C#: A-Star is born

, 23 Jun 2003
 Rate this:
Please Sign up or sign in to vote.
Understand graphs and A* path-finding algorithm with C#

Introduction: what is it supposed to do ?

I dedicated myself to creating a .NET component that allows the building of graphs and the performance of some operations on these structures. In particular A*, the famous path finding algorithm, can be run to find the best path between two places.

First and foremost it should be kept in mind that a graph is defined with :

• A list of nodes
• A list of arcs

Each node is defined with the following data :

• A geographical position in space (co-ordinates X,Y,Z)
• The collection of the outgoing arcs
• The collection of the incoming arcs

Each arc is simply defined with its two extremity nodes. So an arc is oriented from StartNode to EndNode. It is also characterized by a crossing factor named 'Weight'. This value represents the difficulty to reach the ending node from the starting one. Thus, getting through an arc has a cost, which can be basically calculated like this :

`Cost = Distance(StartNode, EndNode) * Weight`

You may be asking "What is the use of this structure ?...". Well, the applications can be various, ranging from road maps and circulation models to a character's mobility in a video game. Now the question is: "How one would move across this graph ?".

What is A* ?

Imagine that you are on a node and that you want to reach another position somewhere else on the graph. Then ask : "Which way will I follow, and why ?". The main factor to take into account here is the cost of moving. It must be minimal. The cost criterion is basically a function of the distance (sum of arcs' lengths). However, it can also be adjusted and varied with other data, which describe for example the slope, the harshness/practicability of the ground. You can even model a traffic jam.

To achieve the best path, there are many algorithms which are more or less effective, depending on the particular case. Efficiency depends not only on the time needed for calculation, but also on the reliability of the result. A* is able to return the best path (if it exists) between two nodes, according to accessibility/orientation and, of course, cost of arcs.

Among the variety of existing algorithms, some do not always actually return the best path, but they can give precedence to speed over accuracy. Efficiency depends on the number of nodes and on their geographical distribution. However in most cases A* turns out to be the most effective, because it combines optimized search with the use of a heuristic.

A heuristic is a function that associates a value with a node to gauge it. One node is considered better than another, if the final point is reached with less effort (e.g. shorter distance).

A* will always return the shortest path if, and only if, the heuristic is admissible; that is to say, if it never overestimates. On the other hand, if the heuristic is not admissible, A* will find a path in less time and with less memory usage, but without the absolute guaranty that it is the best one. Here are three admissible heuristics which correspond to a particular distance between the node of evaluation and the target node :

• Euclidean distance > Sqrt(Dx²+Dy²+Dz²)
• Manhattan distance > |Dx|+|Dy|+|Dz|
• Maximum distance > Max(|Dx|, |Dy|, |Dz|)

These functions give an estimation of the remaining distance for each node that can be explored. Thus the search can be oriented toward the 'best' nodes. For a given node, the sum [Current cost + Heuristic value] is an estimation of the cost of reaching the ending node from the starting node, passing by the current one. This value is used to continuously choose the most promising path.

In practice, the algorithm maintains 2 lists of nodes that are filled and modified during the search :

1. The first one, called `Open`, contains the tracks leading to nodes that can be explored. Initially, there is only the starting node. At each step, the best node of `Open` is taken out. Then, the best successor of this node (according to the heuristic) is added to the list as a new track. One doesn't know where the nodes that are in `Open` lead, because they have not been propagated yet. However, the best one is examined at each new step.
2. The second one, called `Closed`, stores the tracks leading to nodes that have already been explored.

The program is based on a recursive model. The loop is performed as long as `Open` still contains some elements. See the code for details (written with C#, it is sufficient enough to be self explainable).

```public bool SearchPath(Node StartNode, Node EndNode)
{
lock (_Graph)
{
Initialize(StartNode, EndNode);
while ( NextStep() ) {}
return PathFound;
}
}```

Using the code: public interfaces of the component

The `Graph` class gathers a set of methods to manage its data, such as :

• Add/Suppress a node/arc
• Get the nearest/farthest node/arc from a point
• Activate/Inactivate the entire graph
• Empty the graph

From a more practical point of view, here are the methods and properties you can use for the main classes and objects :

