Type-Level Maximum Flow in TypeScript

This challenge involves implementing a maximum flow algorithm at the TypeScript type level. You will represent a directed graph with capacities as a type, and your solution will be a type that computes the maximum flow from a specified source to a sink. This exercise explores the expressive power of TypeScript's type system for solving algorithmic problems.

Problem Description

The goal is to create a TypeScript type, MaxFlow<Graph, Source, Sink>, that statically computes the maximum flow value from a Source node to a Sink node within a given Graph. The graph and its capacities will be defined using TypeScript's type system.

Key Requirements:

1. Graph Representation: The graph will be represented as a tuple of edge objects. Each edge object will define:

   • from: The starting node of the edge.
   • to: The ending node of the edge.
   • capacity: The maximum flow that can pass through this edge.
   • Nodes will be represented by string literals.

2. Maximum Flow Computation: The MaxFlow type should implement a maximum flow algorithm (e.g., Ford-Fulkerson or Edmonds-Karp, but adapted for type-level computation) to find the maximum flow.

3. Output: The MaxFlow type should resolve to a union of numbers, where each number represents a possible flow value. The largest number in this union will be the maximum flow.

4. Iterative Refinement: The type-level computation will likely involve an iterative process of finding augmenting paths and updating residual capacities. This can be achieved using recursive conditional types.

Expected Behavior:

Given a graph definition, a source node, and a sink node, the MaxFlow type should, at compile time, determine the maximum possible flow from the source to the sink.

Edge Cases:

• Graphs with no path from source to sink.
• Graphs with zero capacity edges.
• Graphs with cycles.
• Source and sink being the same node.

Examples

Example 1: A simple path

type SimpleGraph = [
  { from: 'A'; to: 'B'; capacity: 10 },
  { from: 'B'; to: 'C'; capacity: 5 }
];

// Expected: MaxFlow<SimpleGraph, 'A', 'C'> should resolve to 5

Example 2: Multiple paths

type MultiPathGraph = [
  { from: 'S'; to: 'A'; capacity: 10 },
  { from: 'S'; to: 'B'; capacity: 5 },
  { from: 'A'; to: 'T'; capacity: 10 },
  { from: 'B'; to: 'T'; capacity: 5 }
];

// Expected: MaxFlow<MultiPathGraph, 'S', 'T'> should resolve to 15

Example 3: Bottleneck capacity

type BottleneckGraph = [
  { from: 'S'; to: 'A'; capacity: 10 },
  { from: 'S'; to: 'B'; capacity: 10 },
  { from: 'A'; to: 'C'; capacity: 5 },
  { from: 'B'; to: 'C'; capacity: 10 },
  { from: 'C'; to: 'T'; capacity: 10 }
];

// Expected: MaxFlow<BottleneckGraph, 'S', 'T'> should resolve to 15

Example 4: No path

type NoPathGraph = [
  { from: 'A'; to: 'B'; capacity: 10 }
];

// Expected: MaxFlow<NoPathGraph, 'A', 'C'> should resolve to 0

Constraints

• The number of nodes and edges will be relatively small (e.g., up to 10 nodes and 20 edges) to remain within TypeScript's type computation limits.
• Capacities will be positive integers up to 100.
• Node names will be short string literals.
• The MaxFlow type should ideally terminate within a reasonable number of type instantiations.

Notes

• This challenge requires a deep understanding of conditional types, tuple manipulation, and recursive type definitions in TypeScript.
• Consider how to represent and update residual graphs at the type level.
• Think about how to find augmenting paths using type-level pattern matching and filtering.
• The problem is designed to be challenging and may require significant experimentation with TypeScript's type system.
• Success looks like having a MaxFlow type that correctly computes the maximum flow for the provided examples and satisfies the constraints.