296.Best-Meeting-Point

296. Best Meeting Point

题目地址

https://leetcode.com/problems/best-meeting-point/

题目描述

A group of two or more people wants to meet and minimize the total travel distance. You are given a 2D grid of values 0 or 1, where each 1 marks the home of someone in the group. The distance is calculated using Manhattan Distance, where distance(p1, p2) = |p2.x - p1.x| + |p2.y - p1.y|.

Example:
Input: 

1 - 0 - 0 - 0 - 1
|   |   |   |   |
0 - 0 - 0 - 0 - 0
|   |   |   |   |
0 - 0 - 1 - 0 - 0

Output: 6 

Explanation: Given three people living at (0,0), (0,4), and (2,2):
             The point (0,2) is an ideal meeting point, as the total travel distance 
             of 2+2+2=6 is minimal. So return 6.

代码

Approach #1 Sort

Time: O(mn log mn) && Space: O(mn) Since there could be at most m×n points, therefore the time complexity is O(mn log mn) due to sorting.

As long as there is equal number of points to the left and right of the meeting point, the total distance is minimized.

class Solution {
  public int minTotalDistance(int[][] grid) {
        List<Integer> rows = new ArrayList<>();
      List<Integer> cols = new ArrayList<>();
    for (int row = 0; row < grid.length; row++) {
      for (int col = 0; col < grid[0].length; col++) {
        if (grid[row][col] == 1) {
          rows.add(row);
          cols.add(col);
        }
      }
    }
    int row = rows.get(rows.size() / 2);
    Collections.sort(cols);
    int col = cols.get(cols.size() / 2); // median
    return minDistance1D(rows, row) + minDistance1D(cols, col);
  }

  private int minDistance1D(List<Integer> points, int origin) {
    int distance = 0;
    for (int point : points) {
      distance += Math.abs(point - origin);
    }
    return distance;
  }
}

Approach #2 Collect Coordinates in Sorted Order

Time: O(mn) && Space: O(mn)

class Solution {
  public int minTotalDistance(int[][] grid) {
    List<Integer> rows = collectRows(grid);
    List<Integer> cols = collectCols(grid);
    int row = rows.get(rows.size() / 2);
    int col = cols.get(cols.size() / 2);
    return minDistance1D(rows, row) + minDistance1D(cols, col);
  }

  private int minDistance1D(List<Integer> points, int origin) {
    int distance = 0;
    for (int point : points) {
      distance += Math.abs(point - origin);
    }
    return distance;
  }

  private List<Integer> collectRows(int[][] grid) {
    List<Integer> rows = new ArrayList<>();
    for (int row = 0; row < grid.length; row++) {
      for (int col = 0; col < grid[0].length; col++) {
        if (grid[row][col] == 1) {
          rows.add(row);
        }
      }
    }
    return rows;
  }

  private List<Integer> collectCols(int[][] grid) {
    List<Integer> cols = new ArrayList<>();
    for (int col = 0; col < grid[0].length; col++) {
      for (int row = 0; row < grid.length; row++) {
        if (grid[row][col] == 1) {
          cols.add(col);
        }
      }
    }
    return cols;
  }
}

Approach #3 BFS [Time Limit Exceeded]

class Solution {
  public int minTotalDistance(int[][] grid) {
    int minDistance = Integer.MAX_VALUE;
    for (int row = 0; row < grid.length; row++) {
      for (int col = 0; col < grid[0].length; col++) {
        int distance = search(grid, row, col);
        minDistance = Math.min(distance, minDistance);
      }
    }
    return minDistance;
  }

  private int search(int[][] grid, int row, int col) {
    Queue<Point> q = new LinkedList<>();
    int m = grid.length;
    int n = grid[0].length;
    boolean[][] visited = new boolean[m][n];
    q.add(new Point(row, col, 0));

    int totalDistance = 0;
    while (!q.isEmpty()) {
      Point point = q.poll();
      int r = point.row;
      int c = point.col;
      int d = point.distance;
      if (r < 0 || c < 0 || r >= m || c >= n || visited[r][c]) {
        contineu;
      }
      if (grid[r][c] == 1) {
        totalDistance += d;
      }
      visited[r][c] = true;
      q.add(new Point(r+1, c, d+1));
      q.add(new Point(r-1, c, d+1));
      q.add(new Point(r, c+1, d+1));
      q.add(new Point(r, c-1, d+1));
    }
    return totalDistance;
  }

  public class Point {
    int row;
    int col;
    int distance;
    public Point(int row, int col, int distance) {
      this.row = row;
      this.col = col;
      this.distance = distance;
    }
  }
}