What is the maximum and minimum number of edges to be considered in krushkal's algorithm with an example for both cases.
What I thought was since the Krushkal's algorithm is for finding minimum spanning tree the maximum number of edges is (V-1) where V is the number of vertices. Adding one more edge would result in a cycle in the graph. How can we obtain at a minimum value ?
Kruskal's algorithm stops when you've added V - 1 edges to your MST, so this is the minimum that have to be considered. This happens when the lowest value V - 1 edges of your graph do not form a cycle, and they will be added one after the other by the algorithm, after which it will stop.
For example, consider a complete graph with edges with cost 1, which is minimum in the graph, between node 1 and every other node. Make all the other edges have cost 2.
The worst case is when you must inspect every edge (of which there are O(V^2)) until you finally select V - 1. This means that you have to force a lot of cycles to be created before the last edge is added.
Consider a complete graph again. Have the V - 2 edges between node 1 and V - 2 nodes have cost 1, which is minimum in the graph. These will be selected first. Now let node k be the one that is not part of a selected edge, so that is left out of the graph. Have the edge between node k and node 1 have the largest cost. This will cause it to be inspected and added to the MST last, forcing the algorithm to inspect all O(V^2) edges before building the MST.
Remember the Kruskal's algorithm processes edges in increasing order of their cost, rejecting edges that would form a cycle if added to the MST we are building.
A tree of N vertices always has N-1 edges. Consequently you have to consider at least N-1 edges during Kruskal's algorithm. An example may be a graph which is a tree.
Related
Having an undirected graph with v vertices and e edges with each edges having a non-negative weight of at most w, the task is to find(if possible!) a subset of edges that connects all vertices with minimum cost. Also this graph has a root node which in the sub-graph should only be directly connected to exactly k other vertices(so it most have exactly k edges connected to it in the sub-graph). The algorithm should work in O(e⋅log(v)⋅log(w)) time.
So after some thinking I thought that we need to find a minimum spanning tree since it gives us the Minimum-cost subgraph which takes O(e.log(v)) time.However for insuring that the root node has exactly k edges in the resulting tree I couldn't come up with any working idea.
I think that maybe changing the weights of k edges that are connect to the root may enforce their selection,since in the Kruskal's algorithm we first sort the edges, if we pick exactly k edges that are already connected to the edge root in the graph and change their weights to some minimum amount then in the tree construction phase, chances are that this k edges make it to the MST. There are still problems here:
K edges should be picked so that it would be possible for them to make it to an MST.
Given undirected, connected graph G={V,E}, a vertex in V(G), label him v, and a weight function f:E->R+(Positive real numbers), I need to find a MST such that v's degree is minimal. I've already noticed that if all the edges has unique weight, the MST is unique, so I believe it has something to do with repetitive weights on edges. I though about running Kruskal's algorithm, but when sorting the edges, I'll always consider edges that occur on v last. For example, if (a,b),(c,d),(v,e) are the only edges of weight k, so the possible permutations of these edges in the sorted edges array are: {(a,b),(c,d),(v,e)} or {(c,d),(a,b),(v,e)}. I've ran this variation over several graphs and it seems to work, but I couldn't prove it. Does anyone know how to prove the algorithm's correct (Meaning proving v's degree is minimal), or give a contrary example of the algorithm failing?
First note that Kruskal's algorithm can be applied to any weighted graph, whether or not it is connected. In general it results in a minimum-weight spanning forest (MSF), with one MST for each connected component. To prove that your modification of Kruskal's algorithm succeeds in finding the MST for which v has minimal degree, it helps to prove the slightly stronger result that if you apply your algorithm to a possibly disconnected graph then it succeeds in finding the MSF where the degree of v is minimized.
The proof is by induction on the number, k, of distinct weights.
Basis Case (k = 1). In this case weights can be ignored and we are trying to find a spanning forest in which the degree of v is minimized. In this case, your algorithm can be described as follows: pick edges for as long as possible according to the following two rules:
1) No selected edge forms a cycle with previously selected edges
2) An edge involving v isn't selected unless any edge which doesn't
involve v violates rule 1.
