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Problem solution · Java

Greatest Common Divisor Traversal

Greatest Common Divisor Traversal: a Java solution using disjoint set union. Learn the idea, check the complexity, and read the full code, with credit to walkccc LeetCode Solutions.

Technique
Disjoint set union
Source
walkccc LeetCode Solutions
Length
82 lines
Start with the idea.

Try the problem first. If you get stuck, read the approach below, then write your own solution. The full code is at the bottom.

Approach

Disjoint set union

For Greatest Common Divisor Traversal, the implementation maintains connected components and merges them as relationships are processed.

  1. Give each element a component representative.
  2. Merge representatives when a connection is accepted.
  3. Answer connectivity or component queries from the compressed representatives.

Code notes

  • 82 lines of Java from the credited upstream file 2709.java.
  • The implementation visibly relies on sequence storage, hash lookup, ordered lookup.
  • 10 loop blocks detected.

Complexity

Account for every find and union operation; with path compression and ranked merging, the amortized cost is nearly constant per operation.

Check the problem constraints before deciding whether this complexity will pass.

Source

Code and credit

This code comes from walkccc LeetCode Solutions by P.-Y. Chen (walkccc) and is used under the MIT licence.

Open problem ↗Original source ↗
Full codeGreatest Common Divisor Traversal · JavaJava
Use this to learn the idea, then write your own version.
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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