27+Slides--Trees%2C+Intro+to+Automata

27+Slides--Trees%2C+Intro+to+Automata - CS103 HO#27...

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HO#27 Slides--Trees, Intro to Automata 4/20/11 1 CS103 4/20/11 Mathematical Foundations of Computing Midterm Thursday Night (aka Tomorrow!) 7 – 9 pm Cubberley Auditorium Open book, open notes No computers or mobile devices An undirected graph is an ordered pair (V, E) where (i) V is a non-empty set of vertices , and (ii) E is an edge set consisting of unordered pairs of distinct vertices. By convention, we write edges as (u, v), where u, v V and u v. (u, v) and (v, u) are considered to be the same edge. Graphs 1 2 3 4 5 6 7 Unless we say otherwise, we will be talking about undirected graphs. Reference: Introduction to Algorithms, 2 nd Ed. , Thomas Cormen, et al. Graph Terminology 1 2 3 4 5 6 7 If (u, v) is an edge, we say (u, v) is incident on vertices u and v, that u is adjacent to v, and that v is adjacent to u. The degree of a vertex is the number of edges incident on it. A path of length k from a vertex u to a vertex u' in a tree G = (V, E) is a sequence of vertices (v 0 , v 1 , v 2 , . .., v k ) from V such that u = v 0 , u' = v k and (v i-1 , v i ) E for i = 1, 2, . .., k. A path is simple if all its vertices are distinct. A path (v 0 , v 1 , v 2 , . .., v k ) forms a (simple) cycle if k 3, v 0 = v k , and v 1 , v 2 , . .., v k are distinct. A graph with no cycles is acyclic . A graph is connected if there is a path between every pair of vertices. Trees A tree is an acyclic, connected, graph in which one vertex has been designated as the root . 1 2 3 4 5 6 7 We normally draw the root of a tree at the top, and we refer to the vertices as nodes . Trees A tree is an acyclic, connected, graph in which one vertex has been designated as the root . 1 2 3 4 5 6 7 Consider a node x in a tree with root r. Any node on the unique path from r to x is an ancestor of x. If y is an ancestor of x, x is a descendant of y. If the last edge on the path from r to x is (y, x) , then y is the parent of x and x is the child of y. Nodes with the same parent are siblings . A node with no children is a leaf , and a node that is not a leaf is an interior node . The subtree rooted at x is the tree consisting of x, its descendants, and the edges connecting them. Properties of Trees Let G = (V, E) be an undirected graph. The following statements are equivalent: 1. G is a tree. 2. Any two vertices of G are connected by a unique simple path. 3. G is connected, but if any edge in E is removed, the result is not connected. 4. G is connected, and |E| = |V| - 1. 5. G is acyclic, and |E| = |V| - 1. 6. G is acyclic, but if an edge is added to E, the resulting graph has a cycle. 1
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27+Slides--Trees%2C+Intro+to+Automata - CS103 HO#27...

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