928. Minimize Malware Spread II

Description

You are given a network of n nodes represented as an n x n adjacency matrix graph, where the i^th node is directly connected to the j^th node if graph[i][j] == 1.

Some nodes initial are initially infected by malware. Whenever two nodes are directly connected, and at least one of those two nodes is infected by malware, both nodes will be infected by malware. This spread of malware will continue until no more nodes can be infected in this manner.

Suppose M(initial) is the final number of nodes infected with malware in the entire network after the spread of malware stops.

We will remove exactly one node from initial, completely removing it and any connections from this node to any other node.

Return the node that, if removed, would minimize M(initial). If multiple nodes could be removed to minimize M(initial), return such a node with the smallest index.

Example 1:

Input: graph = [[1,1,0],[1,1,0],[0,0,1]], initial = [0,1]
Output: 0

Example 2:

Input: graph = [[1,1,0],[1,1,1],[0,1,1]], initial = [0,1]
Output: 1

Example 3:

Input: graph = [[1,1,0,0],[1,1,1,0],[0,1,1,1],[0,0,1,1]], initial = [0,1]
Output: 1

Constraints:

n == graph.length
n == graph[i].length
2 <= n <= 300
graph[i][j] is 0 or 1.
graph[i][j] == graph[j][i]
graph[i][i] == 1
1 <= initial.length < n
0 <= initial[i] <= n - 1
All the integers in initial are unique.

Solutions

Solution 1: Union-Find

We can use the union-find data structure to merge all nodes that are not in $initial$ and satisfy $graph[i][j] = 1$.

Next, we create a hash table $g$, where $g[i]$ represents the root node of the connected component that is connected to node $i$. We also need a counter $cnt$ to count how many initial nodes each root node is infected by.

For each initially infected node $i$, we traverse all nodes $j$ connected to node $i$. If node $j$ is not in $initial$, we add the root node of node $j$ to the set $g[i]$. At the same time, we count how many initial nodes each root node is infected by and save the result in the counter $cnt$.

Then, we use a variable $ans$ to record the answer, and $mx$ to record the maximum number of infected nodes that can be reduced. Initially, $ans = 0$, $mx = -1$.

We traverse all initially infected nodes. For each node $i$, we traverse all root nodes in $g[i]$. If a root node is only infected by one initial node, we add the size of the connected component where the root node is located to $t$. If $t > mx$ or $t = mx$ and $i < ans$, we update $ans = i$, $mx = t$.

Finally, we return $ans$.

The time complexity is $O(n^2 \times \alpha(n))$, and the space complexity is $O(n^2)$. Where $n$ is the number of nodes, and $\alpha(n)$ is the inverse Ackermann function.

Python3JavaC++GoTypeScript

class UnionFind:
    __slots__ = "p", "size"

    def __init__(self, n: int):
        self.p = list(range(n))
        self.size = [1] * n

    def find(self, x: int) -> int:
        if self.p[x] != x:
            self.p[x] = self.find(self.p[x])
        return self.p[x]

    def union(self, a: int, b: int) -> bool:
        pa, pb = self.find(a), self.find(b)
        if pa == pb:
            return False
        if self.size[pa] > self.size[pb]:
            self.p[pb] = pa
            self.size[pa] += self.size[pb]
        else:
            self.p[pa] = pb
            self.size[pb] += self.size[pa]
        return True

    def get_size(self, root: int) -> int:
        return self.size[root]


class Solution:
    def minMalwareSpread(self, graph: List[List[int]], initial: List[int]) -> int:
        n = len(graph)
        s = set(initial)
        uf = UnionFind(n)
        for i in range(n):
            if i not in s:
                for j in range(i + 1, n):
                    graph[i][j] and j not in s and uf.union(i, j)

        g = defaultdict(set)
        cnt = Counter()
        for i in initial:
            for j in range(n):
                if j not in s and graph[i][j]:
                    g[i].add(uf.find(j))
            for root in g[i]:
                cnt[root] += 1

