Today Notes: Logic/Proofs. Stable Marriage Statement? “3 = 4+1”? Yes. 3? No Logic. P = ⇒ Q ≡ Q = ⇒ P False. Stable Marriage: Quantifiers. ¬∀ x , Q ( x ) ≡ ∃ x , ¬ Q ( x ) . Review for Final.. Improvement Lemma. Optimality/Pessimality. Proofs. Rao’s Cheat Sheet. An an instance, with a stable instance where man 1 and woman 1 Direct. Square of even number is even. are optimal. A million slides. Contrapositive. Square of odd number is odd. There is only one stable marriage! False. Induction. Statement: ∀ n , P ( n ) . Base: P ( 0 ) . Step: P ( k ) = ⇒ P ( k + 1 ) . Simple: ∑ i i = n ( n + 1 ) / 2 . Strengthen: See midterm 1. Graphs Notes: Modular Arithmetic. Notes: Modular + Polynomials Polynomials: a d x d + ··· a + 0 mod p . Graphs: ∑ v d ( v ) = 2 | E | . Prop 1: ≤ d roots. Factoring. Prop 2: d + 1 points gives unique polynomial. Eulerian: All degrees even. Euclid: gcd ( x , y ) = gcd ( x , y − x ) = gcd ( x , y − kx ) Coloring: Degree d graph can be colored with d + 1 colors. Lagrange: 1 at a point, 0 elsewhere. Degree d polynomial suffices. Extended: ax + by = gcd ( x , y ) . Algorithm: Equations: d + 1 unknowns, d + 1 equations. Start with ( 1 ) x +( 0 ) y = x and ( 0 ) x + 1 y = y . Remove vertex. Color remaining. Add vertex. Available color! Modulo prime: inverses gives hope. Can reduce right hand side. By factor of two in two steps. Planar Graph: Euler Formula? Linearly independent from uniqueness. Fermats: a p − 1 = 1 mod p . Proof: Base. Tree. e = v − 1, f = 1. Applications. Proof: Multiplying by a is bijection on { 1 ,... p } . Step: v + f = e + 2. Add edge, adds face. Secret Sharing: Property 2. Large prime for secrecy. RSA: ( N = pq , e ) where e = d − 1 mod ( p − 1 )( q − 1 ) . Max Degree: remove faces from equation using face-edge Erasure Coding: Property 2. Smaller prime for efficiency. Works because: a ( p − 1 )( q − 1 ) = 1 ( mod 1 ) . incidences. Error Correction: Property 2. Public Key Encryption/Signature Scheme. 2 e ≥ 3 f = ⇒ v + 2 e / 3 ≥ e + 2 = ⇒ e ≤ 3 v − 6. Argument that n + 2 k is enough with k errros. Encrypt: x e mod N . Sign: x d mod N . 6-color theorem. 5-color is a recoloring argument. Unique degree n − 1 polynomial that fits at least n + k points. Avoid Attack: add randomness to x . Graphs: Why? � n � Complete: K n . How many edges? . Welsh-Berlekamp: 2 Tree: How many edges? n − 1. No cycles. Linear System from Q ( x ) = P ( x ) E ( x ) with error poly, E ( x ) . Hypercube: d -dimensional. Degree? d . Edges: d 2 d / 2. Divide Q ( x ) by E ( x ) to get P ( x ) .
Notes: Countability/Computability Notes: Counting First there was logic... A statement is a true or false. Countability/Computability. Statements? Countable: bijection with natural numbers or a listing. Counting. 3 = 4 − 1 ? Statement! Countable infinities: pairs of countable sets, rationals... First rule of counting. 3 = 5 ? Statement! all forms of pairs: interleave elements of uncountable sets. Make elt of set with sequence of choices. Multiply. 3 ? Not a statement! Uncountable infinities: real numbers, power set of integers. Second rule of counting. n = 3 ? Not a statement...but a predicate. Diagonalization: Assume list, construct element not on list. Divide with order to get number of unordered. Sometimes. Predicate: Statement with free variable(s). Uncomputability. Stars and Bars. Use bars to group stars into different groups. Example: x = 3 Given a value for x , becomes a statement. Halt: With halt can construct diagonalizer Turing. Inclusion/Exclusion. Predicate? and no Turing = ⇒ no halt. n > 3 ? Predicate: P ( n ) ! Number in union is sum minus the intersection. Concepts: Program can call subroutine! Combinatorial Arguments: Bijection means same number. x = y ? Predicate: P ( x , y ) ! With subroutine can write program. 