```public class Graph
{
public Graph()

public IList Nodes { get }
public IList Arcs { get }

public void Clear()
public bool AddNode(Node NewNode)
public Node AddNode(float x, float y, float z)
public bool AddArc(Arc NewArc)
public Arc AddArc(Node StartNode, Node EndNode, float Weight)
public void Add2Arcs(Node Node1, Node Node2, float Weight)
public bool RemoveNode(Node NodeToRemove)
public bool RemoveArc(Arc ArcToRemove)
public void BoundingBox(out double[] MinPt, out double[] MaxPt)
public Node ClosestNode(double X, double Y, double Z, out double Dist, <BR>                            bool IgnoreFreeWay)
public Arc ClosestArc(double X, double Y, double Z, out double Dist, <BR>                            bool IgnoreFreeWay)
}

public class Node
{
public Node(double PositionX, double PositionY, double PositionZ)

public IList IncomingArcs { get }
public IList OutgoingArcs { get }
public double X { get }
public double Y { get }
public double Z { get }
public void ChangeXYZ(double PositionX, double PositionY, <BR>                          double PositionZ)
public bool Passable { get/set }

public Node[] AccessibleNodes { get }
public Node[] AccessingNodes { get }
public void UntieIncomingArcs()
public void UntieOutgoingArcs()
public void Isolate()
public Arc ArcGoingTo(Node N)
public Arc ArcComingFrom(Node N)
public object Clone()

public static void BoundingBox(IList NodesGroup, out double[] MinPt, <BR>                                   out double[] MaxPt)
public static double EuclidianDistance(Node N1, Node N2)
public static double SquareEuclidianDistance(Node N1, Node N2)
public static double ManhattanDistance(Node N1, Node N2)
public static double MaxDistanceAlongAxis(Node N1, Node N2)

public override string ToString()
public override bool Equals(object O)
public override int GetHashCode()
}

public class Arc
{
public Arc(Node Start, Node End)

public Node StartNode { get/set }
public Node EndNode { get/set }
public double Weight { get/set }
public bool Passable { get/set }

virtual public double Cost { get }
public double Length { get }
virtual protected double CalculateLength()

public override string ToString()
public override bool Equals(object O)
public override int GetHashCode()
}

public class AStar
{
public AStar(Graph G)

public bool SearchPath(Node StartNode, Node EndNode)
public void Initialize(Node StartNode, Node EndNode)
public bool NextStep()

public bool Initialized { get }
public int StepCounter { get }
public bool SearchStarted { get }
public bool SearchEnded { get }
public bool PathFound { get }
public Node[] PathByNodes { get }
public Arc[] PathByArcs { get }
public bool ResultInformation(out int NbArcsOfPath, out double CostOfPath)

public double DijkstraHeuristicBalance { get/set }
public Heuristic ChoosenHeuristic { get/set }
public static Heuristic EuclidianHeuristic { get }
public static Heuristic MaxAlongAxisHeuristic { get }
public static Heuristic ManhattanHeuristic { get }
}
```

Nodes and arcs can be Passable or not. Selecting one of these two states for a node, propagates the same state to outgoing and incoming arcs. Likewise, destroying a node implies the destruction of all the directly linked arcs. In addition, an arc must always be linked to two nodes, whereas a node can be isolated.

Note that these data override `Object.ToString()` as well as `Object.Equals(Object O)`. Moreover they can be serialized.

Simple example of the usage with the console application

```using System;
using System.Collections;
using System.Text;
using System.IO;
using System.Runtime.Serialization;
using System.Runtime.Serialization.Formatters.Binary;
using EMK.Cartography;

namespace EMK.Tests
{
public class TestAStar
{
public static void Main()
{
try
{
Graph G = new Graph();
Node N1 = G.AddNode(0,0,0);
Node N2 = G.AddNode(5,0,0);
Node N3 = G.AddNode(5,5,0);
Node N4 = G.AddNode(5,5,5);
G.AddArc(N1,N2,1);
G.AddArc(N2,N3,1);
G.AddArc(N3,N4,1);
G.AddArc(N1,N3,1);

Console.WriteLine( ListNodesAndArcs(G) );
Console.WriteLine("Best path to reach "+N4+" from "+N1+" :");
AStar AS = new AStar(G);
if ( AS.SearchPath(N1, N4) )
foreach (Arc A in AS.PathByArcs)
Console.WriteLine( A.ToString() );
else Console.WriteLine( "No result !" );

Console.Write("Serialize and Deserialize. ");
Stream StreamWrite = File.Create("GraphSaved.bin");
BinaryFormatter BinaryWrite = new BinaryFormatter();
BinaryWrite.Serialize(StreamWrite, G);
StreamWrite.Close();

