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Graphs 2

1. Strongly Connected Components​

In a directed graph, an SCC is a maximal set of vertices where every vertex can reach every other.

When to use

  • “Mutual reachability” in directed graphs
  • Compressing a directed graph into a DAG of components (condensation graph)
  • 2-SAT (classic application)

Key CP patterns

  • SCC compression turns messy directed graphs into a DAG, enabling DP/topo solutions.
  • Typical algorithms:
    • Kosaraju: two DFS passes (on graph and reversed graph)
    • Tarjan: one DFS pass with low-link values

Complexity

  • O(n + m)

Practice:

2. Union-Find (Disjoint Set Union, DSU)​

DSU maintains a partition of nodes into connected components under merges.

When to use

  • Connectivity under union operations
  • Common use case: “How many components?” after each edge addition
  • Can maintain aggregate properties per component (size, min/max, etc.)

Implementation Details

  • Use path compression + union by size/rank for near-constant time.

Complexity

  • Amortized ~ O(α(n)) per operation (effectively constant)

Practice:

3. Minimum Spanning Trees (MST)​

Given a connected weighted undirected graph, an MST connects all nodes with minimum total edge weight.

When to use

  • “Connect everything as cheaply as possible”
  • Network design / wiring / roads problems
  • Often paired with DSU or a priority queue

Two standard algorithms

  • Kruskal: sort edges by weight, add if it connects two different components (DSU)
    • Great when you already have an edge list and m is manageable.
  • Prim: grow from a start node using a min-heap over frontier edges
    • Great for dense graphs or adjacency-based input.

Why greedy works (intuition) MST uses the cut property: for any cut, the lightest edge crossing it is safe to take.

Complexity

  • Kruskal: O(m log m) (sorting dominates)
  • Prim: O(m log n) with heap

Practice: