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Algorithm of the Week: Graph Breadth First Search

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Since we already know how to represent graphs, we can go further for some very simple approaches of walking through them. Passing by all the vertices of a graph is a fundamental technique for most of the graph algorithms, such as finding shortest/longest paths, etc.

First thing to note is that graphs are not trees, in most of the cases, so walking through them can’t start from a root, as we do with trees. What we must do first is to decide from where to start – in other words – choosing a starting vertex.

BFS Choosing a Starting Point

It’s clear that depending on the starting point we can get different passes through the graph. Thus choosing a starting point can be very important for our algorithm!

After that we need to know how to proceed. There are two approaches mostly known as “breadth first” and “depth first” search. While depth first search start from a vertex and goes as far as possible, then walks back and passes through vertices that haven’t been visited yet, breath first search is an approach of passing through all the neighbors of the node first, and then go to the next level.


We can thing of breadth first search as a “wave” walk through the graph. In other words we go level by level, as shown on the picture below.

BFS Wave

For this very specific graph on the picture we can see how breadth first search walks through the graph level by level!

Initially we mark all vertices as unvisited. A common approach is to create an empty queue where we put the vertices level by level, starting with the initial vertex.

BFS Using a Queue

Using a queue is a typical approach for breadth first search! However this requires more space!


This simple approach is fairly easy to implement. Here’s the PHP implementation in few lines of code.


$g = array(
    0 => array(0, 1, 1, 0, 0, 0),
    1 => array(1, 0, 0, 1, 0, 0),
    2 => array(1, 0, 0, 1, 0, 0),
    3 => array(0, 1, 1, 0, 1, 0),
    4 => array(0, 0, 0, 1, 0, 1),
    5 => array(0, 0, 0, 0, 1, 0),
function init(&$visited, &$graph) 
    foreach ($graph as $key => $vertex) {
        $visited[$key] = 0;
function breadth_first(&$graph, $start, $visited)
    // create an empty queue
    $q = array();
    // initially enqueue only the starting vertex
    array_push($q, $start);
    $visited[$start] = 1;
    echo $start . "\n";
    while (count($q)) {
        $t = array_shift($q);
        foreach ($graph[$t] as $key => $vertex) {
            if (!$visited[$key] && $vertex == 1) {
                $visited[$key] = 1;
                array_push($q, $key);
                echo $key . "\t";
        echo "\n";
$visited = array();
init($visited, $g);
breadth_first($g, 2, $visited);



The complexity of this algorithm clearly is O(n2).


As I said breadth first and depth first searches are used in many practical cases, as finding shortest/minimal paths etc. That is why understanding these basic principles of walking through a graph is crucial for other, more complex, graph algorithms.

Published at DZone with permission of Stoimen Popov, author and DZone MVB. (source)

(Note: Opinions expressed in this article and its replies are the opinions of their respective authors and not those of DZone, Inc.)


Peter Karussell replied on Tue, 2012/09/11 - 4:48pm

Here is BFS (and DFS) for Java - works memory efficient, is customizable and tested! Now I would like to know a speed comparison ;)


BTW1: What kind of complexity you are refering - time or space? Because time compl. is O(|E| + |V|) and space compl. is O(|V|)

BTW2: Nice algorithm posts :)

Wujek Srujek replied on Wed, 2012/09/12 - 1:39am

Shouldn't vertices 1 and 4 in step #4 and #5 be in one step? Both are at depth 2 counting from the starting vertex 2.

John Mclane replied on Sat, 2013/09/28 - 8:48am in response to: Peter Karussell

Hi! Can you please help me? Do you still have the java program for BFS friend search?

Peter Karussell replied on Sat, 2013/09/28 - 12:30pm in response to: John Mclane


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