# Category: Dfs space complexity

## Dfs space complexity

Kazrarisar / 01.04.2021

Artificial Intelligence is the study of building agents that act rationally. Most of the time, these agents perform some kind of search algorithm in the background in order to achieve their tasks.

There are far too many powerful search algorithms out there to fit in a single article. Instead, this article will discuss six of the fundamental search algorithms, divided into two categories, as shown below. Note that there is much more to search algorithms that the chart I have provided above.

However, this article will mostly stick to the above chart, exploring the algorithms given there. The search algorithms in this section have no additional information on the goal node other than the one provided in the problem definition.

Uninformed search is also called Blind search. Depth-first search DFS is an algorithm for traversing or searching tree or graph data structures. The algorithm starts at the root node selecting some arbitrary node as the root node in the case of a graph and explores as far as possible along each branch before backtracking.

The equivalent search tree for the above graph is as follows. The traversal is shown in blue arrows. Time complexity: Equivalent to the number of nodes traversed in DFS. Space complexity: Equivalent to how large can the fringe get. Completeness: DFS is complete if the search tree is finite, meaning for a given finite search tree, DFS will come up with a solution if it exists. Optimality: DFS is not optimal, meaning the number of steps in reaching the solution, or the cost spent in reaching it is high.

Breadth-first search BFS is an algorithm for traversing or searching tree or graph data structures. Time complexity: Equivalent to the number of nodes traversed in BFS until the shallowest solution. Optimality: BFS is optimal as long as the costs of all edges are equal.

In other words, traversing via different edges might not have the same cost. The goal is to find a path where the cumulative sum of costs is least. Cost of each node is the cumulative cost of reaching that node from the root.

Based on UCS strategy, the path with least cumulative cost is chosen. Note that due to the many options in the fringe, the algorithm explores most of them so long as their cost is low, and discards them when a lower cost path is found; these discarded traversals are not shown below. The actual traversal is shown in blue. Then effective depth. Here, the algorithms have information on the goal state, which helps in more efficient searching.

This information is obtained by something called a heuristic. Search Heuristics: In an informed search, a heuristic is a function that estimates how close a state is to the goal state. For examples — Manhattan distance, Euclidean distance, etc.BFS stands for Breadth First Search is a vertex based technique for finding a shortest path in graph.

It uses a Queue data structure which follows first in first out. In BFS, one vertex is selected at a time when it is visited and marked then its adjacent are visited and stored in the queue.

It is slower than DFS. It uses the Stack data structureperforms two stages, first visited vertices are pushed into stack and second if there is no vertices then visited vertices are popped. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below. Writing code in comment? Please use ide. What are the differences between Bellman Ford's and Dijkstra's algorithms? Check out this Author's contributed articles. Load Comments. BFS can be used to find single source shortest path in an unweighted graph, because in BFS, we reach a vertex with minimum number of edges from a source vertex.

BFS considers all neighbors first and therefore not suitable for decision making trees used in games or puzzles. DFS is more suitable for game or puzzle problems. We make a decision, then explore all paths through this decision. And if this decision leads to win situation, we stop.Depth-first search DFS is an algorithm for traversing or searching tree or graph data structures.

The algorithm starts at the root node selecting some arbitrary node as the root node in the case of a graph and explores as far as possible along each branch before backtracking. The time and space analysis of DFS differs according to its application area. Thus, in this setting, the time and space bounds are the same as for breadth-first search and the choice of which of these two algorithms to use depends less on their complexity and more on the different properties of the vertex orderings the two algorithms produce.

For applications of DFS in relation to specific domains, such as searching for solutions in artificial intelligence or web-crawling, the graph to be traversed is often either too large to visit in its entirety or infinite DFS may suffer from non-termination.

In such cases, search is only performed to a limited depth ; due to limited resources, such as memory or disk space, one typically does not use data structures to keep track of the set of all previously visited vertices. When search is performed to a limited depth, the time is still linear in terms of the number of expanded vertices and edges although this number is not the same as the size of the entire graph because some vertices may be searched more than once and others not at all but the space complexity of this variant of DFS is only proportional to the depth limit, and as a result, is much smaller than the space needed for searching to the same depth using breadth-first search.

