June 12, 2022: if there are n*n pieces in an n*n square chessboard, each grid can have exactly one piece. But now some pieces are gathered on a grid, such as 2030100300. The above two-dimensional arra

2022-06-12： stay NN On a square chessboard , Yes NN A chess piece , Then each grid can have exactly one chess piece .
But now some pieces are gathered on a grid , such as ：
2 0 3
0 1 0
3 0 0
The above two-dimensional array represents , altogether 3*3 Lattice ,
But some grids have 2 A chess piece 、 Some have 3 individual 、 Some have 1 individual 、 Some don't ,
Please move the chess pieces , Let each square have a piece ,
Each piece can be played 、 Next 、 Left 、 Move right , Every move counts 1 The price of .
Return the minimum cost .
From Microsoft .

answer 2022-06-12：

km Algorithm , Negative distance .

The code to use rust To write . The code is as follows ：

use rand::Rng;
fn main() {
    
    let len: i32 = 4;
    let test_time: i32 = 1000;
    println!(" Beginning of the test ");
    for _ in 0..test_time {
    
        let mut graph = random_valid_matrix(len);
        let ans1 = min_distance1(&mut graph);
        let ans2 = min_distance2(&mut graph);
        if ans1 != ans2 {
    
            println!(" Something went wrong !");
            println!("ans1 = {}", ans1);
            println!("ans2 = {}", ans2);
            println!("===============");
        }
    }
    println!(" End of test ");
}

//  Violent solution 
//  As logarithm 
fn min_distance1(map: &mut Vec<Vec<i32>>) -> i32 {
    
    let mut n = 0;
    let mut m = 0;
    for i in 0..map.len() as i32 {
    
        for j in 0..map[0].len() as i32 {
    
            n += get_max(0, map[i as usize][j as usize] - 1);
            m += if map[i as usize][j as usize] == 0 {
    
                1
            } else {
    
                0
            };
        }
    }
    if n != m || n == 0 {
    
        return 0;
    }
    let mut nodes: Vec<Vec<i32>> = vec![];
    for i in 0..n {
    
        nodes.push(vec![]);
        for _ in 0..2 {
    
            nodes[i as usize].push(0);
        }
    }
    let mut space: Vec<Vec<i32>> = vec![];
    for i in 0..m {
    
        space.push(vec![]);
        for _ in 0..2 {
    
            space[i as usize].push(0);
        }
    }
    n = 0;
    m = 0;
    for i in 0..map.len() as i32 {
    
        for j in 0..map[0].len() as i32 {
    
            for _k in 2..map[i as usize][j as usize] {
    
                nodes[n as usize][0] = i;
                nodes[n as usize][1] = j;
                n += 1;
            }
            if map[i as usize][j as usize] == 0 {
    
                space[m as usize][0] = i;
                space[m as usize][1] = j;
                m += 1;
            }
        }
    }
    return process1(&mut nodes, 0, &mut space);
}

fn process1(nodes: &mut Vec<Vec<i32>>, index: i32, space: &mut Vec<Vec<i32>>) -> i32 {
    
    let mut ans = 0;
    if index == nodes.len() as i32 {
    
        for i in 0..nodes.len() as i32 {
    
            ans += distance(&mut nodes[i as usize], &mut space[i as usize]);
        }
    } else {
    
        ans = 2147483647;
        for i in index..nodes.len() as i32 {
    
            swap(nodes, index, i);
            ans = get_min(ans, process1(nodes, index + 1, space));
            swap(nodes, index, i);
        }
    }
    return ans;
}

fn swap(nodes: &mut Vec<Vec<i32>>, i: i32, j: i32) {
    
    let tmp = nodes[i as usize].clone();
    nodes[i as usize] = nodes[j as usize].clone();
    nodes[j as usize] = tmp.clone();
}

fn distance(a: &mut Vec<i32>, b: &mut Vec<i32>) -> i32 {
    
    return abs(a[0] - b[0]) + abs(a[1] - b[1]);
}
fn abs(a: i32) -> i32 {
    
    if a < 0 {
    
        -a
    } else {
    
        a
    }
}

//  Formal approach 
// KM Algorithm 
fn min_distance2(map: &mut Vec<Vec<i32>>) -> i32 {
    
    let mut n = 0;
    let mut m = 0;
    for i in 0..map.len() as i32 {
    
        for j in 0..map[0].len() as i32 {
    
            n += get_max(0, map[i as usize][j as usize] - 1);
            m += if map[i as usize][j as usize] == 0 {
    
                1
            } else {
    
                0
            };
        }
    }
    if n != m || n == 0 {
    
        return 0;
    }
    let mut nodes: Vec<Vec<i32>> = vec![];
    for i in 0..n {
    
        nodes.push(vec![]);
        for _ in 0..2 {
    
            nodes[i as usize].push(0);
        }
    }
    let mut space: Vec<Vec<i32>> = vec![];
    for i in 0..m {
    
        space.push(vec![]);
        for _ in 0..2 {
    
            space[i as usize].push(0);
        }
    }
    n = 0;
    m = 0;
    for i in 0..map.len() as i32 {
    
        for j in 0..map[0].len() as i32 {
    
            for _k in 2..=map[i as usize][j as usize] {
    
                nodes[n as usize][0] = i;
                nodes[n as usize][1] = j;
                n += 1;
            }
            if map[i as usize][j as usize] == 0 {
    
