What would the following algorithm look like:
a linear-time algorithm which, given an undirected graph G, and a particular edge e in it, determines whether G has a cycle containing e
I have following Idea:
for each v that belongs to V,
if v is a descendant of e and (e,v) has not been traversed then check following:
if we visited e before v and left v before we left e then
the graph contains cycle
I am not sure if this is your homework so I'll just give a little hint - use the properties of breadth-first search tree (with root in any of the two vertices of the edge e), its subtrees which are determined by neighbors of the root and the edges between those subtrees.
Per comingstorm's hint, an undirected edge is itself a cycle. A<->B back and forth as many times as you like.
Related
It has been proved that the ST generated by kruskal algorithm is a MST. We can call it ST1.
Now the question is: Prove that there is no other ST, maximum weighted edge of which is smaller than the maximum weighted edge in ST1.
Proof by contradiction may help prove this claim.
In an MST of N nodes, each of the (N-1) edges serves the purpose of connecting two subtrees. Assume that there exists two subtrees A and B, and the edge E between them has the largest weight in ST1. This edge E is essential, as it connects the subtrees A and B. Without E, we could not connect A and B. (unless there exists another edge which can do the same)
Now, in order to try and find a contradiction to your claim, let's assume that there existed another edge F between subtrees A and B, and let's say that F has a smaller weight. Then, we would not need to use E and we can connect A and B with a smaller cost.
However, if we did have such an edge F, then the priority queue of edges, which is used in Kruskal's algorithm, would pop that edge F, before E was popped. (i.e., the edges would be stored in a min-heap, and would be popped in increasing order of weight) As E was the first edge to be popped between the nodes included in A and B (i.e., the first one that is able to connect subtrees A and B) we can state that no such edge F exists, and E is the edge with the smallest weight that can connect subtrees A and B. No matter how you construct a spanning tree, when you try to connect the subtrees A and B, there isn't any edge F with a smaller weight that you can use. So, to have a spanning tree, which has to include all N nodes, you have to use that edge E.
can someone explain for me how to solve it i know that we should use DFS but i cant understand what we do after that.
input : undirected graph G and specific v that belong to the G
output : spanning tree that v has k degree
I will suggest the following way.
Here I assume G is connected.
First remove v from the graph, find spanning tree for each of the remaining component.
Now you may have a single spanning tree or a forest depending on the graph, you can add back v and use the edges to connecting v and each of the spanning tree.
Then you will have a spanning tree of G, and there will be three cases.
case 1: degree v > k, in this case, the task is impossible
case 2: degree v = k, you have the answer.
case 3: degree v < k, then you just add unused edges of v. Each time you add an edge you will create a cycle, then you can just choose an edge which does not touch v and remove it.
You keep adding edges until you have your answer or all edges run out.
However, I cannot think of a fast way to query a cycle besides doing bfs/dfs.
Update: There is a faster way for case 3 by Matt, after connecting v to k appropriate neighbors, use Kruskal's or Prim's algorithm to fill in the rest of the spanning tree, starting with the edges from v that you already have.
Here is an algorithm that I am providing, but its pretty much brute force.
Condition for such tree to exist: if the degree of v < k, then such tree does not exist.
else follow the algorithm as:
Pick k vertices out of all the adjacent vertices of v,
1.mark all adjacent vertices of v as VISITED.
2.From each of those adjacent vertices , call DFS and the spanning tree grows.
3.After all DFS is complete,if all vertices of graph G are marked VISITED, then we
have our spanning tree, if a single vertex or more are left UNVISITED, then the
4.pick another set of k vertices.
if v has X as degree and X > k, then step 4 has to be repeated XCk(X choose k) times in the worst case.
We can think this way too:
For the given vertex v among all the neighbor v_i s.t. (v,v_i) in E(G) pick k edges with the smallest weights w(v,v_i), we need O(d*lg(d)) time where deg(v)=d >= k. Add these edges to the spanning tree edge set S. We have to add these edges to S anyway to ensure that the constraint holds.
