Overall, graph search can fall either under the uninformed or the informed category. The difference between the two is that the first one uninformed is naive or blind - meaning it has no knowledge of where the goal could be, while the second one informed uses heuristics to guide the search.

These algorithms can be applied to traverse graphs or trees. To represent such data structures in Python, all we need to use is a dictionary where the vertices or nodes will be stored as keys and the adjacent vertices as values.

Keep in mind that we can represent both directed and undirected graphs easily with a dictionary. In the case our small graph was directed it would maybe look something like this. All of the search algorithms will take a graph and a starting point as input. Having a goal is optional. Depth-First Search will visit the first adjacent vertex of the starting point and then repeat the same process until it reaches the very bottom of the branch and then it will finally start backtracking.

DFS can be implemented using recursion, which is fine for small graphs, or a safer option is iteration. A couple of things we need to do in this algorithm is keep track of which vertices we have visited, and also keep track of the fringe. The fringe or frontier is the collection of vertices that are available for expanding. The BFS algorithm instead of following a branch down to the bottom, will visit all the vertices of the same depth before moving on deeper.

So when choosing which vertex to expand next, it will choose the oldest item on the fringe, unlike DFS which chooses the newest. So by modifying this line. We can make this more efficient though.

To cut down on the cost of pop 0 we can use a double ended queue called deque. This allows us to append items to both ends. This search strategy is for weighted graphs. Each edge has a weight, and vertices are expanded according to that weight; specifically, cheapest node first. As we move deeper into the graph the cost accumulates.

The algorithm needs to know the cost of moving from one vertex to another.We often need to model entities that have connections between them. For example, we might want to know whether two people are friends. If we can identify a group of people all of whom are friends with each other that might tell us something about these people; for example, they might all be members of the same Greek house, or that they might constitute a terrorist cell.

For another example of entities with connections, think of a road map. Each intersection is an entity, and a connection is a road going between two intersections. When you get dressed, there are some articles of clothing that you must put on before other articles, such as socks before shoes. But there are also some articles for which the order that you don them does not matter, such as socks and a shirt. For a road map, we might care not only that a road connects two intersections, but also about the length of the road.

For another example, we might want to know, for any pair of world currencies, the exchange rate from one to the other. Many years ago, mathematicians devised a nice way to model situations with many entities and relationships between pairs of entities: a graph. A graph consists of vertices singular: vertex connected by edges. Each edge connects one vertex to some other vertex. A vertex may have edges to zero, one, or many other vertices. Think of each vertex as representing an entity and each edge as a connection.

Each vertex in this particular graph is labeled by a letter, though in general we can label vertices however we like, including with no label at all.

Just to take an example, vertex A has two edges: one to vertex C and one to vertex E. In some situations, we want directed edgeswhere we care about the edge going from one vertex to another vertex. Directed edges work well in the example about getting dressed, where an edge from a vertex for article X to a vertex for article Y indicates that you have to don X before Y. In other situations, such as the graph drawn above, the edges are undirected. Because the friendship relation is symmetric, if we consider a graph in which vertices represent people and an edge between persons X and Y indicates that X and Y are friends, this graph would have undirected edges.

A graph with undirected edges is an undirected graphand a graph with directed edges is a directed graph. We can always emulate an undirected edge between vertices x and y with directed edges from x to y and from y to x. Edges can be weighted or unweighted in either directed or undirected graphs. You might sometimes hear other names for these structures.

Graphs are sometimes called networksvertices are sometimes called nodes you might recall that when I drew trees, I used the termand edges are sometimes referred to as links or arcs. A few more easy definitions. In the above graph, vertices D and G are adjacent and edge D, G is incident on both of them. The number of edges incident on a vertex in an undirected graph is the degree of the vertex.

In the above graph, the degree of vertex B is 5. In a directed graph, the number of edges leaving a vertex is its out-degree and the number of edges entering a vertex is its in-degree. If we can get from vertex x to vertex y by following a sequence of edges, we say that the vertices along the way, including x and yform a path from x to y.

The length of the path is the number of edges on the path. Note that there is always a path of length 0 from any vertex to itself.Do you know the amount of global air traffic in ? Do you know what the rise has been for air traffic over the past several years?

### Breadth First Search or BFS for a Graph

Well, lets look at some statistics. And, the number of flights rose to 37 million globally in Since there are hundreds and thousands of these flights all around the globe, there are bound to be different routes with multiple stops in between from one place to another.

Every flight has a source and destination of its own and a standard economy seat price associated with it. In such a scenario, it is too confusing to choose what flight would be the best one if we want to go from one place to another. Then there appears a pop up on the website saying that there are other websites that might be offering similar flights at even cheaper rates.

## Depth-First Search and Breadth-First Search in Python

As a developer, if I want to solve this problem, I would build a system to efficiently address the following queries:. As you might guess, there would be potentially thousands of flights as the output of the first two queries. But we can certainly reduce that by providing some other criteria to lessen the output size. For the scope of this article, let us focus on these original queries themselves. Modeling this problem as a graph traversal problem greatly simplifies it and makes the problem much more tractable.