For such applications, DFS also lends itself much better to heuristic methods for choosing a likely-looking branch. When an appropriate depth limit is not known a priori, iterative deepening depth-first search applies DFS repeatedly with a sequence of increasing limits.

In the artificial intelligence mode of analysis, with a branching factor greater than one, iterative deepening increases the running time by only a constant factor over the case in which the correct depth limit is known due to the geometric growth of the number of nodes per level.

DFS may also be used to collect a sample of graph nodes. Iterative deepening is one technique to avoid this infinite loop and would reach all nodes. A convenient description of a depth-first search of a graph is in terms of a spanning tree of the vertices reached during the search. Based on this spanning tree, the edges of the original graph can be divided into three classes: forward edgeswhich point from a node of the tree to one of its descendants, back edgeswhich point from a node to one of its ancestors, and cross edgeswhich do neither.

Sometimes tree edgesedges which belong to the spanning tree itself, are classified separately from forward edges. If the original graph is undirected then all of its edges are tree edges or back edges. An enumeration of the vertices of a graph is said to be a DFS ordering if it is the possible output of the application of DFS to this graph.

It is also possible to use depth-first search to linearly order the vertices of a graph or tree. There are four possible ways of doing this:. Note that repeat visits in the form of backtracking to a node, to check if it has still unvisited neighbors, are included here even if it is found to have none.

Reverse postordering produces a topological sorting of any directed acyclic graph. This ordering is also useful in control flow analysis as it often represents a natural linearization of the control flows. A recursive implementation of DFS: . The order in which the vertices are discovered by this algorithm is called the lexicographic order. These two variations of DFS visit the neighbors of each vertex in the opposite order from each other: the first neighbor of v visited by the recursive variation is the first one in the list of adjacent edges, while in the iterative variation the first visited neighbor is the last one in the list of adjacent edges. The non-recursive implementation is similar to breadth-first search but differs from it in two ways:. This ordering is called the lexicographic depth-first search ordering.

John Reif considered the complexity of computing the lexicographic depth-first search ordering, given a graph and a source.

## Difference Between BFS and DFS

Unsourced material may be challenged and removed. Play media. Leiserson, and Ronald L. Information Processing Letters. Thomas H. CormenCharles E.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' 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 searchwhich instead explores the node branch as far as possible before being forced to backtrack and expand other nodes.

BFS and its application in finding connected components of graphs were invented in by Konrad Zusein his rejected Ph. Moorewho used it to find the shortest path out of a maze,   and later developed by C. Lee into a wire routing algorithm published Input : A graph Graph and a starting vertex root of Graph.

Output : Goal state. The parent links trace the shortest path back to root. This non-recursive implementation is similar to the non-recursive implementation of depth-first searchbut differs from it in two ways:.

Nodes can be labelled as discovered by storing them in a set, or by an attribute on each node, depending on the implementation. The parent attribute of each node 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. The following is an example of the breadth-first tree obtained by running a BFS on German cities starting from Frankfurt :.

This is in addition to the space required for the graph itself, which may vary depending on the graph representation used by an implementation of the algorithm. 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.

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. From Wikipedia, the free encyclopedia.Can someone explain with an example how we can calculate the time and space complexity of both these traversal methods?

Because this is tree traversalwe must touch every node, making this O n where n is the number of nodes in the tree. BFS will have to store at least an entire level of the tree in the queue sample queue implementation.

Best Case in this contextthe tree is severely unbalanced and contains only 1 element at each level and the space complexity is O 1. Worst Case would be storing n - 1 nodes with a fairly useless N-ary tree where all but the root node are located at the second level.

Regardless of the implementation recursive or iterativethe stack implicit or explicit will contain d nodes, where d is the maximum depth of the tree. With a balanced tree, this would be log n nodes. Time complexity is O V where V is the number of nodes,you need to traverse all nodes. Space complecity is O V as well - since at worst case you need to hold all vertices in the queue.