Let G' denote the graph from which v and all incident edges have been removed from G. It is easy to see that the algorithm in this special case works as follows. It starts by creating a spanning forest for G'. Then it takes those trees in the forest that are contained in v's connected component in the original graph G and connects each component to v by a single edge. Since the components connected to v in the second stage can be connected to each other in no other way (since if any connecting edge not involving v exists it would have been selected by rule 2) it is easy to see that the degree of v is minimal.
Inductive Case: Suppose that the result is true for k and G is a weighted graph with k+1 distinct weights and v is a specified vertex in G. Sort the distinct weights in increasing order (so that weight k+1 is the longest of the distinct weights -- say w_{k+1}). Let G' be the sub-graph of G with the same vertex set but with all edges of weight w_{k+1} removed. Since the edges are sorted in the order of increasing weight, note that the modified Kruskal's algorithm in effect starts by applying itself to G'. Thus -- by the induction hypothesis prior to considering edges of weight w_{k+1}, the algorithm has succeeded in constructing an MSF F' of G' for which the degree, d' of v in G' is minimized.
As a final step, modified Kruskal's applied to the overall graph G will merge certain of the trees in F' together by adding edges of weight w_{k+1}. One way to conceptualize the final step is the think of F' as a graph where two trees are connected exactly when there is an edge of weight w_{k+1} from some node in the first tree to some node in the second tree. We have (almost) the basis case with F'. Modified Kruskal's will add edged of weight w_{k+1} until it can't do so anymore -- and won't add an edge connecting to v unless there is no other way to connect to trees in F' that need to be connected to get a spanning forest for the original graph G.
The final degree of v in the resulting MSF is d = d'+d" where d" is the number of edges of weight w_{k+1} added at the final step. Neither d' nor d" can be made any smaller, hence it follows that d can't be made any smaller (since the degree of v in any spanning forest can be written as the sum of the number of edges whose weight is less than w_{k+1} coming into v and the number off edges of weight w_{k+1} coming into v).
QED.
There is still an element of hand-waving in this, especially with the final step -- but Stack Overflow isn't a peer-reviewed journal. Anyway, the overall logic should be clear enough.
One final remark -- it seems fairly clear that Prim's algorithm can be similarly modified for this problem. Have you looked into that?
So I have an exercise that I should prove or disprove that:
1) if e is a minimum weight edge in the connected graph G such that not all edges are necessarily distinct, then every minimum spanning tree of G contains e
2) Same as 1) but now all edge weights are distinct.
Ok so intuitively, I understand that for 1) since not all edge weights are distinct, then it's possible that a vertex has the path with edge e but also another edge e_1 such that if weight(e) = weight (e_1) then there is a spanning tree which does not contain the edge e since the graph is connected. Otherwise if both e_1 and e are in the minimum spanning tree, then there is a cycle
and for 2) since all edge weights are distinct, then of course the minimum spanning tree will contain the edge e since any algorithm will always choose the smaller path.
Any suggestions on how to prove these two though? induction? Not sure how to approach.
Actually in your first proof when you say that if both e and e_1 are in G, then there's a cycle, that's not true, because they're minimal edges, so there doesn't have to be a cycle, and you do need to include them both into the MST, because if |E| > 1 and |V| > 2 then they both have to be there.
Anyways, a counter example for the first one is a complete graph with all edges of the same weight as e, the MST will contain only |V|-1 edges, but you didn't include all the other edges of that same weight, hence you have a contradiction.
As for the second one, if all edges are distinct, then if you remove the minimum edge and want to reconstruct the MST, the only way to go about this is to add a an edge connecting the 2 disjoint sets that were broken up by that minimum-weight edge.
Now suppose that you didn't remove that minimum-weight edge, and added that other edge, now you've created a cycle, and since all edges are distinct the cycle-creating edge will be greater than all of them, hence if you remove any former MST edge from that cycle, it will only increase the size of the MST. Which means that pretty much all former MST edges are critical when all edges have distinct weights.