                space[m as usize][0] = i;
                space[m as usize][1] = j;
                m += 1;
            }
        }
    }
    let mut graph: Vec<Vec<i32>> = vec![];
    for i in 0..n {
    
        graph.push(vec![]);
        for _ in 0..n {
    
            graph[i as usize].push(0);
        }
    }
    for i in 0..n {
    
        for j in 0..n {
    
            graph[i as usize][j as usize] =
                -distance(&mut nodes[i as usize], &mut space[j as usize]);
        }
    }
    return -km(&mut graph);
}

fn get_max<T: Clone + Copy + std::cmp::PartialOrd>(a: T, b: T) -> T {
    
    if a > b {
    
        a
    } else {
    
        b
    }
}

fn get_min<T: Clone + Copy + std::cmp::PartialOrd>(a: T, b: T) -> T {
    
    if a < b {
    
        a
    } else {
    
        b
    }
}

fn km(graph: &mut Vec<Vec<i32>>) -> i32 {
    
    let nn = graph.len() as i32;
    let mut match0: Vec<i32> = vec![];
    let mut lx: Vec<i32> = vec![];
    let mut ly: Vec<i32> = vec![];
    // dfs In the process , Touched points ！
    let mut x: Vec<bool> = vec![];
    let mut y: Vec<bool> = vec![];
    //  Lowered expectations ！
    //  Princess , Make a , Lower the expected value , Keep it to a minimum ！
    let mut slack: Vec<i32> = vec![];
    let mut falsev: Vec<bool> = vec![];
    for _ in 0..nn {
    
        match0.push(0);
        lx.push(0);
        ly.push(0);
        x.push(false);
        y.push(false);
        slack.push(0);
        falsev.push(false);
    }
    let invalid = 2147483647;
    for i in 0..nn {
    
        match0[i as usize] = -1;
        lx[i as usize] = -invalid;
        for j in 0..nn {
    
            lx[i as usize] = get_max(lx[i as usize], graph[i as usize][j as usize]);
        }
        ly[i as usize] = 0;
    }
    for from in 0..nn {
    
        for i in 0..nn {
    
            slack[i as usize] = invalid;
        }
        x = falsev.clone();
        y = falsev.clone();
        // dfs() : from prince , Can we not lower our expectations , The match is successful ！
        //  can ：dfs return true！
        //  You can't ：dfs return false！
        while !dfs(
            from,
            &mut x,
            &mut y,
            &mut lx,
            &mut ly,
            &mut match0,
            &mut slack,
            graph,
        ) {
    
            //  Just now dfs, failed ！
            //  Need to get , Princess's slack Inside , The minimum value of the expected decrease ！
            let mut d = invalid;
            for i in 0..nn {
    
                if !y[i as usize] && slack[i as usize] < d {
    
                    d = slack[i as usize];
                }
            }
            //  Adjust the expectation according to the minimum expectation 
            for i in 0..nn {
    
                if x[i as usize] {
    
                    lx[i as usize] = lx[i as usize] - d;
                }
                if y[i as usize] {
    
                    ly[i as usize] = ly[i as usize] + d;
                }
            }
            x = falsev.clone();
            y = falsev.clone();
            //  Then back while in , Try it again 
        }
    }
    let mut ans = 0;
    for i in 0..nn {
    
        ans += lx[i as usize] + ly[i as usize];
    }
    return ans;
}

// from,  The current Prince 
// x, Did the prince touch it 
// y,  Did the princess touch it 
// lx, All the prince's expectations 
// ly,  All the princess's expectations 
// match, All the princesses , Previous assignments , Men before ！
// slack, Continuous , But the princess without permission , The minimum decrease 
// map, offer , All the prince's offers to the princess 
//  return ,from Prince No , Do not lower expectations can match ！
fn dfs(
    from: i32,
    x: &mut Vec<bool>,
    y: &mut Vec<bool>,
    lx: &mut Vec<i32>,
    ly: &mut Vec<i32>,
    match0: &mut Vec<i32>,
    slack: &mut Vec<i32>,
    map: &mut Vec<Vec<i32>>,
) -> bool {
    
    let nn = map.len() as i32;
    x[from as usize] = true;
    for to in 0..nn {
    
        if !y[to as usize] {
    
            //  Only No dfs The princess who passed , To try 
            let d = lx[from as usize] + ly[to as usize] - map[from as usize][to as usize];
            if d != 0 {
    
                //  If the current road does not meet expectations , Update the princess's slack value 
                slack[to as usize] = get_min(slack[to as usize], d);
            } else {
    
                //  If the current road meets expectations , Try to take it directly , Or grab and let the previous arrangement go upside down 
                y[to as usize] = true;
                if match0[to as usize] == -1
                    || dfs(match0[to as usize], x, y, lx, ly, match0, slack, map)
                {
    
                    match0[to as usize] = from;
                    return true;
                }
            }
        }
    }
    return false;
}

//  In order to test 
fn random_valid_matrix(len: i32) -> Vec<Vec<i32>> {
    
    let mut graph: Vec<Vec<i32>> = vec![];
    for i in 0..len {
    
        graph.push(vec![]);
        for _ in 0..len {
    
            graph[i as usize].push(0);
        }
    }
    let all = len * len;

    for _i in 1..all {
    
        graph[rand::thread_rng().gen_range(0, len) as usize]
            [rand::thread_rng().gen_range(0, len) as usize] += 1;
    }
    return graph;
}

The results are as follows ：

---

Zuo Shen java Code