Given two non-negative integers N and K, return the number of binary strings of length N with at most K consecutive ones
Example:
Input: N = 4, K = 2
Output: 13
Explanation: [
"0000",
"0001",
"0010",
"0011",
"0100",
"0101",
"0110",
"1000",
"1001",
"1010",
"1011",
"1100",
"1101",
]
"0111", "1110" and "1111" are not valid because they have more than 2 consecutive ones
Notes:
K <= N <= 30
The core challenge of this problem is to generate all possible binary strings of length N and count those that have at most K consecutive ones. This problem is significant in various fields such as coding theory, data compression, and error detection. A common pitfall is to generate all possible binary strings and then filter out the invalid ones, which is not efficient.
To solve this problem efficiently, we can use dynamic programming. The idea is to build the solution incrementally and use previously computed results to avoid redundant calculations.
A naive solution would involve generating all possible binary strings of length N and then checking each string to see if it contains more than K consecutive ones. This approach is not optimal because it has a time complexity of O(2^N), which is infeasible for larger values of N.
We can use dynamic programming to optimize the solution. We define a 2D array dp where dp[i][j] represents the number of valid binary strings of length i with j consecutive ones at the end. The final result will be the sum of dp[N][j] for all j from 0 to K.
1. Initialize a 2D array dp with dimensions (N+1) x (K+1) to store the number of valid binary strings.
2. Set dp[0][0] = 1 because there is one valid string of length 0 (the empty string).
3. Iterate over the length of the string from 1 to N.
4. For each length, iterate over the number of consecutive ones from 0 to K.
5. Update the dp array based on the previous values.
6. Sum up the values in dp[N][j] for all j from 0 to K to get the final result.
// Function to count binary strings of length N with at most K consecutive ones
function countBinaryStrings(N, K) {
// Initialize a 2D array dp with dimensions (N+1) x (K+1)
const dp = Array.from({ length: N + 1 }, () => Array(K + 1).fill(0));
// Base case: one valid string of length 0 (the empty string)
dp[0][0] = 1;
// Iterate over the length of the string
for (let i = 1; i <= N; i++) {
// Iterate over the number of consecutive ones
for (let j = 0; j <= K; j++) {
// If the current string ends with a '0', it can be appended to any valid string of length i-1
dp[i][0] += dp[i - 1][j];
// If the current string ends with a '1', it can be appended to any valid string of length i-1 with j-1 consecutive ones
if (j > 0) {
dp[i][j] += dp[i - 1][j - 1];
}
}
}
// Sum up the values in dp[N][j] for all j from 0 to K to get the final result
let result = 0;
for (let j = 0; j <= K; j++) {
result += dp[N][j];
}
return result;
}
// Example usage
const N = 4;
const K = 2;
console.log(countBinaryStrings(N, K)); // Output: 13
The time complexity of this approach is O(N*K) because we iterate over the length of the string and the number of consecutive ones. The space complexity is also O(N*K) due to the 2D array used for dynamic programming.
1. N = 0: The only valid string is the empty string, so the result is 1.
2. K = 0: The only valid strings are those with no consecutive ones, so the result is 1.
3. N = K: All binary strings of length N are valid, so the result is 2^N.
To test the solution comprehensively, we should include a variety of test cases:
We can use testing frameworks like Jest or Mocha to automate the testing process.
1. Break down the problem into smaller subproblems and solve them incrementally.
2. Use dynamic programming to avoid redundant calculations and optimize the solution.
3. Practice solving similar problems to develop problem-solving skills and improve your understanding of algorithms.
In this blog post, we discussed how to solve the problem of counting binary strings of length N with at most K consecutive ones using dynamic programming. We covered the problem definition, approach, algorithm, code implementation, complexity analysis, edge cases, and testing. Understanding and solving such problems is important for developing strong problem-solving skills and improving your knowledge of algorithms.