Find all combinations of k-bit numbers with n bits set where 1 <= n <= k in sorted order
Given a number k, find all the possible combinations of k-bit numbers with n-bits set where 1 <= n <= k. The solution should print all numbers with one set bit first, followed by numbers with two bits set,.. up to the numbers whose all k-bits are set. If two numbers have the same number of set bits, then smaller number should come first.
Examples:
Input: K = 3 Output: 001 010 100 011 101 110 111 Input: K = 4 Output: 0001 0010 0100 1000 0011 0101 0110 1001 1010 1100 0111 1011 1101 1110 1111 Input: K = 5 Output: 00001 00010 00100 01000 10000 00011 00101 00110 01001 01010 01100 10001 10010 10100 11000 00111 01011 01101 01110 10011 10101 10110 11001 11010 11100 01111 10111 11011 11101 11110 11111
We need to find all the possible combinations of k-bit numbers with n set bits where 1 <= n <= k. If we carefully analyze, we see that problem can further be divided into sub-problems. We can find all combinations of length k with n ones by prefixing 0 to all combinations of length k-1 with n ones and 1 to all combinations of length k-1 with n-1 ones. We can use Dynamic Programming to save solutions of sub-problems.
Below is C++ implementation of above idea –
// C++ program find all the possible combinations of // k-bit numbers with n-bits set where 1 <= n <= k #include <iostream> #include <vector> using namespace std; // maximum allowed value of K #define K 16 // DP lookup table vector<string> DP[K][K]; // Function to find all combinations k-bit numbers with // n-bits set where 1 <= n <= k void findBitCombinations(int k) { string str = ""; // DP[k][0] will store all k-bit numbers // with 0 bits set (All bits are 0's) for (int len = 0; len <= k; len++) { DP[len][0].push_back(str); str = str + "0"; } // fill DP lookup table in bottom-up manner // DP[k][n] will store all k-bit numbers // with n-bits set for (int len = 1; len <= k; len++) { for (int n = 1; n <= len; n++) { // prefix 0 to all combinations of length len-1 // with n ones for (string str : DP[len - 1][n]) DP[len][n].push_back("0" + str); // prefix 1 to all combinations of length len-1 // with n-1 ones for (string str : DP[len - 1][n - 1]) DP[len][n].push_back("1" + str); } } // print all k-bit binary strings with // n-bit set for (int n = 1; n <= k; n++) { for (string str : DP[k][n]) cout << str << " "; cout << endl; } } // Driver code int main() { int k = 5; findBitCombinations(k); return 0; } |
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Output:
00000 00001 00010 00100 01000 10000 00011 00101 00110 01001 01010 01100 10001 10010 10100 11000 00111 01011 01101 01110 10011 10101 10110 11001 11010 11100 01111 10111 11011 11101 11110 11111
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