Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

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Theorem: An undirected graph T = (V,E) is a tree if and only if there is a unique simple path between any two of its distinct vertices. Proof: 1. Assume T is a tree, so it has no simple circuits. Since T is connected, for all distinct u,v,V, there is exactly one simple path between u and v. Otherwise, there is another simple path. Combining the two simple paths is a circuit, which is a contradiction that T is a tree. 2. Assume that there is a unique simple path between any two distinct vertices u,v,V. The T is connected. T has no simple circuits since then there would be two simple paths between u and v (thus forming a crcuit), which is a contradiction. Definition: A rooted tree is a tree with one vertex designated as the root and every edge is directed away from the root. Note: Any tree can become a rooted tree by picking the right vertex as the root. 187 Terminology/Definitions: Let T = (V,E) be a rooted tree. Then • If v,V is not a root of, the parent w,V of v is a vertex with an edge directed at v and v is a child of u. • If v i ,V are children of the same u,V, they are siblings. • The ancestors v i ,V of u,V are any vertices in V except the root which are in the path from the root to u. • The descendents v i ,V of u,V are all vertices with u as an ancestor. • A leaf v,V is a vertex with no children. • An internal vertice v,V has children. • A subtree is the subgraph formed from a,V and all of its descendents and the edges incident to these descendents. • The height of a rooted tree T, denoted h(T), is the maximum number of levels (or vertices). • A balanced rooted tree T has all of its leaves at h(T) or h(T)-1. 188
Definition: A m-ary tree is a rooted tree such that every internal vertex has no more than m children. A full m-ary tree is a rooted tree such that every internal vertex has exactly m children. If m = 2, it is a (full) binary tree. Definition: An ordered rooted tree is a rooted tree with an ordering applied to the children of all of the children of the root and the internal vertices. Examples: • Management charts • Directory based file or memory systems Theorem: A tree with n vertices has n41 edges. The proof is by mathematical induction. Theorem: A full m-ary tree with i internal vertices contains n = mi+1 vertices. Proof: There are mi children plus the root. 189 Theorem: A full m-ary tree with • n vertices has i = (n41)/m internal vertices and q = [(m41)n+1]/m leaves. • i internal vertices has n = m+1 vertices and q = (m41)i + 1 leaves. • q leaves has n = (mq41) / (m41) vertices and i = (q41) / (m41) internal vertices. Theorem: There are at most m h leaves in a m-ary tree of height h. The proof uses mathematical induction. Corollary: If an m-ary tree of height h has q leaves, then h < 9log m q:. For a full m-ary and balnced m-ary tree, h = 9log m q:. 190
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Discrete Mathematics University of
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Definition: A proposition is a stat
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We can compound logical operators t
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Theorem: p" (q!r) ' (p"q)!(p"r). Pr
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Propositional logic is pretty limit
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Rules of Inference are used instead
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Sets, Functions, Sequences, and Sum
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Definition: n sets A i are disjoint
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Definition: The composition of n fu
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Some other common summations with c
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Notation: Timing, as a function of
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! $ procedure binary_search(x, " a
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Definition: If a,b,Z and a-0, we sa
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Theorem: There are infinitely many
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Algorithm: Addition of integers pro
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Euclidean Algorithm: Compute gcd(a,
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Sun Tzu’s Puzzle: The a k ,{2, 1,
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• Sometimes P(k) is for a (possib
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Example: Every postage amount < $.1
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Definition: A recursive algorithm s
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Merge sort is a balanced binary tre
Page 43 and 44: Examples: • Consider 3 students i
Page 45 and 46: Theorem: The number of r-combinatio
Page 47 and 48: If we allow repetitions in the perm
Page 49 and 50: Algorithm: Generating the next r-co
Page 51 and 52: Theorem: Let E and F be events in a
Page 53 and 54: Note: For bit strings of length 4,
Page 55 and 56: Theorem: If X i , 1;i;n, are random
Page 57 and 58: Definition: The random variables X
Page 59 and 60: Example: We have 2 boxes. The first
Page 61 and 62: Let E be the event that an incoming
Page 63 and 64: Advanced Counting Principles Defini
Page 65 and 66: Examples: Typical ones include •
Page 67 and 68: Now comes the second case for n = 2
Page 69 and 70: Theorem: Assume {b i },{c i },R. Su
Page 71 and 72: Example (funny integer multiplicati
Page 73 and 74: • In step 4, we have to take care
Page 75 and 76: Definition: The extended binomial c
Page 77 and 78: Example: a n = 8a n41 + 10 n41 with
Page 79 and 80: Relations Definition: A relation on
Page 81 and 82: Representation: The relation R from
Page 83 and 84: ! # 0 0 0 0 # • M R4 = # 1 0 0 0
Page 85 and 86: Definition: A partition of a set S
Page 87 and 88: Examples of Simple Graphs: • A co
Page 89 and 90: Examples: v 1 v 2 • and v 4 v 3 v
Page 91 and 92: Theorem: Let G = (V,E) be a graph w
Page 93: Definition: A coloring of a simple
Page 97 and 98: Definition: A decision tree is a ro
Page 99 and 100: Note: Game trees are another highly
Page 101 and 102: Definition (Postorder Traversal): L
Page 103 and 104: Definition: Let G = (V,E) be a simp
Page 105 and 106: Definition: A minimum spanning tree
Page 107 and 108: Examples: Most circuits are of the
Page 109: Definition: An implicant is sum ter
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Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

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