Question

1102. 得分最高的路径

English Version

题目描述

给定一个 m x n 的整数矩阵 grid，返回从 (0,0) 开始到 (m - 1, n - 1) 在四个基本方向上移动的路径的最大 分数 。

一条路径的 分数 是该路径上的最小值。

• 例如，路径 8 → 4 → 5 → 9 的得分为 4 。

 

示例 1：

输入：grid = [[5,4,5],[1,2,6],[7,4,6]]
输出：4
解释：得分最高的路径用黄色突出显示。 

示例 2：

输入：grid = [[2,2,1,2,2,2],[1,2,2,2,1,2]]
输出：2

示例 3：

输入：grid = [[3,4,6,3,4],[0,2,1,1,7],[8,8,3,2,7],[3,2,4,9,8],[4,1,2,0,0],[4,6,5,4,3]]
输出：3

 

提示：

• m == grid.length
• n == grid[i].length
• 1 <= m, n <= 100
• 0 <= grid[i][j] <= 109

 

解法

方法一：排序 + 并查集

我们先将矩阵的每个元素构建一个三元组 $(v, i, j)$，其中 $v$ 表示元素值，而 $i$ 和 $j$ 分别表示元素在矩阵中的行和列。然后对这些三元组按照元素值从大到小进行排序，存放在列表 $q$ 中。

接下来，我们按顺序从 $q$ 中取出三元组，将其对应的元素值作为路径的分数，并且将该位置标记为已访问。然后我们检查该位置的上下左右四个相邻位置，如果某个相邻位置已经被访问过，那么我们就将该位置与当前位置进行合并。如果发现位置 $(0, 0)$ 和位置 $(m - 1, n - 1)$ 已经被合并，那么我们就可以直接返回当前路径的分数，即为答案。

时间复杂度 $O(m \times n \times (\log (m \times n) + \alpha(m \times n)))$，其中 $m$ 和 $n$ 分别为矩阵的行数和列数。

class Solution:
  def maximumMinimumPath(self, grid: List[List[int]]) -> int:
    def find(x: int) -> int:
      if p[x] != x:
        p[x] = find(p[x])
      return p[x]

    m, n = len(grid), len(grid[0])
    p = list(range(m * n))
    q = [(v, i, j) for i, row in enumerate(grid) for j, v in enumerate(row)]
    q.sort()
    ans = 0
    dirs = (-1, 0, 1, 0, -1)
    vis = set()
    while find(0) != find(m * n - 1):
      v, i, j = q.pop()
      ans = v
      vis.add((i, j))
      for a, b in pairwise(dirs):
        x, y = i + a, j + b
        if (x, y) in vis:
          p[find(i * n + j)] = find(x * n + y)
    return ans

class Solution {
  private int[] p;

  public int maximumMinimumPath(int[][] grid) {
    int m = grid.length, n = grid[0].length;
    p = new int[m * n];
    List<int[]> q = new ArrayList<>();
    for (int i = 0; i < m; ++i) {
      for (int j = 0; j < n; ++j) {
        q.add(new int[] {grid[i][j], i, j});
        p[i * n + j] = i * n + j;
      }
    }
    q.sort((a, b) -> b[0] - a[0]);
    boolean[][] vis = new boolean[m][n];
    int[] dirs = {-1, 0, 1, 0, -1};
    int ans = 0;
    for (int i = 0; find(0) != find(m * n - 1); ++i) {
      int[] t = q.get(i);
      vis[t[1]][t[2]] = true;
      ans = t[0];
      for (int k = 0; k < 4; ++k) {
        int x = t[1] + dirs[k], y = t[2] + dirs[k + 1];
        if (x >= 0 && x < m && y >= 0 && y < n && vis[x][y]) {
          p[find(x * n + y)] = find(t[1] * n + t[2]);
        }
      }
    }
    return ans;
  }

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

class Solution {
public:
  int maximumMinimumPath(vector<vector<int>>& grid) {
    int m = grid.size(), n = grid[0].size();
    vector<tuple<int, int, int>> q;
    vector<int> p(m * n);
    iota(p.begin(), p.end(), 0);
    for (int i = 0; i < m; ++i) {
      for (int j = 0; j < n; ++j) {
        q.emplace_back(grid[i][j], i, j);
      }
    }
    function<int(int)> find = [&](int x) {
      return p[x] == x ? x : p[x] = find(p[x]);
    };
    sort(q.begin(), q.end(), greater<tuple<int, int, int>>());
    int ans = 0;
    int dirs[5] = {-1, 0, 1, 0, -1};
    bool vis[m][n];
    memset(vis, false, sizeof(vis));
    for (auto& [v, i, j] : q) {
      vis[i][j] = true;
      ans = v;
      for (int k = 0; k < 4; ++k) {
        int x = i + dirs[k], y = j + dirs[k + 1];
        if (x >= 0 && x < m && y >= 0 && y < n && vis[x][y]) {
          p[find(x * n + y)] = find(i * n + j);
        }
      }
      if (find(0) == find(m * n - 1)) {
        break;
      }
    }
    return ans;
  }
};

