I need to prove a recursive algorithm. Normally this would be done using some integer value within the code as the base case for induction like when computing a factorial but with a graph traversal I have no idea where to begin. Here is my algorithm. Subscripts didn't convert.
Algorithm
Goal: Traverse a graph creating a depth-first spanning tree, and compute the Last descendent of each vertex that is the descendent vk that has the highest value of k
Input:
A connected graph G with vertices ordered v1, v2, v3 … vn
Output:
A spanning tree T where each vertex in T has had its Last vertex computed
Initialization
Set each vertex to unvisited. Let ak denote a list of all vertices adjacent to vk. Let lk denote the Last descendent of vk. Let ck denote the list of all the children of vk in the spanning tree. Let dk denote the list of all vertices that are descendents of vk in the spanning tree including vk.
dfs(vk){
add vK to T
set v to visited
lk = vk
add vk to dk
foreach(vertex m in ak with lowest value of k){
add m to ck
add dfs(m) to dk
}
foreach(vertex vc in dk){
if( c > k){
lk = vc
}
}
if(k = 1)
return T
else
return dk
}
This is for a group project at school so I don't want the whole proof but a starting point and some direction would be greatly appreciated.
I'm having a hard time understanding your pseudo code. It seems unclear at best, probably the algorithm doesn't even work. Some issues:
The "visited" property is set, but never used.
T is supposed to be a tree. But you only ever add vertices, no edges to it. Without edges, it's certainly not a spanning tree. If you consider all edges from the original graph between the nodes in T part of T, then it will contain cycles and won't be a tree, either.
Why do you calculate lk, if it's never used?
Is the parameter of your function dfs supposed to be k instead of vk? Otherwise k is never set, but it is being used.
I fail to see how your recursion ever terminates. I don't see any guards for the base case (except for maybe the "with lowest value of k" condition in your loop - which I don't understand because I understand from the rest of the code that k is the input parameter of the function and therefore m doesn't depend on it).
So let me tell you about proving recursive algorithms over graphs in general. Apart from the induction over natural numbers that you mentioned, there is also Structural Induction. It has a base case and an inductive step, just like the induction you know. But the base case is usually a trivial component of your data structure and the inductive step proves your proposition for more complex composites, assuming that your proposition holds for its less complex components.
For example, you can prove an algorithm over trees by proving that it works for the leafs (your base case) and by proving that it works for a whole tree, assuming that your algorithm works for the left and right sub-trees of the root (induction step).
Since your graph, other than the tree example above, may contain cycles, it's not automatically guaranteed that the data structure that is passed to your recursive call is less complex than the original one. But your algorithm probably has some way to ensure that the recursive call takes into account only a part of the graph, probably via the "visited" flag. In that case the recursive call has to take into account only the "not visited" subgraph. So you can induce over those unvisited subgraphs. Start from the base case with only one vertex being unvisited. And then inductively add one unvisited node (including its edges) to the unvisited subgraph.
I'm trying to find an efficient algorithm to generate a simple connected graph with given the number of nodes. Something like:
Input:
N - size of generated graph
Output:
simple connected graph G(v,e) with N vertices and S edges, The number of edges should be uniform distribution.
You might want to create a minimal spanning tree first to ensure connectivity. Later, randomly generate two nodes (which are not yet connected) and connect them. Repeat until you have S edges.
For minimal spanning tree, the easiest thing to do is start with a random node as tree. For every remaining node (ordered randomly), connect it to any node in the tree. The way you select the node within the tree (to connect to) defines the distribution of edges/node.
It may be a bit surprising but if you choose (randomly) log(n) edges (where n - maximum number of edges) then you can be almost sure that your graph is connected (a reference). If number of edges is much lower than log(n) you can be almost sure that the graph is disconnected.
How to generate the graph:
GenerateGraph(N,S):
if (log(N)/N) > S) return null // or you can take another action
V <- {0,...,N-1}
E <- {}
possible_edges <- {{v,u}: v,u in V} // all possible edges
while (size(E) < S):
e <- get_random_edge(possible_edges)
E.add(e)
possible_edges.remove(e)
G=(V,E)
if (G is connected)
return G
else
return GenerateGraph(N,S)
Let me know if you need any more guidance.
(Btw. right now I am dealing with exactly the same problem! Let me know if you need to generate some more sophisticated graphs:-))
A very common random graph generation algorithm (used in many academic works) is based on RMat method, or a generalization in the form of Kronecker graphs. It's a simple, iterative process, that uses very few parameters and is easily scalable.
There's a really good explanation here (including why it's better than other methods) -
http://www.graphanalysis.org/SIAM-PP08/Leskovic.pdf
There are versions of both implemented in many graph benchmark suites, for e.g.
