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Problem solution · C++

Maximum Partition Factor

Maximum Partition Factor: a C++ solution using disjoint set union. Learn the idea, check the complexity, and read the full code, with credit to Kamyu LeetCode Solutions.

Technique
Disjoint set union
Source
Kamyu LeetCode Solutions
Length
305 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

Disjoint set union

For Maximum Partition Factor, the implementation maintains connected components and merges them as relationships are processed.

  1. Give each element a component representative.
  2. Merge representatives when a connection is accepted.
  3. Answer connectivity or component queries from the compressed representatives.

Code notes

  • 305 lines of C++ from the credited upstream file maximum-partition-factor.cpp.
  • The implementation visibly relies on sequence storage.
  • 24 loop blocks detected.

Complexity

Account for every find and union operation; with path compression and ranked merging, the amortized cost is nearly constant per operation.

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 codeMaximum Partition Factor · C++C++
Use this to learn the idea, then write your own version.
// Time:  O(n^2 * logn)// Space: O(n^2) // greedy, sort, union find with parityclass Solution {public:    int maxPartitionFactor(vector<vector<int>>& points) {        const auto& dist = [&](auto u, auto v) {            return abs(points[u][0] - points[v][0]) + abs(points[u][1] - points[v][1]);        };         vector<tuple<int, int, int>> sorted_dists;        for (int u = 0; u < size(points); ++u) {            for (int v = u + 1; v < size(points); ++v) {                sorted_dists.emplace_back(dist(u, v), u, v);            }        }        sort(begin(sorted_dists), end(sorted_dists));        UnionFind uf(size(points));        for (const auto& [d, u, v] : sorted_dists) {            if (!uf.union_set(u, v)) {                return d;            }        }        return 0;    } private:    class UnionFind {    public:        UnionFind(int n)            : set_(n)            , rank_(n)            , parity_(n) {            iota(set_.begin(), set_.end(), 0);        }         int find_set(int x) {            vector<int> stk;            while (set_[x] != x) {  // path compression                stk.emplace_back(x);                x = set_[x];            }            while (!empty(stk)) {                const int y = stk.back(); stk.pop_back();                parity_[y] ^= parity_[set_[y]];  // added                set_[y] = x;            }            return x;        }         bool union_set(int x, int y) {            int ox = x, oy = y;            x = find_set(x), y = find_set(y);            if (x == y) {                return parity_[ox] != parity_[oy]; // modified            }            if (rank_[x] > rank_[y]) {                swap(x, y);            }            if (rank_[x] == rank_[y]) {                ++rank_[y];            }            set_[x] = y;  // Union by rank.            parity_[x] = parity_[ox] ^ parity_[oy] ^ 1;            return true;        }     private:        vector<int> set_;        vector<int> rank_;        vector<int> parity_;  // added    };}; // Time:  O(n^2 * logn)// Space: O(n^2)// greedy, sort, union findclass Solution2 {public:    int maxPartitionFactor(vector<vector<int>>& points) {        const auto& dist = [&](auto u, auto v) {            return abs(points[u][0] - points[v][0]) + abs(points[u][1] - points[v][1]);        };         vector<tuple<int, int, int>> sorted_dists;        for (int u = 0; u < size(points); ++u) {            for (int v = u + 1; v < size(points); ++v) {                sorted_dists.emplace_back(dist(u, v), u, v);            }        }        sort(begin(sorted_dists), end(sorted_dists));        vector<int> lookup(size(points), -1);        UnionFind uf(size(points));        for (const auto& [d, u, v] : sorted_dists) {            if (uf.find_set(u) == uf.find_set(v)) {                return d;            }            if (lookup[u] != -1) {                uf.union_set(lookup[u], v);            } else {                lookup[u] = v;            }            if (lookup[v] != -1) {                uf.union_set(lookup[v], u);            } else {                lookup[v] = u;            }        }        return 0;    } private:    class UnionFind {    public:        UnionFind(int n)            : set_(n)            , rank_(n) {            iota(set_.begin(), set_.end(), 0);        }         int find_set(int x) {            vector<int> stk;            while (set_[x] != x) {  // path compression                stk.emplace_back(x);                x = set_[x];            }            while (!empty(stk)) {                const int y = stk.back(); stk.pop_back();                set_[y] = x;            }            return x;        }         bool union_set(int x, int y) {            x = find_set(x), y = find_set(y);            if (x == y) {                return false;            }            if (rank_[x] > rank_[y]) {                swap(x, y);            }            if (rank_[x] == rank_[y]) {                ++rank_[y];            }            set_[x] = y;  // Union