Tuesday, December 15, 2015

LeetCode [317] Shortest Distance from All Buildings


http://algobox.org/shortest-distance-from-all-buildings/
You want to build a house on an empty land which reaches all buildings in the shortest amount of distance. You can only move up, down, left and right. You are given a 2D grid of values 01 or 2, where:
  • Each 0 marks an empty land which you can pass by freely.
  • Each 1 marks a building which you cannot pass through.
  • Each 2 marks an obstacle which you cannot pass through.
Example:
Given three buildings at (0,0)(0,4)(2,2), and an obstacle at (0,2):
The point (1,2) is an ideal empty land to build a house, as the total travel distance of 3+3+1=7 is minimal.
Return 7.
Note:
There will be at least one building. If it is not possible to build such house according to the above rules, return -1.
https://leetcode.com/discuss/76018/share-a-java-implement
https://discuss.leetcode.com/topic/31668/my-bfs-java-solution
Short version: BFS from every building, calculate the distances and find the minimum distance in the end.
Key optimization : we do not go into a land, if it is not accessible by at least one of previous buildings.

The shortest distance from any empty land to a building can be calculated by BFS starting from the building in O(mn)time. Therefore the we can calculate distance from all buildings to empty lands by t rounds of BFS starting from each building. t is the total number of buildings.
In each round, we need maintain two values for every empty land: the distance and the accessibility.
  • dist[i][j] is the empty land (i, j) to all the buildings.
  • grid[i][j] is reused as the accessibility.
What is accessibility? It is the number of buildings that are accessible from the empty land.
In each round of BFS we can maintain these two values. In the end we just need to find the minimum value of dist[i][j] where the accessibility equals t.
One interesting point is that the grid[i][j] can also be used to check if (i, j) is visited in this round. At round k(zero based), those has grid[i][j] == k is the empty land unvisited, visited land will have grid[i][j] == k + 1. Here comes the interesting part. One may ask what if grid[i][j] < k? Answer is we do not go into the lands withgrid[i][j] < k as if it is an obstacle. Why can we do that? Because grid[i][j] < k means it is not accessible by at least one of the buildings in previous rounds. Which means not only this land is not our answer, all the lands accessible from (i, j) is also not our answer.
This might be why it runs faster than many implements. The Java version runs in 13 ms.
int[] dx = {1, 0, -1, 0}, dy = {0, 1, 0, -1}; public int shortestDistance(int[][] grid) { int m = grid.length, n = grid[0].length; int[][] dist = new int[m][n]; // Initialize building list and accessibility matrix `grid` List<Tuple> buildings = new ArrayList<>(); for (int i = 0; i < m; ++i) for (int j = 0; j < n; ++j) { if (grid[i][j] == 1) buildings.add(new Tuple(i, j, 0)); grid[i][j] = -grid[i][j]; } // BFS from every building for (int k = 0; k < buildings.size(); ++k) bfs(buildings.get(k), k, dist, grid, m, n); // Find the minimum distance int ans = -1; for (int i = 0; i < m; ++i) for (int j = 0; j < n; ++j) if (grid[i][j] == buildings.size() && (ans < 0 || dist[i][j] < ans)) ans = dist[i][j]; return ans; } public void bfs(Tuple root, int k, int[][] dist, int[][] grid, int m, int n) { Queue<Tuple> q = new ArrayDeque<>(); q.add(root); while (!q.isEmpty()) { Tuple b = q.poll(); dist[b.y][b.x] += b.dist; for (int i = 0; i < 4; ++i) { int x = b.x + dx[i], y = b.y + dy[i]; if (y >= 0 && x >= 0 && y < m && x < n && grid[y][x] == k) { grid[y][x] = k + 1; q.add(new Tuple(y, x, b.dist + 1)); } } } } class Tuple { public int y; public int x; public int dist; public Tuple(int y, int x, int dist) { this.y = y; this.x = x; this.dist = dist; } }
https://leetcode.com/discuss/74999/java-solution-with-explanation-and-time-complexity-analysis
Traverse the matrix. For each building, use BFS to compute the shortest distance from each '0' to this building. After we do this for all the buildings, we can get the sum of shortest distance from every '0' to all reachable buildings. This value is stored in 'distance[][]'. For example, if grid[2][2] == 0, distance[2][2] is the sum of shortest distance from this block to all reachable buildings. Time complexity: O(number of 1)O(number of 0) ~ O(m^2n^2)
We also count how many building each '0' can be reached. It is stored in reach[][]. This can be done during the BFS. We also need to count how many total buildings are there in the matrix, which is stored in 'buildingNum'.
Finally, we can traverse the distance[][] matrix to get the point having shortest distance to all buildings. O(m*n)
The total time complexity will be O(m^2*n^2), which is quite high!.