BFS benchmark with a subdir with rMat generator - http://www.cs.cmu.edu/~pbbs/benchmarks/breadthFirstSearch.tar
Kronecker graph generator (both c and matlab codes included) - http://www.graph500.org/sites/default/files/files/graph500-2.1.4.tar.bz2
Given an undirected graph, I want to generate all subgraphs which are trees of size N, where size refers to the number of edges in the tree.
I am aware that there are a lot of them (exponentially many at least for graphs with constant connectivity) - but that's fine, as I believe the number of nodes and edges makes this tractable for at least smallish values of N (say 10 or less).
The algorithm should be memory-efficient - that is, it shouldn't need to have all graphs or some large subset of them in memory at once, since this is likely to exceed available memory even for relatively small graphs. So something like DFS is desirable.
Here's what I'm thinking, in pseudo-code, given the starting graph graph and desired length N:
Pick any arbitrary node, root as a starting point and call alltrees(graph, N, root)
alltrees(graph, N, root)
given that node root has degree M, find all M-tuples with integer, non-negative values whose values sum to N (for example, for 3 children and N=2, you have (0,0,2), (0,2,0), (2,0,0), (0,1,1), (1,0,1), (1,1,0), I think)
for each tuple (X1, X2, ... XM) above
create a subgraph "current" initially empty
for each integer Xi in X1...XM (the current tuple)
if Xi is nonzero
add edge i incident on root to the current tree
add alltrees(graph with root removed, N-1, node adjacent to root along edge i)
add the current tree to the set of all trees
return the set of all trees
This finds only trees containing the chosen initial root, so now remove this node and call alltrees(graph with root removed, N, new arbitrarily chosen root), and repeat until the size of the remaining graph < N (since no trees of the required size will exist).
I forgot also that each visited node (each root for some call of alltrees) needs to be marked, and the set of children considered above should only be the adjacent unmarked children. I guess we need to account for the case where no unmarked children exist, yet depth > 0, this means that this "branch" failed to reach the required depth, and cannot form part of the solution set (so the whole inner loop associated with that tuple can be aborted).
So will this work? Any major flaws? Any simpler/known/canonical way to do this?
One issue with the algorithm outlined above is that it doesn't satisfy the memory-efficient requirement, as the recursion will hold large sets of trees in memory.
This needs an amount of memory that is proportional to what is required to store the graph. It will return every subgraph that is a tree of the desired size exactly once.
Keep in mind that I just typed it into here. There could be bugs. But the idea is that you walk the nodes one at a time, for each node searching for all trees that include that node, but none of the nodes that were searched previously. (Because those have already been exhausted.) That inner search is done recursively by listing edges to nodes in the tree, and for each edge deciding whether or not to include it in your tree. (If it would make a cycle, or add an exhausted node, then you can't include that edge.) If you include it your tree then the used nodes grow, and you have new possible edges to add to your search.
To reduce memory use, the edges that are left to look at is manipulated in place by all of the levels of the recursive call rather than the more obvious approach of duplicating that data at each level. If that list was copied, your total memory usage would get up to the size of the tree times the number of edges in the graph.
def find_all_trees(graph, tree_length):
exhausted_node = set([])
used_node = set([])
used_edge = set([])
current_edge_groups = []
def finish_all_trees(remaining_length, edge_group, edge_position):
while edge_group < len(current_edge_groups):
edges = current_edge_groups[edge_group]
while edge_position < len(edges):
edge = edges[edge_position]
edge_position += 1
(node1, node2) = nodes(edge)
if node1 in exhausted_node or node2 in exhausted_node:
continue
node = node1
if node1 in used_node:
if node2 in used_node:
continue
else:
node = node2
used_node.add(node)
used_edge.add(edge)
edge_groups.append(neighbors(graph, node))
if 1 == remaining_length:
yield build_tree(graph, used_node, used_edge)
else:
for tree in finish_all_trees(remaining_length -1
, edge_group, edge_position):
yield tree
edge_groups.pop()
used_edge.delete(edge)
used_node.delete(node)
edge_position = 0
edge_group += 1
for node in all_nodes(graph):
used_node.add(node)
edge_groups.append(neighbors(graph, node))
for tree in finish_all_trees(tree_length, 0, 0):
yield tree
edge_groups.pop()
used_node.delete(node)
exhausted_node.add(node)
Assuming you can destroy the original graph or make a destroyable copy I came up to something that could work but could be utter sadomaso because I did not calculate its O-Ntiness. It probably would work for small subtrees.
do it in steps, at each step:
sort the graph nodes so you get a list of nodes sorted by number of adjacent edges ASC
process all nodes with the same number of edges of the first one
remove those nodes
For an example for a graph of 6 nodes finding all size 2 subgraphs (sorry for my total lack of artistic expression):
Well the same would go for a bigger graph, but it should be done in more steps.
