# Posts tagged as “bit”

Your task is to form an integer array nums from an initial array of zeros arr that is the same size as nums.

Return the minimum number of function calls to make nums from arr.

The answer is guaranteed to fit in a 32-bit signed integer.

Example 1:

Input: nums = [1,5]
Output: 5
Explanation: Increment by 1 (second element): [0, 0] to get [0, 1] (1 operation).
Double all the elements: [0, 1] -> [0, 2] -> [0, 4] (2 operations).
Increment by 1 (both elements)  [0, 4] -> [1, 4] -> [1, 5] (2 operations).
Total of operations: 1 + 2 + 2 = 5.


Example 2:

Input: nums = [2,2]
Output: 3
Explanation: Increment by 1 (both elements) [0, 0] -> [0, 1] -> [1, 1] (2 operations).
Double all the elements: [1, 1] -> [2, 2] (1 operation).
Total of operations: 2 + 1 = 3.


Example 3:

Input: nums = [4,2,5]
Output: 6
Explanation: (initial)[0,0,0] -> [1,0,0] -> [1,0,1] -> [2,0,2] -> [2,1,2] -> [4,2,4] -> [4,2,5](nums).


Example 4:

Input: nums = [3,2,2,4]
Output: 7


Example 5:

Input: nums = [2,4,8,16]
Output: 8


Constraints:

• 1 <= nums.length <= 10^5
• 0 <= nums[i] <= 10^9

## Solution: count 1s

For 5 (101b), we can add 1s for 5 times which of cause isn’t the best way to generate 5, the optimal way is to [+1, *2, +1]. We have to add 1 for each 1 in the binary format. e.g. 11 (1011), we need 3x “+1” op, and 4 “*2” op. Fortunately, the “*2” can be shared/delayed, thus we just need to find the largest number.
e.g. [2,4,8,16]
[0, 0, 0, 0] -> [0, 0, 0, 1] -> [0, 0, 0, 2]
[0, 0, 0, 2] -> [0, 0, 1, 2] -> [0, 0, 2, 4]
[0, 0, 2, 4] -> [0, 1, 2, 4] -> [0, 2, 4, 8]
[0, 2, 4, 8] -> [1, 2, 4, 8] -> [2, 4, 8, 16]
ans = sum{count_1(arr_i)} + high_bit(max(arr_i))

Time complexity: O(n*log(max(arr_i))
Space complexity: O(1)

## Python3

There is a room with n bulbs, numbered from 0 to n-1, arranged in a row from left to right. Initially all the bulbs are turned off.

Your task is to obtain the configuration represented by target where target[i] is ‘1’ if the i-th bulb is turned on and is ‘0’ if it is turned off.

You have a switch to flip the state of the bulb, a flip operation is defined as follows:

• Choose any bulb (index i) of your current configuration.
• Flip each bulb from index i to n-1.

When any bulb is flipped it means that if it is 0 it changes to 1 and if it is 1 it changes to 0.

Return the minimum number of flips required to form target.

Example 1:

Input: target = "10111"
Output: 3
Explanation: Initial configuration "00000".
flip from the third bulb:  "00000" -> "00111"
flip from the first bulb:  "00111" -> "11000"
flip from the second bulb:  "11000" -> "10111"
We need at least 3 flip operations to form target.

Example 2:

Input: target = "101"
Output: 3
Explanation: "000" -> "111" -> "100" -> "101".


Example 3:

Input: target = "00000"
Output: 0


Example 4:

Input: target = "001011101"
Output: 5


Constraints:

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

## Solution: XOR

Flip from left to right, since flipping the a bulb won’t affect anything in the left.
We count how many times flipped so far, and that % 2 will be the state for all the bulb to the right.
If the current bulb’s state != target, we have to flip once.

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

## C++

Winston was given the above mysterious function func. He has an integer array arr and an integer target and he wants to find the values l and r that make the value |func(arr, l, r) - target| minimum possible.

Return the minimum possible value of |func(arr, l, r) - target|.

Notice that func should be called with the values l and r where 0 <= l, r < arr.length.

Example 1:

Input: arr = [9,12,3,7,15], target = 5
Output: 2
Explanation: Calling func with all the pairs of [l,r] = [[0,0],[1,1],[2,2],[3,3],[4,4],[0,1],[1,2],[2,3],[3,4],[0,2],[1,3],[2,4],[0,3],[1,4],[0,4]], Winston got the following results [9,12,3,7,15,8,0,3,7,0,0,3,0,0,0]. The value closest to 5 is 7 and 3, thus the minimum difference is 2.


