Breadth First Search Java

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Breadth First Search Java Code

Here you will get Breadth First Search (BFS) Java program along with example. Breadth First Search is graph traversal algorithm which has many applications in most of the algorithms. We will start with one node and we will explore all the nodes (neighbor nodes) in the same level. Then we should go to next level to explore all nodes in that level. Breadth first search Recursive Java program To write a Java program to recursively do a level order traversal of a binary tree you need to calculate height of the tree and then call method for level order traversal for level 0 to max level of the binary tree.

Breadth First Search; Depth First Search; Breadth First Search (BFS) Algorithm. Breadth first search is a graph traversal algorithm that starts traversing the graph from root node and explores all the neighbouring nodes. Then, it selects the nearest node and explore all the unexplored nodes.

Breadth-first search
Order in which the nodes are expanded
Class	Search algorithm
Data structure	Graph
Worst-case performance	${displaystyle O( V + E )=O(b^{d})}$
Worst-case space complexity	${displaystyle O( V )=O(b^{d})}$

Graph and tree search algorithms
Listings
Related topics

Animated example of a breadth-first search

Breadth-first search (BFS) is an algorithm for traversing or searching tree or graph data structures. It starts at the tree root (or some arbitrary node of a graph, sometimes referred to as a 'search key'^[1]), and explores all of the neighbor nodes at the present depth prior to moving on to the nodes at the next depth level.

It uses the opposite strategy as depth-first search, which instead explores the highest-depth nodes first before being forced to backtrack and expand shallower nodes.

BFS and its application in finding connected components of graphs were invented in 1945 by Konrad Zuse, in his (rejected) Ph.D. thesis on the Plankalkül programming language, but this was not published until 1972.^[2]It was reinvented in 1959 by Edward F. Moore, who used it to find the shortest path out of a maze,^[3]^[4] and later developed by C. Y. Lee into a wire routing algorithm (published 1961).^[5]

1Pseudocode
2Analysis

Pseudocode

Input: A graph Graph and a starting vertexroot of Graph

Output: Goal state. The parent links trace the shortest path back to root

More details

This non-recursive implementation is similar to the non-recursive implementation of depth-first search, but differs from it in two ways:

it uses a queue (First In First Out) instead of a stack and
it checks whether a vertex has been discovered before enqueueing the vertex rather than delaying this check until the vertex is dequeued from the queue.

The Q queue contains the frontier along which the algorithm is currently searching.

Nodes can be labelled as discovered by storing them in a set, or by an attribute on each node, depending on the implementation.

Note that the word node is usually interchangeable with the word vertex.

The parent attribute of each vertex is useful for accessing the nodes in a shortest path, for example by backtracking from the destination node up to the starting node, once the BFS has been run, and the predecessors nodes have been set.

Breadth-first search produces a so-called breadth first tree. You can see how a breadth first tree looks in the following example.

Example

The following is an example of the breadth-first tree obtained by running a BFS on German cities starting from Frankfurt:

An example map of Southern Germany with some connections between cities

The breadth-first tree obtained when running BFS on the given map and starting in Frankfurt

Analysis

Time and space complexity

The time complexity can be expressed as ${displaystyle O( V + E )}$ , since every vertex and every edge will be explored in the worst case. ${displaystyle V }$ is the number of vertices and ${displaystyle E }$ is the number of edges in the graph.Note that ${displaystyle O( E )}$ may vary between ${displaystyle O(1)}$ and ${displaystyle O( V ^{2})}$ , depending on how sparse the input graph is.^[6]

When the number of vertices in the graph is known ahead of time, and additional data structures are used to determine which vertices have already been added to the queue, the space complexity can be expressed as ${displaystyle O( V )}$ , where ${displaystyle V }$ is the cardinality of the set of vertices. This is in addition to the spacerequired for the graph itself, which may vary depending on the graph representation used by an implementation of the algorithm.

When working with graphs that are too large to store explicitly (or infinite), it is more practical to describe the complexity of breadth-first search in different terms: to find the nodes that are at distance d from the start node (measured in number of edge traversals), BFS takes O(b^{d + 1}) time and memory, where b is the 'branching factor' of the graph (the average out-degree).^[7]^:81

Completeness

In the analysis of algorithms, the input to breadth-first search is assumed to be a finite graph, represented explicitly as an adjacency list or similar representation. However, in the application of graph traversal methods in artificial intelligence the input may be an implicit representation of an infinite graph. In this context, a search method is described as being complete if it is guaranteed to find a goal state if one exists. Breadth-first search is complete, but depth-first search is not. When applied to infinite graphs represented implicitly, breadth-first search will eventually find the goal state, but depth-first search may get lost in parts of the graph that have no goal state and never return.^[8]

BFS ordering

An enumeration of the vertices of a graph is said to be a BFS ordering if it is the possible output of the application of BFS to this graph.

