CSC236 Week 4. Larry Zhang

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1 CSC236 Week 4 Larry Zhang 1

2 Announcements PS2 is out Larry s office hours in the reading week: as usual Tuesday 12-2, Wednesday 2-4 2

3 NEW TOPIC Recursion To really understand the math of recursion, and to be able to analyze runtimes of recursive programs. 3

4 ConcepTest 4

5 Recursively Defined Functions The functions that describe the runtime of recursive programs 5

6 Definition of functions The usual way: define a function using a closed-form. Another way: using a recursive definition A function defined in terms of itself. 6

7 Let s work out a few values for T₀(n) and T₁(n) T₀(n) T₁(n) n=1 1 1 n=2 3 3 Maybe T₀(n) and T₁(n) are equivalent to each other. n=3 n=4 n=

8 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. 8

9 Our Goal: Given a recursively defined function, find its equivalent closed-form function 9

10 Method #1 Repeated Substitution 10

11 Repeated substitution: Steps Step 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. 11

12 Example 1 12

13 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): 13

14 Step 2: Guess the recurrence formula after k substitutions The guess: 14

15 Step 3: Consider the base case What we have now: So, let n - k = 1, and solve for k We want this to be T(1), because we know clearly what T(1) is. k = n - 1 This means you need to make n-1 substitutions to reach the base case n=1 15

16 Step 4: plug k back into the guessed formula 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). 16

17 Step 5: Drop the word potential by proving it Prove Use induction! Define predicate: P(n): T₁(n) = T₀(n) Base case: n=1 Induction step: is equivalent to # by def of T(n) # by I.H. 17

18 We have officially found that the closed-form of the recursively defined function is 18

19 Repeated substitution: Steps Step 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. 19

20 ConcepTest 20

21 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

22 Handout, Exercise 1 22

23 Handout, Exercise 2 23

24 Next Example Sometimes you re not given the recursive definition directly, so you need to develop the recursive definition first 24

25 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). 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. 25

26 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. 26

27 Give a recursive definition of f(n): The number of 2-element subsets of n elements. 1. Base case: n = 0 f(0) = 0 27

28 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 28

29 The number of 2-element subsets of n elements. 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! 29

30 The number of 2-element subsets of n elements. 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)! 30

31 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) 31

32 Combining with the base case, the final developed recurrence: f(n): the number of 2-element subsets of n elements. Home exercise: find its closed form using the substitution method. You know what the result should be. 32

33 ConcepTest 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. 33

CSC236 Week 5. Larry Zhang

CSC236 Week 5. Larry Zhang CSC236 Week 5 Larry Zhang 1 Logistics Test 1 after lecture Location : IB110 (Last names A-S), IB 150 (Last names T-Z) Length of test: 50 minutes If you do really well... 2 Recap We learned two types of