        ans, mx = 0, -1
        for i in initial:
            t = sum(uf.get_size(root) for root in g[i] if cnt[root] == 1)
            if t > mx or (t == mx and i < ans):
                ans, mx = i, t
        return ans

class UnionFind {
    private final int[] p;
    private final int[] size;

    public UnionFind(int n) {
        p = new int[n];
        size = new int[n];
        for (int i = 0; i < n; ++i) {
            p[i] = i;
            size[i] = 1;
        }
    }

    public int find(int x) {
        if (p[x] != x) {
            p[x] = find(p[x]);
        }
        return p[x];
    }

    public boolean union(int a, int b) {
        int pa = find(a), pb = find(b);
        if (pa == pb) {
            return false;
        }
        if (size[pa] > size[pb]) {
            p[pb] = pa;
            size[pa] += size[pb];
        } else {
            p[pa] = pb;
            size[pb] += size[pa];
        }
        return true;
    }

    public int size(int root) {
        return size[root];
    }
}

class Solution {
    public int minMalwareSpread(int[][] graph, int[] initial) {
        int n = graph.length;
        boolean[] s = new boolean[n];
        for (int i : initial) {
            s[i] = true;
        }
        UnionFind uf = new UnionFind(n);
        for (int i = 0; i < n; ++i) {
            if (!s[i]) {
                for (int j = i + 1; j < n; ++j) {
                    if (graph[i][j] == 1 && !s[j]) {
                        uf.union(i, j);
                    }
                }
            }
        }
        Set<Integer>[] g = new Set[n];
        Arrays.setAll(g, k -> new HashSet<>());
        int[] cnt = new int[n];
        for (int i : initial) {
            for (int j = 0; j < n; ++j) {
                if (!s[j] && graph[i][j] == 1) {
                    g[i].add(uf.find(j));
                }
            }
            for (int root : g[i]) {
                ++cnt[root];
            }
        }
        int ans = 0, mx = -1;
        for (int i : initial) {
            int t = 0;
            for (int root : g[i]) {
                if (cnt[root] == 1) {
                    t += uf.size(root);
                }
            }
            if (t > mx || (t == mx && i < ans)) {
                ans = i;
                mx = t;
            }
        }
        return ans;
    }
}

class UnionFind {
public:
    UnionFind(int n) {
        p = vector<int>(n);
        size = vector<int>(n, 1);
        iota(p.begin(), p.end(), 0);
    }

    bool unite(int a, int b) {
        int pa = find(a), pb = find(b);
        if (pa == pb) {
            return false;
        }
        if (size[pa] > size[pb]) {
            p[pb] = pa;
            size[pa] += size[pb];
        } else {
            p[pa] = pb;
            size[pb] += size[pa];
        }
        return true;
    }

    int find(int x) {
        if (p[x] != x) {
            p[x] = find(p[x]);
        }
        return p[x];
    }

    int getSize(int root) {
        return size[root];
    }

private:
    vector<int> p, size;
};

class Solution {
public:
    int minMalwareSpread(vector<vector<int>>& graph, vector<int>& initial) {
        int n = graph.size();
        bool s[n];
        memset(s, false, sizeof(s));
        for (int i : initial) {
            s[i] = true;
        }
        UnionFind uf(n);
        for (int i = 0; i < n; ++i) {
            if (!s[i]) {
                for (int j = i + 1; j < n; ++j) {
                    if (graph[i][j] && !s[j]) {
                        uf.unite(i, j);
                    }
                }
            }
        }
        unordered_set<int> g[n];
        int cnt[n];
        memset(cnt, 0, sizeof(cnt));
        for (int i : initial) {
            for (int j = 0; j < n; ++j) {
                if (!s[j] && graph[i][j]) {
                    g[i].insert(uf.find(j));
                }
            }
            for (int root : g[i]) {
                ++cnt[root];
            }
        }
        int ans = 0, mx = -1;
        for (int i : initial) {
            int t = 0;
            for (int root : g[i]) {
                if (cnt[root] == 1) {
                    t += uf.getSize(root);
                }
            }
            if (t > mx || (t == mx && i < ans)) {
                ans = i;
                mx = t;
            }
        }
        return ans;
    }
};

