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Problem solution · C++

Partition Array for Maximum Xor and and

Partition Array for Maximum Xor and and: a C++ solution using direct simulation. Learn the idea, check the complexity, and read the full code, with credit to Kamyu LeetCode Solutions.

Technique
Direct simulation
Source
Kamyu LeetCode Solutions
Length
172 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

Direct simulation

For Partition Array for Maximum Xor and and, the implementation follows the problem’s operations directly while maintaining only the state needed for the next decision.

  1. Translate each rule into one explicit state update.
  2. Maintain the invariant after every processed item.
  3. Return the accumulated state once all relevant input has been handled.

Code notes

  • 172 lines of C++ from the credited upstream file partition-array-for-maximum-xor-and-and.cpp.
  • The implementation visibly relies on sequence storage.
  • 17 loop blocks detected.

Complexity

Count the number and nesting of passes over the input, then include the maintained containers in the memory estimate.

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 codePartition Array for Maximum Xor and and · C++C++
Use this to learn the idea, then write your own version.
// Time:  O(nlogr * 2^n)// Space: O(2^n) // bitmasks, greedyclass Solution {public:    long long maximizeXorAndXor(vector<int>& nums) {        const auto& bit_length = [](int x) {            return (x ? std::__lg(x) : -1) + 1;        };         const int l = bit_length(ranges::max(nums));        const int n = size(nums);        const int full_mask = (1 << n) - 1;        vector<int> and_arr(1 << n);        vector<int> xor_arr(1 << n);        for (int mask = 1; mask < (1 << n); ++mask) {            const auto& lb = mask & -mask;            const auto& i = __builtin_ctz(lb);            and_arr[mask] = (mask ^ lb) ? (and_arr[mask ^ lb] & nums[i]) : nums[i];            xor_arr[mask] = xor_arr[mask ^ lb] ^ nums[i];        }        int64_t result = 0;        vector<int> base(l);  // to improve performance        for (int mask = 1; mask < (1 << n); ++mask) {            const auto& total_and = and_arr[mask];            const auto& total_xor = xor_arr[full_mask ^ mask];            base.assign(l, 0);            for (int remain = full_mask ^ mask; remain; remain &= remain - 1) {                const int i = __builtin_ctz(remain);                int x = nums[i] & ~total_xor;                for (int i = l - 1; i >= 0; --i) {                    if (!(x & (1 << i))) {                        continue;                    }                    if (base[i] == 0) {                        base[i] = x;                        break;                    }                    x ^= base[i];                }            }            int max_xor = 0;            for (int i = l - 1; i >= 0; --i) {                if ((max_xor ^ base[i]) > max_xor) {                    max_xor ^= base[i];                }            }            result = max(result, total_and + total_xor + static_cast<int64_t>(2) * max_xor);        }        return result;    }}; // Time:  O(nlogr * 2^n)// Space: O(2^n)// bitmasks, greedyclass Solution2 {public:    long long maximizeXorAndXor(vector<int>& nums) {        const auto& bit_length = [](int x) {            return (x ? std::__lg(x) : -1) + 1;        };         const int l = bit_length(ranges::max(nums));        const auto& max_xor_subset = [&](const vector<int>& nums) {  // Time: O(nlogr)            vector<int> base(l);            for (auto x : nums) {  // gaussian elimination over GF(2)                for (int i = l - 1; i >= 0; --i) {                    if (!(x & (1 << i))) {                        continue;                    }                    if (base[i] == 0) {                        base[i] = x;                        break;                    }                    x ^= base[i];                }            }            int max_xor = 0;            for (int i = l - 1; i >= 0; --i) {  // greedy                if ((max_xor ^ base[i]) > max_xor) {                    max_xor ^= base[i];                }            }            return max_xor;        };         const int n = size(nums);        const int full_mask = (1 << n) - 1;        vector<int> and_arr(1 << n);        vector<int> xor_arr(1 << n);        for (int mask = 1; mask < (1 << n); ++mask) {            const auto& lb = mask & -mask;            const auto& i = __builtin_ctz(lb);            and_arr[mask] = (mask ^ lb) ? (and_arr[mask ^ lb] & nums[i]) : nums[i];            xor_arr[mask] = xor_arr[mask ^ lb] ^ nums[i];        }        int64_t result = 0;        for (int mask = 1; mask < (1 << n); ++mask) {            const auto& total_and = and_arr[mask];            const auto& total_xor = xor_arr[full_mask ^ mask];            vector<int> needs;            for (int i = 0; i < n; ++i) {                if (!(mask & (1 << i))) {                    needs.emplace_back(nums[i] & ~total_xor);                }            }            result = max(result, total_and + total_xor + static_cast<int64_t>(2) * max_xor_subset(needs));        }        return result;    }}; // Time:  O(nlogr * 2^n)// Space: O(1)// bitmasks, greedyclass Solution3 {public:    long long maximizeXorAndXor(vector<int>& nums) {        const auto& bit_length = [](int x) {            return (x ? std::__lg(x) : -1) + 1;        };         const int l = bit_length(ranges::max(nums));        const auto& max_xor_subset = [&](const vector<int>& nums) {  // Time: O(nlogr)            vector<int> base(l);            for (auto x : nums) {  // gaussian elimination over GF(2)                for (int i = l - 1; i >= 0; --i) {                    if (!(x & (1 << i))) {                        continue;                    }                    if (base[i] == 0) {                        base[i] = x;                        break;                    }                    x ^= base[i];                }            }            int max_xor = 0;            for (int i = l - 1; i >= 0; --i) {  // greedy                if ((max_xor ^ base[i]) > max_xor) {                    max_xor ^= base[i];                }            }            return max_xor;        };         const int n = size(nums);        int64_t result = 0;        for (int mask = 1; mask < (1 << n); ++mask) {            int and_val = -1;            int xor_val = 0;            for (int i = 0; i < n; ++i) {                if (mask & (1 << i)) {                    and_val = (and_val != -1) ? (and_val & nums[i]) : nums[i];                } else {                    xor_val ^= nums[i];                }            }            vector<int> needs;            for (int i = 0; i < n; ++i) {                if (!(mask & (1 << i))) {                    needs.emplace_back(nums[i] & ~xor_val);                }            }            result = max(result, and_val + xor_val + static_cast<int64_t>(2) * max_xor_subset(needs));        }        return result;    }};

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