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

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Integers can be expressed uniquely in any base. Theorem: Let b,Z(b>1). Then if n,N, then there is a unique expression such that n = a k b k + a k-1 b k-1 +…+a 1 b+a 0 , where {a i },k,N 0 , a k -0, and 0;a i 1, is really easy. Just group k bits together and convert to the base 2 k symbol. • Base 10 to any base 2 k is a pain. • Base 2 k to base 10 is also a pain. 56
Algorithm: Addition of integers procedure add(a, b: integers) (a n-1 a n-2 …a 1 a 0 ) 2 := base_2_expansion(a) (b n-1 b n-2 …b 1 b 0 ) 2 := base_2_expansion(b) c := 0 for j := 0 to n-1 d := 7(a j +b j +c)/28 s j := a j +b j +c – 2d c := d s n := c {the binary expansion of the sum is (s k-1 s k-2 …s 1 s 0 ) 2 } Questions: • What is the complexity of this algorithm? • Is this the fastest way to compute the sum? 57 Algorithm: Mutiplication of integers procedure multiply(a, b: integers) (a n-1 a n-2 …a 1 a 0 ) 2 := base_2_expansion(a) (b n-1 b n-2 …b 1 b 0 ) 2 := base_2_expansion(b) for j := 0 to n-1 if b j = 1 then c j := a shifted j places else c j := 0 {c 0 ,c 1 ,…,c n-1 are the partial products} p := 0 for j := 0 to n-1 p := p + c j {p is the value of ab} Examples: • (10) 2 $(11) 2 = (110) 2 . Note that there are more bits than the original integers. • (11) 2 $(11) 2 = (1001) 2 . Twice as many binary digits! 58
Page 1 and 2: Discrete Mathematics University of
Page 3 and 4: Definition: A proposition is a stat
Page 5 and 6: We can compound logical operators t
Page 7 and 8: Theorem: p" (q!r) ' (p"q)!(p"r). Pr
Page 9 and 10: Propositional logic is pretty limit
Page 11 and 12: Rules of Inference are used instead
Page 13 and 14: Sets, Functions, Sequences, and Sum
Page 15 and 16: Definition: n sets A i are disjoint
Page 17 and 18: Definition: The composition of n fu
Page 19 and 20: Some other common summations with c
Page 21 and 22: Notation: Timing, as a function of
Page 23 and 24: ! $ procedure binary_search(x, " a
Page 25 and 26: Definition: If a,b,Z and a-0, we sa
Page 27: Theorem: There are infinitely many
Page 31 and 32: Euclidean Algorithm: Compute gcd(a,
Page 33 and 34: Sun Tzu’s Puzzle: The a k ,{2, 1,
Page 35 and 36: • Sometimes P(k) is for a (possib
Page 37 and 38: Example: Every postage amount < $.1
Page 39 and 40: Definition: A recursive algorithm s
Page 41 and 42: 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 and 94:
Definition: A coloring of a simple
Page 95 and 96:
Definition: A m-ary tree is a roote
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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