This code snippet is a function designed to determine whether...

April 3, 2025 at 10:18 PM

// Complete the 'isPossible' function below. public boolean isPossible() { ArrayList<Boolean> possible = new ArrayList<Boolean>(); for (Vertex v: vertices) { possible.add(CycleDetectionWrapper(v)); } if (possible.contains(false)) return false; else return true; } public boolean CycleDetectionWrapper(Vertex v){ HashMap<Vertex,Edge> record = new HashMap<>(); HashMap<Vertex,Edge> ancestors = new HashMap<>(); record.put(v,null); ancestors.put(v,null); return CycleDetection(record,v,ancestors); } public boolean CycleDetection(HashMap<Vertex,Edge> record, Vertex u, HashMap<Vertex,Edge> ancestors){ for (Edge e: u.outgoing){ Vertex v = opposite(u, e); if(ancestors.containsKey(v)) return true; if(record.containsKey(v)==false){ record.put(v,e); ancestors.put(v,null); if (CycleDetection(record,v,ancestors)==true) return true; ancestors.remove(v); } } return false; }

This code snippet is a function designed to determine whether a directed graph contains any cycles. Here's a detailed breakdown of what the code does:

Main Function: isPossible

  1. Goal: Determines if it is possible for the graph, represented by a collection of vertices (vertices), to be cycle-free.
  2. Process:
    • Creates an ArrayList called possible that will store a Boolean value (true or false) indicating if a cycle exists for each vertex.
    • For each vertex v in the graph:
      • Calls a helper function CycleDetectionWrapper(v) to check if starting from that vertex unveils a cycle.
      • Adds the result (true or false) to the possible list.
    • If the possible list contains any false (i.e., a cycle was found in the graph), the function returns false. Otherwise, it returns true.

Helper Function: CycleDetectionWrapper

  1. Goal: A wrapper function that prepares the necessary data structures for recursive cycle detection.
  2. Process:
    • Initializes two HashMap objects:
      • record tracks all vertices known (visited) so far in the traversal starting from vertex v.
      • ancestors keeps track of vertices in the current recursion stack (these are the "active path" vertices while exploring the potential cycle).
    • Adds the starting vertex v to both mapped structures (it is visited and is an ancestor).
    • Calls the CycleDetection recursive function starting at that vertex to check for any cycles.

Recursive Function: CycleDetection

  1. Goal: Implements the core logic for detecting cycles in a graph using a depth-first traversal approach.
  2. Parameters:
    • record: Tracks all the visited vertices globally, so we don't revisit a vertex unnecessarily.
    • u: The current vertex being explored.
    • ancestors: Tracks the recursion stack to detect back edges (which indicate a cycle in the graph).
  3. Logic:
    • For each outgoing edge from the current vertex u:
      • Determine the vertex v at the opposite end of the edge (opposite(u, e)).
      • If vertex v is already in the ancestors set, that means we found a cycle (a back edge), so return true.
      • If vertex v is not yet visited (i.e., it’s not in record):
        • Mark v as visited by adding it to record.
        • Add it as an ancestor (part of the current recursion stack).
        • Recursively call CycleDetection on this vertex v. If a cycle is detected deeper in the recursion, return true.
        • If no cycle is found, remove v from the ancestors (backtrack).
    • If all outgoing edges are explored and no cycle is detected, return false.

Overall Purpose

The isPossible method checks an entire directed graph (represented by a list of vertices where each vertex contains outgoing edges) for the presence of cycles. Specifically:

  • If any vertex in the graph is found to be part of a cycle when checked using depth-first search (DFS), the function returns false, indicating that cycles exist, and the operation is not "possible."
  • If no vertex participates in a cycle, the function returns true.

This code effectively uses DFS and backtracking to detect cycles in a graph.

Key Concepts Used

  1. Cycle Detection in Directed Graphs:
    • Utilizes DFS and the concept of back edges (edges leading to an ancestor in the recursion stack) to find cycles.
  2. Recursive Approach:
    • The recursive CycleDetection function dives deep into paths and backtracks when no cycle is found.
  3. HashMaps for Record Keeping:
    • record: Tracks globally visited vertices (to prevent revisiting).
    • ancestors: Tracks vertices in the current recursion stack to detect cycles.
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