#Bron-Kerbosch Algorithm in C++

[tpointtech1]

Bron-Kerbosch Algorithm in C++

The Bron-Kerbosch algorithm in C++ is an efficient recursive method for finding all maximal cliques in an undirected graph. It systematically explores subsets of vertices, expanding cliques while maintaining constraints to avoid unnecessary checks. The algorithm uses three sets: candidates, current clique, and excluded vertices. By recursively refining these sets, it identifies all maximal cliques without redundantly revisiting already discovered ones, making it useful for graph analysis and network research.

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[kaygun]

Bron-Kerbosch Algorithm in Clojure

Description of the problem

Assume we have a simple undirected graph $G=(V,E)$. Then a subset $C$ of the set of vertices $V$ is called a clique if for every pair of distinct elements $a,b\in C$ the pair $\{a,b\}$ is an edge in $G$. The set of all cliques of a graph is partially ordered under set inclusion. A clique is called maximal if it is a maximal element in this poset. Today, I am going to implement a version of the Bron-Kerbosch Algorithm which finds the set of all maximal cliques in a graph. I wrote about it earlier but that version was in common lisp, and today I am going to write an implementation from scratch in clojure.

An implementation in clojure

Let us start with a clean namespace with the necessary libraries loaded:

(ns bron-kerbosch (:use clojure.set))

I am going to represent a graph as a set of edges where each edge is a pair of distinct elements chosen from the underlying set of vertices. I am going to need two utility functions for later. The first returns the set of vertices for a given graph, while the second returns the set of neighbors of a given vertex in a given graph.

(defn vertices [graph] (reduce union graph)) (defn neighbors [v graph] (difference (reduce union (filter #(contains? % v) graph)) #{v}))

#'bron-kerbosch/vertices #'bron-kerbosch/neighbors

Finally, the main function, which is implemented as a recursive function:

(defn bron-kerbosch ([graph] (bron-kerbosch graph (vertices graph) #{} #{} #{})) ([graph p x r res] (if (and (empty? x) (empty? p)) (union #{r} res) (loop [ps p xs x cs res] (if (empty? ps) cs (let [v (first ps) nv (neighbors v graph)] (recur (into #{} (rest ps)) (union #{v} xs) (bron-kerbosch graph (intersection ps nv) (intersection xs nv) (union #{v} r) cs))))))))

#'bron-kerbosch/bron-kerbosch

In order to test the code, I am going to need a function that generates a random simple graph:

(defn get-random-graph [n m k] (->> (range n) (mapcat (fn [i] (repeatedly m (fn [] #{i (+ 1 i (rand-int k))})))) distinct (into #{})))

#'bron-kerbosch/get-random-graph

OK. Here is a random graph:

(def random-graph (get-random-graph 12 5 9)) random-graph

#'bron-kerbosch/random-graph #{#{13 5} #{19 11} #{3 11} #{6 3} #{0 1} #{14 10} #{15 11} #{13 9} #{7 16} #{7 6} #{2 10} #{3 5} #{5 10} #{12 10} #{0 4} #{3 10} #{3 12} #{11 16} #{17 11} #{0 7} #{4 8} #{0 3} #{5 14} #{7 13} #{10 8} #{6 11} #{6 8} #{17 10} #{12 5} #{4 10} #{7 2} #{13 11} #{15 8} #{1 6} #{9 10} #{4 2} #{7 9} #{1 3} #{15 9} #{1 10} #{4 9} #{1 9} #{13 10} #{14 8} #{17 8} #{0 5} #{10 18}}

And now, let us test our main function on this random graph:

(bron-kerbosch random-graph)

#{#{3 12 5 10} #{19 11} #{15 11} #{7 16} #{7 6} #{0 4} #{11 16} #{17 11} #{0 7} #{14 10 8} #{5 14 10} #{7 13 9} #{1 6 3} #{1 9 10} #{6 8} #{7 2} #{13 9 10} #{1 3 10} #{13 11} #{15 8} #{0 1 3} #{0 3 5} #{6 3 11} #{15 9} #{17 10 8} #{4 9 10} #{4 10 8} #{4 2 10} #{13 5 10} #{10 18}}