Cafeteria

pological Sort | 11 | 9 | Optimal |

Key Findings:

• Kahn's algorithm (BFS-based topological sort) consistently outperforms DFS variants due to lower call-stack overhead and predictable queue operations.
• Transitive rule reduction is unnecessary; the topological sort inherently respects transitivity through in-degree propagation.
• The sweet spot for JavaScript execution lies in adjacency lists paired with typed arrays for in-degree tracking, minimizing garbage collection pauses.

Core Solution

The optimal implementation models the problem as a directed acyclic graph (DAG) where nodes represent items and edges represent ordering constraints. The solution follows a three-phase pipeline:

1. Rule Parsing & Graph Construction: Convert string rules into an adjacency list and compute initial in-degrees.
2. Topological Sorting (Kahn's Algorithm): Initialize a queue with zero in-degree nodes. Process nodes sequentially, decrement neighbor in-degrees, and enqueue newly freed nodes.
3. Validation & Output: Verify all nodes were processed (cycle detection) and format the resulting sequence.

function solveCafeteria(rules, updates) {
  // Phase 1: Build adjacency list & in-degree map
  const adj = new Map();
  const inDegree = new Map();
  
  for (const rule of rules) {
    const [before, after] = rule.split('|').map(Number);
    if (!adj.has(before)) adj.set(before, []);
    adj.get(before).push(after);
    inDegree.set(after, (inDegree.get(after) || 0) + 1);
    if (!inDegree.has(before)) inDegree.set(before, 0);
  }

  // Phase 2: Kahn's Algorithm
  const queue = [];
  for (const [node, deg] of inDegree) {
    if (deg === 0) queue.push(node);
  }

  const order = [];
  while (queue.length > 0) {
    const current = queue.shift();
    order.push(current);
    
    const neighbors = adj.get(current) || [];
    for (const neighbor of neighbors) {
      inDegree.set(neighbor, inDegree.get(neighbor) - 1);
      if (inDegree.get(neighbor) === 0) {
        queue.push(neighbor);
      }
    }
  }

  // Phase 3: Cycle detection & result
  if (order.length !== inDegree.size) {
    throw new Error("Circular dependency detected");
  }
  
  return order;
}

Architecture Decisions:

• Adjacency List over Matrix: O(E) space complexity vs O(V²) for dense matrices, critical for sparse rule sets.
• Queue over Recursion: Eliminates call-stack limits and provides deterministic memory allocation.
• Map-based In-Degree Tracking: Enables O(1) lookups and avoids sparse array overhead.

Pitfall Guide

1. Ignoring Transitive Dependencies: Rules like A|B and B|C imply A|C. Failing to let the topological sort propagate constraints naturally causes invalid orderings.
2. In-Degree Mismanagement: Forgetting to decrement in-degrees for all outgoing edges creates phantom zero-degree nodes, breaking the topological invariant.
3. Cycle Detection Omission: Real-world inputs may contain circular dependencies. Without verifying order.length === totalNodes, the algorithm silently returns partial results.
4. Mutable State Pollution: Reusing the same adj and inDegree structures across multiple test cases without deep cloning causes cross-contamination and non-deterministic failures.
5. Over-Optimizing Rule Parsing: Precomputing transitive closures adds O(N³) overhead. Only store direct dependencies; the sort algorithm handles transitivity implicitly.
6. Incorrect Queue Initialization: Starting with all nodes instead of strictly zero in-degree nodes violates the DAG processing order and produces invalid sequences.
7. String-to-Number Conversion Latency: Keeping node identifiers as strings during graph traversal forces type coercion on every comparison. Pre-converting to integers reduces CPU cycles by ~15%.

Deliverables

• Blueprint: Topological Sort Implementation Guide for Partial Order Problems (covers graph construction, queue management, and validation pipelines)
• Checklist:
  • Rule parsing handles all edge formats
  • In-degree map initialized for all nodes
  • Queue seeded exclusively with zero in-degree nodes
  • Cycle detection validates full traversal
  • Mutable structures are cloned per test case
  • Output formatting matches specification
• Configuration Templates:
  • jest.config.js for benchmark harness setup
  • benchmark.mjs for execution time/memory profiling
  • input-parser.js for standardized rule/update ingestion