func maximumMinimumPath(grid [][]int) (ans int) {
  m, n := len(grid), len(grid[0])
  p := make([]int, m*n)
  vis := make([][]bool, m)
  q := [][3]int{}
  for i, row := range grid {
    vis[i] = make([]bool, n)
    for j, v := range row {
      p[i*n+j] = i*n + j
      q = append(q, [3]int{v, i, j})
    }
  }
  sort.Slice(q, func(i, j int) bool { return q[i][0] > q[j][0] })
  var find func(int) int
  find = func(x int) int {
    if p[x] != x {
      p[x] = find(p[x])
    }
    return p[x]
  }
  dirs := [5]int{-1, 0, 1, 0, -1}
  for _, t := range q {
    v, i, j := t[0], t[1], t[2]
    ans = v
    vis[i][j] = true
    for k := 0; k < 4; k++ {
      x, y := i+dirs[k], j+dirs[k+1]
      if 0 <= x && x < m && 0 <= y && y < n && vis[x][y] {
        p[find(x*n+y)] = find(i*n + j)
      }
    }
    if find(0) == find(m*n-1) {
      break
    }
  }
  return
}

function maximumMinimumPath(grid: number[][]): number {
  const m = grid.length;
  const n = grid[0].length;
  const p: number[] = Array(m * n)
    .fill(0)
    .map((_, i) => i);
  const q: number[][] = [];
  for (let i = 0; i < m; ++i) {
    for (let j = 0; j < n; ++j) {
      q.push([grid[i][j], i, j]);
    }
  }
  q.sort((a, b) => b[0] - a[0]);
  const find = (x: number): number => {
    if (p[x] !== x) {
      p[x] = find(p[x]);
    }
    return p[x];
  };
  const dirs: number[] = [-1, 0, 1, 0, -1];
  const vis: boolean[][] = Array(m)
    .fill(0)
    .map(() => Array(n).fill(false));
  let ans = 0;
  for (let k = 0; find(0) !== find(m * n - 1); ++k) {
    const [t, i, j] = q[k];
    ans = t;
    vis[i][j] = true;
    for (let d = 0; d < 4; ++d) {
      const [x, y] = [i + dirs[d], j + dirs[d + 1]];
      if (x >= 0 && x < m && y >= 0 && y < n && vis[x][y]) {
        p[find(i * n + j)] = find(x * n + y);
      }
    }
  }
  return ans;
}

struct UnionFind {
  p: Vec<usize>,
  size: Vec<usize>,
}

impl UnionFind {
  fn new(n: usize) -> Self {
    let p: Vec<usize> = (0..n).collect();
    let size = vec![1; n];
    UnionFind { p, size }
  }

  fn find(&mut self, x: usize) -> usize {
    if self.p[x] != x {
      self.p[x] = self.find(self.p[x]);
    }
    self.p[x]
  }

  fn union(&mut self, a: usize, b: usize) {
    let pa = self.find(a);
    let pb = 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];
      }
    }
  }
}

impl Solution {
  pub fn maximum_minimum_path(grid: Vec<Vec<i32>>) -> i32 {
    let m = grid.len();
    let n = grid[0].len();
    let mut uf = UnionFind::new(m * n);
    let mut q: Vec<Vec<i32>> = Vec::new();

    for i in 0..m {
      for j in 0..n {
        q.push(vec![grid[i][j], i as i32, j as i32]);
      }
    }

    q.sort_by(|a, b| b[0].cmp(&a[0]));

    let mut vis: Vec<Vec<bool>> = vec![vec![false; n]; m];
    let dirs: [i32; 5] = [-1, 0, 1, 0, -1];
    let mut ans = 0;
    for k in 0..q.len() {
      if uf.find(0) == uf.find(m * n - 1) {
        break;
      }
      let t = &q[k];
      let (v, i, j) = (t[0], t[1] as usize, t[2] as usize);
      ans = v;
      vis[i][j] = true;
      for d in 0..4 {
        let x = (i as i32) + dirs[d];
        let y = (j as i32) + dirs[d + 1];
        if
          x >= 0 &&
          x < (m as i32) &&
          y >= 0 &&
          y < (n as i32) &&
          vis[x as usize][y as usize]
        {
          uf.union((x as usize) * n + (y as usize), i * n + j);
        }
      }
    }
    ans
  }
}

方法二

class UnionFind:
  __slots__ = ("p", "size")