Stream StreamRead = File.OpenRead("GraphSaved.bin");
BinaryFormatter BinaryRead = new BinaryFormatter();
Graph G2 = (Graph) BinaryRead.Deserialize(StreamRead);
StreamRead.Close();
Console.WriteLine( ListNodesAndArcs(G2) );
}
catch(Exception e)
{
Console.Write( "Error :\n\n"+e.ToString() );
}
}

static private string ListNodesAndArcs(Graph GraphToDescribe)
{
StringBuilder SB = new
StringBuilder("Description of the Graph:\n\tNodes> ");
foreach (Node N in GraphToDescribe.Nodes)
SB.Append( N.ToString()+"; " );
SB.Append("\n\tArcs> ");
foreach (Arc A in GraphToDescribe.Arcs)
SB.Append( A.ToString()+"; " );
return SB.ToString();
}
}
}```
```Description of the Graph:
Nodes> {0;0;0}; {5;0;0}; {5;5;0}; {5;5;5}
Arcs> {0;0;0}-->{5;0;0}; {5;0;0}-->{5;5;0};
{5;5;0}-->{5;5;5}; {0;0;0}-->{5;5;0}
Best path to reach {5;5;5} from {0;0;0} :
{0;0;0}-->{5;5;0}
{5;5;0}-->{5;5;5}
Serialize and Deserialize. Description of the Graph:
Nodes> {0;0;0}; {5;0;0}; {5;5;0}; {5;5;5}
Arcs> {0;0;0}-->{5;0;0}; {5;0;0}-->{5;5;0};
{5;5;0}-->{5;5;5}; {0;0;0}-->{5;5;0}```

Example of use in a graphical environment

The graphical interface aims at bringing the component into play so as to reflect, fairly and simply, what it can do. The application lets you draw, move, erase or inactivate several nodes and arcs. When your graph is complete you just have to click on the 'A*' icon and select the starting and ending nodes with the respective left and right mouse buttons. Then you will automatically see the best way. If you modify the graph, this path will be updated.

If you want to visualize the algorithm's logic, then select the 'Step by Step' mode in the sub-menu of the 'A*' icon. The idea is to give the user a clear view of what happens.

In the option panel ('?' icon), you can change the heuristic and set its influence. The min value (in fact 0) means that only the heuristic will be used. On the contrary, the max value (in fact 1) means that the search will behave in accordance with the Dijkstra algorithm.

Remarks

The program has been validated with a series of adapted tests, gathering various scenarios and interactions. Nevertheless it does not aim to be a perfect application, but an unassuming program that modestly claims to be simple and ergonomic. For information, I had already done this work with C++ for the DLL as well as MFC for the 2D graphical interface. I had also tested it in a 3D context with an OpenGL graphical interface. The next step will be to implement the 3D interface with Direct3D 9.0 for .NET or with one of the OpenGL solutions for this framework.

The source code contains a light but interesting implementation of geometrical tools such as Point3D and Vector3D. They were needed for graphical interactions in the GraphFormer example application (projections on lines, vectors' operations, etc...).

For convenience and performance I used the `SortableList` type for Closed and Open lists. The advantage is that, the best track will always be the first element, since the list remains sorted (cf the article I have written formerly about Automatic Sorted List). Nonetheless it would be very easy to replace it with another list type, such as `ArrayList`, provided you look for the 'best' element (that is to say the lower) at each step.

History

• June the 22nd, 2003 : Translation of the library code and comments from French to English.
• June the 24th, 2003 : Article submission.
• June the 26th, 2003 : No more Flickering thanks to DoubleBuffer Panel style.

License

This article has no explicit license attached to it but may contain usage terms in the article text or the download files themselves. If in doubt please contact the author via the discussion board below.

A list of licenses authors might use can be found here

About the Author

Web Developer
France
Eric Marchesin is working as a software development engineer at Dassault Systèmes, Paris. Dassault Systèmes is a global leader in the market for Product Lifecycle Management using 3D modeling digital technology.
He has also been working for Video Game companies as well as Artificial Intelligence projects.
