CSC236 Week 4Larry Zhang

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Logistics

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NEW TOPIC

RecursionTo really understand the math of recursion,

and to be able to analyze runtimes of recursive programs.

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ConcepTest

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Recursively Defined Functions

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The functions that describe the runtime of recursive programs

Definition of functions

● The usual way: define a function using a closed-form.

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● Another way: using a recursive definition

A function defined in terms of itself.

Let’s work out a few values for T₀(n) and T₁(n)

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T₀(n) T₁(n)n=1

n=2

n=3

n=4

n=5

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7

15

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15

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Maybe T₀(n) and T₁(n) are equivalent to each other.

Every recursively defined function has an equivalent closed-form function.

And we like closed-form functions.

● It is easier to evaluate○ Just need n to know f(n)○ Instead of having to know f(n-1) to know f(n)

● It is easier for telling the growth rate of the function○ T₀(n) is clearly Θ(2ⁿ), while T₁(n) it not clear.

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Our Goal:Given a recursively defined function,

find its equivalent closed-form function

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Method #1Repeated Substitution

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Repeated substitution: StepsStep 1: Substitute a few times to find a pattern

Step 2: Guess the recurrence formula after k substitutions (in terms of k and n)

For each base case:

Step 3: solve for k

Step 4: Plug k back into the formula (from Step 2) to find a potential closed form. (“Potential” because it might be wrong)

Step 5: Prove the potential closed form is equivalent to the recursive definition using induction.

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Example 1

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Step 1

Substitute a few times to find a pattern.

Substitute T(n-1) with T(n-2):

Substitute T(n-2) with T(n-3):

Substitute T(n-3) with T(n-4):

Step 2:

Guess the recurrence formula after k substitutions

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The guess:

Step 3: Consider the base case

What we have now:

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We want this to be T(1), because we know clearly what T(1) is.

So, let n - k = 1, and solve for k

k = n - 1This means you need to make n-1 substitutions to reach the base case n=1

Step 4: plug k back into the guessed formula

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Guessed formula:

For Step 3, we got k = n - 1

Substitute k back in, we get ...

This is the “potential” closed-form of T(n).

Step 5: Drop the word “potential” by proving it

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Prove is equivalent to

Use induction!

Define predicate: P(n): T₁(n) = T₀(n)Base case: n=1

Induction step:

# by I.H.

# by def of T(n)

We have officially found that the closed-form of the recursively defined function

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is

Repeated substitution: StepsStep 1: Substitute a few time to find a pattern

Step 2: Guess the recurrence formula after k substitutions (in terms of k and n)

For each base case:

Step 3: solve for k

Step 4: Plug k back into the formula (from Step 2) to find a potential closed form. (“Potential” because it might be wrong)

Step 5: Prove the potential closed form is equivalent to the recursive definition using induction.

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ConcepTest

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Summary

Find the closed form of a recursively defined function

● Method #1: Repeated substitution

○ It’s nothing tricky, just follow the steps!

○ Take care of all base cases!

○ It’s a bit tedious sometimes, we will learn faster

methods later.21

Handout, Exercise 1

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Handout, Exercise 2

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Next ExampleSometimes you’re not given the recursive definition directly,

so you need to develop the recursive definition first

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Example

Give a recursive definition of

the number of 2-element subsets of n elements.

(Pretend that you never knew “n choose 2”, and let’s develop it from first principles).

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For example, when n=4, we have 4 elements, e₁, e₂, e₃, e₄, the 2-element subsets are (e₁,e₂), (e₁,e₃), (e₁,e₄), (e₂,e₃), (e₂,e₄), (e₃,e₄)There are 6 of them.

Strategy

1. Find the base case(s) that can be evaluated directly

2. Find a way to break the larger problem to smaller problems, so that you can define f(n) in terms of f(m) for some m < n.

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Give a recursive definition of

f(n): The number of 2-element subsets of n elements.

1. Base case:

n = 0

f(0) = 0

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Give a recursive definition of

f(n): The number of 2-element subsets of n elements.

2. Break the larger problem to smaller ones:For the set of n elements, how many 2-element subsets does it have?

Our set looks like:

Break all subsets of S into two parts● the subsets that contain en ● the subsets that do NOT contain en

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The 2-element subsets that contain en

How many?

Two elements …● One has to be en

● The other can be e1, e2, …, en-1

So there are n-1 of such subsets!

The number of 2-element subsets of n elements.

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The 2-element subsets that do NOT contain en

How many?

They are subsets of

The number of 2-element subsets of n-1 elements

It’s f(n-1) !

The number of 2-element subsets of n elements.

Sum up after breaking the larger problem into smaller problems

The number of 2-element subsets of n elements

● number of subsets with en: n-1● number of subsets without en: f(n-1)

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Combining with the base case, the final developed recurrence:

f(n): the number of 2-element subsets of n elements.

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Home exercise: find its closed form using the substitution method. You know what the result should be.

ConcepTest

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Try this: write an efficient program that computes the number of ways to pick up the marbles given the number of marbles in the row.