2022-06-12：在N*N的正方形棋盤中，有N*N個棋子，那麼每個格子正好可以擁有一個棋子。 但是現在有些棋子聚集到一個格子上了，比如： 2 0 3 0 1 0 3 0 0 如上的二維數組代錶，一

2022-06-12：在NN的正方形棋盤中，有NN個棋子，那麼每個格子正好可以擁有一個棋子。
但是現在有些棋子聚集到一個格子上了，比如：
2 0 3
0 1 0
3 0 0
如上的二維數組代錶，一共3*3個格子，
但是有些格子有2個棋子、有些有3個、有些有1個、有些沒有，
請你用棋子移動的方式，讓每個格子都有一個棋子，
每個棋子可以上、下、左、右移動，每移動一步算1的代價。
返回最小的代價。
來自微軟。

答案2022-06-12：

km算法，距離取負數。

代碼用rust編寫。代碼如下：

use rand::Rng;
fn main() {
    
    let len: i32 = 4;
    let test_time: i32 = 1000;
    println!("測試開始");
    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!("出錯了!");
            println!("ans1 = {}", ans1);
            println!("ans2 = {}", ans2);
            println!("===============");
        }
    }
    println!("測試結束");
}

// 暴力解
// 作為對數器
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
    }
}

// 正式方法
// KM算法
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過程中，碰過的點！
    let mut x: Vec<bool> = vec![];
    let mut y: Vec<bool> = vec![];
    // 降低的預期！
    // 公主上，打一個，降低預期的值，只維持最小！
    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王子，能不能不降預期，匹配成功！
        // 能：dfs返回true！
        // 不能：dfs返回false！
        while !dfs(
            from,
            &mut x,
            &mut y,
            &mut lx,
            &mut ly,
            &mut match0,
            &mut slack,
            graph,
        ) {
    
            // 剛才的dfs，失敗了！
            // 需要拿到，公主的slack裏面，預期下降幅度的最小值！
            let mut d = invalid;
            for i in 0..nn {
    
                if !y[i as usize] && slack[i as usize] < d {
    
                    d = slack[i as usize];
                }
            }
            // 按照最小預期來調整預期
            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();
            // 然後回到while裏，再次嘗試
        }
    }
    let mut ans = 0;
    for i in 0..nn {
    
        ans += lx[i as usize] + ly[i as usize];
    }
    return ans;
}

// from, 當前的王子
// x，王子碰沒碰過
// y, 公主碰沒碰過
// lx，所有王子的預期
// ly, 所有公主的預期
// match，所有公主，之前的分配，之前的爺們！
// slack，連過，但沒允許的公主，最小下降的幅度
// map，報價，所有王子對公主的報價
// 返回，from號王子，不降預期能不能配成！
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] {
    
            // 只有沒dfs過的公主，才會去嘗試
            let d = lx[from as usize] + ly[to as usize] - map[from as usize][to as usize];
            if d != 0 {
    
                // 如果當前的路不符合預期，更新公主的slack值
                slack[to as usize] = get_min(slack[to as usize], d);
            } else {
    
                // 如果當前的路符合預期，嘗試直接拿下，或者搶奪讓之前的安排倒騰去
                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;
}

// 為了測試
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;
}

執行結果如下：

---

左神java代碼