His programming experience more specifically includes C/C++, MFC, OpenGL, C# and .NET framework.

Of course I appreciate beautiful and fine algorithms. There's quite an art to creating powerful, effective and ergonomic programs.
Among many other things I enjoy music, sun, passion and generally whatever makes you imagine, travel and dream.

Comments and Discussions

 First PrevNext
 thanks cumthyb 16-Mar-14 3:27
 Re: Great sample!! Mahsa Hassankashi 2-Mar-14 4:04
 Small bug Member 10467940 15-Dec-13 12:25
 Please fix rendering of power of 2 SAKryukov 3-Jul-11 13:48
 hi nghiemvanhung 10-Jun-11 7:06
 custom heuristic AsafMeir 20-Feb-11 2:55
 sorting nodes ghada_ahmed 17-Jul-10 3:25
 euclidean distance metyouba 28-Mar-10 22:18
 How to set and display labels for nodes and arcs? pluto.plutone 23-Jul-09 4:30
 Somewhat naive... petervn 22-Jul-09 13:07
 Need help on C# Programming, nosa oko 21-Nov-08 22:26
 Faster Kilpo 21-Oct-08 20:14
 Re: Faster [modified] chenyingchun 9-Jul-11 23:39
 Well done Fco. Javier Marin 20-Oct-08 16:13
 how to input edges pair?? farhancrazy 5-Sep-08 8:14
 2D Cut optimization software Member 3693033 9-May-08 12:12
 license type Rul 5-Mar-08 1:46
 Re: license type Leinir Turthra 19-Nov-08 6:21
 eric marchesin, are you dead??? murathankocan 2-Mar-08 8:06
 is it possible, for OCR? reymond 18-Jan-08 15:51
 Another method... cplas 7-Jan-08 20:56
 Re: Another method... murathankocan 11-Jan-08 23:00
 Re: Another method... cplas 12-Jan-08 10:52
 It's been 10 years, would have to take a look. The project was for an assignment. Limited to 10 nodes (base 10, but could be any base) as it was just an assignment, but it could find every possible path, where (n^2-n)/2 gives the number of possible interconnections between all nodes. The project was in MFC but below is some CRT code. This was a year out of high school so don't expect anything professional =) Though the instructor (Concorde engineer) definitely liked it. I think it was the maximum flow problem.   Not sure why this piece of crap editor takes out formatting.   ```#include #include #include #include #include // these constants save calculation time const unsigned long power10[10] = {1,10,100,1000,10000,100000,1000000, 10000000,100000000,1000000000};   // define a Boolean type typedef enum BOOL {TRUE, FALSE};   typedef struct _NETWORK { unsigned char nodes; // the 2 connected nodes unsigned long arcFlow; unsigned long arcFloworiginal; } NETWORK;   // function prototypes double permutation(int,int); double numPossiblePaths(char); unsigned long padPath(unsigned long,char); unsigned long convertBase(unsigned long, unsigned char);   /* * This function returns the number of possible paths using * permmutation, not including restrictions posed on arcs and nodes * *----------------------------------------------------------*/ double numPossiblePaths(char nodes) { double sum=0; char i;   for (i=2; i<=(nodes-2); i++) sum += permutation(nodes-2,i);   // nP0 and nP1 are always 1 and n, so save time by doing this return (sum+nodes-1); // or 1+(nodes-2), whatever }   /* * This function returns nPk (permmutation) * *----------------------------------------------------------*/ double permutation(int n, int k) { double sum=0; int i;   if (k>n || n<2) return -1; // return with error if k>n else { // do n! for (i=n; i>1; i--) sum += log(i); // do (n-k)! for (i=(n-k); i>1; i--) sum -= log(i); } // return nPk return exp(sum); }   /* * This function pads a path with the start and end nodes * *----------------------------------------------------------*/ unsigned long padPath(unsigned long num, char nodes) { unsigned char ctr=10;   while ((num / power10[--ctr]) == 0);   num = num + nodes * power10[ctr+1]; num = (num*10)+1;   return num; }   /* * This function takes in a decimal number and a base to convert it to * *----------------------------------------------------------*/ unsigned long convertBase(unsigned long num, unsigned char numBase) { int ctr=0; unsigned long convertedNum=0;   while (num!