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2709. Greatest Common Divisor Traversal 👍

  • Time: $O(k\log^* k + k\log(\log k)$, where $k = \max(\texttt{nums})$
  • Space: $O(n)$
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class UnionFind {
 public:
  UnionFind(int n) : id(n), sz(n, 1) {
    iota(id.begin(), id.end(), 0);
  }

  void unionBySize(int u, int v) {
    const int i = find(u);
    const int j = find(v);
    if (i == j)
      return;
    if (sz[i] < sz[j]) {
      sz[j] += sz[i];
      id[i] = j;
    } else {
      sz[i] += sz[j];
      id[j] = i;
    }
  }

  int getSize(int i) {
    return sz[i];
  }

 private:
  vector<int> id;
  vector<int> sz;

  int find(int u) {
    return id[u] == u ? u : id[u] = find(id[u]);
  }
};

class Solution {
 public:
  bool canTraverseAllPairs(vector<int>& nums) {
    const int n = nums.size();
    const int mx = ranges::max(nums);
    const vector<int> minPrimeFactors = sieveEratosthenes(mx + 1);
    unordered_map<int, int> primeToFirstIndex;
    UnionFind uf(n);

    for (int i = 0; i < n; ++i)
      for (const int primeFactor : getPrimeFactors(nums[i], minPrimeFactors))
        // `primeFactor` already appeared in the previous indices.
        if (const auto it = primeToFirstIndex.find(primeFactor);
            it != primeToFirstIndex.cend())
          uf.unionBySize(it->second, i);
        else
          primeToFirstIndex[primeFactor] = i;

    for (int i = 0; i < n; ++i)
      if (uf.getSize(i) == n)
        return true;

    return false;
  }

 private:
  // Gets the minimum prime factor of i, where 1 < i <= n.
  vector<int> sieveEratosthenes(int n) {
    vector<int> minPrimeFactors(n + 1);
    iota(minPrimeFactors.begin() + 2, minPrimeFactors.end(), 2);
    for (int i = 2; i * i < n; ++i)
      if (minPrimeFactors[i] == i)  // `i` is prime.
        for (int j = i * i; j < n; j += i)
          minPrimeFactors[j] = min(minPrimeFactors[j], i);
    return minPrimeFactors;
  }

  vector<int> getPrimeFactors(int num, const vector<int>& minPrimeFactors) {
    vector<int> primeFactors;
    while (num > 1) {
      const int divisor = minPrimeFactors[num];
      primeFactors.push_back(divisor);
      while (num % divisor == 0)
        num /= divisor;
    }
    return primeFactors;
  }
};

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class UnionFind {
  public UnionFind(int n) {
    id = new int[n];
    sz = new int[n];
    for (int i = 0; i < n; ++i)
      id[i] = i;
    for (int i = 0; i < n; ++i)
      sz[i] = 1;
  }

  public void unionBySize(int u, int v) {
    final int i = find(u);
    final int j = find(v);
    if (i == j)
      return;
    if (sz[i] < sz[j]) {
      sz[j] += sz[i];
      id[i] = j;
    } else {
      sz[i] += sz[j];
      id[j] = i;
    }
  }

  public int getSize(int i) {
    return sz[i];
  }

  private int[] id;
  private int[] sz;

  private int find(int u) {
    return id[u] == u ? u : (id[u] = find(id[u]));
  }
}

class Solution {
  public boolean canTraverseAllPairs(int[] nums) {
    final int n = nums.length;
    final int mx = Arrays.stream(nums).max().getAsInt();
    final int[] minPrimeFactors = sieveEratosthenes(mx + 1);
    Map<Integer, Integer> primeToFirstIndex = new HashMap<>();
    UnionFind uf = new UnionFind(n);

    for (int i = 0; i < n; ++i)
      for (final int primeFactor : getPrimeFactors(nums[i], minPrimeFactors))
        // `primeFactor` already appeared in the previous indices.
        if (primeToFirstIndex.containsKey(primeFactor))
          uf.unionBySize(primeToFirstIndex.get(primeFactor), i);
        else
          primeToFirstIndex.put(primeFactor, i);

    for (int i = 0; i < n; ++i)
      if (uf.getSize(i) == n)
        return true;
    return false;
  }

  // Gets the minimum prime factor of i, where 1 < i <= n.
  private int[] sieveEratosthenes(int n) {
    int[] minPrimeFactors = new int[n + 1];
    for (int i = 2; i <= n; ++i)
      minPrimeFactors[i] = i;
    for (int i = 2; i * i < n; ++i)
      if (minPrimeFactors[i] == i) // `i` is prime.
        for (int j = i * i; j < n; j += i)
          minPrimeFactors[j] = Math.min(minPrimeFactors[j], i);
    return minPrimeFactors;
  }

  private List<Integer> getPrimeFactors(int num, int[] minPrimeFactors) {
    List<Integer> primeFactors = new ArrayList<>();
    while (num > 1) {
      final int divisor = minPrimeFactors[num];
      primeFactors.add(divisor);
      while (num % divisor == 0)
        num /= divisor;
    }
    return primeFactors;
  }
}

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class UnionFind:
  def __init__(self, n: int):
    self.id = list(range(n))
    self.sz = [1] * n

  def unionBySize(self, u: int, v: int) -> None:
    i = self._find(u)
    j = self._find(v)
    if i == j:
      return
    if self.sz[i] < self.sz[j]:
      self.sz[j] += self.sz[i]
      self.id[i] = j
    else:
      self.sz[i] += self.sz[j]
      self.id[j] = i

  def getSize(self, i: int) -> int:
    return self.sz[i]

  def _find(self, u: int) -> int:
    if self.id[u] != u:
      self.id[u] = self._find(self.id[u])
    return self.id[u]


class Solution:
  def canTraverseAllPairs(self, nums: list[int]) -> bool:
    n = len(nums)
    mx = max(nums)
    maxPrimeFactor = self._sieveEratosthenes(mx + 1)
    primeToFirstIndex = collections.defaultdict(int)
    uf = UnionFind(n)

    for i, num in enumerate(nums):
      for prime_factor in self._getPrimeFactors(num, maxPrimeFactor):
        if prime_factor in primeToFirstIndex:
          uf.unionBySize(primeToFirstIndex[prime_factor], i)
        else:
          primeToFirstIndex[prime_factor] = i

    return any(uf.getSize(i) == n for i in range(n))

  def _sieveEratosthenes(self, n: int) -> list[int]:
    """Gets the minimum prime factor of i, where 1 < i <= n."""
    minPrimeFactors = [i for i in range(n + 1)]
    for i in range(2, int(n**0.5) + 1):
      if minPrimeFactors[i] == i:  # `i` is prime.
        for j in range(i * i, n, i):
          minPrimeFactors[j] = min(minPrimeFactors[j], i)
    return minPrimeFactors

  def _getPrimeFactors(self, num: int, minPrimeFactors: list[int]) -> list[int]:
    primeFactors = []
    while num > 1:
      divisor = minPrimeFactors[num]
      primeFactors.append(divisor)
      while num % divisor == 0:
        num //= divisor
    return primeFactors