user
Objective

Develop a program that identifies and lists all maximal cliques within an undirected graph. A *maximal clique* is a subset of vertices where every two distinct vertices are adjacent (i.e., there's an edge connecting them), and the clique cannot be extended by including any adjacent vertex not already in the clique.

Background

In graph theory, a *clique* is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are connected by an edge. A *maximal clique* is a clique that cannot be extended by adding one more adjacent vertex, meaning it's not a subset of a larger clique.

The Bron-Kerbosch algorithm is a recursive backtracking algorithm used to find all maximal cliques in an undirected graph efficiently. The algorithm is particularly effective due to its use of pivot selection, which reduces the number of recursive calls and improves performance.

Program Functionality

1. **Input Representation:**

* The graph is represented as a list of edges, where each edge is a tuple of two vertices (e.g., `('a', 'b')` indicates an undirected edge between vertices `'a'` and `'b'`).

1. **Graph Construction:**

* Convert the list of edges into an adjacency list using a `HashMap<String, HashSet<String>>`. Each key in the hashmap represents a vertex, and its corresponding hash set contains all adjacent vertices (neighbors).

1. **Bron-Kerbosch Algorithm Implementation:**

* **Sets Used:**
  + *R* (Current Clique): *A set representing the current clique being constructed.*
  + *P* (Potential Candidates): *A set of vertices that can be added to* R *to form a larger clique.*
  + *X* (Excluded Vertices): *A set of vertices that have already been processed and should not be reconsidered for the current clique.*

* **Algorithm Steps:**
  + If both *P* and *X* are empty, and *R* contains more than two vertices, *R* is a maximal clique and is added to the list of cliques.
  + Otherwise, select a *pivot vertex* from the union of *P* and *X* that has the maximum number of neighbors. This pivot helps minimize the number of recursive calls.
  + Iterate over all vertices in *P* that are \*\*not\*\* neighbors of the pivot. For each such vertex:
    - Add the vertex to *R*.
    - Update *P* and *X* to include only those vertices that are neighbors of the added vertex.
    - Recursively call the Bron-Kerbosch function with the updated sets.
    - After recursion, move the vertex from *P* to *X* to mark it as processed.

1. **Output:**

* After executing the algorithm, the program prints all maximal cliques with more than two vertices. Each clique is displayed as a comma-separated list of its constituent vertices.

Example

Given the following edges:

```
('a', 'b'), ('b', 'a'), ('a', 'c'), ('c', 'a'), ('b', 'c'), ('c', 'b'),
('d', 'e'), ('e', 'd'), ('d', 'f'), ('f', 'd'), ('e', 'f'), ('f', 'e')

```

**Expected Output:**

```
a, b, c
d, e, f

```

These outputs represent the two maximal cliques in the graph: one comprising vertices `a`, `b`, and `c`, and the other comprising vertices `d`, `e`, and `f`.

Pseudocode

Below is the pseudocode for the Bron-Kerbosch algorithm with pivoting, closely following the Rust implementation provided earlier.

```
FUNCTION BronKerbosch(R, P, X, G, Cliques)
    // R: Current clique (Set)
    // P: Potential candidates to expand the clique (Set)
    // X: Vertices already processed (Set)
    // G: Graph represented as an adjacency list (Dictionary)
    // Cliques: List to store all maximal cliques (List of Lists)

    IF P is empty AND X is empty THEN
        IF size of R > 2 THEN
            SORT R lexicographically
            ADD R to Cliques
        END IF
        RETURN
    END IF

    // Select a pivot vertex from P ∪ X with the maximum degree
    Pivot := vertex in (P ∪ X) with the highest number of neighbors in G

    // Candidates are vertices in P that are not neighbors of the pivot
    Candidates := P - Neighbors(Pivot)

    FOR EACH vertex v IN Candidates DO
        // New clique including vertex v
        NewR := R ∪ {v}

        // New potential candidates are neighbors of v in P
        Neighbors_v := G[v]
        NewP := P ∩ Neighbors_v

        // New excluded vertices are neighbors of v in X
        NewX := X ∩ Neighbors_v

        // Recursive call with updated sets
        BronKerbosch(NewR, NewP, NewX, G, Cliques)

        // Move vertex v from P to X
        P := P - {v}
        X := X ∪ {v}
    END FOR
END FUNCTION

FUNCTION Main()
    // Define input edges as a list of tuples
    Input := [
        ('a', 'b'), ('b', 'a'), ('a', 'c'), ('c', 'a'),
        ('b', 'c'), ('c', 'b'), ('d', 'e'), ('e', 'd'),
        ('d', 'f'), ('f', 'd'), ('e', 'f'), ('f', 'e')
    ]

    // Build the graph as an adjacency list
    Graph := empty Dictionary
    FOR EACH (src, dest) IN Input DO
        IF src NOT IN Graph THEN
            Graph[src] := empty Set
        END IF
        ADD dest TO Graph[src]
    END FOR

    // Initialize R, P, X
    R := empty Set
    P := set of all vertices in Graph
    X := empty Set

    // Initialize list to store cliques
    Cliques := empty List

    // Execute the Bron-Kerbosch algorithm
    BronKerbosch(R, P, X, Graph, Cliques)

    // Sort the cliques for consistent output
    SORT Cliques lexicographically

    // Print each clique
    FOR EACH Clique IN Cliques DO
        PRINT elements of Clique separated by commas
    END FOR
END FUNCTION

```
assistant
```
# Returns all maximal Cliques as sorted List of Lists.
sub BronKerbosch ( @undirected_edges ) {

    #| Adjacency list of the whole Graph, as immutable hash (Map) of immutable Sets.
    my Map $G = @undirected_edges
                .map({ |( .[0,1], .[1,0] ) })
                .classify( {.[0]}, :as{.[1]} )
                .duckmap( *.Set )
                .Map;

    #| Number of neighbors in G
    my Map $degree = $G.map({ .key => .value.elems }).Map;

    return gather BK();

    sub BK (
        Set     :$R         = Set.new,              #= Current clique
        SetHash :$P is copy = SetHash.new($G.keys), #= Potential candidates to expand the clique
        SetHash :$X is copy = SetHash.new,          #= Vertices already processed
    ) {
        if !$P and !$X {
            take $R.keys.sort.cache if $R.elems > 2;
            return;
        }

        my $Pivot = ($P ∪ $X).max({ $degree{$_} }).key;

        my @Candidates = ( $P (-) $G{$Pivot} ).keys;

        for $G{@Candidates}:kv -> $v, $Neighbors_of_v {

            &?ROUTINE.(     # Recursive call with updated sets
                R => ($R ∪ $v),
                P => ($P ∩ $Neighbors_of_v),
                X => ($X ∩ $Neighbors_of_v),
            );

            # Move vertex v from P to X
            $P{$v} = False;
            $X{$v} = True;
        }
    }
}

say .join(",") for sort BronKerbosch <a-b a-c b-c d-e d-f e-f>».split("-");

```

Output:

```
a,b,c
d,e,f
```