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## old comment on redirect

Breadth first recursion now redirects here.

## Request for major editing:

I have been looking at this and also depth-first search and the preliminary explanation has the same meaning. It says that it starts at the root node "or a node", and explores each neighboring node of the "root" fully without making a cycle until it finds the goal. Something needs to be done about this. I'm just a student trying to learn this so I don't think I can really help. Maybe I'll ask my professor to fix this.

No that's not a mistake. Both algorithms have this property. The property you state does not uniquely identify the algorithm. The algorithms use completely different data structures and result in different traversal orders.

I think that since BFS uses a Queue, "push" and "pop" are wrong (stacks, LIFO): a queue uses "enqueue" and "dequeue". Edited. --Folletto 16:00, 21 Jun 2005 (UTC)

"Since all nodes discovered so far have to be saved, the space complexity of breadth-first search is O(|V| + |E|)." I don't see it. Where does the |E| come from?

I agree. The runspace is O(|V|) in the worst case, since that will be the max size of the control queue. —Preceding unsigned comment added by 216.186.140.62 (talk) 23:37, 5 March 2009 (UTC)

## What type of graph does this apply to?

It appears to be assumed that the graphs being dealt with have a root node, but this is not stated explicitly. Presumably any node will do if the graph is undirected, but what happens for a directed graph? If the article deals only with undirected graphs then it might be helpful if this was made clear at the beginning. Autoplunger (talk) 20:05, 30 January 2011 (UTC)

## Pseudocode

The pseudocode looks funny, and a little too close to an actual implementation. I have taken liberty to modify it a little.

You did a poor job in modifying it, language is unclear, look at psuedocode on page for topological sort for good reference. —Preceding unsigned comment added by 97.77.52.94 (talk) 17:21, 22 November 2010 (UTC)

The pseudocode doesn't just look funny, it's wrong. "parent" is never used and nothing is searched for. It's a breadth-first traversal with an unnecessary extra variable.

Also, there's no mention whatsoever of cycles. — Preceding unsigned comment added by 187.163.213.135 (talk) 21:22, 3 July 2016 (UTC)

To clarify: the marking of 'distance' (another unneeded variable) as INFINITE is apparently a check to make sure there are no cycles. But it relies upon there being a collection of unique graph nodes. This is just terrible pseudocode. — Preceding unsigned comment added by 187.163.213.135 (talk) 21:35, 3 July 2016 (UTC)

Can you give a little more detail on your reversion of my pseudocode simplification? Just trying to understand your reasoning. As far as I can understand, adding to a set and setting a value on a node should be the same complexity, both in space and time, if anything it seems like the set should be slower (depending on implementation, of course). Crazy2be (talk) 21:43, 6 August 2017 (UTC)

Sure. Imagine that you have an intractably large state space, and you want to find the shortest path between two points in that state space that are only a few hops apart. If the first step is to flag the entire state space as unvisited, then the whole problem is rendered intractable. Using a "set" does not imply any particular implementation. A big table of flags is also a type of set. By flagging all the states as visited, you are essentially using the "set of all possible states" as a substitute for using a separate "set of visited states". In many cases this might be a good way to implementat it, but it would be unnecessarily specific to imply that combining these sets is always a good design. Headlessplatter (talk) 03:02, 18 August 2017 (UTC)

## Dijsktra's Algorithm vs. BFS

When BFS is used with a priority queue, it is a total algorithm to find the shortest path between any two nodes in a weighted graph. Clearly, therefore, finding the shortest path between two nodes in a weighted graph is an application of BFS.

Somebody deleted this application because they felt that this should be an application of Dijsktra's Algorithm instead. What is the logic here? Clearly it is an application of both DA and BFS. DA is an optimization of the backtracing step of BFS, so you don't have to use back-pointers to remember how you got to the destination. The distinction between BFS and DA therefore has little to do with whether or not the graph is weighted. Finally, BFS is much easier to remember than DA, so BFS is undoubtedly the more popular algorithm to find shortest paths in any graph (weighted or unweighted). Therefore this application should be listed.

