Greedy Algorithms: Chapter 7

3 Algorithms to know 1. Prim’s Minimum Spanning Tree Minimum Spanning Tree 2. Dijkstra’s Shortest path – weighted graph Shortest path – weighted graph 3. Knapsack Problem

Prim’s Algorithm prim( adj, start, parent) { for i = 1 to n key[i] =  ; key[start] = 0; parent[start] = 0; h.copy(key); for i = 1 to n { v = h.delmin(); ref = adj[v]; while (ref != null) { w = ref.ver if (h.isin(w) && ref.weight < h.keyval(w)) { parent[w] = v; h.decrease(w, ref.weight); } ref = ref.next; }}}

Prim’s Algorithm prim( adj, start, parent) { for i = 1 to n  O(n) key[i] =  ; key[start] = 0; parent[start] = 0; h.copy(key);  O(n) for i = 1 to n { v = h.delmin(); ref = adj[v]; while (ref != null) { w = ref.ver if (h.isin(w) && ref.weight < h.keyval(w)) {  O(1) parent[w] = v; h.decrease(w, ref.weight);  O(log n) } ref = ref.next; }}} O(m), m = #of edges

Dijkstra’s Algorithm Finds the shortest path from a start vertex to all the other vertices Finds the shortest path from a start vertex to all the other vertices Single-source shortest path Single-source shortest path

Dijkstra’s Algorithm prim( adj, start, parent) { for i = 1 to n key[i] =  ; key[start] = 0; parent[start] = 0; h.copy(key); for i = 1 to n { v = h.delmin(); ref = adj[v]; while (ref != null) { w = ref.ver if (h.isin(w) && ref.weight < h.keyval(w)) { parent[w] = v; h.decrease(w, ref.weight); } ref = ref.next; }}}