Breadth

Breadth-first search

In graph theory, breadth-first search (BFS) is a graph search algorithm that begins at the root node and explores all the neighboring nodes. Then for each of those nearest nodes, it explores their unexplored neighbor nodes, and so on, until it finds the goal.

How it works

BFS is an uninformed search method that aims to expand and examine all nodes of a graph systematically in search of a solution. In other words, it exhaustively searches the entire graph without considering the goal until it finds it. It does not use a heuristic.

From the standpoint of the algorithm, all child nodes obtained by expanding a node are added to a FIFO queue. In typical implementations, nodes that have not yet been examined for their neighbors are placed in some container (such as a queue or linked list) called "open" and then once examined are placed in the container "closed".

Algorithm (informal)

  1. Put the root node on the queue.
  2. Pull a node from the beginning of the queue and examine it.
    • If the searched element is found in this node, quit the search and return a result.
    • Otherwise push all the (so-far-unexamined) successors (the direct child nodes) of this node into the end of the queue, if there are any.
  3. If the queue is empty, every node on the graph has been examined -- quit the search and return "not found".
  4. Repeat from Step 2.

Note: Using a stack instead of a queue to store the nodes yet to be visited would turn this algorithm into a depth-first search.

Python implementation

Assume we have a graph made up of Node objects, each containing a value and a list of child Node objects: class Node:
   """Simple structure for nodes in a graph."""
   def __init__(self, value, children=[]):
       self.value = value
       self.children = children

Then this function performs a breadth-first search on that graph when passed the graph's root node (a Node instance) and any function to be applied to each node in the graph: from collections import deque def bfs(top_node, visit):

   """Breadth-first search on a graph, starting at top_node."""
   visited = set()
   queue = deque([top_node])
   while queue:
       curr_node = queue.popleft()         # Dequeue
       if curr_node in visited: continue   # Skip visited nodes
       visit(curr_node)                    # Visit the node
       visited.add(curr_node)
       # Enqueue the children
       queue.extend(curr_node.children)

For example, this script finds the sum of the integer values of each node in the graph:

  1. Build an example graph

the_graph = Node(1, [Node(1, [Node(2), Node(3)]),

                    Node(5, [Node(8, [Node(13)]),
                             Node(21, [Node(34), Node(55)])])])

  1. Define a "visit" function

total = 0 def sum_graph(node):

   global total
   total += node.value

  1. Visit the whole graph

bfs(the_graph, sum_graph) print total

C implementation

Algorithm of Breadth-first search:
void BFS(VLink G[], int v) {
  int w;
  VISIT(v);                    /*visit vertex v*/
  visited[v] = 1;              /*mark v as visited : 1 */
  ADDQ(Q,v);
  while(!EMPTYQ(Q)) {
    v = DELQ(Q);               /*Dequeue v*/
    w = FIRSTADJ(G,v);         /*Find first neighbor, return -1 if no neighbor*/
    while(w != -1) {
      if(visited[w] == 0) {
        VISIT(w);              /*visit vertex w*/
        ADDQ(Q,w);             /*Enqueue current visited vertex w*/
        visited[w] = 1;        /*mark w as visited*/
}
      w = NEXTADJ(G,v);        /*Find next neighbor, return -1 if no neighbor*/
} }
}

Main Algorithm of apply Breadth-first search to graph G=(V,E):

void TRAVEL_BFS(VLink G[], int visited[], int n) {
  int i;
  for(i = 0; i < n; i ++) {
    visited[i] = 0;            /* Mark initial value as 0 */
}
  for(i = 0; i < n; i ++)
    if(visited[i] == 0)
      BFS(G,i);
}

C++ implementation

This is the implementation of the above informal algorithm, where the "so-far-unexamined" is handled by the parent array. For actual C++ applications, see the Boost Graph Library.
Suppose we have a struct:
struct Vertex {
 ...
 std::vector out;
 ...
};
and an array of vertices: (the algorithm will use the indices of this array, to handle the vertices)
std::vector graph(vertices);
the algorithm starts from start and returns true if there is a directed path from start to end:
bool BFS(const std::vector& graph, int start, int end) {
  std::queue next;
  std::vector parent(graph.size(), -1) ;
  next.push(start);
  parent[start] = start;
  while (!next.empty()) {
    int u = next.front();
    next.pop();
    // Here is the point where you can examine the u th vertex of graph
    // For example:
    if (u == end) return true;
    for (std::vector::const_iterator j = graph[u].out.begin(); j != graph[u].out.end(); ++j) {
      // Look through neighbors.
      int v = *j;
      if (parent[v] == -1) {
        // If v is unvisited.
        parent[v] = u;
        next.push(v);
} } }
  return false;
}
it also stores the parents of each node, from which you can get the path.

Features

Space Complexity

Since all of the nodes of a level must be saved until their child nodes in the next level have been generated, the space complexity is proportional to the number of nodes at the deepest level. Given a branching factor b and graph depth d the asymptotic space complexity is the number of nodes at the deepest level, O(b^d). When the number of vertices and edges in the graph are known ahead of time, the space complexity can also be expressed as O(|E|+|V|) where |E| is the cardinality of the set of edges (the number of edges), and |V| is the cardinality of the set of vertices. In the worst case the graph has a depth of 1 and all vertices must be stored. Since it is exponential in the depth of the graph, breadth-first search is often impractical for large problems on systems with bounded space.

Time Complexity

Since in the worst case breadth-first search has to consider all paths to all possible nodes the time complexity of breadth-first search is b+b^2+b^3+...+b^d which asymptotically approaches O(b^d). The time complexity can also be expressed as O(|E|+|V|) since every vertex and every edge will be explored in the worst case.

Completeness

Breadth-first search is complete. This means that if there is a solution breadth-first search will find it regardless of the kind of graph. However, if the graph is infinite and there is no solution breadth-first search will diverge.

Optimality

For unit-step cost, breadth-first search is optimal. In general breadth-first search is not optimal since it always returns the result with the fewest edges between the start node and the goal node. If the graph is a weighted graph, and therefore has costs associated with each step, a goal next to the start does not have to be the cheapest goal available. This problem is solved by improving breadth-first search to uniform-cost search which considers the path costs. Nevertheless, if the graph is not weighted, and therefore all step costs are equal, breadth-first search will find the nearest and the best solution.

Applications of BFS

Breadth-first search can be used to solve many problems in graph theory, for example:

Finding connected Components

The set of nodes reached by a BFS (breadth-first search) are the largest connected component containing the start node.

Testing bipartiteness

BFS can be used to test bipartiteness, by starting the search at any vertex and giving alternating labels to the vertices visited during the search. That is, give label 0 to the starting vertex, 1 to all its neighbours, 0 to those neighbours' neighbours, and so on. If at any step a vertex has (visited) neighbours with the same label as itself, then the graph is not bipartite. If the search ends without such a situation occurring, then the graph is bipartite.

Literature

External links

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