Remove vertex v from the graph G along with all edges incident on v. Now run Prim/Kruskal on the graph G \ {v} starting with the edge set S, the algorithm itself will add edges ensuring the acyclic and minimum properties. This will run in O(m*lg(n)).
Assuming d small step 1 & 2 runs in O(m*lg(n)).
I'm having trouble to understand Kosaraju's algorithm for finding the strongly connected components of a directed graph. Here's what I have on my notebook (I'm a student :D):
Start from an arbitrary vertex (label it with #1) and perform a DFS. When you can't go any further, label the last visited vertex with #2, and start another DFS (skipping vertices already labeled), and so on.
Transpose the graph.
Do DFS starting from each vertex in reverse order, those vertices which end visited after each DFS belong to the same SCC.
I have this example:
And after the first step starting from E, the labels are:
E
G
K
J
I
H
F
C
D
B
A
So here comes the thing: Is there a difference for DFS in directed/undirected graphs?
I did a mental test of the first step on my mind ignoring the arrows (just like it was undirected) and only got correct #1 for E (of course) and #2 for G, but #3 fell onto J, not K. So I thought maybe I should respect the arrows, and did a DFS considering that, but after the first pass starting from E, I can't go anywhere from G (which is #2), so I'm stuck there.
Is there anything about DFS on directed graphs that I'm not aware of? I've been taught DFS only on undirected graphs!
Your second step is incomplete. See Wikipedia:
Kosaraju's algorithm works as follows:
Let G be a directed graph and S be an empty stack.
While S does not contain all vertices:
Choose an arbitrary vertex v not in S. Perform a depth-first search starting at v. Each time that depth-first search finishes expanding a vertex u, push u onto S.
Reverse the directions of all arcs to obtain the transpose graph.
While S is nonempty:
Pop the top vertex v from S. Perform a depth-first search starting at v in the transpose graph. The set of visited vertices will give the strongly connected component containing v; record this and remove all these vertices from the graph G and the stack S. Equivalently, breadth-first search (BFS) can be used instead of depth-first search.
So you shouldn't only do something with the last vertex and first vertices, but with each vertex in the DFS.
Also note that you should be backtracking - when you can't go further, you go to the previous vertex and continue from there.
And no, you can't treat it as an undirected graph - the direction of the edges matter significantly.
So, starting from E, you'd, for example, go F, then G, then back to F, then H, then K, then I, then J, then back to I, K, H, F, and finally E, having pushed all visited vertices onto the stack.
Hi so i'm doing some test prep and i need to figure out parts b and c. I know part a is true and i can prove it, but finding the algorithms for part b and c is currently eluding me.
Solve the following for a minimum bottleneck tree where the edge with the maximum cost is referred to as the bottleneck.
(a) Is every minimum-bottleneck
spanning tree of G a minimum-spanning tree of G? Prove your claim.
(b) For a given cost c, give an O(n+m)-time algorithm to
find if the bottleneck cost of a minimum-bottleneck spanning tree
of G is not more than c.
(c) Find an algorithm to find a minimum-bottleneck
spanning tree of G.
thanks in advance to anyone who can help me out
For (b):
Erase every edge in G that costs more than c, then check if the left graph is still connected.
For (c):
Do a binary search on c, using the algorithm that solved (b) as the dividing condition.
Proof of (b):
Let's say the graph we got after deleting edges cost more than c from G is G' .
Then:
If G' is connected, then there must be a spanning tree T in G'. Since no edge in G' costs more than c, we can tell for sure that no edge in T costs more than c. Therefore T is a spanning tree for G' and also G whose bottle neck is at most c
If G' is not connected, then there's no spanning tree in G' at all. Since we know every edge in G- G' costs more than c, and we know that any spanning tree of G will contains at least one edge of G- G', therefore we know there's no edge spanning tree of G whose bottle neck <= c
And of course detecting if a graph is connected costs O(n+m)
Proof of (c):
Say, the algorithm we used in (b) is F(G,c) .
