Farthest node in a graph

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# Farthest node in a graph

Given an undirected tree, we need to find the longest path of this tree where a path is defined as a sequence of nodes. This problem is the same as the diameter of the n-ary tree. We have discussed a simple solution here. In this post, an efficient solution is discussed. We can find the longest path using two BFS s.

Breadth First Search \u0026 Depth First Search - Maximum Depth of N-ary Tree - LeetCode 559.

The idea is based on the following fact: If we start BFS from any node x and find a node with the longest distance from x, it must be an endpoint of the longest path. It can be proved using contradiction. So our algorithm reduces to simple two BFSs. This article is contributed by Utkarsh Trivedi. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.

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The direct built-in function GraphDiameter seems to work normally and quite rapidly for small systems. What I do is as follows: generate a random ER graph with n nodes and edge probability p, check that it is connected, if yes find the diameter of the graph. Which I assume means the search was taking too long and the kernel aborted it. PseudoDiameter finds an approximate graph diameter. It works by starting from a vertex u, and finds a vertex v that is farthest away from u.

This process is repeated by treating v as the new starting vertex, and ends when the graph distance no longer increases. A vertex from the last level set that has the smallest degree is chosen as the final starting vertex u, and a traversal is done to see if the graph distance can be increased.

This graph distance is taken to be the pseudo-diameter. Basically a brute-force approach and thus probably not more efficient than GraphDiameterbut a way to obtain also a path that realizes the diameter:.

## Longest path in an undirected tree

This is just a heuristic but it tends to deliver surprisingly good results. It is based on the intuition that the minimum and the maximum of the first eigenvector of the graph Lapacian tend to be be well separated. So this method computes the first eigenvector v and extracts its minimum and maximum.

Maybe you find something useful in this script by Fan Chung. At least upper bounds for the diameter in terms of eigenvalues can be found there; these could be used to assess the quality of the approximation a posteriorily. Diameter computation is approximately cubic in the number of vertices. Based on a quick fit, I expect a second 5 minute running time for a RandomGraph[BernoulliGraphDistribution[, 0. Now you know how long you need to wait—so try it! Sign up to join this community. The best answers are voted up and rise to the top.

Home Questions Tags Users Unanswered. Efficiently finding furthest two nodes in a graph Ask Question. Asked 1 year, 2 months ago. Active 1 year, 2 months ago. Viewed 2k times. In the light of this, my questions are: What is happening when the abort message is being returned? Is there a way to efficiently obtain the graph diameter even for such large graphs within Mathematica?

Or is the built-in GraphDiameter function known to be the most efficient option within Mathematica? Is asking for the end-nodes corresponding to the found diameter an additional overhead in terms of computation or it can easily be retrieved if a finite diameter has been found? Any feedback on these types of computations on graphs would be much appreciated. Maybe you run out of memory?

Active Oldest Votes. It would be quite useful to have this documented Before I read your message, I tried to reimplement this method, and ran into an issue where VertexDegree is surprisingly slow for large graphs, and has worse than expected complexity cubic in the number of vertices for graphs of constant density.

Just letting you know. Already reported to support. Edit This is just a heuristic but it tends to deliver surprisingly good results.While Cypher is optimized for finding the shortest path between two nodes, with such functionality as shortestPathit does not have the same sort of function for longest path.

In some cases, you may want this, and not the shortest route. The problem with this is that it will give you all paths, one for each hop, until you get to the leaf node.

What you want is just the longest path to see how the parent and leaf child are connected. To do this efficiently, do the following:. What the above query is doing: The variable length If not using an acyclic tree structure, you may have several paths between two nodes, and you may want to get just the longest.

We can do this by ordering by path length and only taking the longest path:. With a fairly well connected graph, variable-length path queries like this may get increasingly expensive. Here are some tips to keep these performant. Knowledge Base Search.

Go Back. Achieving longestPath Using Cypher. Longest path when there are multiple paths present If not using an acyclic tree structure, you may have several paths between two nodes, and you may want to get just the longest. Tips for making these queries more efficient With a fairly well connected graph, variable-length path queries like this may get increasingly expensive.

Constrain relationship type and direction — If possible, use only the relevant types needed, and use a directed relationship. This may cut down on the paths followed during expansion. Supply an upper bound for the variable-length pattern — Patterns without bounds may get out of hand in a well connected graph.