Does the opposite of Kruskal's algorithm for minimum spanning tree work for it? I mean, choosing the max weight (edge) every step?
Any other idea to find maximum spanning tree?
Yes, it does.
One method for computing the maximum weight spanning tree of a network G –
due to Kruskal – can be summarized as follows.
Sort the edges of G into decreasing order by weight. Let T be the set of edges comprising the maximum weight spanning tree. Set T = ∅.
Add the first edge to T.
Add the next edge to T if and only if it does not form a cycle in T. If
there are no remaining edges exit and report G to be disconnected.
If T has n−1 edges (where n is the number of vertices in G) stop and
output T . Otherwise go to step 3.
Source: https://web.archive.org/web/20141114045919/http://www.stats.ox.ac.uk/~konis/Rcourse/exercise1.pdf.
From Maximum Spanning Tree at Wolfram MathWorld:
"A maximum spanning tree is a spanning tree of a weighted graph having maximum weight. It can be computed by negating the weights for each edge and applying Kruskal's algorithm (Pemmaraju and Skiena, 2003, p. 336)."
If you invert the weight on every edge and minimize, do you get the maximum spanning tree? If that works you can use the same algorithm. Zero weights will be a problem, of course.
Although this thread is too old, I have another approach for finding the maximum spanning tree (MST) in a graph G=(V,E)
We can apply some sort Prim's algorithm for finding the MST. For that I have to define Cut Property for the maximum weighted edge.
Cut property: Let say at any point we have a set S which contains the vertices that are in MST( for now assume it is calculated somehow ). Now consider the set S/V ( vertices not in MST ):
Claim: The edge from S to S/V which has the maximum weight will always be in every MST.
Proof: Let's say that at a point when we are adding the vertices to our set S the maximum weighted edge from S to S/V is e=(u,v) where u is in S and v is in S/V. Now consider an MST which does not contain e. Add the edge e to the MST. It will create a cycle in the original MST. Traverse the cycle and find the vertices u' in S and v' in S/V such that u' is the last vertex in S after which we enter S/V and v' is the first vertex in S/V on the path in cycle from u to v.
Remove the edge e'=(u',v') and the resultant graph is still connected but the weight of e is greater than e' [ as e is the maximum weighted edge from S to S/V at this point] so this results in an MST which has sum of weights greater than original MST. So this is a contradiction. This means that edge e must be in every MST.
Algorithm to find MST:
Start from S={s} //s is the start vertex
while S does not contain all vertices
do
{
for each vertex s in S
add a vertex v from S/V such that weight of edge e=(s,v) is maximum
}
end while
Implementation:
we can implement using Max Heap/Priority Queue where the key is the maximum weight of the edge from a vertex in S to a vertex in S/V and value is the vertex itself. Adding a vertex in S is equal to Extract_Max from the Heap and at every Extract_Max change the key of the vertices adjacent to the vertex just added.
So it takes m Change_Key operations and n Extract_Max operations.
Extract_Min and Change_Key both can be implemented in O(log n). n is the number of vertices.
So This takes O(m log n) time. m is the number of edges in the graph.
Let me provide an improvement algorithm:
first construct an arbitrary tree (using BFS or DFS)
then pick an edge outside the tree, add to the tree, it will form a cycle, drop the smallest weight edge in the cycle.
continue doing this util all the rest edges are considered
Thus, we'll get the maximum spanning tree.
This tree satisfies any edge outside the tree, if added will form a cycle and the edge outside <= any edge weights in the cycle
In fact, this is a necessary and sufficient condition for a spanning tree to be maximum spanning tree.
Pf.
Necessary: It's obvious that this is necessary, or we could swap edge to make a tree with a larger sum of edge weights.
Sufficient: Suppose tree T1 satisfies this condition, and T2 is the maximum spanning tree.
