Primov algoritam

U ovom ćete tutorijalu naučiti kako djeluje Primov algoritam. Također ćete pronaći radne primjere Primova algoritma na C, C ++, Javi i Pythonu.

Primov algoritam minimalni je obuhvaćajući algoritam stabla koji uzima graf kao ulaz i pronalazi podskup rubova tog grafa koji

čine drvo koje uključuje svaki vrh
ima najmanji zbroj težina među svim stablima koja se mogu oblikovati iz grafikona

Kako radi Primov algoritam

Podpada pod klasu algoritama zvanih pohlepni algoritmi koji pronalaze lokalni optimum u nadi da će pronaći globalni optimum.

Polazimo od jednog vrha i nastavljamo dodavati rubove s najmanjom težinom dok ne postignemo cilj.

Koraci za implementaciju Priminog algoritma su sljedeći:

Inicijalizirajte stablo minimalnog raspona s nasumično odabranim vrhom.
Pronađite sve rubove koji drvo povezuju s novim vrhovima, pronađite minimum i dodajte ga na stablo
Ponavljajte korak 2 dok ne dobijemo minimalno rastegnuto stablo

Primjer Priminog algoritma

Započnite s ponderiranim grafom

Odaberite vrh

Odaberite najkraći rub ovog vrha i dodajte ga

Odaberite najbliži vrh koji još nije u rješenju

Odaberite najbliži rub koji još nije u rješenju, ako postoji više mogućnosti, odaberite jedan nasumce

Ponavljajte dok ne imati rastegnuto drvo

Pseudokod Primova algoritma

Pseudokod za primov algoritam pokazuje kako stvaramo dva skupa vrhova U i VU. U sadrži popis vrhova koji su posjećeni, a VU popis vrhova koji nisu. Jedan po jedan pomičemo vrhove iz skupa VU u skup U povezujući rub najmanje težine.

 T = ∅; U = ( 1 ); while (U ≠ V) let (u, v) be the lowest cost edge such that u ∈ U and v ∈ V - U; T = T ∪ ((u, v)) U = U ∪ (v)

Primjeri Pythona, Java i C / C ++

Iako se koristi matrični prikaz susjedstva grafova, ovaj se algoritam također može implementirati pomoću Popisa susjedstva kako bi se poboljšala njegova učinkovitost.

Python Java C C ++

 # Prim's Algorithm in Python INF = 9999999 # number of vertices in graph V = 5 # create a 2d array of size 5x5 # for adjacency matrix to represent graph G = ((0, 9, 75, 0, 0), (9, 0, 95, 19, 42), (75, 95, 0, 51, 66), (0, 19, 51, 0, 31), (0, 42, 66, 31, 0)) # create a array to track selected vertex # selected will become true otherwise false selected = (0, 0, 0, 0, 0) # set number of edge to 0 no_edge = 0 # the number of egde in minimum spanning tree will be # always less than(V - 1), where V is number of vertices in # graph # choose 0th vertex and make it true selected(0) = True # print for edge and weight print("Edge : Weight") while (no_edge  G(i)(j): minimum = G(i)(j) x = i y = j print(str(x) + "-" + str(y) + ":" + str(G(x)(y))) selected(y) = True no_edge += 1

 // Prim's Algorithm in Java import java.util.Arrays; class PGraph ( public void Prim(int G()(), int V) ( int INF = 9999999; int no_edge; // number of edge // create a array to track selected vertex // selected will become true otherwise false boolean() selected = new boolean(V); // set selected false initially Arrays.fill(selected, false); // set number of edge to 0 no_edge = 0; // the number of egde in minimum spanning tree will be // always less than (V -1), where V is number of vertices in // graph // choose 0th vertex and make it true selected(0) = true; // print for edge and weight System.out.println("Edge : Weight"); while (no_edge < V - 1) ( // For every vertex in the set S, find the all adjacent vertices // , calculate the distance from the vertex selected at step 1. // if the vertex is already in the set S, discard it otherwise // choose another vertex nearest to selected vertex at step 1. int min = INF; int x = 0; // row number int y = 0; // col number for (int i = 0; i < V; i++) ( if (selected(i) == true) ( for (int j = 0; j G(i)(j)) ( min = G(i)(j); x = i; y = j; ) ) ) ) ) System.out.println(x + " - " + y + " : " + G(x)(y)); selected(y) = true; no_edge++; ) ) public static void main(String() args) ( PGraph g = new PGraph(); // number of vertices in grapj int V = 5; // create a 2d array of size 5x5 // for adjacency matrix to represent graph int()() G = ( ( 0, 9, 75, 0, 0 ), ( 9, 0, 95, 19, 42 ), ( 75, 95, 0, 51, 66 ), ( 0, 19, 51, 0, 31 ), ( 0, 42, 66, 31, 0 ) ); g.Prim(G, V); ) )