Let ${displaystyle G=(V,E)}$ be a graph with ${displaystyle n}$ vertices. Recall that ${displaystyle N(v)}$ is the set of neighbors of ${displaystyle v}$ .For ${displaystyle sigma =(v_{1},dots ,v_{m})}$ be a list of distinct elements of ${displaystyle V}$ , for ${displaystyle vin Vsetminus {v_{1},dots ,v_{m}}}$ , let ${displaystyle nu _{sigma }(v)}$ be the least ${displaystyle i}$ such that ${displaystyle v_{i}}$ is a neighbor of ${displaystyle v}$ , if such a ${displaystyle i}$ exists, and be ${displaystyle infty }$ otherwise.

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 ${displaystyle 1<ileq n}$ , ${displaystyle v_{i}}$ is the vertex ${displaystyle win Vsetminus {v_{1},dots ,v_{i}-1}}$ such that ${displaystyle nu _{(v_{1},dots ,v_{i-1})}(w)}$ is minimal. Equivalently, ${displaystyle sigma }$ is a BFS ordering if, for all ${displaystyle 1leq i<j<kleq n}$ with ${displaystyle v_{i}in N(v_{k})setminus N(v_{j})}$ , there exists a neighbor ${displaystyle v_{m}}$ of ${displaystyle v_{j}}$ such that ${displaystyle m<i}$ .

Applications

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

Copying garbage collection, Cheney's algorithm
Finding the shortest path between two nodes u and v, with path length measured by number of edges (an advantage over depth-first search)^[9]
(Reverse) Cuthill–McKee mesh numbering
Ford–Fulkerson method for computing the maximum flow in a flow network
Serialization/Deserialization of a binary tree vs serialization in sorted order, allows the tree to be re-constructed in an efficient manner.
Construction of the failure function of the Aho-Corasick pattern matcher.
Testing bipartiteness of a graph.

References

^'Graph500 benchmark specification (supercomputer performance evaluation)'. Graph500.org, 2010. Archived from the original on 2015-03-26. Retrieved 2015-03-15.
^Zuse, Konrad (1972), Der Plankalkül (in German), Konrad Zuse Internet Archive. See pp. 96–105 of the linked pdf file (internal numbering 2.47–2.56).
^Moore, Edward F. (1959). 'The shortest path through a maze'. Proceedings of the International Symposium on the Theory of Switching. Harvard University Press. pp. 285–292. As cited by Cormen, Leiserson, Rivest, and Stein.
^Skiena, Steven (2008). 'Sorting and Searching'. The Algorithm Design Manual. Springer. p. 480. doi:10.1007/978-1-84800-070-4_4. ISBN978-1-84800-069-8.
^Lee, C. Y. (1961). 'An Algorithm for Path Connections and Its Applications'. IRE Transactions on Electronic Computers.
^Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001) [1990]. '22.2 Breadth-first search'. Introduction to Algorithms (2nd ed.). MIT Press and McGraw-Hill. pp. 531–539. ISBN0-262-03293-7.
^Russell, Stuart; Norvig, Peter (2003) [1995]. Artificial Intelligence: A Modern Approach (2nd ed.). Prentice Hall. ISBN978-0137903955.
^Coppin, B. (2004). Artificial intelligence illuminated. Jones & Bartlett Learning. pp. 79–80.
^Aziz, Adnan; Prakash, Amit (2010). '4. Algorithms on Graphs'. Algorithms for Interviews. p. 144. ISBN978-1453792995.

Knuth, Donald E. (1997), The Art of Computer Programming Vol 1. 3rd ed., Boston: Addison-Wesley, ISBN978-0-201-89683-1

External links

Breadth First Search Java Code

Wikimedia Commons has media related to Breadth-first search.

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Breadth First Search (BFS) algorithm traverses a graph in a breadthward motion and uses a queue to remember to get the next vertex to start a search, when a dead end occurs in any iteration.

As in the example given above, BFS algorithm traverses from A to B to E to F first then to C and G lastly to D. It employs the following rules.

Rule 1 − Visit the adjacent unvisited vertex. Mark it as visited. Display it. Insert it in a queue.
Rule 2 − If no adjacent vertex is found, remove the first vertex from the queue.
Rule 3 − Repeat Rule 1 and Rule 2 until the queue is empty.

Step	Traversal	Description
1	Initialize the queue.
2	We start from visiting S (starting node), and mark it as visited.
3	We then see an unvisited adjacent node from S. In this example, we have three nodes but alphabetically we choose A, mark it as visited and enqueue it.
4	Next, the unvisited adjacent node from S is B. We mark it as visited and enqueue it.
5	Next, the unvisited adjacent node from S is C. We mark it as visited and enqueue it.
6	Now, S is left with no unvisited adjacent nodes. So, we dequeue and find A.
7	From A we have D as unvisited adjacent node. We mark it as visited and enqueue it.

At this stage, we are left with no unmarked (unvisited) nodes. But as per the algorithm we keep on dequeuing in order to get all unvisited nodes. When the queue gets emptied, the program is over.

The implementation of this algorithm in C programming language can be seen here.

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