Hi, I think I was the one who deleted that passage. Why I did it? BFS simply does not find a shortest path in a weighted graph! It finds the path that has the fewest number of edges, but if edges have edge weights, BFS will find an arbitrary expensive path of n edges instead of the cheap path that has n+1 edges. Therefore BFS is not an application for finding a shortest path in a weighted graph simply because it computes absolute nonsense if run on a weighted graph. BFS will always find the path that has the fewest number of nodes which just happens to be the shortest path if all weights are the same. You certainly can modify BFS to use a priority queue instead of a normal queue so that it then really finds a shortest path. But then DFS is the same as BFS just with a stack instead of a queue. So you would have to add the part about shortest paths in the DFS article, too. All three algorithms differ (mainly) only by just the datastructure used to save the nodes in, but their output is absolutely different. --Regnaron 21:14, 26 September 2006 (UTC)

The original formulation of BFS did not restrict the queue to be a FIFO queue. If you use BFS with a FIFO queue on a weighted graph then you're effectively discarding the edge weights. That's why you get a different result from the algorithm with a priority queue versus FIFO queue: you're changing the input. DFS is completely different because you get a result that you won't get merely by changing the inputs. BFS has been used with a priority queue in scheduling simulators long before Dijkstra's algorithm got its name. It is not limited to unweighted graphs!

If you use BFS with a FIFO queue on a weighted graph then you're effectively discarding the edge weights. That's why you get a different result from the algorithm with a priority queue versus FIFO queue: you're changing the input. And exactly that result you get is wrong. It is not a shortest path but the path with fewest nodes. If you use it with a priority queue, the result is correct, but then, where is the difference to Dijkstras algorithm? Find the vertex that is first in the queue, update its value and never look at it again. Just the same as Dijkstra. --Regnaron 05:43, 4 October 2006 (UTC)

Your question, "Where is the difference to Dijkstra's Algorithm?" gets to the heart of the matter. BFS with PQ is easy to remember, is immediately clear to anybody who has learned BFS, and is the oldest and most popular algorithm for finding all shortest paths in a weighted graph. By contrast, Dijkstra's Algorithm is obscure, known only in elite circles, difficult to memorize, and named after a professor. The differences between the algorithms have nothing to do with the problem they are solving, and everything to do with name-dropping elitism and obfuscation.

What you seem to be suggesting is that "Dijkstra's Algorithm solves the problem, therefore BFS with PQ does not". BFS and PQ long predate Dijkstra's Algorithm: they were used in the 1940s back in Alan Turing's days when lattice theory, AI, and scheduling systems were first developed for computers. So your statement should be turned around. What problem is Dijkstra's Algorithm solving that BFS and PQ did not already solve? If anything needs to be deleted, it's the Dijkstra's Algorithm entry, not the BFS entry.

BFS is what we call a blind search algorithm, cause it uses no more information than links between nodes. On the other hand are informed search algorithms, such as Best-first search. This last algorithm effectively uses a priority queue to handle the visiting nodes. So modifying BFS with such queue results in a different class of algorithm: BestFS. By the way this class of algorithm is still not optimal. You need to move forward to A* search algorithm (and in the process finding an admissible heuristic) to solve the "shortest" path in a weighted graph. So I'm with Regnaron in this matter. —Preceding unsigned comment added by 200.55.140.181 (talk) 01:46, 16 June 2008 (UTC)

## Usage in 2D grids for computer games

It is worth mentioning that when BFS is used in that manner, the neighbour list should be created such that north, east, south and west get priority over north-east, south-east, south-west and north-west. The reason for this is that BFS tends to start searching in a diagonal manner rather than adjacent, and the path found will not be the correct one. BFS should first search adjacent nodes, then diagonal nodes.

I think a better explanation or a reference is required here. As I understand it the resulting path will be zig-zagged if diagonal moves are prioritized over moves to adjacent grid point. The resulting path should still be correct.

This section is mostly unintelligible and describes an obscure application in a very special field. There are thousands of equally obscure applications which are only interesting to their practitioners and add no additional insights to BFS. Suggest to remove it. --Mbetter 07:59, 17 August 2007 (UTC)
I agree this is not helpful and is incorrect as it is currently worded. As Mbetter points out the topic is not especially important to BFS. I am removing it. Add it as a bullet to the applications section if you really feel it is worth mentioning. Jludwig 18:22, 2 October 2007 (UTC)

## C++ implementation

In the sample implementation, why do we use an std::vector<int> to keep track of key-value pairs? The method we currently use wastes more space, runs slower (when attempting to recover path data later on), makes reading the code less clear, and is less versatile (e.g., in expanding the function for a situation where graph size is unknown beforehand) than the much more elegant solution for such an associative array—an std::map<int,int>. I will change the code accordingly if no reasonable objections appear within a week or so. --76.189.119.174 00:41, 10 August 2007 (UTC)