Assuming:
Z number of edges of most ramificated node
M desired subtree size
S number of steps
Ns number of nodes in step
assuming quicksort for sorting nodes
Worst case:
S*(Ns^2 + MNsZ)
Average case:
S*(NslogNs + MNs(Z/2))
Problem is: cannot calculate the real omicron because the nodes in each step will decrease depending how is the graph...
Solving the whole thing with this approach could be very time consuming on a graph with very connected nodes, however it could be paralelized, and you could do one or two steps, to remove dislocated nodes, extract all subgraphs, and then choose another approach on the remainder, but you would have removed a lot of nodes from the graph so it could decrease the remaining run time...
Unfortunately this approach would benefit the GPU not the CPU, since a LOT of nodes with the same number of edges would go in each step.... and if parallelization is not used this approach is probably bad...
Maybe an inverse would go better with the CPU, sort and proceed with nodes with the maximum number of edges... those will be probably less at start, but you will have more subgraphs to extract from each node...
Another possibility is to calculate the least occuring egde count in the graph and start with nodes that have it, that would alleviate the memory usage and iteration count for extracting subgraphs...
Unless I'm reading the question wrong people seem to be overcomplicating it.
This is just "all possible paths within N edges" and you're allowing cycles.
This, for two nodes: A, B and one edge your result would be:
AA, AB, BA, BB
For two nodes, two edges your result would be:
AAA, AAB, ABA, ABB, BAA, BAB, BBA, BBB
I would recurse into a for each and pass in a "template" tuple
N=edge count
TempTuple = Tuple_of_N_Items ' (01,02,03,...0n) (Could also be an ordered list!)
ListOfTuple_of_N_Items ' Paths (could also be an ordered list!)
edgeDepth = N
Method (Nodes, edgeDepth, TupleTemplate, ListOfTuples, EdgeTotal)
edgeDepth -=1
For Each Node In Nodes
if edgeDepth = 0 'Last Edge
ListOfTuples.Add New Tuple from TupleTemplate + Node ' (x,y,z,...,Node)
else
NewTupleTemplate = TupleTemplate + Node ' (x,y,z,Node,...,0n)
Method(Nodes, edgeDepth, NewTupleTemplate, ListOfTuples, EdgeTotal
next
This will create every possible combination of vertices for a given edge count
What's missing is the factory to generate tuples given an edge count.
You end up with a list of possible paths and the operation is Nodes^(N+1)
If you use ordered lists instead of tuples then you don't need to worry about a factory to create the objects.
If memory is the biggest problem you can use a NP-ish solution using tools from formal verification. I.e., guess a subset of nodes of size N and check whether it's a graph or not. To save space you can use a BDD (http://en.wikipedia.org/wiki/Binary_decision_diagram) to represent the original graph's nodes and edges. Plus you can use a symbolic algorithm to check if the graph you guessed is really a graph - so you don't need to construct the original graph (nor the N-sized graphs) at any point. Your memory consumption should be (in big-O) log(n) (where n is the size of the original graph) to store the original graph, and another log(N) to store every "small graph" you want.
Another tool (which is supposed to be even better) is to use a SAT solver. I.e., construct a SAT formula that is true iff the sub-graph is a graph and supply it to a SAT solver.
For a graph of Kn there are approximately n! paths between any two pairs of vertices. I haven't gone through your code but here is what I would do.
Select a pair of vertices.
Start from a vertex and try to reach the destination vertex recursively (something like dfs but not exactly). I think this would output all the paths between the chosen vertices.
You could do the above for all possible pairs of vertices to get all simple paths.
It seems that the following solution will work.
Go over all partitions into two parts of the set of all vertices. Then count the number of edges which endings lie in different parts (k); these edges correspond to the edge of the tree, they connect subtrees for the first and the second parts. Calculate the answer for both parts recursively (p1, p2). Then the answer for the entire graph can be calculated as sum over all such partitions of k*p1*p2. But all trees will be considered N times: once for each edge. So, the sum must be divided by N to get the answer.
Your solution as is doesn't work I think, although it can be made to work. The main problem is that the subproblems may produce overlapping trees so when you take the union of them you don't end up with a tree of size n. You can reject all solutions where there is an overlap, but you may end up doing a lot more work than needed.
Since you are ok with exponential runtime, and potentially writing 2^n trees out, having V.2^V algorithms is not not bad at all. So the simplest way of doing it would be to generate all possible subsets n nodes, and then test each one if it forms a tree. Since testing whether a subset of nodes form a tree can take O(E.V) time, we are potentially talking about V^2.V^n time, unless you have a graph with O(1) degree. This can be improved slightly by enumerating subsets in a way that two successive subsets differ in exactly one node being swapped. In that case, you just have to check if the new node is connected to any of the existing nodes, which can be done in time proportional to number of outgoing edges of new node by keeping a hash table of all existing nodes.
