Dijkstra Algorithm for single-source shortest path (1)

Dijkstra implemented in this articleAlgorithmThe simplest method is brute-force search. The time complexity is O (V ^ 2). The priority queue method will be used later to modify the breadth-first traversal to implement the Dijkstra algorithm, the time complexity of this algorithm will be improved.

CodeInstance:

/* <Br/> reference: http://baike.baidu.com/view/7839.htm <br/> algorithm flow: <br/> In the following description, S is the source, W [U, v] is the length of the edge between the vertex u and v, and the result is stored in DIST [] <br/> 1. initialization: the distance from the source Dist [s] is set to 0, the distance from other points is set to infinite, and the status of all points is set to not extended. <Br/> 2. Loop n-1 times: <br/> In a point that has not been extended, obtain the U point with the smallest distance and set its status to extended. <Br/> execute relax (u, v) for each vertex v adjacent to U, that is, if Dist [u] + W [U, v] <Dist [v], <br/> change Dist [v] to a shorter distance from Dist [u] + W [U, V]. In this case, the first node is u in the shortest path of point v. <Br/> 3. End. In this case, for any U, DIST [u] is the distance from S to u. <Br/> direct implementation: (the method used in this instance) <br/> the simplest implementation method is to find the nearest point in each cycle, then, any method is used to update the adjacent edge. The time complexity is obviously O (n2) <br/> */<br/> # include <fstream> <br/> # include <cstring> <br/> # include <iostream> <br/> # include <stdlib. h> <br/> using namespace STD; <br/> const int maxnum = 1000000; // maximum edge weight <br/> int N; // number of nodes <br/> int Dist [501]; // the shortest path to node 1 <br/> bool State [501]; // indicates the status of the node that has been searched <br/> int data [501] [501]; // adjacent matrix <br/> // find the node with the smallest weight <br/> in T findmin () <br/>{< br/> int minnode = 0, min = maxnum, I; <br/> for (I = 1; I <= N; I ++) <br/> If (Dist [I] <min )&&(! State [I]) <br/>{< br/> min = DIST [I]; <br/> minnode = I; <br/>}< br/> return minnode; // If minnode is set to 0, no minimum node is found. <br/>}< br/> void Dijkstra () <br/>{< br/> cout <"Number of Input Points:" <Endl; <br/> CIN> N; // number of input vertices <br/> // construct an adjacent matrix. The distance between the vertex and itself is infinite. <Br/> cout <"input right of the adjacent matrix:" <Endl; <br/> for (INT p = 1; P <= N; P ++) <br/> for (INT q = 1; q <= N; q ++) <br/>{< br/> // In> data [p] [Q]; <br/> CIN> data [p] [Q]; <br/> If (data [p] [Q] = 0) <br/> data [p] [Q] = maxnum; <br/>}< br/> // initialization <br/> for (INT I = 1; I <= N; I ++) <br/> {<br/> State [I] = 0; // initialization, indicates that all vertices have not been extended <br/> Dist [I] = data [1] [I]; // data [1] [I] indicates that 1 is the source point <br/>}< br/> State [1] = true; <br/> int done = 1; <br/> while (done <n) // loop n-1 times <br/> {<Br/> int node = findmin (); <br/> If (node! = 0) <br/>{< br/> done ++; // Add 1 to the number of points found <br/> State [node] = true; // indicates that the shortest path from node 1 to node has been found. <br/> for (INT I = 1; I <= N; I ++) // update the PATH value of a vertex not found <br/> If (Dist [I]> Dist [node] + data [node] [I]) & (! State [I]) <br/> Dist [I] = DIST [node] + data [node] [I]; <br/>}< br/> else <br/> break; <br/>}< br/> int main () <br/> {<br/> Dijkstra (); <br/> for (int K = 1; k <= N; k ++) <br/> {<br/> If (Dist [k] = maxnum) <br/> cout <-1; <br/> else <br/> cout <Dist [k]; <br/> If (k = N) <br/> cout <Endl; // final output line feed <br/> else <br/> cout <""; // if it does not reach the end, it is separated by null characters <br/>}< br/> system ("pause"); <br/> return 0; <br/>}< br/>/* <br/> Number of Input Points: <br/> 5 <br/> enter the right of the adjacent matrix: <br/> 0 4 2 0 0 <br/> 0 0 3 2 3 <br/> 0 1 0 4 5 <br/> 0 0 0 0 0 <br/> 0 0 0 1 0 <br/> */

PS: 2011-6-15

Modified:

