Aim:
Program for finding shortest path
using Dijikshtras algorithm.
Theory:
Dijkstra's algorithm, conceived by Dutch computer scientist Edsger
Dijkstra in 1956 and published in 1959, is a graph search algorithm that solves
the single-source shortest path problem for a graph with nonnegative edge path
costs, producing a shortest path tree. This algorithm is often used in routing
and as a subroutine in other graph algorithms.
For a given source vertex (node) in the graph, the algorithm finds the path
with lowest cost (i.e. the shortest path) between that vertex and every other
vertex. It can also be used for finding costs of shortest paths from a single
vertex to a single destination vertex by stopping the algorithm once the
shortest path to the destination vertex has been determined. For example, if
the vertices of the graph represent cities and edge path costs represent
driving distances between pairs of cities connected by a direct road,
Dijkstra's algorithm can be used to find the shortest route between one city
and all other cities. As a result, the shortest path first is widely used in
network routing protocols, most notably IS-IS and OSPF (Open Shortest Path
First).
Dijkstra's
original algorithm does not use a min-priority queue and runs in O(|V|2).
The idea of this algorithm is also given in (Leyzorek et al. 1957). The common
implementation based on a min-priority queue implemented by a Fibonacci heap
and running in O(|E| + |V| log |V|)
is due to (Fredman & Tarjan 1984). This is asymptotically the fastest known
single-source shortest-path algorithm for arbitrary directed graphs with
unbounded nonnegative weights.
Let
the node at which we are starting be called the initial node. Let the distance
of node Y be the distance from the initial node to Y. Dijkstra's
algorithm will assign some initial distance values and will try to improve them
step by step.
- Assign to every node a tentative
distance value: set it to zero for our initial node and to infinity for
all other nodes.
- Mark all nodes unvisited. Set the
initial node as current. Create a set of the unvisited nodes called the unvisited
set consisting of all the nodes except the initial node.
- For the current node, consider all of
its unvisited neighbors and calculate their tentative distances.
For example, if the current node A is marked with a distance of 6, and the
edge connecting it with a neighbor B has length 2, then the distance to B
(through A) will be 6+2=8. If this distance is less than the previously
recorded distance, then overwrite that distance. Even though a neighbor
has been examined, it is not marked as visited at this time, and it
remains in the unvisited set.
- When we are done considering all of
the neighbors of the current node, mark the current node as visited and
remove it from the unvisited set. A visited node will never be
checked again; its distance recorded now is final and minimal.
- If the destination node has been
marked visited (when planning a route between two specific nodes) or if
the smallest tentative distance among the nodes in the unvisited set
is infinity (when planning a complete traversal), then stop. The algorithm
has finished.
- Set the unvisited node marked with
the smallest tentative distance as the next "current node" and
go back to step 3.
Suppose
you want to find the shortest path between two intersections on a city map, a
starting point and a destination. The order is conceptually simple: to start,
mark the distance to every intersection on the map with infinity. This is done
not to imply there is an infinite distance, but to note that that intersection
has not yet been visited; some variants of this method simply leave the
intersection unlabeled. Now, at each iteration, select a current
intersection. For the first iteration the current intersection will be the
starting point and the distance to it (the intersection's label) will be zero.
For subsequent iterations (after the first) the current intersection will be
the closest unvisited intersection to the starting point—this will be easy to
find.
From the
current intersection, update the distance to every unvisited intersection that
is directly connected to it. This is done by determining the sum of the
distance between an unvisited intersection and the value of the current
intersection, and relabeling the unvisited intersection with this value if it
is less than its current value. In effect, the intersection is relabeled if the
path to it through the current intersection is shorter than the previously
known paths. To facilitate shortest path identification, in pencil, mark the
road with an arrow pointing to the relabeled intersection if you label/relabel
it, and erase all others pointing to it. After you have updated the distances
to each neighboring intersection, mark the current intersection as visited
and select the unvisited intersection with lowest distance (from the starting
point) -- or lowest label—as the current intersection. Nodes marked as visited
are labeled with the shortest path from the starting point to it and will not
be revisited or returned to.
Continue
this process of updating the neighboring intersections with the shortest
distances, then marking the current intersection as visited and moving onto the
closest unvisited intersection until you have marked the destination as
visited. Once you have marked the destination as visited (as is the case with
any visited intersection) you have determined the shortest path to it, from the
starting point, and can trace your way back, following the arrows in reverse.
Of
note is the fact that this algorithm makes no attempt to direct
"exploration" towards the destination as one might expect. Rather,
the sole consideration in determining the next "current" intersection
is its distance from the starting point. In some sense, this algorithm
"expands outward" from the starting point, iteratively considering
every node that is closer in terms of shortest path distance until it reaches
the destination. When understood in this way, it is clear how the algorithm
necessarily finds the shortest path, however it may also reveal one of the
algorithm's weaknesses: its relative slowness in some topologies.
