Hi I'm going about implementing a Graph data structure for a course (the graph isn't part of the requirement, I've chosen to use it to approach the problem) and my first thought was to implement it using an adjacency list, because that requires less memory, and I don't expect to have that many edges in my graph.
But then it occurred to me. I can implement an adjacency list Graph data structure using a Map (HashMap to be specific). Instead of the list of vertices I'll have a Map of vertices, which then hold a short list of edges to vertices.
This seems to be the way to go for me. But I was wondering if anyone can see any drawbacks that a student such as I might have missed in using a HashMap for this? (unfortunately I recall being very tired whilst we were going over HashMaps...so my knowledge of them is less than all the other data structures I know of.) So I want to be sure.
By the way I'm using Java.
The two primary ways of representing a graph are:
with the adjacency list (as you mentioned)
with the adjacency matrix
Since you will not have too many edges (i.e. the adjacency matrix representing your graph would be sparse), I think your decision to use the list instead of the matrix is a good one since, as you said, it will indeed take up less space since no space is wasted to represent the absent edges. Furthermore, the Map approach seems to be logical, as you can map each Node of your graph to the list of Nodes to which it is connected. Another alternative would be to have each Node object contain, as a data field, the list of nodes to which it is connected. I think either of these approaches could work well. I've summed it up below.
First approach (maintain the map):
Map<Node, Node[]> graph = new HashMap<Node, Node[]>();
Second approach (data built into Node class):
public class Node {
private Node[] adjacentNodes;
public Node(Node[] nodes) { adjacentNodes = nodes; }
public Node[] adjacentNodes() { return adjacentNodes; }
}
Graphs are traditionally represented either via an adjacency list or an adjacency matrix (there are other ways that are optimized for certain graph formats, such as if the node id's are labeled sequentially and/or you know the number of nodes/edges ahead of time, but I won't get into that).
Picking between an adjacency list and an adjacency matrix depends on your needs. Clearly, an adjacency matrix will take up more space than an adjacency list (matrix will always take (# of nodes)^2 whereas a list will take (# of nodes + # of edges), but if your graph is "small" then it doesn't really make a difference.
Another concern is how many edges you have (is your graph sparse or dense)? You can find the density of your graph by taking the # of edges you have and dividing it by:
n(n-1) / 2
Where "n" is the number of nodes of the graph. The above equation finds the total # of possible edges in an "n" node UNDIRECTED graph. If the graph is directed, remove the " / 2".
Something else to think of is if efficient edge membership is important. An adjacency list can detect edge membership easily (O(1)) since it's just an array lookup - for an adjacency list if the "list" is stored as something other than a HashSet it will be much slower since you will have to look through the entire edgelist. Or maybe you keep the edgelist's sorted and you can just do a binary search, but then edge insertion takes longer. Maybe your graph is very sparse, and adjacency matrix is using too much memory, so you have to use an adjacency list. Lot's of things to think about.
There's a lot more concerns that may relate to your project, I just list a few.
In general, assuming your graph isn't very complex or "big" (in the sense of millions of nodes), a HashMap where the key is the node ID and the value is a Set or some other collection of node ID's indicating neighbors of the key node is fine, I've done this for 400,000+ node graphs on an 8gb machine. A HashMap based implementation will probably be easiest to implement.
Related
I've done a fair bit of reading around this, and know that discussions regarding this algorithm in Java have been semi-frequent. My issue with implementing Dijkstra's algorithm in Java is simply that I'm not sure how to prepare my data.
I have a set of coordinates within an array, and a set of 1s and 0s in a matrix that represent whether there is a path between the points that the coordinates represent. My question is, how do I present this information so that I can search for the best path with Dijkstra? I have seen many people create a "Node" class, but they never seem to store the coordinates within that Node. Is there some standardized way of creating this kind of structure (I suppose it's a graph?) that I am simply missing?
Any help would be appreciated.