This has been changed accordingly. --76.189.119.174 03:16, 18 August 2007 (UTC)

## Connectedness, search and traversal

The article is vague about stating the problem that BFS is supposed to solve. Is the goal to search for a path to a given vertex, or to traverse all vertices? The distinction is important for instance if the graph is not connected. (The link to "graph search algorithms" actually leads to a page about traversal!) --Mbetter 08:10, 17 August 2007 (UTC)

## Complexity

The sections on complexity are slightly inconsistent and worded incorrectly. I am changing them to correct the reason for the time and space complexity and to reflect the common notation. Jludwig 18:26, 2 October 2007 (UTC)

The section on complexity (and the given complexity) is deceiving to the reader and doesn't obey usual conventions outside of this particular algorithm. If the graph has multiple connected components (which could cause O(|E| + |V|) to not be equivalent to O(|E|)), the given code will fail to explore all the nodes and thereby run in time O(|E|) for the subgraph it does explore. If the graph has a single connected component, the code will run in O(|E|) which is equivalent in this case to O(|E| + |V|). This is true regardless of what the convention in CLRS is. To illustrate my point, name any other example of an algorithm running in O(n^2 + n) time being the standard convention over O(n^2). 12.32.116.227 (talk) 20:56, 3 August 2018 (UTC)

## Python Implementation

I think the Python implementation should store visited nodes in a set rather than a list, because testing membership in a set is O(1) rather than O(n) with the list. —Preceding unsigned comment added by 65.28.233.233 (talk) 08:30, 16 June 2008 (UTC)

• Wait, Python set test membership is worst case O(n)! BFS is a linear-time problem, this seems to me to destroy that property. Also, why is a huge sprawling, poorly coded Python implementation the code snippet for something so basic and important as BFS? — Preceding unsigned comment added by 94.146.192.229 (talk) 16:37, 23 April 2018 (UTC)

I believe that the Python implementation is still not correct because pop(0) on a list is linear time. In the worst case, the length of the queue is O(|V|), which makes the complexity of this implementation O(|V|^2). Also, checking to see whether the child is already in the queue is currently O(n) and needs to be O(1).

I will wait a few days to see if anyone contradicts me; otherwise I will go ahead and fix it (or at least make a note of the problem).--AllenDowney

## Pseudo code

It seems to be wrong. "Otherwise enqueue any successors (the direct child nodes) that have not yet been examined." But successors might have been already enqueued earlier (and not examined yet). So we should color them grey when enqueue, and enqueue only white nodes. Poemich (talk) 11:41, 3 May 2009 (UTC) kij —Preceding unsigned comment added by 164.78.252.57 (talk) 08:58, 4 August 2009 (UTC)

## Merger_proposal

Merging the Implementation in to the main document would make more sense. And avoid deletion.

No. There's a good reason we don't clutter up the main document with large numbers of implementations. See Wikipedia talk:WikiProject Computer science#Source code written by editors. —David Eppstein (talk) 13:40, 11 May 2009 (UTC)

## Breadth-first search implementation at AfD

See Wikipedia:Articles for deletion/Breadth-first search implementation. As usual for AfD discussions, please leave a detailed comment explaining your opinion rather than just a keep or delete without an explanation. —David Eppstein (talk) 21:59, 12 May 2009 (UTC)

<harry_ftw> afaik, there's no such thing as a "breath first search"

I tried a Google search for that, hoping to find something about underwater rescue techniques, but sadly it's all just spelling mistakes. —David Eppstein (talk) 06:47, 18 May 2009 (UTC)

## Completeness

What in the world is "completeness" (and "proof of completeness")? Is "completeness" a well-established term in some field? --Robin (talk) 04:14, 17 November 2009 (UTC)

## Example with connected components

Would be great with an example showing what happens with Connected_component_(graph_theory) --Andreas (talk) 11:40, 18 March 2010 (CET)

## C# implementation appears to be incorrect

When checking whether to enqueue a new node, the posted version appears to be check if the node is already in the queue by comparing it with every other node in the queue (an O(n) operation). This implementation doesn't have the same characteristics or behavior as the Java implementation, or the description in Cormen, Rivest. Could someone who knows C# fix this? 128.101.35.83 (talk) 20:14, 26 April 2010 (UTC)

