Leetcode: Campus Bikes II

On a campus represented as a 2D grid, there are N workers and M bikes, with N <= M. Each worker and bike is a 2D coordinate on this grid.

We assign one unique bike to each worker so that the sum of the Manhattan distances between each worker and their assigned bike is minimized.

The Manhattan distance between two points p1 and p2 is Manhattan(p1, p2) = |p1.x - p2.x| + |p1.y - p2.y|.

Return the minimum possible sum of Manhattan distances between each worker and their assigned bike.
Example 1:



Input: workers = [[0,0],[2,1]], bikes = [[1,2],[3,3]]
Output: 6
Explanation: 
We assign bike 0 to worker 0, bike 1 to worker 1. The Manhattan distance of both assignments is 3, so the output is 6.
Example 2:


Input: workers = [[0,0],[1,1],[2,0]], bikes = [[1,0],[2,2],[2,1]]
Output: 4
Explanation: 
We first assign bike 0 to worker 0, then assign bike 1 to worker 1 or worker 2, bike 2 to worker 2 or worker 1. Both assignments lead to sum of the Manhattan distances as 4.
 

Note:

0 <= workers[i][0], workers[i][1], bikes[i][0], bikes[i][1] < 1000
All worker and bike locations are distinct.
1 <= workers.length <= bikes.length <= 10

 

Basic: Backtracking + pruning

class Solution {
    int minDis = Integer.MAX_VALUE;
    
    public int assignBikes(int[][] workers, int[][] bikes) {
        dfs(workers, bikes, new boolean[bikes.length], 0, 0);    
        return minDis;
    }
    
    public void dfs(int[][] workers, int[][] bikes, boolean[] visited, int pos, int distance) {
        if (pos == workers.length) {
            minDis = Math.min(minDis, distance);
            return;
        }
        if (distance > minDis) return;
        for (int i = 0; i < visited.length; i ++) {
            if (visited[i]) continue;
            visited[i] = true;
            dfs(workers, bikes, visited, pos + 1, distance + manhattanDis(workers[pos], bikes[i]));
            visited[i] = false;
        }
    }
    
    public int manhattanDis(int[] worker, int[] bike) {
        return Math.abs(worker[0] - bike[0]) + Math.abs(worker[1] - bike[1]);
    }
}

 

DP: refer to https://leetcode.com/problems/campus-bikes-ii/discuss/305218/DFS-%2B-Pruning-And-DP-Solution

state : dp[i][s] = the min distance for first i workers to build the state s ,
transit: dp[i][s] = min(dp[i][s], dp[i - 1][prev] + dis(worker[i -1], bike[j)) | 0 < j <m, prev = s ^ (1 << j)
init:dp[0][0] = 0;
result: dp[n][s] s should have n bit

  public int assignBikes(int[][] workers, int[][] bikes) {
        int n = workers.length;
        int m = bikes.length;
        int[][] dp = new int[n + 1][1 << m];
        for (int[] d : dp) {
            Arrays.fill(d, Integer.MAX_VALUE / 2);
        }
        dp[0][0] = 0;
        int min = Integer.MAX_VALUE;
        for (int i = 1; i <= n; i++) {
            for (int s = 1; s < (1 << m); s++) {
                for (int j = 0; j < m; j++) {
                    if ((s & (1 << j)) == 0) { // s is current state after the operation of taking bike at j, so s at j should be 1 already 
                        continue;
                    }
                    int prev = s ^ (1 << j);   // previously s at j should be 0
                    dp[i][s] = Math.min(dp[i - 1][prev] + dis(workers[i - 1], bikes[j]), dp[i][s]) ;
                    if (i == n) {
                        min = Math.min(min, dp[i][s]);
                    }
                }
            }
        }
        return min;
    }
  
    public int dis(int[] p1, int[] p2) {
        return Math.abs(p1[0] - p2[0]) + Math.abs(p1[1] - p2[1]);
    }