=0) // dont bother converting when quotient is 0 { convertedNum = convertedNum+((num % numBase) * power10[ctr]); num /= numBase; // go down 1 digit in old base ctr++; // go up 1 digit in new base }   return convertedNum; }   /*==========================================================*/ int main(void) { char nodes=6; // number of nodes in network unsigned long countTo=0, // number to count in decimal ctr,ctr2,ctr3,n,n2,n3, // general counters and holders path, // a path networkFlow=0, // initlally network flow is 0 smallestArcCapacity; // Duh. NETWORK network[34] = {1}; // holds network information // this value changes along the process BOOL nodePath,validPath=TRUE; // of path selection, and determines whether // a path makes it into the path array. network[0].nodes = 21; network[0].arcFlow = 2; network[1].nodes = 31; network[1].arcFlow = 6; network[2].nodes = 41; network[2].arcFlow = 3; network[3].nodes = 12; network[3].arcFlow = 0; network[4].nodes = 42; network[4].arcFlow = 1; network[5].nodes = 52; network[5].arcFlow = 4; network[6].nodes = 13; network[6].arcFlow = 0; network[7].nodes = 43; network[7].arcFlow = 3; network[8].nodes = 63; network[8].arcFlow = 2; network[9].nodes = 14; network[9].arcFlow = 0; network[10].nodes = 24; network[10].arcFlow = 1; network[11].nodes = 34; network[11].arcFlow = 3; network[12].nodes = 54; network[12].arcFlow = 1; network[13].nodes = 64; network[13].arcFlow = 3; network[14].nodes = 25; network[14].arcFlow = 4; network[15].nodes = 45; network[15].arcFlow = 1; network[16].nodes = 65; network[16].arcFlow = 6; network[17].nodes = 36; network[17].arcFlow = 0; network[18].nodes = 46; network[18].arcFlow = 0; network[19].nodes = 56; network[19].arcFlow = 0;     // get the number to count to countTo = (unsigned long) pow(nodes,nodes-2);   // start the path building for (ctr=1; ctr1 || (n%10)<=1) validPath = FALSE; n /= 10; // go down 1 digit } while (n!=0 && validPath!=FALSE); // stop converting when quotient is 0 // add the start and end nodes to path. e.g. if path is 423 and // there are 6 nodes, the path becomes 64231 path = padPath(path,nodes);   // check if path is proper by checking the restrictions given // by the user if (validPath==TRUE) { ctr2 = 0; smallestArcCapacity = 255; nodePath = TRUE; // variable used to check path at each node while ((n=(path % power10[ctr2+2] / power10[ctr2]))>9 && (nodePath!=FALSE)) { nodePath = FALSE; // check each path with the information for (ctr3=0; ctr3<20; ctr3++) // given by user // check if nodes correct and arc flow capacity>0 if ((n == network[ctr3].nodes) && (network[ctr3].arcFlow != 0)) { nodePath = TRUE; // in the noetwork array, path is bad if (network[ctr3].arcFlow < smallestArcCapacity) smallestArcCapacity = network[ctr3].arcFlow; } ctr2++; // as soon as path is know to be bad, loop quits } validPath = nodePath; // result of path , bad or good }   // if path is valid, decrease all flows on path by smallestArcFlow // and increase in oppositte direction if (validPath==TRUE) { ctr2 = 0;   while ((n=(path % power10[ctr2+2]) / power10[ctr2])>9) { ctr3 = 0;   while (n != network[ctr3++].nodes); network[ctr3-1].arcFlow -= smallestArcCapacity;   ctr3 = 0;   while ((((n%10)*10)+(n/10)) != network[ctr3++].nodes); network[ctr3-1].arcFlow += smallestArcCapacity; ctr2++; // as soon as path is know to be bad, loop quits } networkFlow += smallestArcCapacity; // Duh. }   if (validPath==TRUE) cout << path << " " << smallestArcCapacity << endl; } cout << networkFlow; return 0; }```
 Re: Another method... Member 4758703 5-Feb-08 17:14
 Please help me for Senoir Project SalmaAlnajim 7-Jan-08 9:04
 Last Visit: 31-Dec-99 19:00     Last Update: 28-Dec-14 7:20 Refresh 12345 Next »

General    News    Suggestion    Question    Bug    Answer    Joke    Rant    Admin

Use Ctrl+Left/Right to switch messages, Ctrl+Up/Down to switch threads, Ctrl+Shift+Left/Right to switch pages.

| Advertise | Privacy | Terms of Use | Mobile
Web04 | 2.8.141223.1 | Last Updated 24 Jun 2003
Article Copyright 2003 by Eric Marchesin
Everything else Copyright © CodeProject, 1999-2014
Layout: fixed | fluid