Then for the edges T1 ∪ T2, there're T1-only edges, T2-only edges, T1 ∩ T2 edges, if we add a T1-only edge(x1, xk) to T2, we know it will form a cycle, and we claim, in this cycle there must exist one T2-only edge that has the same edge weights as (x1, xk). Then we can exchange these edges will produce a tree with one more edge in common with T2 and has the same sum of edge weights, repeating doing this we'll get T2. so T1 is also a maximum spanning tree.
Prove the claim:
suppose it's not true, in the cycle we must have a T2-only edge since T1 is a tree. If none of the T2-only edges has a value equal to that of (x1, xk), then each of T2-only edges makes a loop with tree T1, then T1 has a loop leads to a contradiction.
This algorithm taken from UTD professor R. Chandrasekaran's notes. You can refer here: Single Commodity Multi-terminal Flows
Negate the weight of original graph and compute minimum spanning tree on the negated graph will give the right answer. Here is why: For the same spanning tree in both graphs, the weighted sum of one graph is the negation of the other. So the minimum spanning tree of the negated graph should give the maximum spanning tree of the original one.
Only reversing the sorting order, and choosing a heavy edge in a vertex cut does not guarantee a Maximum Spanning Forest (Kruskal's algorithm generates forest, not tree). In case all edges have negative weights, the Max Spanning Forest obtained from reverse of kruskal, would still be a negative weight path. However the ideal answer is a forest of disconnected vertices. i.e. a forest of |V| singleton trees, or |V| components having total weight of 0 (not the least negative).
Change the weight in a reserved order(You can achieve this by taking a negative weight value and add a large number, whose purpose is to ensure non-negative) Then run your family geedy-based algorithm on the minimum spanning tree.
I'm writing an algorithm for finding the second min cost spanning tree. my idea was as follows:
Use kruskals to find lowest MST.
Delete the lowest cost edge of the MST.
Run kruskals again on the entire graph.
return the new MST.
My question is: Will this work? Is there a better way perhaps to do this?
You can do it in O(V2). First compute the MST using Prim's algorithm (can be done in O(V2)).
Compute max[u, v] = the cost of the maximum cost edge on the (unique) path from u to v in the MST. Can be done in O(V2).
Find an edge (u, v) that's NOT part of the MST that minimizes abs(max[u, v] - weight(u, v)). Can be done in O(E) == O(V2).
Return MST' = MST - {the edge that has max[u, v] weight} + {(u, v)}, which will give you the second best MST.
Here's a link to pseudocode and more detailed explanations.
Consider this case:
------100----
| |
A--1--B--3--C
| |
| 3
| |
2-----D
The MST consists of A-B-D-C (cost 6). The second min cost is A-B-C-D (cost 7). If you delete the lowest cost edge, you will get A-C-B-D (cost 105) instead.
So your idea will not work. I have no better idea though...
You can do this -- try removing the edges of the MST, one at a time from the graph, and run the MST, taking the min from it. So this is similar to yours, except for iterative:
Use Kruskals to find MST.
For each edge in MST:
Remove edge from graph
Calculate MST' on MST
Keep track of smallest MST
Add edge back to graph
Return the smallest MST.
This is similar to Larry's answer.
After finding MST,
For each new_edge =not a edge in MST
Add new_edge to MST.
Find the cycle that is formed.
Find the edge with maximum weight in
cycle that is not the non-MST edge
you added.
Record the weight increase as W_Inc
= w(new_edge) - w(max_weight_edge_in_cycle).
If W_Inc < Min_W_Inc_Seen_So_Far Then
Min_W_Inc_Seen_So_Far = W_Inc
edge_to_add = new_edge
edge_to_remove = max_weight_edge_in_cycle
Solution from following link.
http://web.mit.edu/6.263/www/quiz1-f05-sol.pdf
slight edit to your algo.
Use kruskals to find lowest MST.
for all edges i of MST
Delete edge i of the MST.
Run kruskals again on the entire graph.
loss=cost new edge introduced - cost of edge i
return MST for which loss is minimum
Here is an algorithm which compute the 2nd minimum spanning tree in O(n^2)
First find out the mimimum spanning tree (T). It will take O(n^2) without using heap.
