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October Meeting 25, 2005
Minimum Spanning Trees
When a graph is connected, we may ask how many edges we can delete before it stops being connected. Depending on the edges we remove, this may happen sooner or later. The slowest strategy is to remove edges until the graph becomes a tree. Here we study the somewhat more difcult problem of removing edges with a maximum total weight. The remaining graph is then a tree with minimum total weight. Applications that motivate this question can be found in life support systems modeled as graphs or networks, such as telephone, power supply, sewer systems, etc. etc. Free trees. An undirected graph (U, T ) is a free tree if it is connected and contains no cycle. We could impose a hierarchy by declaring any one vertex as the root and thus obtain a rooted tree. Here, we have no use for a hierarchical organization and exclusively deal with free trees. The
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Minimum spanning trees. For the remainder of this section, we assume that we also have a weighting function, w : E R. The weight of subgraph is then the total weight of its edges, w(T ) = eT w(e). A minimum spanning tree, or MST of G is a spanning tree that minimizes the weight. The denitions are illustrated in Figure 61 which shows a graph of solid edges with a minimum spanning tree of bold edges. A generic algorithm
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Figure 61: The bold edges form a spanning tree of weight 0.9 + 1.2 + 1.3 + 1.4 + 1.1 + 1.2 + 1.6 + 1.9 = 10.6.
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for constructing an MST grows a tree by adding more and more edges. Let A E be a subset of some MST of a connected graph (V, E). An edge uv E A is safe for A if A {uv} is also subset of some MST. The generic algorithm adds safe edges until it arrives at an MST. A = ; while (V, A) is not a spanning tree do nd a safe edge uv; A = A {uv} endwhile . As long as A is a proper subset of an MST there are safe edges. Specically, if (V, T ) is an MST and A T then all edges in T A are safe for A. The algorithm will therefore succeed in constructing an MST. The only thing that is not yet clear is how to nd safe edges quickly.
Figure 60: Adding the edge dg to the tree creates a single cycle with vertices d, g, h, f, e, a.
number of edges of a free tree is always one less than the number of vertices. Whenever we add a new edge (connecting two old vertices) we create exactly one cycle. This cycle can be destroyed by deleting any one of its edges, and we get a new free tree, as in Figure 60. Let (V, E) be a connected and undirected graph. A subgraph is another graph (U, T ) with U V and T E. It is a spanning tree if it is a free tree with U = V .
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Cuts. To develop a mechanism for identifying safe edges, we dene a cut, which is a partition of the vertex set into two complementary sets, V = W (V W ). It is crossed by an edge uv E if u W and v V W , and it respects an edge set A if A contains no crossing edge. The denitions are illustrated in Figure 62.
component. The complementary set V W contains all other vertices, and crossing edges connect the component with its complement. Prims algorithm. Prims algorithm chooses safe edges to grow the tree as a single component from an arbitrary rst vertex s. Similar to Dijkstras algorithm, the vertices that do not yet belong to the tree are stored in a priority queue. For each vertex i outside the tree, we dene its priority V [i].d equal to the minimum weight of any edge that connects i to a vertex in the tree. If there is no such edge then V [i].d = . In addition to the priority, we store the index of the other endpoint of the minimum weight edge. We rst initialize this information. V [s].d = 0; V [s]. = 1; I NSERT(s); forall vertices i = s do V [i].d = ; I NSERT(i) endfor . The main algorithm expands the tree by one edge at a time. It uses marks to distinguish vertices in the tree from vertices outside the tree. while priority queue is non-empty do i = E XTRACT M IN ; mark i; forall neighbors j of i do if j is unmarked and w(ij) < V [j].d then V [j].d = w(ij); V [j]. = i endif endfor endwhile . After running the algorithm, the MST can be recovered from -elds the of the vertices. The algorithm together with its initialization phase performs n = card V insertions into the priority queue, n extractmin operations, and at most m = card E decreasekey operations. Using the Fibonacci heap implementation we get a running time of O(n log n + m), which is the same as for constructing the shortest-path tree with Dijkstras algorithm. Kruskals algorithm. Kruskals algorithm is another implementation of the generic algorithm. It adds edges in a sequence of non-decreasing weight. At any moment, the chosen edges form a collection of trees. These trees merge to form larger and fewer trees, until they eventually combine into a single tree. The algorithm uses a priority queue for the edges and a set system for the vertices. In this context, the term system is just another word for set,
Figure 62: The vertices inside and outside the shaded regions form a cut that respects the collection of solid edges. The dotted edges cross the cut.
C UT L EMMA . Let A be subset of an MST and consider a cut W (V W ) that respects A. If uv is a crossing edge with minimum weight then uv is safe for A. P ROOF. Consider a minimum spanning tree (V, T ) with A T . If uv T then we are done. Otherwise, let T = T {uv}. Because T is a tree, there is a unique path from u to v in T . We have u W and v V W , so the path switches at least once between the two sets. Suppose it switches along xy, as in Figure 63. Edge xy
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Figure 63: Adding uv creates a cycle and deleting xy destroys the cycle.
crosses the cut, and since A contains no crossing edges we have xy A. Because uv has minimum weight among crossing edges we have w(uv) w(xy). Dene T = T {xy}. Then (V, T ) is a spanning tree and because w(T ) = w(T ) w(xy) + w(uv) w(T ) it is a minimum spanning tree. The claim follows because A {uv} T . A typical application of the Cut Lemma takes a component of (V, A) and denes W as the set of vertices of that
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but we will use it exclusively for sets whose elements are themselves sets. Implementations of the set system will be discussed in the next lecture. Initially, A = , the priority queue contains all edges, and the system contains a singleton set for each vertex, C = {{u} | u V }. The algorithm nds an edge with minimum weight that connects two components dened by A. We set W equal to the vertex set of one component and use the Cut Lemma to show that this edge is safe for A. The edge is added to A and the process is repeated. The algorithm halts w...