So, as a first step, let us define our graph. Directed because every flight will have a designated source and a destination. These carry a lot of meaning. Cyclic because it is very possible to follow a bunch of flights starting from a given location and ending back at the same location. Weighted because every flight has a cost associated with it which would be the economy class flight ticket for this article.

And finally, a forest because we might have multiple connected components. It is not necessary that all the cities in the world have some sort of flight network between them. So, the graph can be disconnected, and hence a forest. The vertices, Vwould be the locations all over the world wherever there are working airports. The edges, Ewould be representative of all the flights constituting the air traffic. Now that we have an idea about how to model the flight network as a graph, let us move on and solve the first common query that a user might have.

As someone who likes to explore different places, you would want to know what all destinations are reachable from your local airport. Again, there would be additional criteria here to reduce the results of this query.

But to keep things simple, we will simply try and find all the locations reachable from our local airport. We will initialize the BFS queue with the given location as the starting point. We then perform the breadth first traversal, and keep going until the queue is empty or until the maximum number of stops have been exhausted.

Note: If you are not familiar with the breadth first search or depth first search, I would recommend going through this article before continuing. We also need to look at how the BFS algorithm would end up giving us all the destinations reachable from a given source.

Performing bfs on the city of Los Angeles would give us the following destinations which are reachable:. In case we have a humongous flight network, which we would have in a production scenario, then we would not ideally want to explore all the reachable destinations from a given starting point. This is a use case if the flight network is very small or pertains only to a few regions in the United States.Graph theory and in particular the graph ADT abstract data-type is widely explored and implemented in the field of Computer Science and Mathematics.

One of the most popular areas of algorithm design within this space is the problem of checking for the existence or shortest path between two or more vertices in the graph. Properties such as edge weighting and direction are two such factors that the algorithm designer can take into consideration. In this post I will be exploring two of the simpler available algorithms, Depth-First and Breath-First search to achieve the goals highlighted below:.

So as to clearly discuss each algorithm I have crafted a connected graph with six vertices and six incident edges. The resulting graph is undirected with no assigned edge weightings, as length will be evaluated based on the number of path edges traversed. There are two popular options for representing a graph, the first being an adjacency matrix effective with dense graphs and second an adjacency list effective with sparse graphs. I have opted to implement an adjacency list which stores each node in a dictionary along with a set containing their adjacent nodes.

As the graph is undirected each edge is stored in both incident nodes adjacent sets. This has been purposely included to provide the algorithms with the option to return multiple paths between two desired nodes. The first algorithm I will be discussing is Depth-First search which as the name hints at, explores possible vertices from a supplied root down each branch before backtracking.

This property allows the algorithm to be implemented succinctly in both iterative and recursive forms. Below is a listing of the actions performed upon each visit to a node. The implementation below uses the stack data-structure to build-up and return a set of vertices that are accessible within the subjects connected component.

The second implementation provides the same functionality as the first, however, this time we are using the more succinct recursive form. Due to a common Python gotcha with default parameter values being created only once, we are required to create a new visited set on each user invocation.

Another Python language detail is that function variables are passed by reference, resulting in the visited mutable set not having to reassigned upon each recursive call. We are able to tweak both of the previous implementations to return all possible paths between a start and goal vertex.

The implementation below uses the stack data-structure again to iteratively solve the problem, yielding each possible path when we locate the goal. Using a generator allows the user to only compute the desired amount of alternative paths. Unfortunately the version of Pygments installed on the server at this time does not include the updated keyword combination. An alternative algorithm called Breath-First search provides us with the ability to return the same results as DFS but with the added guarantee to return the shortest-path first.

## Breadth First Search (BFS) Algorithm with EXAMPLE

This algorithm is a little more tricky to implement in a recursive manner instead using the queue data-structure, as such I will only being documenting the iterative approach. The actions performed per each explored vertex are the same as the depth-first implementation, however, replacing the stack with a queue will instead explore the breadth of a vertex depth before moving on.

This behavior guarantees that the first path located is one of the shortest-paths present, based on number of edges being the cost factor. Similar to the iterative DFS implementation the only alteration required is to remove the next item from the beginning of the list structure instead of the stacks last.

This implementation can again be altered slightly to instead return all possible paths between two vertices, the first of which being one of the shortest such path. As we are using a generator this in theory should provide similar performance results as just breaking out and returning the first matching path in the BFS implementation.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Code Review Stack Exchange is a question and answer site for peer programmer code reviews.

It only takes a minute to sign up. As your mileage may vary building a traversal list, finding the first node that satisfies a condition, etc. Note that this alternative iteration also takes care of the bug mentioned in this answer.

I agree with Mathias Ettinger's use of set s and deque s, with two changes:. Otherwise the root may be revisited eg test case below where 1 points back to 0. The answers that have been posted including the accepted answer are wrong because they don't explore the vertex completely before moving onto the next vertex. For the below adjacency matrix. The output should be 1, 2, 4, 5, 7, 6, 3or 1, 5, 4, 2, 3, 6, 7etc.

