Question

1632. Rank Transform of a Matrix

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Description

Given an m x n matrix, return _a new matrix _answer_ where _answer[row][col]_ is the __rank of _matrix[row][col].

The rank is an integer that represents how large an element is compared to other elements. It is calculated using the following rules:

• The rank is an integer starting from 1.
• If two elements p and q are in the same row or column, then:
  • If p < q then rank(p) < rank(q)
  • If p == q then rank(p) == rank(q)
  • If p > q then rank(p) > rank(q)
• The rank should be as small as possible.

The test cases are generated so that answer is unique under the given rules.

 

Example 1:

Input: matrix = [[1,2],[3,4]]
Output: [[1,2],[2,3]]
Explanation:
The rank of matrix[0][0] is 1 because it is the smallest integer in its row and column.
The rank of matrix[0][1] is 2 because matrix[0][1] > matrix[0][0] and matrix[0][0] is rank 1.
The rank of matrix[1][0] is 2 because matrix[1][0] > matrix[0][0] and matrix[0][0] is rank 1.
The rank of matrix[1][1] is 3 because matrix[1][1] > matrix[0][1], matrix[1][1] > matrix[1][0], and both matrix[0][1] and matrix[1][0] are rank 2.

Example 2:

Input: matrix = [[7,7],[7,7]]
Output: [[1,1],[1,1]]

Example 3:

Input: matrix = [[20,-21,14],[-19,4,19],[22,-47,24],[-19,4,19]]
Output: [[4,2,3],[1,3,4],[5,1,6],[1,3,4]]

 

Constraints:

• m == matrix.length
• n == matrix[i].length
• 1 <= m, n <= 500
• -109 <= matrix[row][col] <= 109

Solutions

Solution 1

class UnionFind:
  def __init__(self, n):
    self.p = list(range(n))
    self.size = [1] * n

  def find(self, x):
    if self.p[x] != x:
      self.p[x] = self.find(self.p[x])
    return self.p[x]

  def union(self, a, b):
    pa, pb = self.find(a), self.find(b)
    if pa != pb:
      if self.size[pa] > self.size[pb]:
        self.p[pb] = pa
        self.size[pa] += self.size[pb]
      else:
        self.p[pa] = pb
        self.size[pb] += self.size[pa]

  def reset(self, x):
    self.p[x] = x
    self.size[x] = 1


class Solution:
  def matrixRankTransform(self, matrix: List[List[int]]) -> List[List[int]]:
    m, n = len(matrix), len(matrix[0])
    d = defaultdict(list)
    for i, row in enumerate(matrix):
      for j, v in enumerate(row):
        d[v].append((i, j))
    row_max = [0] * m
    col_max = [0] * n
    ans = [[0] * n for _ in range(m)]
    uf = UnionFind(m + n)
    for v in sorted(d):
      rank = defaultdict(int)
      for i, j in d[v]:
        uf.union(i, j + m)
      for i, j in d[v]:
        rank[uf.find(i)] = max(rank[uf.find(i)], row_max[i], col_max[j])
      for i, j in d[v]:
        ans[i][j] = row_max[i] = col_max[j] = 1 + rank[uf.find(i)]
      for i, j in d[v]:
        uf.reset(i)
        uf.reset(j + m)
    return ans

class UnionFind {
  private int[] p;
  private int[] size;

  public UnionFind(int n) {
    p = new int[n];
    size = new int[n];
    for (int i = 0; i < n; ++i) {
      p[i] = i;
      size[i] = 1;
    }
  }

  public int find(int x) {
    if (p[x] != x) {
      p[x] = find(p[x]);
    }
    return p[x];
  }

  public void union(int a, int b) {
    int pa = find(a), pb = find(b);
    if (pa != pb) {
      if (size[pa] > size[pb]) {
        p[pb] = pa;
        size[pa] += size[pb];
      } else {
        p[pa] = pb;
        size[pb] += size[pa];
      }
    }
  }

  public void reset(int x) {
    p[x] = x;
    size[x] = 1;
  }
}

class Solution {
  public int[][] matrixRankTransform(int[][] matrix) {
    int m = matrix.length, n = matrix[0].length;
    TreeMap<Integer, List<int[]>> d = new TreeMap<>();
    for (int i = 0; i < m; ++i) {
      for (int j = 0; j < n; ++j) {
        d.computeIfAbsent(matrix[i][j], k -> new ArrayList<>()).add(new int[] {i, j});
      }
    }
    int[] rowMax = new int[m];
    int[] colMax = new int[n];
    int[][] ans = new int[m][n];
    UnionFind uf = new UnionFind(m + n);
    int[] rank = new int[m + n];
    for (var ps : d.values()) {
      for (var p : ps) {
        uf.union(p[0], p[1] + m);
      }
      for (var p : ps) {
        int i = p[0], j = p[1];
        rank[uf.find(i)] = Math.max(rank[uf.find(i)], Math.max(rowMax[i], colMax[j]));
      }
      for (var p : ps) {
        int i = p[0], j = p[1];
        ans[i][j] = 1 + rank[uf.find(i)];
        rowMax[i] = ans[i][j];
        colMax[j] = ans[i][j];
      }
      for (var p : ps) {
        uf.reset(p[0]);
        uf.reset(p[1] + m);
      }
    }
    return ans;
  }
}

