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When representing a sparse graph with an adjacency matrix, why use linked list as structure to contain edges?

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.

Simplest algorithm to find 4-cycles in an undirected graph

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.

Finding closest coordinates in an array?

I have two ArrayList, Double data type,
1.latitudes
2. longitudes,
each has over 200 elements
say i give a random test coordinates, say (1.33, 103.4), the format is [latitude, longitude]
is there any algorithm to easily find closest point,
or do i have to brute force calculate every possible point, find hypotenuse, and then compare over 200 hypotenuses to return the closest point? thanks

Sort the array of points along one axis. Then, locate the point in the array closest to the required point along this axis and calculate the distance (using whatever metric is appropriate to the problem topology and scale).
Then, search along the array in both directions until the distance to these points is greater than the best result so far. The shortest distance point is the answer.
This can result in having to search the entire array, and is a form of Branch and bound constrained by the geometry of the problem. If the points are reasonably evenly distributed around the point you are searching for, then the scan will not require many trials.
Alternate spatial indices (like quad-trees) will give better results, but your small number of points would make the setup cost in preparing the index much larger than a simple sort. You will need to track the position changes caused by the sort as your other array will not be sorted the same way. If you change the data into a single array of points, then the sort will reorder entire points at the same time.

If your arrays are sorted, you can use binary search to find a position of a requested point in array. After you find index, you should check four near by points to find the closest.
1)Suppose you have two sorted arrays longitudes-wise and latitudes-wise
2)You search first one and find two nearby points
3)Then you search second one and find two more points
4)Now you have from two to four points(results might intersect)
5)These points will form a square around destination point
6)Find the closest point

it's not true that closest lat (or long) value should be choosen to search over the long (or lat) axis, in fact you could stay on a lat (or long) line but far away along the long (or lat) value
so best way is to calculate all distances and sort them

Using a Map for a Graph

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.

Implementing a graph in either C++ or Java [closed]

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Closed 11 years ago.
What's the best ways of implementing a graph in either C++ or Java? In C++, I was thinking about using a 2d array to do it. In java, I was considering an arrayList.

Firstly, language choices aren't the most massive issue in the world, in my opinion. Unless you have a requirement to use a specific language or on portability, using either C++ or Java will be sufficient. Having said that, your question seems somewhat homework-ish. Are you using Prim's algorithm for your MST implementation?
As other answers have already said, there are a few ways to represent graphs. The two that I'm most familiar with for representing graphs are:
An Adjacency Matrix, which is a 2D array where each "row" and "column" is a node, and a value at that position of the matrix indicates an edge (or an edge weight) and a null-value (or a 0-value, or some other sentinel/meaningful value) indicates no edge
An Adjacency List, which is a 2D array (kinda), where the i-th list is the list of all the nodes that are connected to (adjacent to) node i. At times, you may also choose to make the list a list of pairs of node names and edge weights, if your graph is directed/weighted.
In the adjacency list article on Wikipedia (linked above) there is a section on tradeoffs between the two representations.
On the subject of the MST algorithm:
You will probably get better performance using an adjacency list, out of the top of my head, but that's only theoretically (I think?). Implementation-wise, there are things such as locality of reference to take into account. I would personally prefer, for ease of coding, however, to use an adjacency matrix (I just personally find them easier to work with, especially on weighted graphs), unless there's a need for really good performance.
Adjacency Matrix (C++):
vector<vector<int> > adj_Mat(n, vector<int>(n, 0));
where n is the number of nodes in the graph. Then adj_Mat[i][j] is the weight of the edge between nodes i and j.
Adjacency List (C++):
vector<vector<pair<int, int> > > adj_list(n);
Then, if the i-th node has an edge weight of k to node j, I'd do something like this (assuming the graph is directed)
adj_list[i].push_back(pair<int, int>(j, k));
Now, my C++ is really hacky because I tend to use it for hacking random code in coding competitions, so this isn't really good code but it's really basic ways to code up these data representations.
Sorry about the massive rant, and I hope it does help.

There are 2 conventional ways to represent a graph.
Adjacency matrix: meaning if you got n nodes then you got a matrix of n by n with matrix[i, j] indicates if there is an edge between the nodes (usually with a boolean or int if you want weights)
Adjacency list: have an collection for each node which indicates the nodes that it has edges to.
For the first approach a 2d array is the best, for the second I would go with an HashSet (or HashMap if you want weights)
When choosing one over the other you need to consider the graph size, the number of edges vs number of nodes and what algorithm you are going to run on it.

Check out the Boost Graph Library. There is a learning phase, but once you get it you will never look back.

The most efficient way to implement graphs G=(V,E) depends on the properties of these graphs and what kind of operations you tend to perform on them. Generally speaking, for sparse graphs, i.e. graphs where |E| << |V|^2, one usually chooses adjacency lists (slower, but memory-efficient). For graphs with |E| =~ |V|^2, adjacency matrices are usually better (faster, but memory-intensive).
Depending on what exactly you intend to do with your graphs, other approaches might be more performant.

C++ is faster than Java. And if you want to create a good graph, you should make your own type with fields you need (for example weight of graph's nodes). You also should create a Node class, which contains it's attributes and list of other nodes to which it's connected.
In Graph you should have list of it's nodes, its attributes and functions (for example bool isFlat()).
I think it's a better way to implement graph than 2D array or ArrayList.

If you need fast algorithm use array-based approach in C++.
But if you write algorithm, that should be read by other people use OOP-based approach in Java.