Type-Level Strongly Connected Components in TypeScript

This challenge asks you to implement a type-level algorithm in TypeScript to identify strongly connected components (SCCs) within a directed graph. This is a foundational problem in graph theory with applications in areas like dependency analysis, compiler optimization, and state machine modeling, where understanding cyclical relationships is crucial.

Problem Description

You need to create a set of TypeScript conditional types that can take a representation of a directed graph and output its strongly connected components. A strongly connected component is a subgraph where for any two nodes U and V in the subgraph, there's a path from U to V and a path from V to U. The graph will be represented using a tuple of tuples, where each inner tuple represents an edge.

Requirements:

• Graph Representation: The input graph will be represented as a tuple of tuples: [[NodeA, NodeB], [NodeB, NodeC], ...], where NodeX is a string literal representing a node.
• Output Format: The output should be a tuple of tuples, where each inner tuple represents an SCC. Each SCC tuple will contain the string literal node names belonging to that component. The order of SCCs and nodes within an SCC does not matter.
• Type-Level Computation: All computations must be performed purely at the TypeScript type level. No runtime JavaScript code should be involved in the SCC calculation itself.
• Handling Cycles: The algorithm must correctly identify SCCs, including single-node SCCs for nodes that are not part of any cycle.
• Handling Disconnected Graphs: The solution should work correctly for graphs with multiple disconnected components.

Expected Behavior:

Given a type representing a graph, the output type should be a tuple of tuples, where each inner tuple lists the nodes within a single SCC.

Examples

Example 1:

type Graph1 = [
  ["A", "B"],
  ["B", "C"],
  ["C", "A"],
  ["B", "D"]
];

// Expected Output:
// [
//   ["A", "B", "C"],
//   ["D"]
// ]

Explanation: Nodes A, B, and C form a cycle (A -> B -> C -> A), making them a single SCC. Node D is reachable from B but has no outgoing edges to A, B, or C, so it forms its own SCC.

Example 2:

type Graph2 = [
  ["X", "Y"],
  ["Y", "Z"],
  ["Z", "X"],
  ["A", "B"],
  ["B", "A"]
];

// Expected Output:
// [
//   ["X", "Y", "Z"],
//   ["A", "B"]
// ]

Explanation: The graph consists of two separate cycles, each forming its own SCC.

Example 3: (Single node SCCs)

type Graph3 = [
  ["P", "Q"],
  ["R", "S"]
];

// Expected Output:
// [
//   ["P"],
//   ["Q"],
//   ["R"],
//   ["S"]
// ]

Explanation: There are no cycles. Each node forms its own SCC because there's no path back to itself or any other node in a cyclic manner.

Example 4: (Self-loop)

type Graph4 = [
  ["M", "N"],
  ["N", "N"] // Self-loop
];

// Expected Output:
// [
//   ["M"],
//   ["N"]
// ]

Explanation: Node N has a self-loop, making it an SCC of size 1. Node M can reach N, but N cannot reach M, so M is its own SCC.

Constraints

• Node names are string literals (e.g., "A", "Node1").
• The graph is represented as a tuple of tuples of string literals.
• The number of nodes and edges will be within reasonable limits for TypeScript's type system (e.g., not exceeding 100 nodes or 200 edges to avoid excessively long compile times).
• The solution must be entirely type-level; no runtime code execution for the core SCC algorithm.

Notes

• This is a challenging problem that requires a deep understanding of TypeScript's conditional types, recursive type inference, and tuple manipulation.
• You will likely need to build helper types for tasks such as:
  • Extracting all unique nodes from the graph.
  • Finding direct neighbors of a node.
  • Performing reachability analysis (type-level DFS/BFS).
  • Breaking down the graph into subgraphs.
• Consider algorithms like Tarjan's algorithm or Kosaraju's algorithm as inspiration for the logic, but adapt them to the type system. You might need to manage state (like visited nodes, discovery times, low-link values) using intermediate types.
• The order of nodes within an SCC and the order of SCCs in the output tuple do not matter. However, the output should be a tuple of tuples.
• Start by breaking down the problem into smaller, manageable type-level operations.