.. This file is part of the OpenDSA eTextbook project. See .. http://opendsa.org for more details. .. Copyright (c) 2012-2020 by the OpenDSA Project Contributors, and .. distributed under an MIT open source license. .. avmetadata:: :title: Minimal Cost Spanning Trees :author: Cliff Shaffer :institution: Virginia Tech :satisfies: Prim's algorithm; MCST :requires: graph shortest path :topic: Graphs :keyword: Graph; Minimal-Cost Spanning Tree :naturallanguage: en :programminglanguage: N/A :description: Introduces the concept of minimal-cost spanning trees, and presents Prim's algorithm for constructing them. Minimal Cost Spanning Trees =========================== Minimal Cost Spanning Trees --------------------------- The :term:`minimal-cost spanning tree` (MCST) problem takes as input a connected, undirected graph :math:`\mathbf{G}`, where each edge has a distance or weight measure attached. The MCST is the graph containing the vertices of :math:`\mathbf{G}` along with the subset of :math:`\mathbf{G}` 's edges that (1) has minimum total cost as measured by summing the values for all of the edges in the subset, and (2) keeps the vertices connected. Applications where a solution to this problem is useful include soldering the shortest set of wires needed to connect a set of terminals on a circuit board, and connecting a set of cities by telephone lines in such a way as to require the least amount of cable. The MCST contains no cycles. If a proposed MCST did have a cycle, a cheaper MCST could be had by removing any one of the edges in the cycle. Thus, the MCST is a free tree with :math:`|\mathbf{V}| - 1` edges. The name "minimum-cost spanning tree" comes from the fact that the required set of edges forms a tree, it spans the vertices (i.e., it connects them together), and it has minimum cost. Figure :num:`Figure #MCSTdgm` shows the MCST for an example graph. .. _MCSTdgm: .. inlineav:: MCSTCON dgm :links: AV/Graph/MCSTCON.css :scripts: AV/Graph/MCSTCON.js :align: justify :keyword: Graphs; Minimal-Cost Spanning Trees A graph and its MCST. All edges appear in the original graph. Those edges drawn with heavy lines indicate the subset making up the MCST. Note that edge :math:`(C, F)` could be replaced with edge :math:`(D, F)` to form a different MCST with equal cost. .. TODO:: :type: Slideshow Replace the previous diagram with a slideshow illustrating the concept of MCST. Prim's Algorithm ~~~~~~~~~~~~~~~~ The first of our two algorithms for finding MCSTs is commonly referred to as :term:`Prim's algorithm`. Prim's algorithm is very simple. Start with any Vertex :math:`N` in the graph, setting the MCST to be :math:`N` initially. Pick the least-cost edge connected to :math:`N`. This edge connects :math:`N` to another vertex; call this :math:`M`. Add Vertex :math:`M` and Edge :math:`(N, M)` to the MCST. Next, pick the least-cost edge coming from either :math:`N` or :math:`M` to any other vertex in the graph. Add this edge and the new vertex it reaches to the MCST. This process continues, at each step expanding the MCST by selecting the least-cost edge from a vertex currently in the MCST to a vertex not currently in the MCST. Prim's algorithm is quite similar to Dijkstra's algorithm for finding the single-source shortest paths. The primary difference is that we are seeking not the next closest vertex to the start vertex, but rather the next closest vertex to any vertex currently in the MCST. Thus we replace the lines:: if (D[w] > (D[v] + G.weight(v, w))) D[w] = D[v] + G.weight(v, w); in Djikstra's algorithm with the lines:: if (D[w] > G.weight(v, w)) D[w] = G.weight(v, w); in Prim's algorithm. The following code shows an implementation for Prim's algorithm that searches the distance matrix for the next closest vertex. .. codeinclude:: Graphs/Prim :tag: Prims For each vertex :math:`I`, when :math:`I` is processed by Prim's algorithm, an edge going to :math:`I` is added to the MCST that we are building. Array ``V[I]`` stores the previously visited vertex that is closest to Vertex `I`. This information lets us know which edge goes into the MCST when Vertex :math:`I` is processed. The implementation above also contains calls to ``AddEdgetoMST`` to indicate which edges are actually added to the MCST. .. inlineav:: primCON ss :long_name: Prim's Minimum Cost Spanning Tree Algorithm Slideshow :links: AV/Graph/primCON.css :scripts: AV/Graph/primCON.js :output: show :keyword: Graphs; Minimal-Cost Spanning Trees; Prim's Algorithm Prim's Algorithm Alternative Implementation ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Alternatively, we can implement Prim's algorithm using a :term:`priority queue` to find the next closest vertex, as shown next. As with the priority queue version of Dijkstra's algorithm, the :term:`heap` stores ``DijkElem`` objects. .. codeinclude:: Graphs/PrimPQ :tag: PrimsPQ .. TODO:: :type: Slideshow Implement a slideshow demonstrating the Priority Queue version of Prim's algorithm Prim's algorithm is an example of a greedy algorithm. At each step in the ``for`` loop, we select the least-cost edge that connects some marked vertex to some unmarked vertex. The algorithm does not otherwise check that the MCST really should include this least-cost edge. This leads to an important question: Does Prim's algorithm work correctly? Clearly it generates a spanning tree (because each pass through the ``for`` loop adds one edge and one unmarked vertex to the spanning tree until all vertices have been added), but does this tree have minimum cost? **Theorem:** Prim's algorithm produces a minimum-cost spanning tree. **Proof:** We will use a proof by contradiction. Let :math:`\mathbf{G} = (\mathbf{V}, \mathbf{E})` be a graph for which Prim's algorithm does *not* generate an MCST. Define an ordering on the vertices according to the order in which they were added by Prim's algorithm to the MCST: :math:`v_0, v_1, ..., v_{n-1}`. Let edge :math:`e_i` connect :math:`(v_x, v_i)` for some :math:`x < i` and :math:`i \leq 1`. Let :math:`e_j` be the lowest numbered (first) edge added by Prim's algorithm such that the set of edges selected so far *cannot* be extended to form an MCST for :math:`\mathbf{G}`. In other words, :math:`e_j` is the first edge where Prim's algorithm "went wrong." Let :math:`\mathbf{T}` be the "true" MCST. Call :math:`v_p (p