There are two main options:
1. You can use an adjacency matrix in which rows an columns represent your nodes. The value matrix[x, y] must be the weight(e.g. distance/cost etc.) to travel from x to y. You could use the Euclidian distance to calculate these values from your coordinate array;
2. You can implement a couple of classes (Node, Edge - or just Node with a internal Map to another node and the weight as a map value) - it is a graph indeed.
When representing graphs in memory in a language like Java, either an adjacency matrix is used (for dense graphs) or an adjacency list for sparse graphs.
So say we represent the latter like
Map<Integer, LinkedList<Integer>> graph;
The integer key represents the vertex and LinkedList contains all the other vertexes it points to.
Why use a LinkedList to represent the edges? Couldn't an int[] or ArrayList work just as fine, or is there a reason why you want to represent the edges in a way that maintains the ordering such as
2 -> 4 -> 1 -> 5
Either an int[] or ArrayList could also work.
I wouldn't recommend an int[] right off the bat though, as you'll need to cater for resizing in case you don't know all the sizes from the start, essentially simulating the ArrayList functionality, but it might make sense if memory is an issue.
A LinkedList might be slightly preferable since you'd need to either make the array / ArrayList large enough to handle the maximum number of possible edges, or resize it as you go, where-as you don't have this problem with a LinkedList, but then again, creating the graph probably isn't the most resource-intensive task for most applications.
Bottom line - it's most likely going to make a negligible difference for most applications - just pick whichever one you feel most comfortable with (unless of course you need to do access-by-index a lot, or something which one of the two performs a lot better than the other).
Algorithms 4th Edition by Sedgewick and Wayne proposes the following desired performance characteristics for a graph implementation that is useful for most graph-processing applications:
Space usage proportional to V + E
Constant time to add an edge
Time proportional to the degree of v to iterate through vertices adjacent to v (constant time per adjacent vertex processed)
Using a linked list to represent the vertices adjacent to each vertex has all these characteristics. Using an array instead of a linked list would result in either (1) or (2) not being achieved.
I have an input text file containing a line for each edge of a simple undirected graph. The file contains reciprocal edges, i.e. if there's a line u,v, then there's also the line v,u.
I need an algorithm which just counts the number of 4-cycles in this graph. I don't need it to be optimal because I only have to use it as a term of comparison. If you can suggest me a Java implementation, I would appreciate it for the rest of my life.
Thank you in advance.
Construct the adjacency matrix M, where M[i,j] is 1 if there's an edge between i and j. M² is then a matrix which counts the numbers of paths of length 2 between each pair of vertices.
The number of 4-cycles is sum_{i<j}(M²[i,j]*(M²[i,j]-1)/2)/2. This is because if there's n paths of length 2 between a pair of points, the graph has n choose 2 (that is n*(n-1)/2) 4-cycles. We sum only the top half of the matrix to avoid double counting and degenerate paths like a-b-a-b-a. We still count each 4-cycle twice (once per pair of opposite points on the cycle), so we divide the overall total by another factor of 2.
If you use a matrix library, this can be implemented in a very few lines code.
Detecting a cycle is one thing but counting all of the 4-cycles is another. I think what you want is a variant of breadth first search (BFS) rather than DFS as has been suggested. I'll not go deeply into the implementation details, but note the important points.
1) A path is a concatenation of edges sharing the same vertex.
2) A 4-cycle is a 4-edge path where the start and end vertices are the same.
So I'd approach it this way.
Read in graph G and maintain it using Java objects Vertex and Edge. Every Vertex object will have an ArrayList of all of the Edges that are connected to that Vertex.
The object Path will contain all of the vertexes in the path in order.
PathList will contain all of the paths.
Initialize PathList to all of the 1-edge paths which are exactly all of edges in G. BTW, this list will contain all of the 1-cycles (vertexes connected to themselves) as well as all other paths.
Create a function that will (pseudocode, infer the meaning from the function name)
PathList iterate(PathList currentPathList)
{
newPathList = new PathList();
for(path in currentPathList.getPaths())
{
for(edge in path.lastVertexPath().getEdges())
{
PathList.addPath(Path.newPathFromPathAndEdge(path,edge));
}
}
return newPathList;
}
Replace currentPathList with PathList.iterate(currentPathList) once and you will have all of the 2-cyles, call it twice and you will have all of the 3 cycles, call it 3 times and you will have all of the 4 cycles.
