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generalized pigeonhole principle permutations and

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Unformatted text preview: xists a constant L and a natural number n0 such that L |g(n)| <= |f(n)| holds for all n >= n0. In other words, f(n) = Ω(g(n)) if and only if g(n) = O(f(n)). 27 Big Θ We define f(n) = Θ(g(n)) if and only if there exist constants L and U and a natural number n0 such that L|g(n)| <= |f(n)| <= U|g(n)| holds for all n >= n0. In other words, f(n) = Θ(g(n)) if and only if f(n) = Ω(g(n)) and f(n) = O(g(n)). 28 Counting You need to know - the basic counting principles (product rule, sum rule,...) - (generalized) pigeonhole principle - permutations and combinations - binomial coefficients - binomial theorem - counting with repeated elements 29 Pigeonhole Principle In any cocktail party n >=2 people, there must be at least two people who have the same number of friends (assuming that the friends relation is symmetric and irreflexive). The number of friends of each person ranges between 0 and n-1. Case 1: Everyone has at least one friend. If everyone has at least one friend, then each person has between 1 to n-1 friends. Since we have n people, and just n-1 different values, there must be two partygoers that have the same number of friends by the pigeonhole principle. Case 2: Someone has no friends. If someone lacks any friends, then that person is a stranger to all other guests. Because friend is symmetric, the highest value anyone else could have is n - 2, that is, everyone has between 0 to n – 2 friends. Since we have n people, and just n-1 different values, there must be two partygoers that have the same number of friends by the pigeonhole principle. Solving Recurrences You need to be able to solve recurrences by - solving characteristic equations (for homogeneous linear recurrences of degree 2) - by inspecting, guessing, and verifying a solution - by applying the master theorem 31 Relations You need to know - the basic properties of relations (reflexive, symmetric, antisymmetric, transitive, ...) - how to show that a relation is an equivalence relation - the congruence relation mod m - how to show that a relation is a partial order - what an order lattice is 32 Formal Languages You need to - be familiar with the Chomsky Hierarchy - be able to determine the language associated with a grammar - know the characterization of regular languages as languages accepted by finite state automata or languages described by regular expressions - be familiar with finite state machines and finite state automata 33 Hints Read the textbook and the class notes. Study old exams, quizzes, homeworks. Drill using odd numbered exercises. Scan through review sections of the textbook. Get enough sleep!! 34...
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