Isn't the time complexity O(#buildings * m^2 * n^2)? BFS's time complexity is O(|V||E|). Here we have mn vertices and the edges are proportional to m*n, so every BFS is O(m^2 * n^2). We do a BFS for every building so the total time complexity is O(#buildings * m^2 * n^2)?

The time complexity for BFS/DFS is O(|V|+|E|), not O(|V||E|). In this problem, every vertex has up to 4 edges (left, right, up, down), so |E| ~ 4|V|. Thus, you have overall O(|V|) = O(mn) for a BFS. This has been proven for all sparse graphs like this problem. Now, we do a BFS for each building, so the overall complexity is O(#buildings*(mn)). In worst case, every vertex is a building. So the number of buildings is also upper bounded by O(mn), and thus you have O((mn)(mn)) = O(m^2n^2). This is a very loose bound since when every vertex is a building, we don't even need to do a BFS (nowhere to go).

public int shortestDistance(int[][] grid) { if (grid == null || grid[0].length == 0) return 0; final int[] shift = new int[] {0, 1, 0, -1, 0}; int row = grid.length, col = grid[0].length; int[][] distance = new int[row][col]; int[][] reach = new int[row][col]; int buildingNum = 0; for (int i = 0; i < row; i++) { for (int j =0; j < col; j++) { if (grid[i][j] == 1) { buildingNum++; Queue<int[]> myQueue = new LinkedList<int[]>(); myQueue.offer(new int[] {i,j}); boolean[][] isVisited = new boolean[row][col]; int level = 1; while (!myQueue.isEmpty()) { int qSize = myQueue.size(); for (int q = 0; q < qSize; q++) { int[] curr = myQueue.poll(); for (int k = 0; k < 4; k++) { int nextRow = curr[0] + shift[k]; int nextCol = curr[1] + shift[k + 1];
// if reach[nextRow][nextCol] < buildingNum, ignore it.
if (nextRow >= 0 && nextRow < row && nextCol >= 0 && nextCol < col && grid[nextRow][nextCol] == 0 && !isVisited[nextRow][nextCol]) { //The shortest distance from [nextRow][nextCol] to thic building // is 'level'. distance[nextRow][nextCol] += level; reach[nextRow][nextCol]++; isVisited[nextRow][nextCol] = true; myQueue.offer(new int[] {nextRow, nextCol}); } } } level++; } } } } int shortest = Integer.MAX_VALUE; for (int i = 0; i < row; i++) { for (int j = 0; j < col; j++) { if (grid[i][j] == 0 && reach[i][j] == buildingNum) { shortest = Math.min(shortest, distance[i][j]); } } } return shortest == Integer.MAX_VALUE ? -1 : shortest; }

static final int[] s={-1,0,1,0,-1};

class Move{
    int x;
    int steps;
    boolean[][] visited;
    Move(int p, int s){
        x=p;
        steps=s;
    }
}

public int shortestDistance(int[][] grid) {
    if(grid==null || grid.length==0 || grid[0].length==0) return 0;
    int m = grid.length, n=grid[0].length, total=0, min=Integer.MAX_VALUE;
    int[][] distance= new int[m][n];
    int[][] reach= new int[m][n];
    Deque<Move> queue = new ArrayDeque<>();
    for(int i=0; i<m; i++){
        for(int j=0; j<n; j++){
            if(grid[i][j]==1){
                Move mv = new Move(i*n+j, 0);
                mv.visited = new boolean[m][n];
                mv.visited[i][j]=true;
                queue.offer(mv);
                total++;
            }
        }
    }
    while(!queue.isEmpty()){
        Move mv = queue.poll();
        int x = mv.x/n, y=mv.x%n;
        for(int i=0; i<4; i++){
            int p = x+s[i], q=y+s[i+1];
            if(p>=0 && p<m && q>=0 && q<n && !mv.visited[p][q] && grid[p][q]==0){
                Move newMv = new Move(p*n+q, mv.steps+1);
                newMv.visited=mv.visited;
                newMv.visited[p][q]=true;
                distance[p][q]+=mv.steps+1;
                if(distance[p][q]<min) {
                    if(++reach[p][q]==total) min = distance[p][q];
                    queue.offer(newMv);
                }
            }
        }
    }
    return (min==Integer.MAX_VALUE)?-1: min;
}
http://www.cnblogs.com/yrbbest/p/5068730.html
https://leetcode.com/discuss/74380/my-bfs-java-solution
    private final int[][] directions = {{0, 1}, {1, 0}, {-1, 0}, {0, -1}};
    
