444. Sequence Reconstruction 🔒

Description

You are given an integer array nums of length n where nums is a permutation of the integers in the range [1, n]. You are also given a 2D integer array sequences where sequences[i] is a subsequence of nums.

Check if nums is the shortest possible and the only supersequence. The shortest supersequence is a sequence with the shortest length and has all sequences[i] as subsequences. There could be multiple valid supersequences for the given array sequences.

For example, for sequences = [[1,2],[1,3]], there are two shortest supersequences, [1,2,3] and [1,3,2].
While for sequences = [[1,2],[1,3],[1,2,3]], the only shortest supersequence possible is [1,2,3]. [1,2,3,4] is a possible supersequence but not the shortest.

Return true if nums is the only shortest supersequence for sequences, or false otherwise.

A subsequence is a sequence that can be derived from another sequence by deleting some or no elements without changing the order of the remaining elements.

Example 1:

Input: nums = [1,2,3], sequences = [[1,2],[1,3]]
Output: false
Explanation: There are two possible supersequences: [1,2,3] and [1,3,2].
The sequence [1,2] is a subsequence of both: [1,2,3] and [1,3,2].
The sequence [1,3] is a subsequence of both: [1,2,3] and [1,3,2].
Since nums is not the only shortest supersequence, we return false.

Example 2:

Input: nums = [1,2,3], sequences = [[1,2]]
Output: false
Explanation: The shortest possible supersequence is [1,2].
The sequence [1,2] is a subsequence of it: [1,2].
Since nums is not the shortest supersequence, we return false.

Example 3:

Input: nums = [1,2,3], sequences = [[1,2],[1,3],[2,3]]
Output: true
Explanation: The shortest possible supersequence is [1,2,3].
The sequence [1,2] is a subsequence of it: [1,2,3].
The sequence [1,3] is a subsequence of it: [1,2,3].
The sequence [2,3] is a subsequence of it: [1,2,3].
Since nums is the only shortest supersequence, we return true.

Constraints:

n == nums.length
1 <= n <= 10⁴
nums is a permutation of all the integers in the range [1, n].
1 <= sequences.length <= 10⁴
1 <= sequences[i].length <= 10⁴
1 <= sum(sequences[i].length) <= 10⁵
1 <= sequences[i][j] <= n
All the arrays of sequences are unique.
sequences[i] is a subsequence of nums.

Solutions

Solution 1: Topological Sorting

We can first traverse each subsequence seq. For each pair of adjacent elements $a$ and $b$, we establish a directed edge $a \to b$. At the same time, we count the in-degree of each node, and finally add all nodes with an in-degree of $0$ to the queue.

When the number of nodes in the queue is equal to $1$, we take out the head node $i$, remove $i$ from the graph, and decrease the in-degree of all adjacent nodes of $i$ by $1$. If the in-degree of the adjacent nodes becomes $0$ after decreasing, add these nodes to the queue. Repeat the above operation until the length of the queue is not $1$. At this point, check whether the queue is empty. If it is not empty, it means there are multiple shortest supersequences, return false; if it is empty, it means there is only one shortest supersequence, return true.

The time complexity is $O(n + m)$, and the space complexity is $O(n + m)$. Where $n$ and $m$ are the number of nodes and edges, respectively.

Python3JavaC++GoTypeScript

class Solution:
    def sequenceReconstruction(
        self, nums: List[int], sequences: List[List[int]]
    ) -> bool:
        n = len(nums)
        g = [[] for _ in range(n)]
        indeg = [0] * n
        for seq in sequences:
            for a, b in pairwise(seq):
                a, b = a - 1, b - 1
                g[a].append(b)
                indeg[b] += 1
        q = deque(i for i, x in enumerate(indeg) if x == 0)
        while len(q) == 1:
            i = q.popleft()
            for j in g[i]:
                indeg[j] -= 1
                if indeg[j] == 0:
                    q.append(j)
        return len(q) == 0

class Solution {
    public boolean sequenceReconstruction(int[] nums, List<List<Integer>> sequences) {
        int n = nums.length;
        int[] indeg = new int[n];
        List<Integer>[] g = new List[n];
        Arrays.setAll(g, k -> new ArrayList<>());
        for (var seq : sequences) {
            for (int i = 1; i < seq.size(); ++i) {
                int a = seq.get(i - 1) - 1, b = seq.get(i) - 1;
                g[a].add(b);
                ++indeg[b];
            }
        }
        Deque<Integer> q = new ArrayDeque<>();
        for (int i = 0; i < n; ++i) {
            if (indeg[i] == 0) {
                q.offer(i);
            }
        }
        while (q.size() == 1) {
            int i = q.poll();
            for (int j : g[i]) {
                if (--indeg[j] == 0) {
                    q.offer(j);
                }
            }
        }
        return q.isEmpty();
    }
}

class Solution {
public:
    bool sequenceReconstruction(vector<int>& nums, vector<vector<int>>& sequences) {
        int n = nums.size();
        vector<int> indeg(n);
        vector<int> g[n];
        for (auto& seq : sequences) {
            for (int i = 1; i < seq.size(); ++i) {
                int a = seq[i - 1] - 1, b = seq[i] - 1;
                g[a].push_back(b);
                ++indeg[b];
            }
        }
        queue<int> q;
        for (int i = 0; i < n; ++i) {
            if (indeg[i] == 0) {
                q.push(i);
            }
        }
        while (q.size() == 1) {
            int i = q.front();
            q.pop();
            for (int j : g[i]) {
                if (--indeg[j] == 0) {
                    q.push(j);
                }
            }
        }
        return q.empty();
    }
};

func sequenceReconstruction(nums []int, sequences [][]int) bool {
    n := len(nums)
    indeg := make([]int, n)
    g := make([][]int, n)
    for _, seq := range sequences {
        for i, b := range seq[1:] {
            a := seq[i] - 1
            b -= 1
            g[a] = append(g[a], b)
            indeg[b]++
        }
    }
    q := []int{}
    for i, x := range indeg {
        if x == 0 {
            q = append(q, i)
        }
    }
    for len(q) == 1 {
        i := q[0]
        q = q[1:]
        for _, j := range g[i] {
            indeg[j]--
            if indeg[j] == 0 {
                q = append(q, j)
            }
        }
    }
    return len(q) == 0
}

function sequenceReconstruction(nums: number[], sequences: number[][]): boolean {
    const n = nums.length;
    const g: number[][] = Array.from({ length: n }, () => []);
    const indeg: number[] = Array(n).fill(0);
    for (const seq of sequences) {
        for (let i = 1; i < seq.length; ++i) {
            const [a, b] = [seq[i - 1] - 1, seq[i] - 1];
            g[a].push(b);
            ++indeg[b];
        }
    }
    const q: number[] = indeg.map((v, i) => (v === 0 ? i : -1)).filter(v => v !== -1);
    while (q.length === 1) {
        const i = q.pop()!;
        for (const j of g[i]) {
            if (--indeg[j] === 0) {
                q.push(j);
            }
        }
    }
    return q.length === 0;
}

444. Sequence Reconstruction 🔒

Description

Solutions

Solution 1: Topological Sorting

Comments