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Problem solution · C++

Count Ways to Choose Coprime Integers from Rows

Count Ways to Choose Coprime Integers from Rows: a C++ solution using dynamic programming. Learn the idea, check the complexity, and read the full code, with credit to Kamyu LeetCode Solutions.

Technique
Dynamic programming
Source
Kamyu LeetCode Solutions
Length
88 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Dynamic programming

For Count Ways to Choose Coprime Integers from Rows, the implementation records answers for smaller states and reuses them to build the requested result without repeating work.

  1. Define precisely what one DP state represents.
  2. Establish the base cases before transitions are evaluated.
  3. Process states in dependency order and combine only already-known values.

Code notes

  • 88 lines of C++ from the credited upstream file count-ways-to-choose-coprime-integers-from-rows.cpp.
  • The implementation visibly relies on sequence storage, hash lookup, cached states.
  • 12 loop blocks detected.

Complexity

Multiply the number of reachable states by the work performed for each transition, then include the stored state table in memory usage.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from Kamyu LeetCode Solutions by kamyu104 and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeCount Ways to Choose Coprime Integers from Rows · C++C++
Use this to learn the idea, then write your own version.
// Time:  O(n * (m + rlogr)), r = max(max(row) for row in mat)// Space: O(r) // dp, number theory, mobius function, principle of inclusion-exclusionclass Solution {public:    int countCoprime(vector<vector<int>>& mat) {        static const int MOD = 1e9 + 7;         const auto& linear_sieve_of_eratosthenes = [](int n) {  // Time: O(n), Space: O(n)            vector<int> spf(n + 1, -1);            vector<int> primes;            for (int i = 2; i <= n; ++i) {                if (spf[i] == -1) {                    spf[i] = i;                    primes.emplace_back(i);                }                for (const auto& p : primes) {                    if (i * p > n || p > spf[i]) {                        break;                    }                    spf[i * p] = p;                }            }            return spf;        };         // https://www.geeksforgeeks.org/program-for-mobius-function-set-2/        const auto& mobius = [](const auto& spf) {  // Time: O(n), Space: O(n)            vector<int> mu(size(spf));            for (int i = 1; i < size(mu); ++i) {                mu[i] = i == 1 ? 1 : (spf[i / spf[i]] == spf[i] ? 0 :  -mu[i / spf[i]]);            }            return mu;        };         int mx = 0;        for (const auto& row : mat) {            mx = max(mx, ranges::max(row));        }        const auto& mu = mobius(linear_sieve_of_eratosthenes(mx));        vector<int64_t> dp(mx + 1, 1);        for (const auto& row : mat) {            unordered_map<int, int> cnt;            for (const auto& x : row) {                ++cnt[x];            }            for (int i = 1; i <= mx; ++i) {                int total = 0;                for (int j = i; j <= mx; j += i) {                    total = (total + cnt[j]) % MOD;                }                dp[i] = (dp[i] * total) % MOD;            }                    }        int result = 0;        for (int i = 1; i <= mx; ++i) {            result = (result + (((dp[i] * mu[i]) % MOD + MOD) % MOD)) % MOD;        }        return result;    }};  // Time:  O(n * m * rlogr)// Space: O(r)// dp, number theoryclass Solution2 {public:    int countCoprime(vector<vector<int>>& mat) {        static const int MOD = 1e9 + 7;         unordered_map<int, int> dp;        dp[0] = 1;        for (const auto& row : mat) {            unordered_map<int, int> new_dp;            for (const auto& x : row) {                for (const auto& [g, c] : dp) {                    const auto& ng = gcd(g, x);                    new_dp[ng] = (new_dp[ng] + c) % MOD;                }            }            dp = move(new_dp);        }        return dp[1];    }};

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