The next question is how do you enumerate all the subsets of a given size
such that no more than one element is swapped between succesive subsets. I'll leave that as an exercise for you to figure out :)
I think there is a good algorithm (with Perl implementation) at this site (look for TGE), but if you want to use it commercially you'll need to contact the author. The algorithm is similar to yours in the question but avoids the recursion explosion by making the procedure include a current working subtree as a parameter (rather than a single node). That way each edge emanating from the subtree can be selectively included/excluded, and recurse on the expanded tree (with the new edge) and/or reduced graph (without the edge).
This sort of approach is typical of graph enumeration algorithms -- you usually need to keep track of a handful of building blocks that are themselves graphs; if you try to only deal with nodes and edges it becomes intractable.
This algorithm is big and not easy one to post here. But here is link to reservation search algorithm using which you can do what you want. This pdf file contains both algorithms. Also if you understand russian you can take a look to this.
So you have a graph with with edges e_1, e_2, ..., e_E.
If I understand correctly, you are looking to enumerate all subgraphs which are trees and contain N edges.
A simple solution is to generate each of the E choose N subgraphs and check if they are trees.
Have you considered this approach? Of course if E is too large then this is not viable.
EDIT:
We can also use the fact that a tree is a combination of trees, i.e. that each tree of size N can be "grown" by adding an edge to a tree of size N-1. Let E be the set of edges in the graph. An algorithm could then go something like this.
T = E
n = 1
while n<N
newT = empty set
for each tree t in T
for each edge e in E
if t+e is a tree of size n+1 which is not yet in newT
add t+e to newT
T = newT
n = n+1
At the end of this algorithm, T is the set of all subtrees of size N. If space is an issue, don't keep a full list of the trees, but use a compact representation, for instance implement T as a decision tree using ID3.
I think problem is under-specified. You mentioned that graph is undirected and that subgraph you are trying to find is of size N. What is missing is number of edges and whenever trees you are looking for binary or you allowed to have multi-trees. Also - are you interested in mirrored reflections of same tree, or in other words does order in which siblings are listed matters at all?
If single node in a tree you trying to find allowed to have more than 2 siblings which should be allowed given that you don't specify any restriction on initial graph and you mentioned that resulting subgraph should contain all nodes.
You can enumerate all subgraphs that have form of tree by performing depth-first traversal. You need to repeat traversal of the graph for every sibling during traversal. When you'll need to repeat operation for every node as a root.
Discarding symmetric trees you will end up with
N^(N-2)
trees if your graph is fully connected mesh or you need to apply Kirchhoff's Matrix-tree theorem
I have a DAG representing a list of properties. These properties are such that if a>b, then a has a directed edge to b. It is transitive as well, so that if a>b and b>c, then a has a directed edge to c.
However, the directed edge from a to c is superfluous because a has a directed edge to b and b has a directed edge to c. How can I prune all these superfluous edges? I was thinking of using a minimum spanning tree algorithm, but I'm not really sure what is the appropriate algorithm to apply in this situation
I suppose I could do a depth first search from each node and all its outgoing edges and compare if it can reach certain nodes without using certain edges, but this seems horribly inefficient and slow.
After the algorithm is complete, the output would be a linear list of all the nodes in an order that is consistent with the graph. So if a has three directed edges to b,c, and d. b and c also each of which has a directed edge to d, the output could be either abcd or acbd.
This is called the transitive reduction problem. Formally speaking, you are looking for a minimal (fewest edges) directed graph, the transitive closure of which is equal to the transitive closure of the input graph. (The diagram on the above Wikipedia link makes it clear.)
Apparently there exists an efficient algorithm for solving this problem that takes the same time as for producing a transitive closure (i.e. the more common inverse problem of adding transitive links instead of removing them), however the link to the 1972 paper by Aho, Garey, and Ullman costs $25 to download, and some quick googling didn't turn up any nice descriptions.
EDIT: Scott Cotton's graphlib contains a Java implementation! This Java library looks to be very well organised.
Actually, after looking around a little more, I think a Topologicalsort is what I'm really after here.
If these are already n nodes with directed edges:
Starting from any point M, loop all its child edge, select the biggest child (like N), remove other edges, the complexity should be o(n) . If no N exists (no child edge, goto step 3).
start from N, repeat step 1.
start from point M, select the smallest parent node ( like T), remove others' edges.
start from T, repeat step 3.....
Actually it's just a ordering algorithm, and the totally complexity should be o(0.5n^2).
One problem is that if we want loop one node's parent nodes, then we need more memory to log edge so we can trace back from child to parent. This can be improved in the step 3 where we choose one node from the left nodes bigger than M, this means we need to keep a list of nodes to know what nodes are left..