The idea is that the current vertex should be completely explored before moving onto the next vertex. In addition, connected vertices can occur in any order. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 3 years, 8 months ago.

Active 19 days ago. Viewed 44k times. Apollo Apollo 1 1 gold badge 2 2 silver badges 5 5 bronze badges. Active Oldest Votes. OP uses list. For some reason, I didn't realize OP used 0 as an argument. Oct 2 '18 at Saurabh Saurabh 1 1 silver badge 7 7 bronze badges. The Overflow Blog. The Overflow How many jobs can be done at home?

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Related 5. Hot Network Questions. Question feed.Breadth-first search BFS is an algorithm used for traversing graph data structures.

What is this exploration strategy? BFS starts with a node, then it checks the neighbours of the initial node, then the neighbours of the neighbours, and so on. BFS is an AI search algorithm, that can be used for finding solutions to a problem. Indeed, several AI problems can be solved by searching through a great number of solutions. The reasoning process, in these cases, can be reduced to performing a search in a problem space.

The solution path is a sequence of admissible moves. The main goal for this article is to explain how breadth-first search works and how to implement this algorithm in Python.

In particular, in this tutorial I will:. As you might have understood by now, BFS is inherently tied with the concept of a graph. The edges are undirected and unweighted. Distance between two nodes will be measured based on the number of edges separating two vertices.

It is possible to represent a graph in a couple of ways: with an adjacency matrix that can be implemented as a 2-dimensional list and that is useful for dense graphs or with an adjacency list useful for sparse graphs. In this tutorial, I use the adjacency list. The keys of the dictionary represent nodes, the values have a list of neighbours.

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BFS visits all the nodes of a graph connected component following a breadthward motion. In other words, BFS starts from a node, then it checks all the nodes at distance one from the starting node, then it checks all the nodes at distance two and so on. In order to remember the nodes to be visited, BFS uses a queue. The algorithm can keep track of the vertices it has already checked to avoid revisiting them, in case a graph had one or more cycles.

In FIFO queues, the oldest first entry is processed first. The process is similar to what happens in queues at the post office. Who arrives first is served first. The answer is pretty simple. The algorithm checks all the nodes at a given depth distance from the entry pointbefore moving to the level below. As you can note, queue already has a node to be checked, i. The next step is to implement a loop that keeps cycling until queue is empty.

At each iteration of the loop, a node is checked. Tip: To make the code more efficient, you can use the deque object from the collections module instead of a list, for implementing queue. We have a functioning BFS implementation that traverses a graph. Now on to a more challenging task: finding the shortest path between two nodes. For this task, the function we implement should be able to accept as argument a graph, a starting node e.

If the algorithm is able to connect the start and the goal nodes, it has to return the path. The nice thing about BFS is that it a lways returns the shortest path, even if there is more than one path that links two vertices. There are a couple of main differences between the implementations of BDF for traversing a graph and for finding the shortest path. First, in case of the shortest path application, we need for the queue to keep track of possible paths implemented as list of nodes instead of nodes.

Second, when the algorithm checks for a neighbour node, it needs to check whether the neighbour node corresponds to the goal node. For example, if a path exists that connects two nodes in a graph, BFS will always be capable of identifying it — given the search space is finite. Completeness is a nice-to-have feature for an algorithm, but in case of BFS it comes to a high cost.

The execution time of BFS is fairly slow, because the time complexity of the algorithm is exponential. In the case of problems which translate into huge graphs, the high memory requirements make the use of BFS unfeasible.It fans away from the starting node by visiting the next node of the lowest weight and continues to do so until the next node of the lowest weight is the end node.

The weights in this example are given by the numbers on the edges between nodes. We choose the lowest cost option, to visit node B at a cost of 2. We then have the following options:. The next lowest cost item is visiting C from X, so we try that and then we are left with the above options, as well as:.

For each destination node that we visit, we note the possible next destinations and the total weight to visit that destination. If a destination is one we have seen before and the weight to visit is lower than it was previously, this new weight will take its place. For example. We continue evaluating until the destination node weight is the lowest total weight of all possible options.

And we can work backwards through this path to get all the nodes on the shortest path from X to Y. Once we have reached our destination, we continue searching until all possible paths are greater than 11; at that point we are certain that the shortest path is We will be using it to find the shortest path between two nodes in a graph.

Now we need to implement our algorithm. At our starting node Xwe have the following choice: Visit A next at a cost of 7 Visit B next at a cost of 2 Visit C next at a cost of 3 Visit E next at a cost of 4 We choose the lowest cost option, to visit node B at a cost of 2. For example Visiting A from X is a cost of 7 But visiting A from X via B is a cost of 5 Therefore we note that the shortest route to X is via B We only need to keep a note of the previous destination node and the total weight to get there.

In this case, we will end up with a note of: The shortest path to Y being via G at a weight of 11 The shortest path to G is via H at a weight of 9 The shortest path to H is via B at weight of 7 The shortest path to B is directly from X at weight of 2 And we can work backwards through this path to get all the nodes on the shortest path from X to Y.

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