class UnionFind {
public:
  UnionFind(int n) {
    p = vector<int>(n);
    size = vector<int>(n, 1);
    iota(p.begin(), p.end(), 0);
  }

  void unite(int a, int b) {
    int pa = find(a), pb = find(b);
    if (pa != pb) {
      if (size[pa] > size[pb]) {
        p[pb] = pa;
        size[pa] += size[pb];
      } else {
        p[pa] = pb;
        size[pb] += size[pa];
      }
    }
  }

  int find(int x) {
    if (p[x] != x) {
      p[x] = find(p[x]);
    }
    return p[x];
  }

  void reset(int x) {
    p[x] = x;
    size[x] = 1;
  }

private:
  vector<int> p, size;
};

class Solution {
public:
  vector<vector<int>> matrixRankTransform(vector<vector<int>>& matrix) {
    int m = matrix.size(), n = matrix[0].size();
    map<int, vector<pair<int, int>>> d;
    for (int i = 0; i < m; ++i) {
      for (int j = 0; j < n; ++j) {
        d[matrix[i][j]].push_back({i, j});
      }
    }
    vector<int> rowMax(m);
    vector<int> colMax(n);
    vector<vector<int>> ans(m, vector<int>(n));
    UnionFind uf(m + n);
    vector<int> rank(m + n);
    for (auto& [_, ps] : d) {
      for (auto& [i, j] : ps) {
        uf.unite(i, j + m);
      }
      for (auto& [i, j] : ps) {
        rank[uf.find(i)] = max({rank[uf.find(i)], rowMax[i], colMax[j]});
      }
      for (auto& [i, j] : ps) {
        ans[i][j] = rowMax[i] = colMax[j] = 1 + rank[uf.find(i)];
      }
      for (auto& [i, j] : ps) {
        uf.reset(i);
        uf.reset(j + m);
      }
    }
    return ans;
  }
};

type unionFind struct {
  p, size []int
}

func newUnionFind(n int) *unionFind {
  p := make([]int, n)
  size := make([]int, n)
  for i := range p {
    p[i] = i
    size[i] = 1
  }
  return &unionFind{p, size}
}

func (uf *unionFind) find(x int) int {
  if uf.p[x] != x {
    uf.p[x] = uf.find(uf.p[x])
  }
  return uf.p[x]
}

func (uf *unionFind) union(a, b int) {
  pa, pb := uf.find(a), uf.find(b)
  if pa != pb {
    if uf.size[pa] > uf.size[pb] {
      uf.p[pb] = pa
      uf.size[pa] += uf.size[pb]
    } else {
      uf.p[pa] = pb
      uf.size[pb] += uf.size[pa]
    }
  }
}

func (uf *unionFind) reset(x int) {
  uf.p[x] = x
  uf.size[x] = 1
}

func matrixRankTransform(matrix [][]int) [][]int {
  m, n := len(matrix), len(matrix[0])
  type pair struct{ i, j int }
  d := map[int][]pair{}
  for i, row := range matrix {
    for j, v := range row {
      d[v] = append(d[v], pair{i, j})
    }
  }
  rowMax := make([]int, m)
  colMax := make([]int, n)
  ans := make([][]int, m)
  for i := range ans {
    ans[i] = make([]int, n)
  }
  vs := []int{}
  for v := range d {
    vs = append(vs, v)
  }
  sort.Ints(vs)
  uf := newUnionFind(m + n)
  rank := make([]int, m+n)
  for _, v := range vs {
    ps := d[v]
    for _, p := range ps {
      uf.union(p.i, p.j+m)
    }
    for _, p := range ps {
      i, j := p.i, p.j
      rank[uf.find(i)] = max(rank[uf.find(i)], max(rowMax[i], colMax[j]))
    }
    for _, p := range ps {
      i, j := p.i, p.j
      ans[i][j] = 1 + rank[uf.find(i)]
      rowMax[i], colMax[j] = ans[i][j], ans[i][j]
    }
    for _, p := range ps {
      uf.reset(p.i)
      uf.reset(p.j + m)
    }
  }
  return ans
}