Then we have
If F(G,c) = True for some c, then F(G,c') = True for all c' that have c'>=c
If F(G,c) = False for some c, then F(G,c') = False for all c' that have c'<=c
So we can binary search on c :)
Ans. a)False,every minimum bottleneck spanning tree of graph G is not a minimum spanning tree of G.
b)To check whether the value of minimum bottleneck spanning tree is atmost c,you can apply depth first search by selecting any vertex from the set of vertices in graph G.
***Algorithm:***
check_atmostvalue(Graph G,int c)
{
for each vertex v belongs to V[G] do {
visited[v]=false;
}
DFS(v,c); //v is any randomly choosen vertex in Graph G
for each vertex v belongs to V[G] do {
if(visited[v]==false) then
return false;
}
return true;
}
DFS(v,c)
{
visited[v]=true;
for each w adjacent to v do {
if(visited[w]=false and weight(v,w)<=c) then
DFS(w,c);
}
visited[w]=true;
}
This algorithm works in O(V+E) in the worst case as running timr for depth first search DFS is O(V+E).
This problem can be solved by simply finding the MST of the graph. This based on the following claim:
MST is a MBST for a connected graph.
For a MST, choose the maximum edge e in the MST and the edge e divides the MST into two sets S and T. Then from the cut property, edge e must be the minimum weight among those edges that connects S and T.
Then for a MBST, there must be some edge e' that connect S and T. Then w(e') must be no less than w(e). Thus we know that MST must be a MBST.
However, there is another way to determine the minimum bottleneck. We don't need to computer the MBST. In your question, you actually implies the monotocity of the minimum bottleneck. Therefor we can use binary search combined with the connectivity algorithm to find the minimum bottle neck. I haves seen the use of monocity in other cases. I am a bit amazed that the similar technique can be used here!
I have the following problem on my homework:
Give an O(n+m) algorithm to find that whether an edge e would be a part of the MST of a graph
(We are allowed to get help from others on this assignment, so this isn't cheating.)
I think that I could do a BFS and find if this edge is a edge between two layers and if so whether this edge was the smallest across those layers. But what could I say when this edge is not a tree edge of the BFS tree?
As a hint, if an edge is not the heaviest edge in any cycle that contains it, there is some MST that contains that edge. To see this, consider any MST. If the MST already contains the edge, great! We're done. If not, then add the edge into the MST. This creates a cycle in the graph. Now, find the heaviest edge in that cycle and remove it from the graph. Everything is now still connected (because if two nodes used to be connected by a path that went across that edge, now they can be connected by just going around the cycle the other way). Moreover, since the cost of the edge was deleted wasn't any smaller than the cost of the edge in question (because the edge isn't the heaviest edge in the cycle), the cost of this tree can't be any greater than before. Since we started with an MST, we must therefore end with an MST.
Using this property, see if you can find whether the edge is the heaviest edge on any cycle in linear time.
We will solve this using MST cycle property, which says that, "For any cycle C in the graph, if the weight of an edge e of C is larger than the weights of all other edges of C, then this edge cannot belong to an MST."
Now, run the following O(E+V) algorithm to test if the edge E connecting vertices u and v will be a part of some MST or not.
Step 1
Run dfs from one of the end-points(either u or v) of the edge E considering only those edges that have weight less than that of E.
Step 2
Case 1
If at the end of this dfs, the vertices u and v get connected, then edge E cannot be a part of some MST. This is because in this case there definitely exists a cycle in the graph with the edge E having the maximum weight and it cannot be a part of the MST(from the cycle property).
Case 2
But if at the end of the dfs u and v stay disconnected, then edge E must be the part of some MST as in this case E is never the maximum weight edge of the cycles that it is a part of.
Find if there are any paths that are cheaper than the current one (u,v) that creates a cycle to u and v. If yes, then (u,v) is not on the mst. Otherwise it is. This can be proved by the cut property and the cycle property.