PDST Information Workshops – Autumn 2018 Prim's Algorithm
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PDST Information Workshops – Autumn 2018 Prim’s Algorithm Prim’s algorithm was developed in 1930 by Czech mathematician Vojtěch Jarník and later rediscovered and republished by computer scientists Robert C. Prim in 1957 and Edsger W. Dijkstra in 1959. Prim's algorithm is a greedy algorithm that finds a minimum spanning tree (MST) for a weighted undirected graph. This means it finds a subset of the edges that forms a tree that includes every vertex, where the total weight of all the edges in the tree is minimized. The algorithm operates by building this tree one vertex at a time, from an arbitrary starting vertex, at each step adding the cheapest possible connection from the tree to another vertex. Criteria for creating MST using Prims Algorithm You can choose any starting vertex Continue by selecting the next lowest vertex attached to the tree. You can only select a vertex once. You cannot select a vertex that would produce a cycle (closed loop) The number of edges in the MST is calculated by: #edges = (#vertices, -1). Vojtěch Jarník, Robert C Prim & Edsger W Dijkstra Page | 1 PDST Information Workshops – Autumn 2018 Prim’s Algorithm (Practical Example 1) A telecommunications company is contracted to connect six towns in Cork and South Tipperary with high speed fibre optic cabling. Determine the lowest price to complete the project if the length of protective piping and fibre optic cable is estimated to cost €10,000 per kilometre. Details of the network framework is shown in Figure 1 Cahir 30 40 Mallow Fermoy 45 Macroom 35 30 40 25 Cork Midleton 30 30 55 Bandon Figure 1 https://commons.wikimedia.org/wiki/File: 1 Page | 2 PDST Information Workshops – Autumn 2018 Instructions 1. Select any random town (vertex) as the starting point (e.g. Mallow). Figure 2. Cahir 30 40 Mallow Fermoy 45 Macroom 35 30 40 25 Cork 30 Midleton 30 55 Bandon Figure 2 2. Select the shortest distance (Edge) from this point to the next unvisited town. You can only visit a town (Vertex) once. Figure 3. Cahir 30 40 Mallow Fermoy 45 Macroom 35 30 40 25 Cork Midleton 30 30 55 Bandon Figure 3 Page | 3 PDST Information Workshops – Autumn 2018 3. Continue to select the shortest distances to the next unvisited town without completing a cycle (acyclic graph). Figure 4. Cahir 30 40 Mallow Fermoy 45 Macroom 35 30 40 25 Cork 30 Midleton 30 55 Bandon Figure 4 4. Continue to select shortest distances to the next unvisited town (vertex & edge) without completing a cycle. Figure 5. Cahir 30 40 Mallow Fermoy 45 Macroom 35 30 40 25 Cork 30 Midleton 30 55 Bandon Figure 5 Page | 4 PDST Information Workshops – Autumn 2018 5. Continue to select unvisited vertex, but graph must remain acyclic. Figure 6. Cahir 30 40 Mallow Fermoy 45 Macroom 35 30 40 25 Cork 30 Midleton 30 55 Bandon Figure 6 6. In figure 7, Cork to Mallow cannot be used as it would form a cycle as would selecting Bandon to Midleton. Therefore Bandon to Macroom (30 Km’s) is chosen as the shortest remaining distance. Cahir 30 40 Mallow Fermoy 45 Macroom 35 30 40 25 Cork 30 Midleton 30 55 Bandon Figure 7 Page | 5 PDST Information Workshops – Autumn 2018 In figure 8 the only remaining option is Fermoy to Cahir as any other route selected would form a cycle. This completed the graph. Cahir 30 40 Mallow Fermoy 45 Macroom 35 30 40 25 Cork 30 Midleton 30 55 Bandon Figure 8 Formula for the total number of edges is: #edges = (#vertex, -1) #edges = (7 -1) #edges = 6 Cahir 30 40 Mallow Fermoy Macroom 30 25 Cork 30 Midleton 30 Bandon Figure 9 Page | 6 PDST Information Workshops – Autumn 2018 The shortest length of fibre optic cabling & therefore the lowest cost to connect all towns in the project is €1,850,000. From To Distance (km) Cost Macroom Bandon 30 €300,000 Bandon Cork 30 €300,000 Cork Midleton 25 €250,000 Midleton Fermoy 30 €300,000 Fermoy Mallow 30 €300,000 Fermoy Cahir 40 €400,000 Total 185 €1,850,000 https://commons.wikimedia.org/wiki Page | 7 .