Advanced Counting Techniques CSC-2259 Discrete Structures Konstantin Busch - LSU1

Recurrence Relations Konstantin Busch - LSU2 Sequence Recurrence relation: For any

Konstantin Busch - LSU3 Example: Solutions to recurrence relation: Recurrence relation

Konstantin Busch - LSU4 $10,000 bank deposit %11 interest :amount after years Example:

Konstantin Busch - LSU5 Fibonacci sequence Example:

Konstantin Busch - LSU6 Towers of HanoiExample: bar1bar2bar3 Goal: move all discs to bar3 Rule: not allowed to put larger discs on top of smaller discs discs

Konstantin Busch - LSU7 bar1bar2bar3 move recursively discs to bar2Step 1:

Konstantin Busch - LSU8 bar1bar2bar3 move largest disc to bar3Step 2:

Konstantin Busch - LSU9 bar1bar2bar3 move recursively discs to bar3Step 3:

Konstantin Busch - LSU10 :total disc moves 2 recursive calls with discs (steps 1&3) movement of largest disc (step 2) one move for one disc

Konstantin Busch - LSU11

Solving Linear Recurrence Relations Konstantin Busch - LSU12 Linear homogeneous recurrence relation of degree : Constant coefficients:

Konstantin Busch - LSU13 A sequence (solution) satisfying the relation is uniquely determined by the initial values: (these are different constants than the coefficients)

Konstantin Busch - LSU14 if an only if Solution to recurrence relation: divide both sides with characteristic equation

Konstantin Busch - LSU15 characteristic equation: factorize with roots characteristic roots: Multiple possible solutions:

Konstantin Busch - LSU16 The solutions may not satisfy the initial conditions

Konstantin Busch - LSU17 Theorem:Recurrence relation of degree 2 has unique solution where are solutions to the characteristic equation, and are constants that depend on initial conditions

Konstantin Busch - LSU18 Characteristic Equation Roots: Proof:

Konstantin Busch - LSU19 First compute from initial conditions

Konstantin Busch - LSU20

Konstantin Busch - LSU21 Prove by induction that Basis cases: true for the specific choices of

Konstantin Busch - LSU22 Inductive hypothesis: for all Inductive step: for prove that assume that

Konstantin Busch - LSU23 By inductive hypothesis:

Konstantin Busch - LSU24 By recurrence relation definition Inductive hypothesis End of Proof

Konstantin Busch - LSU25 Fibonacci sequence Example: Has solution: Characteristic roots:

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Konstantin Busch - LSU28 Degree recurrence relation: has unique solution where are solutions to the characteristic equation, and are constants that depend on initial conditions

Konstantin Busch - LSU29 Example: Solution: Characteristic equation: Roots:

Konstantin Busch - LSU30 Final solution:

Recurrence Relations for Divide and Conquer Algorithms Konstantin Busch - LSU31 Typical divide and conquer algorithm: Input of size Divide into sub-problems each of size Combine sub-problems with cost

Konstantin Busch - LSU32 Divide an conquer recurrence relation: Cost of subproblem of size

Konstantin Busch - LSU33 Examples: Binary search: Merge Sort: Fast Matrix Multiplication (Stassen’s Alg.):

Konstantin Busch - LSU34 Theorem: if then

Konstantin Busch - LSU35 Proof:

Konstantin Busch - LSU36

Konstantin Busch - LSU37 End of Proof

Konstantin Busch - LSU38 Theorem: if then

Konstantin Busch - LSU39 Proof: From previous theorem

Konstantin Busch - LSU40 Case:

Konstantin Busch - LSU41 Case: End of Proof

Konstantin Busch - LSU42 Example: Binary search:

Konstantin Busch - LSU43 Master Theorem: if then

Konstantin Busch - LSU44 Example: Merge Sort:

Generating Functions Konstantin Busch - LSU45 Find number of solutions for: Answer:

Konstantin Busch - LSU46 Alternative solution choices for

Konstantin Busch - LSU47 Alternative solution is the total number of solutions to equation

Konstantin Busch - LSU48 Another problem: Find total number of solutions which satisfy:

Konstantin Busch - LSU49 Alternative solution choices for

Konstantin Busch - LSU50 Alternative solution is the total number of solutions to equation

Konstantin Busch - LSU51 Generating function: generating function for sequence

Konstantin Busch - LSU52 Solve recurrence relation Generating functions can also be used to solve recurrence relations Example:

Konstantin Busch - LSU53 Let be the generating function for sequence

Konstantin Busch - LSU54

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Konstantin Busch - LSU56

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Konstantin Busch - LSU58 Solution to recurrence relation

Inclusion-Exclusion Konstantin Busch - LSU59

Konstantin Busch - LSU60

Konstantin Busch - LSU61 Principle of Inclusion-Exclusion:

Konstantin Busch - LSU62 Proof: We want to prove that: an arbitrary element is counted exactly one time in the expression of the theorem

Konstantin Busch - LSU63 Suppose is a member of exactly sets: Then is counted in the terms:

is counted times Konstantin Busch - LSU64 In sum: (since belongs exactly to sets)

is counted times Konstantin Busch - LSU65 In sum: (since belongs exactly to sets)

is counted times Konstantin Busch - LSU66 In sum: (since belongs exactly to sets)

is counted times Konstantin Busch - LSU67 In sum: (since belongs exactly to sets)

Konstantin Busch - LSU68 Thus, in the expression of the theorem is counted so many times:

Konstantin Busch - LSU69 End of Proof From binomial expansion we have that: Thus, is counted exactly one time

Konstantin Busch - LSU70 Example:Find the number of primes between 1…100 If a number is composite and between 1…100 then it must be divided by a prime which is at most : 2, 3, 5, 7

Konstantin Busch - LSU71 :the set of primes between 1…100 :composites between 1…100 divided by 2 :composites between 1…100 divided by 3 :the set of composites between 1…100 :composites between 1…100 divided by 5 :composites between 1…100 divided by 7

Konstantin Busch - LSU72 From the principle of inclusion-exclusion:

Konstantin Busch - LSU73

Konstantin Busch - LSU74 Number of primes between 1…100: