1600 - 1699
1680 - Concatenation of Consecutive Binary Numbers (Medium)
Problem Link
https://leetcode.com/problems/concatenation-of-consecutive-binary-numbers/
Problem Statement
Given an integer n, return the decimal value of the binary string formed by concatenating the binary representations of1ton*in order, modulo*1e9 + 7.
Example 1:
Input: n = 1
Output: 1
Explanation: "1" in binary corresponds to the decimal value 1.Example 2:
Input: n = 3
Output: 27
Explanation: In binary, 1, 2, and 3 corresponds to "1", "10", and "11".
After concatenating them, we have "11011", which corresponds to the decimal value 27.Example 3:
Input: n = 12
Output: 505379714
Explanation: The concatenation results in "1101110010111011110001001101010111100".
The decimal value of that is 118505380540.
After modulo 1e9 + 7, the result is 505379714.Constraints:
1 <= n <= 10^5
Approach 1: Bit Manipulation
// Time Complexity: O(N)
// Space Complexity: O(1)
class Solution {
public:
// the idea is to use bit manipulation to set the current number based on the previous number
// for example,
// n = 1, ans = 0b1
// n = 2 (10), we need to shift 2 bits of the previous ans to the left and add `n`
// i.e. 1 -> 100 (shift 2 bits to the left) -> 110 (set `10`). ans = 0b110
// n = 3 (11), we need to shift 2 bits of the previous ans to the left and add `n`
// i.e 110 -> 11000 (shift 2 bits to the left) -> 11011 (set `11`). ans = 0b11011
// n = 4 (100), we need to shift 3 bits of the previous ans to the left and add `n`
// i.e. 11011 -> 11011000 (shift 3 bits to the left) -> 11011100 (set `100). ans = 0b11011100
// so now we can see a pattern here
// we need to shift `l` bits of the previous ans to the left and add the current `i`
// how to know `l`? it is not difficult to see `x` only increases when we meet power of 2
int concatenatedBinary(int n) {
// `l` is the bit length to be shifted
int M = 1e9 + 7, l = 0;
// use long here as it potentially could overflow for int
long ans = 0;
for (int i = 1; i <= n; i++) {
// i & (i - 1) means removing the rightmost set bit
// e.g. 100100 -> 100000
// 000001 -> 000000
// 000000 -> 000000
// after removal, if it is 0, then it means it is power of 2
// as all power of 2 only contains 1 set bit
// if it is power of 2, we increase the bit length `l`
if ((i & (i - 1)) == 0) l += 1;
// (ans << l) means shifting the orginal answer `l` bits to th left
// (x | i) means using OR operation to set the bit
// e.g. 0001 << 3 = 0001000
// e.g. 0001000 | 0001111 = 0001111
ans = ((ans << l) | i) % M;
}
return ans;
}
};