + |2-3| + |2-5| = 0 + 1 + 3 = 4,
result[1] = |3-2| + |3-3| + |3-5| = 1 + 0 + 2 = 3,
result[2] = |5-2| + |5-3| + |5-5| = 3 + 2 + 0 = 5.


Example 2:

Input: nums = [1,4,6,8,10]
Output: [24,15,13,15,21]


Constraints:

• 2 <= nums.length <= 105
• 1 <= nums[i] <= nums[i + 1] <= 104

## Solution: Prefix Sum

Let s[i] denote sum(num[i] – num[j]) 0 <= j <= i
s[i] = s[i – 1] + (num[i] – num[i – 1]) * i
Let l[i] denote sum(nums[j] – nums[i]) i <= j < n
l[i] = l[i + 1] + (nums[i + 1] – num[i]) * (n – i – 1)
ans[i] = s[i] + l[i]

e.g. 1, 3, 7, 9
s[0] = 0
s[1] = 0 + (3 – 1) * 1 = 2
s[2] = 2 + (7 – 3) * 2 = 10
s[3] = 10 + (9 – 7) * 3 = 16
l[3] = 0
l[2] = 0 + (9 – 7) * 1 = 2
l[1] = 2 + (7 – 3) * 2 = 10
l[0] = 10 + (3 – 1) * 3 = 16

ans = [16, 12, 12, 16]

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

## 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)