    public int shortestDistance(int[][] grid) {
        if(grid == null || grid.length == 0) {
            return Integer.MAX_VALUE;
        }
        int rowNum = grid.length;
        int colNum = grid[0].length;
        int[][] distance = new int[rowNum][colNum];
        int[][] canReachBuildings = new int[rowNum][colNum];
        int buildingNum = 0; 
        
        for(int i = 0; i < rowNum; i++) {
            for(int j = 0; j < colNum; j++) {
                if(grid[i][j] != 0) {
                    distance[i][j] = Integer.MAX_VALUE;
                }
                if(grid[i][j] == 1) {     // find out all buildings
                    buildingNum++;
                    updateDistance(grid, distance, canReachBuildings, i, j);
                }
            }
        }
        
        int min = Integer.MAX_VALUE;
        for(int i = 0; i < rowNum; i++) {
            for(int j = 0; j < colNum; j++) {
                if(canReachBuildings[i][j] == buildingNum) {
                    min = Math.min(distance[i][j], min);
                }
            }
        }
        
        return min == Integer.MAX_VALUE ? -1 : min;
    }
    
    private void updateDistance(int[][] grid, int[][] distance, int[][] canReachBuildings, int row, int col) {
        Queue<int[]> queue = new LinkedList<>();
        queue.offer(new int[]{row, col});
        boolean[][] visited = new boolean[grid.length][grid[0].length];
        visited[row][col] = true;
        int dist = 0;
        int curLevel = 1;
        int nextLevel = 0;
        
        while(!queue.isEmpty()) {
            int[] position = queue.poll();
            distance[position[0]][position[1]] += dist;
            curLevel--;
            for(int[] direction : directions) {
                int x = position[0] + direction[0];
                int y = position[1] + direction[1];
                if(x < 0 || x >= grid.length || y < 0 || y >= grid[0].length || grid[x][y] != 0) {
                    continue;
                }
                if(!visited[x][y]) {
                    queue.offer(new int[]{x, y}); 
                    nextLevel++;
                    visited[x][y] = true;
                    canReachBuildings[x][y]++;
                }
            }
            if(curLevel == 0) {
                curLevel = nextLevel;
                nextLevel = 0;
                dist++;
            }
        }
    }
http://buttercola.blogspot.com/2016/01/leetcode-shortest-distance-from-all.html
A BFS problem. Search from each building and calculate the distance to the building. One thing to note is an empty land must be reachable by all buildings. To achieve this, maintain an array of counters. Each time we reach an empty land from a building, increase the counter. Finally, a reachable point must have the counter equaling to the number of buildings. 
    public int shortestDistance(int[][] grid) {
        if (grid == null || grid.length == 0) {
            return 0;
        }
         
        int m = grid.length;
        int n = grid[0].length;
         
        int[][] dist = new int[m][n];
        int[][] reach = new int[m][n];
        // step 1: BFS and calcualte the min dist from each building
        int numBuildings = 0;
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                if (grid[i][j] == 1) {
                    boolean[][] visited = new boolean[m][n]; // no need declare here
                    Queue<Integer> queue = new LinkedList<>();
                    shortestDistanceHelper(i, j, 0, dist, reach, grid, visited, queue);
                    numBuildings++;
                }
            }
        }
         