Output THE Dist [I] Changes in the process

# Include <fstream> <br/> # include <cstring> <br/> # include <iostream> <br/> # include <stdlib. h> <br/> using namespace STD; <br/> const int maxnum = 1000000; // maximum edge weight <br/> int N; // number of nodes <br/> int Dist [501]; // the shortest path to node 1 <br/> bool State [501]; // indicates the status of the node that has been searched <br/> int data [501] [501]; // adjacent matrix <br/> // find the node with the smallest weight <br/> int findmin () <br/> {<br/> int minnode = 0, min = maxnum, I; <br/> for (I = 1; I <= N; I ++) <br/> If (Dist [I] <min) &&(! State [I]) <br/>{< br/> min = DIST [I]; <br/> minnode = I; <br/>}< br/> return minnode; // If minnode is set to 0, no minimum node is found. <br/>}< br/> void Dijkstra () <br/>{< br/> cout <"Number of Input Points:" <Endl; <br/> CIN> N; // number of input vertices <br/> // construct an adjacent matrix. The distance between the vertex and itself is infinite. <Br/> cout <"input right of the adjacent matrix:" <Endl; <br/> for (INT p = 1; P <= N; P ++) <br/> for (INT q = 1; q <= N; q ++) <br/>{< br/> // In> data [p] [Q]; <br/> CIN> data [p] [Q]; <br/> If (data [p] [Q] = 0) <br/> data [p] [Q] = maxnum; <br/>}< br/> // initialization <br/> for (INT I = 1; I <= N; I ++) <br/> {<br/> State [I] = 0; // initialization, indicates that all vertices have not been extended <br/> Dist [I] = data [1] [I]; // data [1] [I] indicates that 1 is the source point <br/>}< br/> cout <"Dist [I] in the output process" <Endl; <br/> for (INT I = 1; I <= N; I ++) <br/> {<B R/> cout <Dist [I] <""; <br/>}< br/> cout <Endl; <br/> State [1] = true; <br/> int done = 1; <br/> while (done <n) // loop n-1 times <br/>{< br/> int node = findmin (); <br/> If (node! = 0) <br/>{< br/> done ++; // Add 1 to the number of points found <br/> State [node] = true; // indicates that the shortest path from node 1 to node has been found. <br/> for (INT I = 1; I <= N; I ++) // update the PATH value of a vertex not found <br/> If (Dist [I]> Dist [node] + data [node] [I]) & (! State [I]) <br/> Dist [I] = DIST [node] + data [node] [I]; <br/> for (Int J = 1; j <= N; j ++) <br/> cout <Dist [J] <""; <br/> cout <Endl; <br/>}< br/> else <br/> break; <br/>}< br/> int main () <br/>{< br/> Dijkstra (); <br/> cout <"output the final Dist [I] value.-1 indicates the Source Vertex" <Endl; <br/> for (int K = 1; k <= N; k ++) <br/> {<br/> If (Dist [k] = maxnum) <br/> cout <-1; <br/> else <br/> cout <Dist [k]; <br/> If (k = N) <br/> cout <Endl; // final output line feed <br/> else <br/> cout <""; // if it does not reach the end, it is separated by null characters <br/>}< br/> system ("pause"); <br/> return 0; <br/>}< br/>/* <br/> Number of Input Points: <br/> 5 <br/> enter the right of the adjacent matrix: <br/> 0 4 2 0 0 <br/> 0 0 3 2 3 <br/> 0 1 0 4 5 <br/> 0 0 0 0 0 <br/> 0 0 0 1 0 <br/> Dist [I] During output <br/> 1000000 4 2 1000000 1000000 <br/> 1000000 3 2 6 7 <br/> 1000000 3 2 5 6 <br/> 1000000 3 2 5 6 <br/> 1000000 3 2 5 6 <br/> output the final Dist [I] value, -1 indicates the source point <br/>-1 3 2 5 6 <br/> press any key to continue... <br/> */

The Dijkstra algorithm calculates the single-source shortest path, which is equivalent to finding the shortest path of all other points that can be obtained from S.

Therefore, perform the following steps:

1. Dist [I] is used to mark the shortest path from S to I. The Initialization is only Dist [s] = 0, and other Dist [I] = maxweitht, indicating that the weight is infinite.

2. Because the source point has been found, you also need to traverse the remaining V-1 points to find the shortest path.

3. From Dist [I], find that the I point is not accessed and the minimum value of DIST is used as the next vertex minnode.

4. if the value of DIST [minnode] plus the value on the edge of the reachable vertex J is smaller than the value of DIST [J], the DIST [J] is updated, indicating that the pre-order node of J is minnode.