算法(02)并查集

并查集也叫作不相交集合(Disjoint Set)

并查集有2个核心操作

• (Find):查找元素所在的集合(这里的集合并不是特指Set这种数据结构，是指广义的数据集合)
• 合并(Union):将两个元素所在的集合合并为一个集合

有2种常见的实现思路

• Quick Find
  ✓ 查找(Find)的时间复杂度:O(1)
  ✓ 合并(Union)的时间复杂度:O(n)
• Quick Union
  ✓查找(Find)的时间复杂度:O(logn)，可以优化至 O(𝛼(𝑛)) ，α(𝑛) < 5
  ✓合并(Union)的时间复杂度:O(logn)，可以优化至 O(𝛼(𝑛)) ，α(𝑛) < 5

UnionFind.java

public abstract class UnionFind {
    protected int[] parents;
    
    public UnionFind(int capacity) {
        if (capacity < 0) {
            throw new IllegalArgumentException("capacity must be >= 1");
        }
        
        parents = new int[capacity];
        for (int i = 0; i < parents.length; i++) {
            parents[i] = i;
        }
    }
    
    /**
     * 查找v所属的集合（根节点）
     * @param v
     * @return
     */
    public abstract int find(int v);

    /**
     * 合并v1、v2所在的集合
     */
    public abstract void union(int v1, int v2);
    
    /**
     * 检查v1、v2是否属于同一个集合
     */
    public boolean isSame(int v1, int v2) {
        return find(v1) == find(v2);
    }
    
    protected void rangeCheck(int v) {
        if (v < 0 || v >= parents.length) {
            throw new IllegalArgumentException("v is out of bounds");
        }
    }
}

UnionFind_QF.java

public class UnionFind_QF extends UnionFind {
    public UnionFind_QF(int capacity) {
        super(capacity);
    }

    /*
     * 父节点就是根节点
     */
    public int find(int v) {
        rangeCheck(v);
        return parents[v];
    }

    /**
     * 将v1所在集合的所有元素，都嫁接到v2的父节点上
     */
    public void union(int v1, int v2) {
        int p1 = find(v1);
        int p2 = find(v2);
        if (p1 == p2) return;
        
        for (int i = 0; i < parents.length; i++) {
            if (parents[i] == p1) {
                parents[i] = p2;
            }
        }
    }
}

UnionFind_QU.java

public class UnionFind_QU extends UnionFind {

    public UnionFind_QU(int capacity) {
        super(capacity);
    }

    /**
     * 通过parent链条不断地向上找，直到找到根节点
     */
    public int find(int v) {
        rangeCheck(v);
        while (v != parents[v]) {
            v = parents[v];
        }
        return v;
    }

    /**
     * 将v1的根节点嫁接到v2的根节点上
     */
    public void union(int v1, int v2) {
        int p1 = find(v1);
        int p2 = find(v2);
        if (p1 == p2) return;
        parents[p1] = p2;
    }

}

UnionFind_QU_R.java

/**
 * Quick Union - 基于rank的优化
 */
public class UnionFind_QU_R extends UnionFind_QU {
    private int[] ranks;

    public UnionFind_QU_R(int capacity) {
        super(capacity);

        ranks = new int[capacity];
        for (int i = 0; i < ranks.length; i++) {
            ranks[i] = 1;
        }
    }
    
    public void union(int v1, int v2) {
        int p1 = find(v1);
        int p2 = find(v2);
        if (p1 == p2) return;
        
        if (ranks[p1] < ranks[p2]) {
            parents[p1] = p2;
        } else if (ranks[p1] > ranks[p2]) {
            parents[p2] = p1;
        } else {
            parents[p1] = p2;
            ranks[p2] += 1;
        }
    }
}

UnionFind_QU_S.java

/**
 * Quick Union - 基于size的优化
 */
public class UnionFind_QU_S extends UnionFind_QU {
    private int[] sizes;

    public UnionFind_QU_S(int capacity) {
        super(capacity);
        
        sizes = new int[capacity];
        for (int i = 0; i < sizes.length; i++) {
            sizes[i] = 1;
        }
    }

    /**
     * 将v1的根节点嫁接到v2的根节点上
     */
    public void union(int v1, int v2) {
        int p1 = find(v1);
        int p2 = find(v2);
        if (p1 == p2) return;
        
        if (sizes[p1] < sizes[p2]) {
            parents[p1] = p2;
            sizes[p2] += sizes[p1];
        } else {
            parents[p2] = p1;
            sizes[p1] += sizes[p2];
        }
    }

}

UnionFind_QU_R_PC.java

/**
 * Quick Union - 基于rank的优化 - 路径压缩(Path Compression)
 *
 */
public class UnionFind_QU_R_PC extends UnionFind_QU_R {

    public UnionFind_QU_R_PC(int capacity) {
        super(capacity);
    }
    
    @Override
    public int find(int v) { // v == 1, parents[v] == 2
        rangeCheck(v);
        if (parents[v] != v) {
            parents[v] = find(parents[v]);
        }
        return parents[v];
    }
}

UnionFind_QU_R_PH.java

/**
 * Quick Union - 基于rank的优化 - 路径减半(Path Halving)
 *
 */
public class UnionFind_QU_R_PH extends UnionFind_QU_R {

    public UnionFind_QU_R_PH(int capacity) {
        super(capacity);
    }
    
    @Override
    public int find(int v) { 
        rangeCheck(v);
        while (v != parents[v]) {
            parents[v] = parents[parents[v]];
            v = parents[v];
        }
        return v;
    }
}

UnionFind_QU_R_PS.java

/**
 * Quick Union - 基于rank的优化 - 路径分裂(Path Spliting)
 *
 */
public class UnionFind_QU_R_PS extends UnionFind_QU_R {

    public UnionFind_QU_R_PS(int capacity) {
        super(capacity);
    }
    
    @Override
    public int find(int v) { 
        rangeCheck(v);
        while (v != parents[v]) {
            int p = parents[v];
            parents[v] = parents[parents[v]];
            v = p;
        }
        return v;
    }
}

泛型的并查集 GenericUnionFind.java

public class GenericUnionFind<V> {
    private Map<V, Node<V>> nodes = new HashMap<>();

    public void makeSet(V v) {
        if (nodes.containsKey(v)) return;
        nodes.put(v, new Node<>(v));
    }
    
    /**
     * 找出v的根节点
     */
    private Node<V> findNode(V v) {
        Node<V> node = nodes.get(v);
        if (node == null) return null;
        while (!Objects.equals(node.value, node.parent.value)) {
            node.parent = node.parent.parent;
            node = node.parent;
        }
        return node;
    }
    
    public V find(V v) {
        Node<V> node = findNode(v);
        return node == null ? null : node.value;
    }
    
    public void union(V v1, V v2) {
        Node<V> p1 = findNode(v1);
        Node<V> p2 = findNode(v2);
        if (p1 == null || p2 == null) return;
        if (Objects.equals(p1.value, p2.value)) return;
        
        if (p1.rank < p2.rank) {
            p1.parent = p2;
        } else if (p1.rank > p2.rank) {
            p2.parent = p1;
        } else {
            p1.parent = p2;
            p2.rank += 1;
        }
    }
    
    public boolean isSame(V v1, V v2) {
        return Objects.equals(find(v1), find(v2));
    }
    
    private static class Node<V> {
        V value;
        Node<V> parent = this;
        int rank = 1;
        Node(V value) {
            this.value = value;
        }
    }
}