Example 2:

Input: arr = [1000000,1000000,1000000], target = 1
Output: 999999
Explanation: Winston called the func with all possible values of [l,r] and he always got 1000000, thus the min difference is 999999.


Example 3:

Input: arr = [1,2,4,8,16], target = 0
Output: 0


Constraints:

• 1 <= arr.length <= 10^5
• 1 <= arr[i] <= 10^6
• 0 <= target <= 10^7

## Solution: Brute Force w/ Optimization

Try all possible [l, r] range with pruning.
1. for a given l, we extend r, since s & x <= s, if s becomes less than target, we can stop the inner loop.
2. Case 1, s = arr[l] & … & arr[n-1], s > target,
Let s’ = arr[l+1] & … & arr[n-1], s’ >= s,
if s > target, then s’ > target, we can stop outer loop as well.
Case 2, inner loop stops at r, s = arr[l] & … & arr[r], s <= target, we continue with l+1.

Time complexity: O(n)? on average, O(n^2) in worst case.
Space complexity: O(1)

## C++

You are given a tree with n nodes numbered from 0 to n-1 in the form of a parent array where parent[i] is the parent of node i. The root of the tree is node 0.

Implement the function getKthAncestor(int node, int k) to return the k-th ancestor of the given node. If there is no such ancestor, return -1.

The k-th ancestor of a tree node is the k-th node in the path from that node to the root.

Example:

Input:
["TreeAncestor","getKthAncestor","getKthAncestor","getKthAncestor"]
[[7,[-1,0,0,1,1,2,2]],[3,1],[5,2],[6,3]]

Output:


[null,1,0,-1]

Explanation: TreeAncestor treeAncestor = new TreeAncestor(7, [-1, 0, 0, 1, 1, 2, 2]); treeAncestor.getKthAncestor(3, 1); // returns 1 which is the parent of 3 treeAncestor.getKthAncestor(5, 2); // returns 0 which is the grandparent of 5 treeAncestor.getKthAncestor(6, 3); // returns -1 because there is no such ancestor

Constraints:

• 1 <= k <= n <= 5*10^4
• parent[0] == -1 indicating that 0 is the root node.
• 0 <= parent[i] < n for all 0 < i < n
• 0 <= node < n
• There will be at most 5*10^4 queries.

## Solution: LogN ancestors

1. Build the tree from parent array
2. Traverse the tree
3. For each node stores up to logn ancestros, 2^0-th, 2^1-th, 2^2-th, …

When k comes in, each node take the highest bit h out, and query its 2^h’s ancestors with k <- (k – 2^h). There will be at most logk recursive query. When it ends? k == 0, we found the ancestors which is the current node. Or node == 0 and k > 0, we already at root which doesn’t have any ancestors so return -1.

Time complexity:
Construction: O(nlogn)
Query: O(logk)

Space complexity:
O(nlogn)

DP method

## Solution 2: Binary Search

credit: Ziwu Zhou

Construction: O(n)

Traverse the tree in post order, for each node record its depth and id (visiting order).
For each depth, store all the nodes and their ids.

Query: O(logn)

Get the depth and id of the node, if k > d, return -1.
Use binary search to find the first node at depth[d – k] that has a id greater than the query’s one That node is the k-th ancestor of the node.

## C++

Given a number s in their binary representation. Return the number of steps to reduce it to 1 under the following rules:

• If the current number is even, you have to divide it by 2.
• If the current number is odd, you have to add 1 to it.

It’s guaranteed that you can always reach to one for all testcases.

Example 1:

Input: s = "1101"
Output: 6
Explanation: "1101" corressponds to number 13 in their decimal representation.
Step 1) 13 is odd, add 1 and obtain 14.
Step 2) 14 is even, divide by 2 and obtain 7.
Step 3) 7 is odd, add 1 and obtain 8.
Step 4) 8 is even, divide by 2 and obtain 4.
Step 5) 4 is even, divide by 2 and obtain 2.
Step 6) 2 is even, divide by 2 and obtain 1.


Example 2:

Input: s = "10"
Output: 1
Explanation: "10" corressponds to number 2 in their decimal representation.
Step 1) 2 is even, divide by 2 and obtain 1.


Example 3:

Input: s = "1"
Output: 0


Constraints:

• 1 <= s.length <= 500
• s consists of characters ‘0’ or ‘1’
• s[0] == '1'

## Solution: Simulation

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

## Python3

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