Repeat for every edge e in T. =O(n^2)
Lets say current tree edge is e. This tree edge will divide the tree into two trees, lets say T1 and T-T1. e=(u,v) where u is in T1 and v is in T-T1. =O(n^2)
Repeat for every vertex v in T-T1. =O(n^2)
Select edge e'=(u,v) for all v in T-T1 and e' is in G (original graph) and it is minimum
Calculate the weight of newly formed tree. Lets say W=weight(T)-weight(e)+weight(e')
Select the one T1 which has a minimum weight
Your approach will not work, as it might be the case that min. weight edge in the MST is a bridge (only one edge connecting 2 parts of graph) so deleting this edge from the set will result in 2 new MST as compared to one MST.
based on #IVlad's answer
Detailed explanation of the O(V² log V) algorithm
Find the minimum spanning tree (MST) using Kruskal's (or Prim's) algorithm, save its total weight, and for every node in the MST store its tree neighbors (i.e. the parent and all children) -> O(V² log V)
Compute the maximum edge weight between any two vertices in the minimum spanning tree. Starting from every vertex in the MST, traverse the entire tree with a depth- or breadth-first search by using the tree node neighbor lists computed earlier and store the maximum edge weight encountered so far at every new vertex visited. -> O(V²)
Find the second minimum spanning tree and its total weight. For every edge not belonging to the original MST, try disconnecting the two vertices it connects by removing the tree edge with the maximum weight in between the two vertices, and then reconnecting them with the currently considered vertex (note: the MST should be restored to its original state after every iteration). The total weight can be calculated by subtracting the weight of the removed edge and adding that of the added one. Store the minimum of the total weights obtained.
To practice you could try the competitive programming problem UVa 10600 - ACM Contest and Blackout, which involves finding the second minimum spanning tree in a weighted graph, as asked by the OP. My implementation (in modern C++) can be found here.
MST is a tree which has the minimum weight total of all edges of the graph. Thus, 2nd minimum mst will have the 2nd minimum total weight of all edges in the graph.
let T -> BEST_MST ( sort the edges in the graph , then find MST using kruskal algorithm)
T ' -> 2nd best MST
let's say T has 7 edges , now to find T ' we will one by one remove one of those 7 edges and find a replacement for that edge ( cost of that edge definitely will be greater than the edge we just removed from T ).
let's say original graph has 15 edges
our best MST ( T ) has 7 edges
and 2nd best MST ( T ' ) will also going to have 7 edges only
How to find T '
there are 7 edges in T , now for all those 7 edges remove them one by one and find replacement for those edges .
let's say edges in MST ( T ) --> { a,b,c,d,e,f,g }
let's say our answer will be 2nd_BEST_MST and initially it has infinte value ( i know it doesn't sounds good , let's just assume it for now ).
for all edges in BEST_MST :
current_edge = i
find replacement for that edge, replacement for that edge will definitely going to have have weight more than the ith edge ( one of 7 edges )
how we will going to find the replacement for that edge , using Kruskul algorithm ( we are finding the MST again , so we will use kruskal algorithm only , but this we don't have to sort edges again , because we did it when we were finding the BEST_MST ( T ).
NEW_MST will be generated
2nd_best_MST = min( NEW_MST , 2nd_best_MST )
return 2nd_best_MST
ALGORITHM
let' say orignal graph has 10 edges
find the BEST_MST ( using kruskal algo) and assume BEST_MST has only 6 edges
bow there are 4 edges remaining which is not in the BEST_MST ( because their weight value is large and one of those edges will give us our 2nd_Best_MST
for each edge 'X' not present in the BEST_MST ( i.e. 4 edges left ) add that edge in out BEST_MST which will create the cycle
find the edge 'K' with the maximum weight in the cycle ( other than newly_added_edge 'X' )
remove edge 'K' temporarily which will form a new spanning tree
calculate the difference in weight and map the weight_difference with the edge 'X' .
repeat step 4 for all those 4 edges and return the spanning tree with the smallest weight difference to the BEST_MST.