Setting an upper bound can help constrain the work the query needs to perform. Use all or none in the WHERE clause following the MATCH — If predicates need to be evaluated during expansion, and they must apply to all nodes or relationships in the path, use all or none on the nodes or relationships from the path to evaluate during expansion rather than filtering after all paths are found.

Use APOC path expanders for special cases or restrictions — For certain restrictions, such as not repeating nodes during expansion, you may want to use path expander procs from APOC Procedures to enforce different configurations of uniqueness during expansion.

Relevant for Neo4j Versions: 3. Relevant keywords cypher, path, apoc. Author Dave Gordon. Applicable versions 3. Keywords cypher longestpath neo4jThe problem requires finding the farthest node. The solution given was to solve it using HLD.

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But I was thinking if there is some other way of doing it. I came across Centroid Decomposition. So I though of decomposing the tree into the levels l1,l2,l Then sort each level in dfs order.

Then for a particular node a the farthest node b would be the first element in the lowest level of the centroid graph. This level should not contain node a. Auto comment: topic has been updated by dragonzurfer previous revisionnew revisioncompare.

The problem is invisible right now, but I think it can be solved by simple DFS to calc the farthest node from each node, so dist [ i ] will contains the distance to the farthest node from node i.

Farthest node from any vertex of tree is actually one of ends of its diameter, which you can compute using two dfses, and then compute distances to vertices from ends using another two dfses. Doesn't matter. There is a nice article about tree centroids here. They're also usefull when you need to find if two trees are isomorphic. Mhammad1 has explained above. It's a tree, there is only one path between two nodes. So, you can — for sure — use a BFS or a DFS to get the farthest node and it's, actually, the most effective way if there is only a single query per graph.

Enter Register. Before contest Codeforces Round Div. Grand Prix of Korea. Lecture 5. Farthest node b from a node a in a tree. Comments Write comment?

The problem is invisible right now, but I think it can be solved by simple DFS to calc the farthest node from each node, so dist [ i ] will contains the distance to the farthest node from node i when you consider the farthest node from node uit could be found in the subtree of node u or outside, to calculate the farthest node from node u in its subtree is trivial, but outside is a little bit confusing, to calculate this value, for each node keep a vector vec [ u ] contains the farthest distance node in the subtree of node v where v is a direct child of uthen when you do DFSyou can pass a parameter called MaxUp that holds the farthest node from outside subtree of node uthis can be done fast by calculating prefix and suffix max on vector vec.Problem Description Given an arbitrary unweighted rooted tree which consists of N nodes.

The goal of the problem is to find largest distance between two nodes in a tree. Distance between two nodes is a number of edges on a path between the nodes there will be a unique path between any pair of nodes since it is a tree.

Exactly one of the i's will have A[i] equal to -1, it will be root node. Sign Up using. Log In using. Toggle navigation. The nodes will be numbered 0 through N - 1. First and only argument is an integer array A of size N. Return a single integer denoting the largest distance between two nodes in a tree. Output 1: 3. See Expected Output.

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### Largest Distance between nodes of a Tree

Sign up. Like Us Follow Us Email. Click here to start solving coding interview questions.If they want to feed tight end Rob Gronkowski in his annual homecoming game in Western New York, they can do that, too.

That's less of an indictment on the Buffalo defense, which has been up and down (sometimes very down) all year but has overall surpassed low preseason expectations.

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Rather, it's a commentary on the fact that the Pats are firing on all cylinders after winning seven straight. The guess here is that this game becomes an "anything you can do, I can do better" matchup between the Carolina Panthers' No.

When these two teams played in September, Thomas had five catches for 50 yards and one touchdown on the first drive alone. He didn't see many targets the rest of the way as the Saints were largely in command of the matchup and went to the ground, but Thomas showed he can beat the Panthers secondary. As for Funchess, he is facing a Saints team that is missing safety Marcus Williams and may not have cornerback Marshon Lattimore, who is questionable with an ankle injury.

Ken Crawley is back from an abdomen injury, but as he told Josh Katzenstein of NOLA. Philadelphia Eagles at Seattle SeahawksIf the Seattle offensive line can hold off the Philadelphia Eagles' front seven long enough for quarterback Russell Wilson to do work, then the 'Hawks can protect their house and pull off the upset.

