Reconstructing the shortest path in a graph in BFS

Question

I am implementing a function in ruby using BFS, and I would like to know how to print the shortest path in an undirected graph, between a start value and end value

Graph In this example the graph has ten vertices. Each node represents a vertex and it has an array that stores adjacent nodes (edges).

$ irb
graph.to_s
1. 1 -> [2 3 4 5 8 9 10]
2. 2 -> [1 4 5 6 7 8 10]
3. 3 -> [1 4 6]
4. 4 -> [1 2 3 6 7 8 9 10]
5. 5 -> [1 2 8 9 10]
6. 6 -> [2 3 4 7 8 9 10]
7. 7 -> [2 4 6 8 9]
8. 8 -> [1 2 4 5 6 7 9 10]
9. 9 -> [1 4 5 6 7 8]
10. 10 -> [1 2 4 5 6 8]
=> [1, 2, 3, 4, 5, 6, 7, 8, 9, 10]

Expected Output

2,1,9 or 2,4,9, etc.

BFS

def bfs_shortest_path(graph, start=2, search=9)
    if graph.nodes[start].nil? || graph.nodes[search].nil?
        return nil
    end
    visited = Set.new
    search_queue = Queue.new
    search_queue.enq(start)
    while !search_queue.empty? do      
        current_node_key = search_queue.deq  
        current_node = graph.nodes[current_node_key]         
        visited.add(current_node_key)
        if current_node.value == search
            return current_node # I would like to return the PATH Eg. 2,1,9
        end
        adjacent_nodes_array = current_node.adjacent_nodes.map{|x| x.value}
        adjacent_nodes_array.each do |value|
            if !visited.include?(value)                   
                search_queue.enq(value)
                graph.nodes[value].concat_to_path(current_node.path_from_start)                
            end
        end        
    end
end

Node

class Node
    attr_reader :value
    attr_reader :adjacent_nodes
    def initialize(value)
        @value = value
        @adjacent_nodes = []
    end

    def add_edge(node)
        @adjacent_nodes.push(node)
    end

    def to_s
        "#{@value} -> [#{@adjacent_nodes.map(&:value).sort.join(" ")}]"
    end
end

Graph

class Graph
    attr_reader :nodes    
    def initialize
        @nodes = {}
    end

    def add_node(node)        
        @nodes[node.value] = node
    end

    def add_edge(node1,node2)
        if @nodes[node1.value].adjacent_nodes.map(&:value).include? (node2.value)        
            puts "#{node1.value} and #{node2.value} already have an edge"
        elsif node1.value == node2.value
            puts "node1.value == node2.value"
        else
            @nodes[node1.value].add_edge(@nodes[node2.value])
            @nodes[node2.value].add_edge(@nodes[node1.value])
        end
    end

    def to_s
        @nodes.keys.sort.each_with_index do |key,index|
            puts "#{index + 1}. #{@nodes[key].to_s}" 
        end
    end
end

Generating a Graph

def generate_random_graph
    g = Graph.new
    [*1..10].shuffle.each do |node_value|
        g.add_node(Node.new(node_value))
    end
    40.times do 
        key1 = g.nodes.keys.sample
        key2 = g.nodes.keys.sample
        g.add_edge(g.nodes[key1],g.nodes[key2])
    end
    return g
end

Test

graph = generate_random_graph    
graph.to_s
bfs_shortest_path(graph,2,9)

Answer 1

Thanks for the comments

Working Version

class Node
    attr_reader :value
    attr_reader :adjacent_nodes
    attr_reader :path_from_start
    def initialize(value)
        @value = value
        @adjacent_nodes = []
        @path_from_start = []
    end

    def add_edge(node)
        @adjacent_nodes.push(node)
    end

    def add_to_path(value)
        @path_from_start.push(value)
    end

    def concat_to_path(value_array)
        @path_from_start.concat(value_array)
    end

    def to_s
        "#{@value} -> [#{@adjacent_nodes.map(&:value).sort.join(" ")}]"
    end
end

class Graph
    attr_reader :nodes    
    def initialize
        @nodes = {}
    end