        // step 2: caluclate the minimum distance
        int minDist = Integer.MAX_VALUE;
        for (int i = 0; i < m; i++) {
            for (int j = 0; j < n; j++) {
                if (grid[i][j] == 0 && reach[i][j] == numBuildings) {
                    minDist = Math.min(minDist, dist[i][j]);
                }
            }
        }
         
        return minDist == Integer.MAX_VALUE ? -1 : minDist;
    }
     
    private void shortestDistanceHelper(int x, int y, int currDist,
                                        int[][] dist, int[][] reach, int[][] grid,
                                        boolean[][] visited, Queue<Integer> queue) {
        fill(x, y, x, y, currDist, dist, reach, grid, visited, queue);
         
        int m = grid.length;
        int n = grid[0].length;
         
        while (!queue.isEmpty()) {
            int size = queue.size();
            currDist++;
            for (int sz = 0; sz < size; sz++) {
                int cord = queue.poll();
                int i = cord / n;
                int j = cord % n;
                 
                fill(x, y, i - 1, j, currDist, dist, reach, grid, visited, queue);
                fill(x, y, i + 1, j, currDist, dist, reach, grid, visited, queue);
                fill(x, y, i, j - 1, currDist, dist, reach, grid, visited, queue);
                fill(x, y, i, j + 1, currDist, dist, reach, grid, visited, queue);
            }
        }
    }
     
    private void fill(int origX, int origY, int x, int y, int currDist,
                      int[][] dist, int[][] reach, 
                      int[][] grid, boolean[][] visited, Queue<Integer> queue) {
         
        int m = dist.length;
        int n = dist[0].length;
        if (x < 0 || x >= m || y < 0 || y >= n || visited[x][y]) {
            return;
        }
         
        if ((x != origX || y != origY) && grid[x][y] != 0) {
            return;
        }
         
        visited[x][y] = true;
         
        dist[x][y] += currDist;
        reach[x][y]++;
         