If the Seahawks falter in that regard, then it could be a long day for the Seattle offense. The other issue is the Seattle running game, which simply hasn't been able to get going this year. According to Football Outsiders, Seattle ranks just 28th in the league in adjusted line yards. The problem is the Eagles' defensive line is ranked first by a long shot in that category.

Ultimately, the Eagles' front seven seems to have a big edge over the Seattle front, and that should be the difference. Pittsburgh Steelers at Cincinnati BengalsIt seems like the Pittsburgh Steelers and Cincinnati Bengals engage in knockdown, drag-'em-out games against each other when they face off. Such is the reputation of the AFC North.

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Don't expect many offensive fireworks in this contest. Both defenses are stout and capable of shutting down the opposing offenses. The Bengals' attack has been hit or miss all season, although they showed signs of life on the ground thanks to rookie Joe Mixon's best day as a professional.

Green is still a problem for any team he faces, and tight end Tyler Kroft has done well in Tyler Eifert's stead. Although the Bengals are at home and the more desperate team as they sit one game out of the playoffs, it's hard to bet against Pittsburgh, who has won six straight and has been lights out on both sides of the ball.

Even if wideout Antonio Brown doesn't go (he didn't practice on Saturday and is questionable, per Jeremy Fowler of ESPN. Their exploits should lead to a close Steeler win. But now it gets even trickier and Joe Root and co as they look to complete the hardest task of all: winning the Ashes in Australia. According to Oddschecker on November 15. Lads, these games last five days. Away from home they will have to be a strong, tight-knit group and will need big leaders. Enter Ben Stokes and Alastair Cook.

Simon Heffer Premium 07 Dec 2017, 2:39pm Comment: England should fear a 5-0 - Australia have not even played their best yet Mitchell Johnson Premium 07 Dec 2017, 10:17am Who will win the 2017 Ashes.

Our Ashes-ometer on whether Australia or England are on track for success 07 Dec 2017, 10:17am Can England avoid another Ashes whitewash Down Under. England triumphed in the last series in 2015, defeating their opponents 3-2 on home soil. What are the odds of an England win. Or a series draw. Simon Heffer 07 Dec 2017, 2:39pm Comment: England should fear a 5-0 - Australia have not even played their best yet Mitchell Johnson 07 Dec 2017, 10:17am Who will win the 2017 Ashes.The Gunners played some terrific football against Manchester United last week and they are unlikely to come up against a keeper as inspired as David De Gea was on Sunday.

The Saints are in decent form themselves, but I fancy Arsenal in what promises to be an open entertaining contest. Liverpool are high in confidence, have pace and power in abundance and it could be a very long afternoon for the Blue half of Merseyside. Romelu Lukaku scored in both games against City for Everton last season and if the striker were to get the winning goal in a Manchester Derby, it would be a pivotal moment in his United career.

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For more tips from Michael read his column here. By continuing, your consent is assumed. WatchThe Seattle Seahawks hit the road this weekend for a Week 14 matchup with the Jacksonville Jaguars. WatchThe Seattle Seahawks took advantage of being able to practice outside all week and are "fired up" for the upcoming game against Jacksonville on Sunday.

Brady Henderson at ESPN. But the Seahawks' pass protection has improved quite a bit with Duane Brown at left tackle. According to Pro Football Focus, Seattle allowed pressure on 39. The Seahawks' 24-10 win over Philadelphia last week was another reminder that they shouldn't be written off in their injury-weakened state. An improved offensive line, a running game that has shown signs of life and a defense that has still been good without Richard Sherman and Kam Chancellor will be enough on Sunday in Jacksonville.

The defense -- which leads the NFL in sacks, pass defense and scoring -- has its hands full with Wilson, Jimmy Graham and Doug Baldwin, but there are enough playmakers at all levels to limit the big plays that Wilson creates with his legs.

However, the Jags' offense is going to have a hard time moving the ball consistently. Leonard Fournette is still dealing with an ankle injury and has run for just 226 yards and one touchdown on 77 carries (2. The offensive line is banged up too. The biggest issue, though, are the receivers dealing with the half of the 'Legion of Boom' still standing. Rookie Keelan Colehas been up and down all season and has battled drops. Rookie Dede Westbrook is playing in just his fourth game. Marqise Lee is the Jaguars' top option, but he has battled rib and knee injuries over the past six weeks and leads the NFL with seven drops.