Graphs and Trees This handout: Trees Minimum Spanning Tree Problem

Terminology of Graphs: Cycles, Connectivity and Trees A path that begins and ends at the same node is called a cycle. Example: Two nodes are connected if there is a path between them. A graph is connected if every pair of its nodes is connected. A graph is acyclic if it doesn’t have any cycle. A graph is called a tree if it is connected and acyclic.

Examples/Applications of Trees Family tree Evolutionary tree Possibility tree Prime number factorization tree Spanning tree of a telecom network (considered later in this handout) Hydrocarbon molecules

Properties of Trees Definition: Let T be a tree. If T has at least 3 nodes, then A node of degree 1 in T is called a leaf (or a terminal node). A node of degree greater than 1 is called an internal node. Lemma: Every tree with more than one node has at least one leaf. Theorem: For any positive integer n, any tree with n nodes has n-1 edges. Proof by induction (blackboard).

Properties of Trees Lemma: If G is any connected graph, C is a cycle in G, and one of the edges of C is removed from G, then the graph that remains is still connected. Theorem: For any positive integer n, if G is a connected graph with n vertices and n-1 edges, then G is a tree. if G is an acyclic graph with n vertices and n-1 edges, then G is a tree. Proofs on blackboard.

Rooted Trees Definition: A rooted tree is a tree in which one vertex is distinguished from the others and is called the root. The level of a vertex is the number of edges along the unique path between it and the root. The height of a rooted tree is the maximum level to any vertex of the tree. The children of a vertex v are those vertices that are adjacent to v and one level farther away from the root than v. When every vertex in a rooted tree has at most two children, the tree is called a binary tree. r a b c d

Minimum Spanning Tree Problem Given: Graph G=(V, E), |V|=n Cost function c: E  R . Goal: Find a minimum-cost spanning tree for V i.e., find a subset of arcs E*  E which connects any two nodes of V with minimum possible cost. Example: Min. span. tree: G*=(V,E*) G=(V,E) 2 3 4 5 7 8 2 3 4 5 7 8 e b c d a e b c d a Red bold arcs are in E*

Algorithm for solving the Minimum Spanning Tree Problem Initialization: Select any node arbitrarily, connect to its nearest node. Repeat Identify the unconnected node which is closest to a connected node Connect these two nodes Until all nodes are connected Note: Ties for the closest node are broken arbitrarily.

The algorithm applied to our example Initialization: Select node a to start. Its closest node is node b. Connect nodes a and b. Iteration 1: There are two unconnected node closest to a connected node: nodes c and d (both are 3 units far from node b). Break the tie arbitrarily by connecting node c to node b. 2 3 4 5 7 8 e b c d a Red bold arcs are in E*; thin arcs represent potential links. 2 3 4 5 7 8 e b c d a

The algorithm applied to our example Iteration 2: The unconnected node closest to a connected node is node d (3 far from node b). Connect nodes b and d. Iteration 3: The only unconnected node left is node e. Its closest connected node is node c (distance between c and e is 4). Connect node e to node c. All nodes are connected. The bold arcs give a min. spanning tree. 2 3 4 5 7 8 e b c d a 2 3 4 5 7 8 e b c d a