Type-Level Graph Traversal: Depth-First Search in TypeScript

This challenge focuses on implementing a graph traversal algorithm, specifically Depth-First Search (DFS), entirely at the type level in TypeScript. This means the entire graph structure and the traversal logic will be defined using TypeScript's type system, allowing for compile-time validation and exploration of graph properties without runtime execution. This is useful for understanding advanced type system capabilities and for scenarios where compile-time guarantees about graph structure and reachability are paramount.

Problem Description

Your task is to implement a type-level Depth-First Search (DFS) in TypeScript. You will define a way to represent a graph using types, and then create a type that, given a graph and a starting node, can determine the sequence of nodes visited by a DFS traversal.

What needs to be achieved:

1. Graph Representation: Define a type system for representing directed graphs. A graph will consist of nodes and edges. An edge connects one node to another.
2. DFS Implementation: Create a type-level function (a mapped type or conditional type) that performs DFS.
3. Output: The output of your DFS type should be a tuple of node identifiers representing the order in which nodes are visited during the DFS.

Key Requirements:

• Nodes: Nodes can be represented by string literals or numbers. For simplicity, let's assume string literals for node identifiers.
• Edges: Edges are directed. An edge from A to B means B is a neighbor of A.
• Visited Tracking: The DFS type must correctly handle visited nodes to avoid infinite loops in cyclic graphs.
• Order of Visitation: The output tuple should reflect the order in which nodes are first encountered.

Expected Behavior:

Given a graph and a starting node, the DFS type should explore as far as possible along each branch before backtracking. The sequence of visited nodes should be a tuple.

Edge Cases to Consider:

• Empty Graph: A graph with no nodes or edges.
• Disconnected Graph: A graph where not all nodes are reachable from the starting node.
• Cyclic Graph: A graph containing cycles.
• Starting Node Not in Graph: Handle the case where the specified start node is not part of the graph definition.

Examples

Example 1: Simple Linear Graph

Input Graph Representation:

type LinearGraph = {
  'A': ['B'];
  'B': ['C'];
  'C': [];
};

Input Start Node: 'A'

Output:

type LinearGraphDFS = ['A', 'B', 'C'];

Explanation: Starting from 'A', we visit 'B', then 'C'. Since 'C' has no outgoing edges, we backtrack.

Example 2: Branching Graph

Input Graph Representation:

type BranchingGraph = {
  'A': ['B', 'C'];
  'B': ['D'];
  'C': ['E'];
  'D': [];
  'E': [];
};

Input Start Node: 'A'

Output:

type BranchingGraphDFS = ['A', 'B', 'D', 'C', 'E'];

Explanation: Starting from 'A', we go to 'B', then 'D'. Backtrack to 'B', then to 'A'. Then go to 'C', then 'E'.

Example 3: Cyclic Graph

Input Graph Representation:

type CyclicGraph = {
  'A': ['B'];
  'B': ['C', 'A']; // Cycle B -> A
  'C': ['D'];
  'D': [];
};

Input Start Node: 'A'

Output:

type CyclicGraphDFS = ['A', 'B', 'C', 'D'];

Explanation: Starting from 'A', visit 'B'. From 'B', visit 'C', then 'D'. Backtrack to 'C', then to 'B'. From 'B', we see 'A' as a neighbor, but 'A' has already been visited, so we don't revisit it. Backtrack to 'A'.

Constraints

• Node identifiers will be string literals (e.g., 'A', 'Node1').
• The graph representation will be a mapped type where keys are node identifiers and values are tuples of adjacent node identifiers.
• The number of nodes and edges will be manageable within TypeScript's type system inference limits (avoid extremely large graphs that might cause excessive compilation times or stack overflows).
• The solution must be entirely at the type level. No runtime code should be necessary for the core DFS logic.

Notes

• Consider how to represent the "visited" set at the type level. This is crucial for handling cycles.
• Recursive conditional types will likely be a key tool for implementing the traversal logic.
• You might need helper types to manage the state of the traversal (e.g., current node, visited nodes, accumulated path).
• Think about the order in which neighbors are processed. The examples assume a left-to-right processing order of the neighbor tuples.