There was a lot of weirdness in that implementation, beyond what you noted. I fixed that specific problem, and also changed it so that it's essentially the same as the Java implementation. -Xodarap00 (talk) 21:58, 9 May 2010 (UTC)

## Example explosion

We only need one implementation. As it is, both languages conflate language features with explanation of the algorithm. We should either have only a pseudocode implementation, or maybe Python or some other reads-mostly-like-pseudocode language if it can be made sufficiently close to pseudocode. The fact that java.util.LinkedList is in the example suggests to me that the example says more about Java than about BFS. —Ben FrantzDale (talk) 12:53, 11 May 2010 (UTC)

I agree. —David Eppstein (talk) 14:38, 11 May 2010 (UTC)

## Testing Bipartiteness

Doesn't this algorithm for testing bipartiteness assume that every node is reachable from the start node? I don't think it will work correctly for graphs that aren't connected. — Preceding unsigned comment added by 164.107.120.19 (talk) 20:00, 5 April 2013 (UTC)

I concur. The pseudo-code that's given is specific to BFS for a tree graph. It doesn't work correctly for graphs with multiple components, and doesn't agree with the text. Frankly, this article is a hot mess. I'll edit this when I get home and can refer to Cormen et al. 163.173.9.3 (talk) 09:37, 29 May 2013 (UTC)

## White, Grey and Black

The use of a colour attribute which is WHITE, GREY or BLACK, which I assume originated with the Corman et al. algorithms book, is useful for proving things about the algorithm. But it isn't used for anything in this article and it isn't needed for correctness of the algorithm. The pseudocode is still valid if "if adj.color == WHITE" is replaced by "if adj.distance == INFINITY" and then all statements involving colour are removed. So at the moment it only serves to complicate something simple. Who will object if I make this simplification? McKay (talk) 04:35, 28 July 2015 (UTC)

I won't object. —David Eppstein (talk) 06:11, 29 July 2015 (UTC)

Felipe Sobreira Abrahão (talk · contribs) Could you please explain your edit [1] ? I don't understand what your infinity means. And especially, why you deleted the condition about minimality ? Arthur MILCHIOR (talk) 18:32, 23 March 2017 (UTC)

It was very poorly stated indeed. Thank you and David Eppstein (talk · contribs) for noticing it. My editing was intended to do the disambiguation between minimalities within the very definition of function ${\displaystyle \nu _{(v_{1},\dots ,v_{i-1})}}$ and the one for function ${\displaystyle \nu _{(v_{1},\dots ,v_{i-1})}(w)}$ in respect to other possible enumerations ${\displaystyle \sigma '=(v_{1},v'_{2}\dots ,v'_{n})}$ for which ${\displaystyle \nu _{(v_{1},v'_{2}\dots ,v'_{i-1})}(w)\neq \infty }$ also holds. So that, I could have been better stated the respective paragraph as:

"Let ${\displaystyle \sigma =(v_{1},\dots ,v_{n})}$ be an enumeration of the vertices of ${\displaystyle V}$. The enumeration ${\displaystyle \sigma }$ is said to be a BFS ordering (with source ${\displaystyle v_{1}}$) if, for all ${\displaystyle 1, ${\displaystyle v_{i}}$ is the vertex ${\displaystyle w\in V\setminus \{v_{1},\dots ,v_{i}-1\}}$ such that

${\displaystyle \nu _{(v_{1},\dots ,v_{i-1})}(w)=min\{\,\nu _{(v_{1},v'_{2},\dots ,v'_{i-1})}(w)\mid \sigma '=(v_{1},v'_{2}\dots ,v'_{n}){\text{ is a enumeration of }}V\,\land \,\nu _{(v_{1},v'_{2}\dots ,v'_{i-1})}(w)\neq \infty \,\land \,(v_{1},v'_{2},\dots ,v'_{i-1}){\text{ is a BFS ordering of }}\{v_{1},\dots ,v_{i-1}\}\,\}}$
Equivalently, ${\displaystyle \sigma }$ is a BFS ordering if, for all ${\displaystyle 1\leq i with ${\displaystyle v_{i}\in N(v_{j})\setminus N(v_{k})}$, there exists a neighbor ${\displaystyle v_{m}}$ of ${\displaystyle v_{j}}$ such that ${\displaystyle m."