        queue.offer(x * n + y);
    }
https://leetcode.com/discuss/92508/11ms-simple-java-soluton-that-beats-100%25-with-explanation
public int shortestDistance(int[][] grid) { int[][] d = new int[grid.length][grid[0].length];//sum of distance from all buildings int[][] cc = new int[grid.length][grid[0].length];//reachable by how many buildings int count = 0; //find how many buildings for(int i = 0;i<grid.length;i++){ for(int j = 0;j<grid[0].length;j++){ if(grid[i][j] == 1) { count++; } } } for(int i = 0;i<grid.length;i++){ for(int j = 0;j<grid[0].length;j++){ if(grid[i][j] == 1) { //bfs each building, keep on updating empty spot's distance value boolean x = bfs(grid, i, j, d, new boolean[grid.length][grid[0].length], count, cc); if(!x) return -1;//if this building can not reach all other buildings return -1 } } } //find the smallest distance, need be an empty spot, and a spot can be reached by all buildings int res = Integer.MAX_VALUE; for(int i = 0;i<grid.length;i++){ for(int j = 0;j<grid[0].length;j++){ if(d[i][j] != 0 && cc[i][j] == count) res = Math.min(res, d[i][j]); } } return res == Integer.MAX_VALUE ? -1 : res ; } //nothing much, just bfs, but updates how many building can an empty spot reach, and //if current building can reach all other buildings, otherwise returns false, solution returns -1 private boolean bfs(int[][]grid, int i, int j, int[][] d, boolean[][] used, int count, int[][] cc) { Queue<Integer> is = new LinkedList<>(); Queue<Integer> js = new LinkedList<>(); Queue<Integer> ds = new LinkedList<>(); is.add(i); js.add(j); ds.add(0); used[i][j] = true; int[] xx = new int[]{-1,0,1,0}; int[] yy = new int[]{0,-1,0,1}; int c = 1;//used to check whether this house can reach all houses int neigh = 0; //used to check whether it can reach any empty land while(!is.isEmpty()){ int x = is.poll(); int y = js.poll(); int dis = ds.poll(); for(int k =0 ;k<4;k++){ int newx = x + xx[k]; int newy = y + yy[k]; if(newx >=0 && newx < grid.length && newy >=0 && newy<grid[0].length && !used[newx][newy]) { if(grid[newx][newy] == 0) { is.add(newx); js.add(newy); ds.add(dis+1); used[newx][newy] = true; d[newx][newy] += dis+1; cc[newx][newy]++; neigh++; } else if(grid[newx][newy] == 1) { c++; used[newx][newy] = true; } } } } return c == count && neigh > 0; }
https://leetcode.com/discuss/74441/share-my-easy-java-solution-with-comments
private class Point { public int i; public int j; public int steps; public Point(int i, int j, int steps){ this.i = i; this.j = j; this.steps = steps; } } public int shortestDistance(int[][] grid) { int height = grid.length, width = grid[0].length; int min = Integer.MAX_VALUE, houses = 0; // Count all existing houses in grid for(int i=0; i<height; i++){ for(int j=0; j<width; j++){ if(grid[i][j] == 1) houses++; } } // Start from all empty cells one by one for(int i=0; i<height; i++){ for(int j=0; j<width; j++){ if(grid[i][j] == 0){ int sum = search(i, j, houses, grid); min = Math.min(min, sum); } } } // Can't find all existing houses if(min == Integer.MAX_VALUE) return -1; else return min; } public int search(int i, int j, int houses, int[][] grid){ int height = grid.length, width = grid[0].length; boolean[][] marked = new boolean[height][width]; int houseCount = 0, sum = 0; Queue<Point> q = new LinkedList<>(); q.add(new Point(i, j, 0)); marked[i][j] = true; while(!q.isEmpty()){ Point point = q.poll(); // Traverse four surrounding cells for(int m=-1; m<=1; m++){ for(int n=-1; n<=1; n++){ // Current point itself or out of boundary if((Math.abs(m) == Math.abs(n)) || !(point.i+m >=0 && point.i+m<height && point.j+n>=0 && point.j+n<width) || marked[point.i+m][point.j+n]) continue; int cell = grid[point.i+m][point.j+n]; // Find a unmarked house if(cell == 1){ houseCount++; sum += point.steps+1; marked[point.i+m][point.j+n] = true; } else if(cell == 0){ q.add(new Point(point.i+m, point.j+n, point.steps+1)); marked[point.i+m][point.j+n] = true; } } } } // Can't find all existing houses if(houseCount < houses) return Integer.MAX_VALUE; else return sum; } }
https://evanyang.gitbooks.io/leetcode/content/LeetCode/shortest_distance_from_all_buildings.html
-- good naming
https://leetcode.com/discuss/74526/java-and-python-solutions-with-bfs
public int shortestDistance(int[][] grid) { if (grid == null || grid.length == 0 || grid[0] == null || grid[0].length == 0) return -1; List<int[]> emptyLands = new ArrayList<>(); List<int[]> buildings = new ArrayList<>(); int nrow = grid.length; int ncol = grid[0].length; int[][] dists = new int[nrow][ncol]; int[][] visitedNums = new int[nrow][ncol]; int minimalDist = Integer.MAX_VALUE; // Find Buildings and Empty Lands for (int i = 0; i < nrow; i++) { for (int j = 0; j < ncol; j++) { if (grid[i][j] == 0) { emptyLands.add(new int[]{i, j}); } else if (grid[i][j] == 1) { buildings.add(new int[]{i, j}); } } } //BFS for each Building for (int[] indices: buildings) { Queue<int[]> queue = new ArrayDeque<>(); boolean[][] visited = new boolean[nrow][ncol]; queue.offer(new int[]{indices[0], indices[1], 0}); while (!queue.isEmpty()) { int[] current = queue.poll(); int x = current[0], y = current[1], dist = current[2]; if (x+1 < nrow && grid[x+1][y] == 0 && !visited[x+1][y]) { dists[x+1][y] += dist + 1; queue.offer(new int[]{x+1, y, dist+1}); visited[x+1][y] = true; visitedNums[x+1][y]++; } if (x-1 >= 0 && grid[x-1][y] == 0 && !visited[x-1][y]) { dists[x-1][y] += dist + 1; queue.offer(new int[]{x-1, y, dist+1}); visited[x-1][y] = true; visitedNums[x-1][y]++; } if (y+1 < ncol && grid[x][y+1] == 0 && !visited[x][y+1]) { dists[x][y+1] += dist + 1; queue.offer(new int[]{x, y+1, dist+1}); visited[x][y+1] = true; visitedNums[x][y+1]++; } if (y-1 >= 0 && grid[x][y-1] == 0 && !visited[x][y-1]) { dists[x][y-1] += dist + 1; queue.offer(new int[]{x, y-1, dist+1}); visited[x][y-1] = true; visitedNums[x][y-1]++; } } } //Find the Empty Land with smallest total dist int size = buildings.size(); for (int[] indices: emptyLands) { int x = indices[0], y = indices[1]; if (dists[x][y] < minimalDist && visitedNums[x][y] == size) { minimalDist = dists[x][y]; } } return minimalDist == Integer.MAX_VALUE ? -1 : minimalDist; }
http://www.cnblogs.com/icecreamdeqinw/p/5048375.html

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