# Time complexity of recursive dfs

The only catch here is, unlike trees, graphs may contain cycles, so a node might be visited twice. To avoid processing a node more than once, use a boolean visited array.

The recursive implementation of DFS is already discussed: previous post. Modifiation of the above Solution : Note that the above implementation prints only vertices that are reachable from a given vertex. For example, if the edges and are removed, then the above program would only print 0. To print all vertices of a graph, call DFS for every unvisited vertex.

DFS int s traverses vertices. Graph::Graph int V. DFS 0. Graph int V.

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This class represents a directed graph using adjacency. Initially mark all verices as not visited. Create a stack for DFS. Push the current source node. Pop a vertex from stack and print it. Stack may contain same vertex twice. Get all adjacent vertices of the popped vertex s.

If a adjacent has not been visited, then push it. Driver program to test methods of graph class. DFS 0.Adrian Sampson shows how to develop depth-first search dfs and breadth-first search bfs.

Both algorithms are used to traverse a graph, "visiting" each of its nodes in an orderly fashion. He assumes you are familiar with the idea.

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He also figures out the time complexity of these algorithms. Finally, he shows you how to implement a DFS walk of a graph. In this 2.

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Read it here: dfs01develop. Precondition: u is not visited. We urge you to first draw the outline of the graph as done in the video, for that will help you develop the procedure. Compare your result to the procedure we developed, note differences, and do the task again until you understand the development and can do it at any time. We left the notion of visiting and testing whether a node has been visited abstract.

Figure out how to implement the set of visited nodes. You can add parameters to the method if you want. We ask you to develop an abstract version of the DFS procedure that does not have the precondition that u is not visited. The specification is given below. We will use our own solution in the next video. We determine the exact number of times each statement of procedure dfs1 is executed. The number of recursive calls turns out to be very large, and we show how to eliminate most of them 3.

Read it here: dfs02analyze. In just over 4 minutes, we develop a non-recursive version of DFS. It maintains a stack of nodes, akin to the stack of frames for incompleted calls on the recursive DFS procedure. We show two ways of implementing the stack.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.

## Recursion Time Complexity || Analysis of Recursion

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.

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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.

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A recursive implementation of DFS: [5].IDDFS is optimal like breadth-first searchbut uses much less memory; at each iteration, it visits the nodes in the search tree in the same order as depth-first search, but the cumulative order in which nodes are first visited is effectively breadth-first. This implementation of IDDFS does not account for already-visited nodes and therefore does not work for undirected graphs.

If the goal node is found, then DLS unwinds the recursion returning with no further iterations. Otherwise, if at least one node exists at that level of depth, the remaining flag will let IDDFS continue. Another solution could use sentinel values instead to represent not found or remaining level results. IDDFS combines depth-first search's space-efficiency and breadth-first search's completeness when the branching factor is finite.

If a solution exists, it will find a solution path with the fewest arcs. Since iterative deepening visits states multiple times, it may seem wasteful, but it turns out to be not so costly, since in a tree most of the nodes are in the bottom level, so it does not matter much if the upper levels are visited multiple times.

The main advantage of IDDFS in game tree searching is that the earlier searches tend to improve the commonly used heuristics, such as the killer heuristic and alpha-beta pruningso that a more accurate estimate of the score of various nodes at the final depth search can occur, and the search completes more quickly since it is done in a better order.

For example, alpha-beta pruning is most efficient if it searches the best moves first. A second advantage is the responsiveness of the algorithm. When used in an interactive setting, such as in a chess -playing program, this facility allows the program to play at any time with the current best move found in the search it has completed so far.

This can be phrased as each depth of the search co recursively producing a better approximation of the solution, though the work done at each step is recursive. This is not possible with a traditional depth-first search, which does not produce intermediate results.

The time complexity of IDDFS in a well-balanced tree works out to be the same as breadth-first search, i. Then we have. The higher the branching factor, the lower the overhead of repeatedly expanded states, [1] : 6 but even when the branching factor is 2, iterative deepening search only takes about twice as long as a complete breadth-first search. Since IDDFS, at any point, is engaged in a depth-first search, it need only store a stack of nodes which represents the branch of the tree it is expanding.

In general, iterative deepening is the preferred search method when there is a large search space and the depth of the solution is not known. Iterative deepening prevents this loop and will reach the following nodes on the following depths, assuming it proceeds left-to-right as above:.

It still sees C, but that it came later. Also it sees E via a different path, and loops back to F twice. For this graph, as more depth is added, the two cycles "ABFE" and "AEFB" will simply get longer before the algorithm gives up and tries another branch. Similar to iterative deepening is a search strategy called iterative lengthening search that works with increasing path-cost limits instead of depth-limits.

It expands nodes in the order of increasing path cost; therefore the first goal it encounters is the one with the cheapest path cost.It's often possible to compute the time complexity of a recursive function by formulating and solving a recurrence relation. We also show how to analyze recursive algorithms that depend on the size and shape of a data structure.

To find the time complexity for the Sum function can then be reduced to solving the recurrence relation. The very same method can be used also for more complex recursive algorithms. Formulating the recurrences is straightforward, but solving them is sometimes more difficult.

Then the recurrences become. The master theorem is a recipe that gives asymptotic estimates for a class of recurrence relations that often show up when analyzing recursive algorithms. In fact, you can use repeated substitution in the same way as in the previous examples.

Instead, we can count the work performed for each piece of the data structure visited by the algorithm. To compute the time complexity, we can use the number of calls to DFS as an elementary operation: the if statement and the mark operation both run in constant time, and the for loop makes a single call to DFS for each iteration.

Say we have a graph where each node is connected to all the rest. We start at some node, visit it and then push all it's neighbors onto the stack. Now we pop the next node and put all of it's neighbors onto the stack. We repeat until we visit all the nodes. Let's pretend that the node that finds itself on top of the stack is not visited yet in each iteration best case scenario for this graph.

In this case we visited all the nodes in V moves, but for each of them we pushed V -1 nodes on the stack which means that all the edges are pushed on the stack and the complexity is O E. A few notes.

I'm arguing that the complexity is LESS than that so this proof that only looks at the best scenario for a worst case graph is fine. I'm also assuming that E is always larger than V. The explosion here is due to the fact that we keep stacking up useless nodes that will never be processed.

What if we instead just recurse? The advantage is that we can check if we're done before each recursive call. Since there's no explicit stack and I'm still only visiting nodes I haven't seen before, I don't see how I can exceed the complexity of O V. That check still contributes to the run time. For each node you visit, you need to see which of its neighbors still need to be visited, which means checking each adjacent edge.

## Graph – Depth First Search using Recursion

Active 3 years, 5 months ago. Viewed 2k times. The usual explanation I've seen goes as follows: Say we implement a DFS using an explicit stack for simplicity.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Stack Overflow for Teams is a private, secure spot for you and your coworkers to find and share information. Can someone explain with an example how we can calculate the time and space complexity of both these traversal methods?

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.

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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? Ask Question. Asked 8 years, 3 months ago. Active 2 years, 3 months ago.

Viewed 77k 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. This is not necessery for trees. Active Oldest Votes. I'm not sure if this answer is right?

However, the stack implementation only uses O bd space where b is the branching factor and d is the depth. However, bd! Thus, I think the space complexity can be bounded more tightly by O bd rather than saying O V. Note that the answer says it's worst case behavior. So why is space complexity ever O N? The unvisited stack or queue will never contain all nodes. Where am I going wrong? 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.