Another issue is that the last sentence

" Equivalently, ${\displaystyle \sigma }$ is a BFS ordering if, for all ${\displaystyle 1\leq i with ${\displaystyle v_{i}\in N(v_{j})\setminus N(v_{k})}$, there exists a neighbor ${\displaystyle v_{m}}$ of ${\displaystyle v_{j}}$ such that ${\displaystyle m. "

doesn't hold for ${\displaystyle i=1}$. As a counter-example, one may take ${\displaystyle i=1,\,j=2,k=5\,}$, ${\displaystyle \nu _{i}=vertex\,3,\,\nu _{j}=vertex\,2,\,\nu _{k}=vertex\,5}$ in the example presented in The breadth-first search algorithm.

Felipe Sobreira Abrahão (talk) 22:38, 9 May 2017 (UTC)

Some IP editor is repeatedly trying to replace the pseudocode with (modulo typos and syntax) the following simplified algorithm:

def BFS(G,start):
V = set()
Q = queue()
Q.enqueue(start)
while Q:
v = Q.dequeue()
for w in G[v]:
if w not in V:
Q.enqueue(w)

This is a bad algorithm. Because it does not check whether a vertex w has already been enqueued before enqueueing it again, it ends up adding each vertex multiple times, where the number of times is equal to the number of shortest paths from the start to the vertex. This could end up being exponential in the size of the input graph. The current pseudocode could probably be significantly simplified, but this is not the way to do it. —David Eppstein (talk) 19:22, 20 December 2017 (UTC)

That is a good algorithm. The set V of visited nodes prevents the problem of "adding each vertex multiple times". It's a pseudocode an not an implementation (like the python one). It's in sync whith the DFS article (https://en.wikipedia.org/wiki/Depth-first_search) 186.130.99.217 (talk) 15:00, 26 June 2018 (UTC)
Incorrect. Look at where in the pseudocode membership in V is set tested, and think: if w has not yet been dequeued, what is preventing another copy of w from being enqueued? —David Eppstein (talk) 16:07, 26 June 2018 (UTC)

Great catch, David! What are your thoughts of adding an additional set to test queue membership to this style of pseudocode; I think it is much easier to follow than the current code.

def BFS(G, start):
visited = set()
prev_queued = set()
Q = queue()

Q.enqueue(start)

while not Q.empty():
v = Q.dequeue()
for w in G[v]:
if w not in visited and w not in prev_queued:
Q.enqueue(w)
prev_queued.add(w)

Themichaelyang (talk) 00:40, 4 October 2018 (UTC)

It's an improvement over the current mess, definitely, but are you really using visited? It seems that the test for w not in visited and w not in prev_queued always gives exactly the same results as if you merely tested prev_queued. And in case you need them after the end of the algorithm, the two sets are exactly the same at that point. So wouldn't it be simpler as:

def BFS(G, start):
queued = {start}
Q = queue(queued)

while not Q.empty():
v = Q.dequeue()
for w in G[v]:
if w not in queued:
Q.enqueue(w)
queued.add(w)

David Eppstein (talk) 00:54, 4 October 2018 (UTC)

Whoops, another great catch! I hadn't even noticed that, but it seems obvious now. I'll update the pseudocode section with this snippet and a remarks about the incorrect implementation next time I get a chance. Michael Yang (talk) 19:58, 4 October 2018 (UTC)

## Simplifying the Python code

The code has closed_set to remember which nodes have been dequeued, but if vertices are put into closed_set when they are enqueued instead of when they come off the queue, the two tests "if child in closed_set: continue" and "if child not in open_set" can be replaced by "if child not in closed_set". That would also fix an existing bug when the graph has a loop (while a node is having its neighbours processed it is in neither open_set nor closed_set so it will be enqueued again). McKay (talk) 06:23, 16 April 2018 (UTC)

Revisiting this page after a while, and now the pseudocode section contains a very confusing snippet of "Python" code (in actuality, it is not valid Python code). The new code also includes a very strange problem object parameter, which makes very little sense. Why not separate the inputs into actual parameters? I think for clarity purposes, the original pseudocode was much easier to follow, and made less assertions about the syntax and style of a particular implementation. Themichaelyang (talk) 00:01, 4 October 2018 (UTC)