In the
following algorithm, the code
u := vertex in Q with smallest dist[], searches for the vertex u in the vertex set
Q that has the least dist[u] value. That vertex is removed from the
set Q and returned to the user. dist_between(u, v) calculates the
length between the two neighbor-nodes u and v. The variable alt on line 15 is the length of the path from
the root node to the neighbor node v if it were to go through u. If this path is
shorter than the current shortest path recorded for v, that current path is replaced with this alt path. The previous array is populated with a pointer to the
"next-hop" node on the source graph to get the shortest route to the
source. 1 function Dijkstra(Graph, source):
2 for each vertex v in Graph: // Initializations
3 dist[v] := infinity ; // Unknown distance function from source to v
4 previous[v] := undefined ; // Previous node in optimal path from source
5 end for ;
6 dist[source] := 0 ; // Distance from source to source
7 Q := the set of all nodes in Graph ; // All nodes in the graph are unoptimized - thus are in Q
8 while Q is not empty: // The main loop
9 u := vertex in Q with smallest distance in dist[] ;
10 if dist[u] = infinity:
11 break ; // all remaining vertices are inaccessible from source
12 end if ;
13 remove u from Q ;
14 for each neighbor v of u: // where v has not yet been removed from Q.
15 alt := dist[u] + dist_between(u, v) ;
16 if alt < dist[v]: // Relax (u,v,a)
17 dist[v] := alt ;
18 previous[v] := u ;
19 decrease-key v in Q; // Reorder v in the Queue
20 end if ;
21 end for ;
22 end while ;
23 return dist[] ;
24 end Dijkstra.
If we
are only interested in a shortest path between vertices source and target, we can
terminate the search at line 13 if u
= target. Now we can
read the shortest path from source to target by iteration:1 S := empty sequence
2 u := target
3 while previous[u] is defined:
4 insert u at the beginning of S
5 u := previous[u]
6 end while ;
Now
sequence S is the list of vertices constituting one of the shortest paths from source to target, or the
empty sequence if no path exists.
A more
general problem would be to find all the shortest paths between source and target (there might
be several different ones of the same length). Then instead of storing only a
single node in each entry of
previous[] we would store all nodes satisfying the relaxation condition. For example,
if both r and source connect to target and both of them lie on different shortest paths
through target (because the edge cost is the same in both cases), then we would add both r and source to previous [target]. When the
algorithm completes, previous[] data structure will actually describe a graph that is a subset of the
original graph with some edges removed. Its key property will be that if the
algorithm was run with some starting node, then every path from that node to
any other node in the new graph will be the shortest path between those nodes
in the original graph, and all paths of that length from the original graph
will be present in the new graph. Then to actually find all these short paths
between two given nodes we would use a path finding algorithm on the new graph,
such as depth-first search.
Coclusion :
Dijkstra's original algorithm does
not use a min-priority queue and runs in O(|V|2).
PROGRAM:
//Program
for dijkstra's algorithm
import
java.util.*;
class
Dijkstra
{
public static void main(String[]
args)
{
Scanner scr = new
Scanner(System.in);
int weight[][],s,d,sp=0,n,visit[],finall=0,smalldist;
int i,j,f,INFINITY=5000,x,newdist,k=0,current,distcurr;
System.out.println("\nNo.
of nodes in graph");
n=scr.nextInt();
weight=new int[n][n];
visit=new int[n];
int distance[]=new
int[n];
int precede[]=new
int[n];
int path[]=new int[n];
System.out.println("\nEnter cost matrix");
for (i=0; i<n; i++)
for (j=0; j<n; j++)
{
System.out.println("\nIs
"+(i+1)+"--"+(j+1)+" present?(1/0): ");
f=scr.nextInt();
if
(i==j||f!=1)
{
weight[i][j]=INFINITY;
}
else
{
System.out.println("\nEnter
cost "+(i+1)+"--"+(j+1)+" : ");
weight[i][j]=scr.nextInt();
}
}
System.out.print("\nEnter
the source node : ");
s=scr.nextInt();
System.out.print("\nEnter
the destination node : ");
d=scr.nextInt();
for (i=0;i<n;i++)
{
distance[i]=INFINITY;
precede[i]=INFINITY;
}
distance[s]=0;
current=s;
visit[current]=1;
while(current!=d)
{
distcurr=distance[current];
smalldist=INFINITY;
for(i=0;
i<n; i++)
{
if(visit[i]==0)
{
newdist=distcurr+weight[current][i];
if(newdist<distance[i])
{
distance[i]=newdist;
precede[i]=current;
}
if(distance[i]<
smalldist)
{
smalldist=distance[i];
k=i;
}
}
}
current = k;
visit[current]=1;
}
i=d;
path[finall]=d;
finall++;
while(precede[i] != s)
{
j = precede[i];
i = j;
path[finall] = i;
finall++;
}
path[finall] = s;
System.out.println("\nThe
shortest path followed is : \n\n");
for(i=finall;i>0;i--)
System.out.print("\t\t(
"+ path[i]+" ->"+path[i-1]+" ) with cost = ");
System.out.print(weight[path[i]][path[i-1]]+"\n\n");
System.out.println("For
the Total Cost = "+distance[d]);
}
}
Output:
No. of
nodes in graph: 3
Enter cost matrix
Is
1--1 present?(1/0):
0
Is
1--2 present?(1/0):
2
Is
1--3 present?(1/0):
10
Is
2--1 present?(1/0):
3
Is
2--2 present?(1/0):
0
Is
2--3 present?(1/0):
1
Enter cost
2--3 :
3
Is
3--1 present?(1/0):
0
Is
3--2 present?(1/0):
0
Is
3--3 present?(1/0):
0
Enter
the source node : 1
Enter
the destination node : 3
The
shortest path followed is :
1
-> 2 -> 3
For
the Total Cost = 5
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