Search through all of the paths and find the 4-cycles by checking
Path.firstVertex().isEqualTo(path.lastVertex())
Depth-first search, DFS-this is what you need
Construct an adjacency matrix, as prescribed by Anonymous on Jan 18th, and then find all the cycles of size 4.
It is an enumeration problem. If we know that the graph is a complete graph, then we know off a generating function for the number of cycles of any length. But for most of other graphs, you have to find all the cycles to find the exact number of cycles.
Depth first search with backtracking should be the ideal strategy. Implement it with each node as the starting node, one by one. Keep track of visited nodes. If you run out of nodes without finding a cycle of size 4, just backtrack and try a different route.
Backtrack is not ideal for larger graphs. For example, even a complete graph of order 11 is a little to much for backtracking algorithms. For larger graphs you can look for a randomized algorithm.
I need to generate a graph using integer arrays. Edges of the graphs are kept as edges[e][2] where e is the number of edges.
I need my graph to be connected, i.e. you should be able to traverse from all nodes to all nodes.
edges[0] = {0,5} means an edge connects node 0 and node 5.
Could you suggest an algorithm, please?
And please keep in mind that i will generate graphs with millions of nodes, so that it is better if algorithm complexity isn't too high.
If each node is directly connected to each node, don't store all the edges ;)
If each node is reachable from each other node, but not necessarily directly, use an adjacency matrix. That's the simplest way if you need to use integer arrays.
If the matrix is sparse, I would store it differently, though. The best encoding depends on what graph algorithms you want to use it for. The wikipedia article on sparce matrices) lists the major ones.
I want to know how to implement a DFA as a linked list in C/C++/Java.
since every state can have several branches, you probably need more than one linked list. that means, every state has an array of n linked lists. so it's more like a tree structure with cycles than a simple linked list.
This is definitely possible, but would be grossly inefficient. What you would do is to simply store all your states in a link list, and then each state would need to keep a transition table. The transition table would look something like:
'a' -> 2
'b' -> 5
where your alphabet is {a,b}, and 2 and 5 are the states stored at position 2 and 5 in the linked list. As I said, this is definitely NOT how you would want to implement a DFA, but it is possible.
The first thing that came up in my mind is that,
create a class/struct called state with two array components. one for the states that can reach our state and one for the ones that are reachable from our state.
Then create a linked list whose elements are your states.
here's my implementation of this class
class state
{
private:
string stateName;
vector<state> states_before_me;
vector<state> states_after_me;
state* next;
//methods of this state
}
Single linked list couldn't represent the DFA efficiently. You can think DFA as a directed weighted graph data structure as states are vertices, transitions are edges, transition symbols are weights. There are two main method to implement graph structure.
i) Adjacency list: It basically has V(Number of vertices) linked lists. Each link list contains vertices which has edge to corresponding vertex. If we have vertices (1,2,3) and edges (1,2),(1,3),(2,1),(2,3),(3,3) corresponding adjanceny list is:
1->2->3
2->1->3
3->3
ii) Adjacency matrix: It is a VxV matrix with every entry at (i,j) symbolize an edge from i to j. The same example above represented like(1 means there is edge, 0 mean there is not):
1 2 3
1 0 1 1
2 1 0 1
3 0 0 1
But you must make little changes to these because your graph is weighted.
For list implementation you can change vertices in linklist to a struct which contains vertex and the weight of the edge connecting these vertices.
For matrix implementation you can place the weights directly into matrix instead of 0,1 values.
If you don't want to deal with the implementation of graph class there is libraries like Boost Graph Library which contains the two implementation and all the important graph algorithms DFS to Dijkstra's shortest path algorithm. You can look it up from http://www.boost.org/doc/libs/1_47_0/libs/graph/doc/index.html.