by rank.            return true;        }     private:        vector<int> set_;        vector<int> rank_;    };}; // Time:  O(n^2 * logn)// Space: O(n^2)// binary search, bfs, coordinate compressionclass Solution3 {public:    int maxPartitionFactor(vector<vector<int>>& points) {        static const int INF = numeric_limits<int>::max();         const auto& binary_search_right = [](auto left, auto right, const auto& check) {            while (left <= right) {                const auto mid = left + (right - left) / 2;                if (!check(mid)) {                    right = mid - 1;                } else {                    left = mid + 1;                }            }            return right;        };         const auto& dist = [&](auto u, auto v) {            return abs(points[u][0] - points[v][0]) + abs(points[u][1] - points[v][1]);        };         const auto& is_bipartite = [&](auto d) {            vector<int> lookup(size(points), -1);            const auto& bfs = [&](auto u) {                if (lookup[u] != -1) {                    return true;                }                lookup[u] = 0;                vector<int> q = {u};                while (!empty(q)) {                    vector<int> new_q;                    for (const auto& u : q) {                        for (int v = 0; v < size(points); ++v) {                            if (!(v != u && dist(v, u) < d)) {                                continue;                            }                            if (lookup[v] != -1) {                                if (lookup[v] != lookup[u] ^ 1) {                                    return false;                                }                                continue;                            }                            lookup[v] = lookup[u] ^ 1;                            new_q.emplace_back(v);                        }                    }                    q = move(new_q);                }                return true;            };             for (int u = 0; u < size(points); ++u) {                if (!bfs(u)) {                    return false;                }            }            return true;        };         vector<int> sorted_dists;        for (int u = 0; u < size(points); ++u) {            for (int v = u + 1; v < size(points); ++v) {                sorted_dists.emplace_back(dist(u, v));            }        }        sorted_dists.emplace_back(INF);        sort(begin(sorted_dists), end(sorted_dists));        auto it = unique(begin(sorted_dists), end(sorted_dists));        sorted_dists.erase(it, end(sorted_dists));        int left = 0, right = size(sorted_dists) - 1;        const auto& result = binary_search_right(left, right, [&](auto i) { return is_bipartite(sorted_dists[i]); });        return sorted_dists[result] != INF ? sorted_dists[result] : 0;    }}; // Time:  O(n^2 * logr)// Space: O(n)// binary search, bfsclass Solution4 {public:    int maxPartitionFactor(vector<vector<int>>& points) {        const auto& binary_search_right = [](auto left, auto right, const auto& check) {            while (left <= right) {                const auto mid = left + (right - left) / 2;                if (!check(mid)) {                    right = mid - 1;                } else {                    left = mid + 1;                }            }            return right;        };         const auto& dist = [&](auto u, auto v) {            return abs(points[u][0] - points[v][0]) + abs(points[u][1] - points[v][1]);        };         const auto& is_bipartite = [&](auto d) {            vector<int> lookup(size(points), -1);            const auto& bfs = [&](auto u) {                if (lookup[u] != -1) {                    return true;                }                lookup[u] = 0;                vector<int> q = {u};                while (!empty(q)) {                    vector<int> new_q;                    for (const auto& u : q) {                        for (int v = 0; v < size(points); ++v) {                            if (!(v != u && dist(v, u) < d)) {                                continue;                            }                            if (lookup[v] != -1) {                                if (lookup[v] != lookup[u] ^ 1) {                                    return false;                                }                                continue;                            }                            lookup[v] = lookup[u] ^ 1;                            new_q.emplace_back(v);                        }                    }                    q = move(new_q);                }                return true;            };             for (int u = 0; u < size(points); ++u) {                if (!bfs(u)) {                    return false;                }            }            return true;        };         int mx = 0;        for (int u = 0; u < size(points); ++u) {            for (int v = u + 1; v < size(points); ++v) {                mx = max(mx, dist(u, v));            }        }        int left = 0, right = mx + 1;        const auto& result = binary_search_right(left, right, is_bipartite);        return result != mx + 1 ? result : 0;    }};

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