The DSU Template

The core data structure is a single array.

parent: [0, 1, 2, 3, ..., n-1]

Initially parent[i] = i — every element is its own root, in its own singleton tree.

Naive find

find(x) walks parent pointers up to the root:

def find(x):
    while parent[x] != x:
        x = parent[x]
    return x

The root is the element whose parent is itself.

Naive union

union(x, y) finds the roots of x and y. If they’re different, make one root point at the other:

def union(x, y):
    rx, ry = find(x), find(y)
    if rx == ry:
        return False            # already in same set
    parent[rx] = ry
    return True                 # union happened

Returning True/False is a useful convention — it tells the caller whether anything actually merged, which doubles as a cycle-detection signal.

The naive code, end to end

struct DSU {
    vector<int> parent;
    DSU(int n) : parent(n) {
        iota(parent.begin(), parent.end(), 0);
    }
    int find(int x) {
        while (parent[x] != x) x = parent[x];
        return x;
    }
    bool unite(int x, int y) {
        int rx = find(x), ry = find(y);
        if (rx == ry) return false;
        parent[rx] = ry;
        return true;
    }
    bool connected(int x, int y) { return find(x) == find(y); }
};

class DSU:
    def __init__(self, n):
        self.parent = list(range(n))
 
    def find(self, x):
        while self.parent[x] != x:
            x = self.parent[x]
        return x
 
    def union(self, x, y):
        rx, ry = self.find(x), self.find(y)
        if rx == ry:
            return False
        self.parent[rx] = ry
        return True
 
    def connected(self, x, y):
        return self.find(x) == self.find(y)

class DSU {
    int[] parent;
    DSU(int n) {
        parent = new int[n];
        for (int i = 0; i < n; i++) parent[i] = i;
    }
    int find(int x) {
        while (parent[x] != x) x = parent[x];
        return x;
    }
    boolean union(int x, int y) {
        int rx = find(x), ry = find(y);
        if (rx == ry) return false;
        parent[rx] = ry;
        return true;
    }
    boolean connected(int x, int y) {
        return find(x) == find(y);
    }
}

That’s it. Functionally correct, easy to read, and dangerously slow on a bad input.

The worst case — a degenerate chain

Build a DSU on {0, 1, ..., n-1}, then call:

union(0, 1)
union(1, 2)
union(2, 3)
...
union(n-2, n-1)

After each union, the smaller-indexed root gets re-parented to the larger one:

After union(0, 1):  0 → 1
After union(1, 2):  0 → 1 → 2
After union(2, 3):  0 → 1 → 2 → 3
...
After union(n-2, n-1):  0 → 1 → 2 → ... → n-1

The tree is a linked list. Now find(0) walks all n parent pointers — O(n).

m operations on a degenerate DSU cost O(m · n). That’s worse than naive renaming with a dict. We need to fix this.

Two complementary fixes

The two optimizations attack the problem from different angles:

1. Union by rank (or size) — when uniting two trees, attach the smaller tree under the larger root. Trees stay balanced; max depth grows like log n instead of n.
2. Path compression — every time find(x) walks up to the root, re-point every visited node directly to the root. Future finds on those nodes are O(1).

Each one alone gives a meaningful speedup. Together they’re devastating — amortized O(α(n)) per operation. That’s the next page.

Common variants of the template

A few small extensions show up repeatedly:

• Component count. Keep an integer components initialized to n. Decrement it whenever union returns True. At the end, components is the number of connected groups.
• Component sizes. Maintain a size[] array alongside parent[]. When uniting, the new root’s size is the sum of the two old roots’ sizes. Use cases: largest component, query “how big is x’s group?”.
• Weighted DSU. Store an offset weight[x] = “distance from x to its parent.” Maintained through find and union. Powers parity-check and equality-with-offset problems.

Quick check

Next: the two optimizations and the amortized analysis that makes DSU shockingly fast.