 // Prim's Algorithm in C #include #include #define INF 9999999 // number of vertices in graph #define V 5 // create a 2d array of size 5x5 //for adjacency matrix to represent graph int G(V)(V) = ( (0, 9, 75, 0, 0), (9, 0, 95, 19, 42), (75, 95, 0, 51, 66), (0, 19, 51, 0, 31), (0, 42, 66, 31, 0)); int main() ( int no_edge; // number of edge // create a array to track selected vertex // selected will become true otherwise false int selected(V); // set selected false initially memset(selected, false, sizeof(selected)); // set number of edge to 0 no_edge = 0; // the number of egde in minimum spanning tree will be // always less than (V -1), where V is number of vertices in //graph // choose 0th vertex and make it true selected(0) = true; int x; // row number int y; // col number // print for edge and weight printf("Edge : Weight"); while (no_edge < V - 1) ( //For every vertex in the set S, find the all adjacent vertices // , calculate the distance from the vertex selected at step 1. // if the vertex is already in the set S, discard it otherwise //choose another vertex nearest to selected vertex at step 1. int min = INF; x = 0; y = 0; for (int i = 0; i < V; i++) ( if (selected(i)) ( for (int j = 0; j G(i)(j)) ( min = G(i)(j); x = i; y = j; ) ) ) ) ) printf("%d - %d : %d", x, y, G(x)(y)); selected(y) = true; no_edge++; ) return 0; )

 // Prim's Algorithm in C++ #include #include using namespace std; #define INF 9999999 // number of vertices in grapj #define V 5 // create a 2d array of size 5x5 //for adjacency matrix to represent graph int G(V)(V) = ( (0, 9, 75, 0, 0), (9, 0, 95, 19, 42), (75, 95, 0, 51, 66), (0, 19, 51, 0, 31), (0, 42, 66, 31, 0)); int main() ( int no_edge; // number of edge // create a array to track selected vertex // selected will become true otherwise false int selected(V); // set selected false initially memset(selected, false, sizeof(selected)); // set number of edge to 0 no_edge = 0; // the number of egde in minimum spanning tree will be // always less than (V -1), where V is number of vertices in //graph // choose 0th vertex and make it true selected(0) = true; int x; // row number int y; // col number // print for edge and weight cout << "Edge" << " : " << "Weight"; cout << endl; while (no_edge < V - 1) ( //For every vertex in the set S, find the all adjacent vertices // , calculate the distance from the vertex selected at step 1. // if the vertex is already in the set S, discard it otherwise //choose another vertex nearest to selected vertex at step 1. int min = INF; x = 0; y = 0; for (int i = 0; i < V; i++) ( if (selected(i)) ( for (int j = 0; j G(i)(j)) ( min = G(i)(j); x = i; y = j; ) ) ) ) ) cout << x << " - " << y << " : " << G(x)(y); cout << endl; selected(y) = true; no_edge++; ) return 0; )

Primov vs Kruskalov algoritam

Kruskalov algoritam je još jedan popularan algoritam minimalnog raspona koji se koristi drugom logikom za pronalaženje MST-a grafa. Umjesto da krene od vrha, Kruskalov algoritam sortira sve rubove od male težine do visokih i nastavlja dodavati najniže rubove, zanemarujući one rubove koji stvaraju ciklus.