1023. Binary String With Substrings Representing 1 To N

Given a binary string S (a string consisting only of ‘0’ and ‘1’s) and a positive integer N, return true if and only if for every integer X from 1 to N, the binary representation of X is a substring of S.


Example 1:


S = "0110", N = 3



Example 2:


S = "0110", N = 4





  1. 1 <= S.length <= 1000
  2. 1 <= N <= 10^9


I was thinking there could be an amazing algo to deal with it.
Well, the solution is quite straight forward, just follow your mind. I didn’t even give it a try during the contest.
It’s kind of unexpected for a median level question.
For the time complexity, here’s a great analysis from lee215:
Time Complexity
  1. Prove I, check number of substring

Pick two indices, there are at most S^2 substrings,
so S can contains at most S^2 integers
Time complexity upper bound O(S^2)

  1. Prove II, Check the continuous digits
    Meanwhile I know the interviewer and my reader won’t be satisfied,
    as they want no more “cheat”.

Here I have a brief demonstration to give the time complexity an acceptable upper bound.

Have a look at the number 1001 ~ 2000 and their values in binary.

1001 0b1111101001
1002 0b1111101010
1003 0b1111101011

1997 0b11111001101
1998 0b11111001110
1999 0b11111001111
2000 0b11111010000

The number 1001 ~ 2000 have 1000 different continuous 10 digits.
The string of length S has at most S - 9 different continuous 10 digits.
So S <= 1000N <= 2000.

So S * 2 is a upper bound for N.
If N > S * 2, we can return false directly.

It’s the same to prove with the numbers 512 ~ 1511, or even smaller range.

Time complexity upper bound O(S)

And finally, the code:
class Solution {

     * @param String $S
     * @param Integer $N
     * @return Boolean
    function queryString($S, $N) {
        while($N > 1){
            $binary = decbin($N);
            if(strpos($S, $binary) === false){
                return false;
            $N --;
        return true;


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