type unionFind struct {
    p, size []int
}

func newUnionFind(n int) *unionFind {
    p := make([]int, n)
    size := make([]int, n)
    for i := range p {
        p[i] = i
        size[i] = 1
    }
    return &unionFind{p, size}
}

func (uf *unionFind) find(x int) int {
    if uf.p[x] != x {
        uf.p[x] = uf.find(uf.p[x])
    }
    return uf.p[x]
}

func (uf *unionFind) union(a, b int) bool {
    pa, pb := uf.find(a), uf.find(b)
    if pa == pb {
        return false
    }
    if uf.size[pa] > uf.size[pb] {
        uf.p[pb] = pa
        uf.size[pa] += uf.size[pb]
    } else {
        uf.p[pa] = pb
        uf.size[pb] += uf.size[pa]
    }
    return true
}

func (uf *unionFind) getSize(root int) int {
    return uf.size[root]
}

func minMalwareSpread(graph [][]int, initial []int) int {
    n := len(graph)
    s := make([]bool, n)
    for _, i := range initial {
        s[i] = true
    }
    uf := newUnionFind(n)
    for i := range graph {
        if !s[i] {
            for j := i + 1; j < n; j++ {
                if graph[i][j] == 1 && !s[j] {
                    uf.union(i, j)
                }
            }
        }
    }
    g := make([]map[int]bool, n)
    for _, i := range initial {
        g[i] = map[int]bool{}
    }
    cnt := make([]int, n)
    for _, i := range initial {
        for j := 0; j < n; j++ {
            if !s[j] && graph[i][j] == 1 {
                g[i][uf.find(j)] = true
            }
        }
        for root := range g[i] {
            cnt[root]++
        }
    }
    ans, mx := 0, -1
    for _, i := range initial {
        t := 0
        for root := range g[i] {
            if cnt[root] == 1 {
                t += uf.getSize(root)
            }
        }
        if t > mx || t == mx && i < ans {
            ans, mx = i, t
        }
    }
    return ans
}

class UnionFind {
    p: number[];
    size: number[];
    constructor(n: number) {
        this.p = Array(n)
            .fill(0)
            .map((_, i) => i);
        this.size = Array(n).fill(1);
    }

    find(x: number): number {
        if (this.p[x] !== x) {
            this.p[x] = this.find(this.p[x]);
        }
        return this.p[x];
    }

    union(a: number, b: number): boolean {
        const [pa, pb] = [this.find(a), this.find(b)];
        if (pa === pb) {
            return false;
        }
        if (this.size[pa] > this.size[pb]) {
            this.p[pb] = pa;
            this.size[pa] += this.size[pb];
        } else {
            this.p[pa] = pb;
            this.size[pb] += this.size[pa];
        }
        return true;
    }

    getSize(root: number): number {
        return this.size[root];
    }
}

function minMalwareSpread(graph: number[][], initial: number[]): number {
    const n = graph.length;
    const s = new Set(initial);
    const uf = new UnionFind(n);
    for (let i = 0; i < n; ++i) {
        if (!s.has(i)) {
            for (let j = i + 1; j < n; ++j) {
                if (graph[i][j] && !s.has(j)) {
                    uf.union(i, j);
                }
            }
        }
    }
    const g: Set<number>[] = Array.from({ length: n }, () => new Set());
    const cnt: number[] = Array(n).fill(0);
    for (const i of initial) {
        for (let j = 0; j < n; ++j) {
            if (graph[i][j] && !s.has(j)) {
                g[i].add(uf.find(j));
            }
        }
        for (const root of g[i]) {
            ++cnt[root];
        }
    }
    let ans = 0;
    let mx = -1;
    for (const i of initial) {
        let t = 0;
        for (const root of g[i]) {
            if (cnt[root] === 1) {
                t += uf.getSize(root);
            }
        }
        if (t > mx || (t === mx && i < ans)) {
            [ans, mx] = [i, t];
        }
    }
    return ans;
}

928. Minimize Malware Spread II

Description

Solutions

Solution 1: Union-Find

Comments