Time complexity is again O Vyou need to traverse all nodes. Space complexity - depends on the implementation, a recursive implementation can have a O h space complexity [worst case], where h is the maximal depth of your tree. Using an iterative solution with a stack is actually the same as BFS, just using a stack instead of a queue - so you get both O V time and space complexity. What is the time and space complexity of a breadth first and depth first tree traversal?

Also, how does recursive solution to depth first traversal affect the time and space complexity? DFS and BFS time complexity: O n Because this is tree traversalwe must touch every node, making this O n where n is the number of nodes in the tree. BFS space complexity: O n BFS will have to store at least an entire level of the tree in the queue sample queue implementation.

DFS space complexity: O d Regardless of the implementation recursive or iterativethe stack implicit or explicit will contain d nodes, where d is the maximum depth of the tree.

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Computer Science Stack Exchange is a question and answer site for students, researchers and practitioners of computer science. It only takes a minute to sign up. The same is said in this Wikibook page on Uninformed Search. Now the "infobox" of the Wikipedia article on DFS presents the following for the space complexity of the algorithm:.

I think it may depend on the implementation, so I would appreciate an explanation of the space complexity for the different known implementations. It depends on what exactly you call DFS. Consider for example the algorithm DFS-iterative described in Wikipediaand suppose that you run it on a tree so that you don't have to keep track of which nodes you have already visited.

The algorithm operates as follows:. You can check that this is the pint in time in which the size of the stack is maximized. Yuval Filmus' reply refers to this case indeed. Admittedly, however, this is not always the case. For example, in the Pancake puzzle, the branching factor grows with the length of the optimal solution and they both have similar values.

To see this just simply tweak the expansion procedure of any node: Instead of inserting all descendants of the current node as suggested by Yuval Filmus insert them in order. You first generate the first descendant and immediately after you proceed in depth-first order as a matter of fact, you do not need all the other descendants at this point. In case you ever backtrack to this node, then its descendant has been removed from the stack. Next, generate the second descendant and proceed in depth-first order again.

In case you backtrack to this node, then generate the third and so on until there are no more descendants.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Ask Question. Asked 3 years, 3 months ago. Active 3 years, 3 months ago. Viewed 10k times.

Why do I think this is the case? So, what's the correct space complexity of DFS? So seems to be argueing about different statements of the second kind, which need not be contradictory at all. Wait: what do you mean: result in a tree? Find me a definition where it revisits the nodes, then we can discuss about it. Active Oldest Votes. The algorithm operates as follows: Start at the root.

Time complexity is O V where V is the number of nodes,you need to traverse all nodes. Space complecity is O V as well - since at worst case you need to hold all vertices in the queue. Time complexity is again O Vyou need to traverse all nodes.

Space complexity - depends on the implementation, a recursive implementation can have a O h space complexity [worst case], where h is the maximal depth of your tree. Using an iterative solution with a stack is actually the same as BFS, just using a stack instead of a queue - so you get both O V time and space complexity. Because this is tree traversalwe must touch every node, making this O n where n is the number of nodes in the tree.

BFS will have to store at least an entire level of the tree in the queue sample queue implementation. Best Case in this contextthe tree is severely unbalanced and contains only 1 element at each level and the space complexity is O 1. Worst Case would be storing n - 1 nodes with a fairly useless N-ary tree where all but the root node are located at the second level.

Regardless of the implementation recursive or iterativethe stack implicit or explicit will contain d nodes, where d is the maximum depth of the tree. With a balanced tree, this would be log n nodes. In DFS the amount of time needed is proportional to the depth and branching factor. For DFS the total amount of time needed is given by.

It is the amount of space or memory required for getting a solution DFS stores only current path it is pursuing. Hence the space complexity is a linear function of the depth. Learn more. What is the time and space complexity of a breadth first and depth first tree traversal?

Time Complexity of Depth First Search (DFS) Algorithm

Ask Question. Asked 8 years ago. Active 2 years ago. Viewed 72k times. Also, how does recursive solution to depth first traversal affect the time and space complexity? Frank Q. There are quite decent explanations available on wikipedia: en. 