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

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

  def union(self, a: int, b: int) -> bool:
    pa, pb = self.find(a), self.find(b)
    if pa == pb:
      return False
    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]
    return True


class Solution:
  def maximumMinimumPath(self, grid: List[List[int]]) -> int:
    m, n = len(grid), len(grid[0])
    uf = UnionFind(m * n)
    q = [(v, i, j) for i, row in enumerate(grid) for j, v in enumerate(row)]
    q.sort()
    ans = 0
    vis = set()
    dirs = (-1, 0, 1, 0, -1)
    while uf.find(0) != uf.find(m * n - 1):
      v, i, j = q.pop()
      ans = v
      vis.add((i, j))
      for a, b in pairwise(dirs):
        x, y = i + a, j + b
        if (x, y) in vis:
          uf.union(x * n + y, i * n + j)
    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];
      }
    }
  }
}

class Solution {
  public int maximumMinimumPath(int[][] grid) {
    int m = grid.length, n = grid[0].length;
    UnionFind uf = new UnionFind(m * n);
    List<int[]> q = new ArrayList<>();
    for (int i = 0; i < m; ++i) {
      for (int j = 0; j < n; ++j) {
        q.add(new int[] {grid[i][j], i, j});
      }
    }
    q.sort((a, b) -> b[0] - a[0]);
    boolean[][] vis = new boolean[m][n];
    int[] dirs = {-1, 0, 1, 0, -1};
    int ans = 0;
    for (int i = 0; uf.find(0) != uf.find(m * n - 1); ++i) {
      int[] t = q.get(i);
      vis[t[1]][t[2]] = true;
      ans = t[0];
      for (int k = 0; k < 4; ++k) {
        int x = t[1] + dirs[k], y = t[2] + dirs[k + 1];
        if (x >= 0 && x < m && y >= 0 && y < n && vis[x][y]) {
          uf.union(x * n + y, t[1] * n + t[2]);
        }
      }
    }
    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];
  }

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

class Solution {
public:
  int maximumMinimumPath(vector<vector<int>>& grid) {
    int m = grid.size(), n = grid[0].size();
    vector<tuple<int, int, int>> q;
    UnionFind uf(m * n);
    for (int i = 0; i < m; ++i) {
      for (int j = 0; j < n; ++j) {
        q.emplace_back(grid[i][j], i, j);
      }
    }
    sort(q.begin(), q.end(), greater<tuple<int, int, int>>());
    int ans = 0;
    int dirs[5] = {-1, 0, 1, 0, -1};
    bool vis[m][n];
    memset(vis, false, sizeof(vis));
    for (auto& [v, i, j] : q) {
      vis[i][j] = true;
      ans = v;
      for (int k = 0; k < 4; ++k) {
        int x = i + dirs[k], y = j + dirs[k + 1];
        if (x >= 0 && x < m && y >= 0 && y < n && vis[x][y]) {
          uf.unite(x * n + y, i * n + j);
        }
      }
      if (uf.find(0) == uf.find(m * n - 1)) {
        break;
      }
    }
    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 maximumMinimumPath(grid [][]int) (ans int) {
  m, n := len(grid), len(grid[0])
  uf := newUnionFind(m * n)
  vis := make([][]bool, m)
  q := [][3]int{}
  for i, row := range grid {
    vis[i] = make([]bool, n)
    for j, v := range row {
      q = append(q, [3]int{v, i, j})
    }
  }
  sort.Slice(q, func(i, j int) bool { return q[i][0] > q[j][0] })
  dirs := [5]int{-1, 0, 1, 0, -1}
  for _, t := range q {
    v, i, j := t[0], t[1], t[2]
    ans = v
    vis[i][j] = true
    for k := 0; k < 4; k++ {
      x, y := i+dirs[k], j+dirs[k+1]
      if 0 <= x && x < m && 0 <= y && y < n && vis[x][y] {
        uf.union(x*n+y, i*n+j)
      }
    }
    if uf.find(0) == uf.find(m*n-1) {
      break
    }
  }
  return
}

class UnionFind {
  private p: number[];
  private size: number[];

  constructor(n: number) {
    this.p = Array(n)
      .fill(0)
      .map((_, i) => i);
    this.size = Array(n).fill(1);
  }

  find(x: number): number {
    if (this.p[x] !== x) {
      this.p[x] = this.find(this.p[x]);
    }
    return this.p[x];
  }

  union(a: number, b: number): boolean {
    const pa = this.find(a);
    const pb = this.find(b);
    if (pa === pb) {
      return false;
    }
    if (this.size[pa] > this.size[pb]) {
      this.p[pb] = pa;
      this.size[pa] += this.size[pb];
    } else {
      this.p[pa] = pb;
      this.size[pb] += this.size[pa];
    }
    return true;
  }
}

function maximumMinimumPath(grid: number[][]): number {
  const m = grid.length;
  const n = grid[0].length;
  const q: number[][] = [];
  for (let i = 0; i < m; ++i) {
    for (let j = 0; j < n; ++j) {
      q.push([grid[i][j], i, j]);
    }
  }
  q.sort((a, b) => b[0] - a[0]);
  const dirs: number[] = [-1, 0, 1, 0, -1];
  const vis: boolean[][] = Array(m)
    .fill(0)
    .map(() => Array(n).fill(false));
  let ans = 0;
  const uf = new UnionFind(m * n);
  for (let k = 0; uf.find(0) !== uf.find(m * n - 1); ++k) {
    const [t, i, j] = q[k];
    ans = t;
    vis[i][j] = true;
    for (let d = 0; d < 4; ++d) {
      const [x, y] = [i + dirs[d], j + dirs[d + 1]];
      if (x >= 0 && x < m && y >= 0 && y < n && vis[x][y]) {
        uf.union(i * n + j, x * n + y);
      }
    }
  }
  return ans;
}