2 n = ∑ i � n � . x + y ? No. An expression, not a statement. Reduce from Halt: i Quantifiers: Transform instance of halt to instance of problem X. Note: for sample spaces, usually first rule of counting is easier. ( ∀ x ) P ( x ) . For every x , P ( x ) is true. Concept: Programs are text. Can change text. for events, may need second or others. ( ∃ x ) P ( x ) . There exists an x , where P ( x ) is true. Computability/Enumerability. Can run programs and see! ( ∀ n ∈ N ) , n 2 ≥ n . Can enumerate halting programs. ( ∀ x ∈ R )( ∃ y ∈ R ) y > x . Connecting Statements ..and then proofs... ...jumping forward.. Direct: P = ⇒ Q ⇒ a 2 is even. Example: a is even = Approach: What is even? a = 2 k a 2 = 4 k 2 . A ∧ B , A ∨ B , ¬ A . What is even? Contradiction in induction: a 2 = 2 ( 2 k 2 ) You got this! contradict place where induction step doesn’t hold. Integers closed under multiplication! Propositional Expressions and Logical Equivalence a 2 is even. Well Ordering Principle. Stable Marriage: ( A = ⇒ B ) ≡ ( ¬ A ∨ B ) Contrapositive: P = ⇒ Q or ¬ Q = ⇒ ¬ P . first day where women does not improve. ¬ ( A ∨ B ) ≡ ( ¬ A ∧¬ B ) Example: a 2 is odd = ⇒ a is odd. first day where any man rejected by optimal women. ⇒ a 2 is even. Contrapositive: a is even = Proofs: truth table or manipulation of known formulas. Do not exist. Contradiction: P ( ∀ x )( P ( x ) ∧ Q ( x )) ≡ ( ∀ x ) P ( x ) ∧ ( ∀ x ) Q ( x ) ¬ P = ⇒ false ¬ P = ⇒ R ∧¬ R Useful for prove something does not exist: √ Example: rational representation of 2 does not exist. Example: finite set of primes does not exist. Example: rogue couple does not exist.
...and then induction... Stable Marriage: a study in definitions and WOP . TMA. Traditional Marriage Algorithm: Each Day: P ( 0 ) ∧ (( ∀ n )( P ( n ) = ⇒ P ( n + 1 ) ≡ ( ∀ n ∈ N ) P ( n ) . n -men, n -women. All men propose to favorite woman who has not yet rejected Thm: For all n ≥ 1, 8 | 3 2 n − 1. Each person has completely ordered preference list him. contains every person of opposite gender. Every woman rejects all but best men who proposes. Induction on n . Base: 8 | 3 2 − 1. Pairing. Useful Algorithmic Definitions: Set of pairs ( m i , w j ) containing all people exactly once. Man crosses off woman who rejected him. Induction Hypothesis: Assume P ( n ) : True for some n . How many pairs? n . Woman’s current proposer is “on string.” (3 2 n − 1 = 8 d ) People in pair are partners in pairing. “Propose and Reject.” : Either men propose or women. But not both. Induction Step: Prove P ( n + 1 ) Rogue Couple in a pairing. Traditional propose and reject where men propose. 3 2 n + 2 − 1 = 9 ( 3 2 n ) − 1 (by induction hypothesis) A m j and w k who like each other more than their partners Key Property: Improvement Lemma: = 9 ( 8 d + 1 ) − 1 Stable Pairing. Every day, if man on string for woman, = 72 d + 8 Pairing with no rogue couples. = ⇒ any future man on string is better. = 8 ( 9 d + 1 ) Does stable pairing exist? Stability: No rogue couple. Divisible by 8. rogue couple (M,W) No, for roommates problem. = ⇒ M proposed to W = ⇒ W ended up with someone she liked better than M . Not rogue couple! Optimality/Pessimal ...Graphs... Graph Algorithm: Eulerian Tour Optimal partner if best partner in any stable pairing. Not necessarily first in list. G = ( V , E ) Possibly no stable pairing with that partner. V - set of vertices. Thm: Every connected graph where every vertex has even degree E ⊆ V × V - set of edges. Man-optimal pairing is pairing where every man gets optimal partner. has an Eulerian Tour; a tour which visits every edge exactly once. Directed: ordered pair of vertices. Thm: TMA produces male optimal pairing, S . Algorithm: First man M to lose optimal partner. Adjacent, Incident, Degree. Take a walk using each edge at most once. Better partner W for M . In-degree, Out-degree. Property: return to starting point. Different stable pairing T . Proof Idea: Even degree. Thm: Sum of degrees is 2 | E | . TMA: M asked W first! There is M ′ who bumps M in TMA. Edge is incident to 2 vertices. Recurse on connected components. Degree of vertices is total incidences. W prefers M ′ . Put together. M ′ likes W at least as much as optimal partner. Property: walk visits every component. Pair of Vertices are Connected: Proof Idea: Original graph connected. Not first bump. If there is a path between them. M ′ and W is rogue couple in T . Connected Component: maximal set of connected vertices. Thm: woman pessimal. Connected Graph: one connected component. Man optimal = ⇒ Woman pessimal. Woman optimal = ⇒ Man pessimal.
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