    def add_node(node)        
        @nodes[node.value] = node
    end

    def add_edge(node1,node2)
        if @nodes[node1.value].adjacent_nodes.map(&:value).include? (node2.value)        
            puts "#{node1.value} and #{node2.value} already have an edge"
        elsif node1.value == node2.value
            puts "node1.value == node2.value"
        else
            @nodes[node1.value].add_edge(@nodes[node2.value])
            @nodes[node2.value].add_edge(@nodes[node1.value])
        end
    end

    def to_s
        @nodes.keys.sort.each_with_index do |key,index|
            puts "#{index + 1}. #{@nodes[key].to_s}" 
        end
    end
end


def generate_random_graph
    g = Graph.new
    [*1..10].shuffle.each do |node_value|
        g.add_node(Node.new(node_value))
    end
    40.times do 
        key1 = g.nodes.keys.sample
        key2 = g.nodes.keys.sample
        g.add_edge(g.nodes[key1],g.nodes[key2])
    end
    return g
end

def bfs(graph, start_node_value=2, search_value=9)
    if graph.nodes[start_node_value].nil? || graph.nodes[search_value].nil?
        return nil
    end
    visited = Set.new
    search_queue = Queue.new
    search_queue.enq(graph.nodes[start_node_value])    
    while !search_queue.empty? do                
        current_node = search_queue.deq        
        visited.add(current_node)
        if current_node.value == search_value
            return current_node
        end
        current_node.adjacent_nodes.each do |node|
            if !visited.include?(graph.nodes[node.value])                
                search_queue.enq(graph.nodes[node.value])
            end
        end        
    end
end

def bfs_shortest_path(graph, start=2, search=9)
    if graph.nodes[start].nil? || graph.nodes[search].nil?
        return nil
    end
    visited = Set.new
    visited.add(start)
    search_queue = Queue.new
    search_queue.enq(start)
    while !search_queue.empty? do      
        current_node_key = search_queue.deq  
        current_node = graph.nodes[current_node_key]                 
        current_node.add_to_path(current_node.value)
        if current_node.value == search
            return current_node.path_from_start
        end
        adjacent_nodes_array = current_node.adjacent_nodes.map{|x| x.value}
        adjacent_nodes_array.each do |value|
            if !visited.include?(value)                   
                search_queue.enq(value)
                visited.add(value)
                graph.nodes[value].concat_to_path(current_node.path_from_start)                
            end
        end        
    end
end


def test_graph
    graph = generate_random_graph    
    graph.to_s
    bfs_shortest_path(graph,2,9)
end

Answer 2

To keep a history of where you've been so you can reconstruct a path, instead of a visited set, use a hash. The hash tracks which node is the predecessor of each visited node. When you push a neighbor onto the queue, add the current node we're moving away from as the parent/predecessor by came_from[neighbor] = current.

This hash also works for eliminating cycles as visited does.

When you reach the goal, you can rebuild the path by repeatedly keying into this came_from hash until you run out of predecessors. Reverse the array (linear time) and return it as the final path.

Here's a minimal, runnable example you can adapt to your class structure:

def reconstruct_path(tail, came_from)
  path = []

  while !tail.nil?
    path << tail
    tail = came_from[tail]
  end

  path.reverse
end

def bfs(graph, start, goal)
  q = Queue.new
  q.enq(start)
  came_from = {start => nil}
  
  while !q.empty?
    curr = q.deq
    
    if graph.key? curr
      return reconstruct_path(goal, came_from) if curr == goal

      graph[curr].each do |neighbor|
        if !came_from.key?(neighbor)
          came_from[neighbor] = curr
          q.enq(neighbor)
        end
      end
    end
  end
end

graph = {
    "A" => ["B", "C"],
    "B" => ["A", "F"],
    "C" => ["D"],
    "D" => ["E", "F"],
    "E" => [],
    "F" => []
}
=begin

+----+
v    |
A--->B--->F
|         ^
V         |
C--->D----+
     |
     v
     E
   
=end
p bfs(graph, "A", "F") # => ["A", "B", "F"]
p bfs(graph, "A", "E") # => ["A", "C", "D", "E"]
p bfs(graph, "B", "E") # => ["B", "A", "C", "D", "E"]