017-prim algorithm-greedy-"algorithmic design techniques and analysis" M.H.A study notes

Basic ideas:

Define node Set U, V (U = Node set that has been selected to join MST, V = not selected)

1. Choose a node to join U

2. Select an edge with the least edge, and his two nodes belong to U, V, and the node belonging to V is added to u

3. Repeat execution 2 until v null

Pseudo code:

C++Code:

int g[mnx][mnx];int N, m;int d[mnx];//naïve prim, complexity O (| V|^2) | v|: Points, | E|: The number of sides int prim () {memset (d, 0x3f, sizeof D);//Initialize int ret = d[1] = 0;  First d[1] into 0for (int i = 1; I <= n; ++i) {int u = -1;for (int j = 1; j <= N; ++j)   //Find D[u] the smallest uif ((U = =-1 | | d[u  ] > D[j]) && d[j] = 1) u = j;ret + D[u];d [u] = -1;for (int j = 1; j <= N; ++j)  //Update and U adjacency node D[j] value d[j] = Min (D[j], g[u][j]);} return ret;}

Algorithm Analysis:

The main cost is to find the edge with the least edge, the second cycle of this step consumes θ(n2), so the time complexity of the algorithm is θ(n2).

Heap Optimization Improvements:

We use the small top heap to complete the search for the minimum edge, as with the Dijkstra algorithm, the algorithm has a total of n-1 inserts,n-1 Delete, M-n+1 times siftup operation. The total time complexity is O(mlogn).

Pseudo code:

C++Code:

int FST[MNX], nxt[mxe], Cost[mxe], to[mxe], e;void init () {memset (FST,-1, sizeof FST); e = 0;} void Add (int u, int v, int c) {To[e] = V, nxt[e] = Fst[u], cost[e] = c, fst[u] = e++;} struct node {int u, dis;node (int u, int dis): U (u), dis (dis) {}bool operator < (const node &b) const {return dis ; b.dis;}};/ /heap optimization, Complexity O (| E|log| v|), dense graph with slower int primheap () {memset (d, 0x3f, sizeof D);d [1] = 0;priority_queue<node> q;q.push (node (1,0));//Select first Section First Point int ret = 0;while (!q.empty ()) {int u = q.top (). U;int dd = Q.top (). Dis;q.pop (); if (d[u]! = DD) continue;//If the value before the update is not taken , continue off ret + = Dd;d[u] = -1;for (int j = fst[u]; ~j; j = nxt[j]) {int v = to[j], c = cost[j];//Update if (D[v] > C && Amp D[V]! =-1) {D[v] = C;q.push (node (v, c));}}} return ret;}

017-prim algorithm-greed-algorithmic design techniques and analysis M.H.A learning notes