Recursion I and II. Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois 1

Transcription

1 Recursion I and II Discrete Structures (CS 173) Madhusudan Parthasarathy, University of Illinois 1

2 Recursive definitions of functions 2

3 One simple form In other words, f(n) is defined in terms of f(n-1) 3

4 In other words, f(n) is defined in terms of f(n-1),.. f(n-k), and with k base cases 4

5 The primary principle of recursion 5

6 Proof Notice that there is at least one function that satisfies the recursive definition. For any n, replace f(n) by its definition successively till you get to the base case(s). And then substitute the definition of the base cases. This gives a function g that clearly satisfies the recursive definitions. And then prove that the above function g is the only function that satisfies the recursive definitions: Prove that for any g that satisfies the recursive definitions, g is equal to g. Or prove by contradiction (take the smallest value where g and g differ and argue they cannot differ as they agree on smaller values). 6

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8 Recursive definitions of functions can be used to compute the function (provided the function combining the values of the function on smaller values, i.e., m, is computable). How? Recursion. Functional prog languages give you such a framework. 8

9 Induction and Recursion Very similar In induction, you prove a property. In recursive definitions, you define a function. In many of the induction problems you have already done, you can turn the question from proving a property to finding an object. And you can turn your proof into a recursive function that computes the object. Let s look at some examples. 9

10 Example you already solved Two players A and B take turns adding numbers from 1 through 8, and the first person to get to the target (9n+1). A starts game. Prove A has a winning strategy. 10

11 Example you already solved Two players A and B take turns adding numbers from 1 through 8, and the first person to get to the target (9n+1). A starts game. Prove A has a winning strategy. Construct winning strategy for A. 11

12 Example you already solved Return of the Guild of Parity Milliners. Prove that Linus can maintain the code of the cult. Show how Linus can maintain the code of the cult every day. 12

13 Example you already solved Hail to the orange and blue Compute the station Show there exists a station you can start with so that when you go clockwise from there, number of orange stations you see is at least as many as the number of blue stations you see. 13

14 Example Show that any connected graph with at least two vertices is weakly 2-colorable. (you can color the graph with 2 colors such that for every vertex u, there is *some* neighbor v of u such that u and v have different colors). Compute the weak 2-coloring. 14

15 Examples Show every densely connected graph has a Hamiltonian path. Find the Hamiltonian path, given any densely connected graph. 15

16 Example: Tiling using triominoes 16

17 Proving properties of rec def functions: Induction! How do you prove a property P of a recursively defined fn f? By induction, closely following the structure of the recursive def. 17

18 Example: the Mars question Assume we have one person living on Mars in Year 1. Every year, the number of people living on Mars doubles and we send one more person to Mars. How many people live in Mars in Year n? 18

19 Proving closed forms of rec def fns Example: T: Nat - {0} -> Nat T(1) = 1 T(n) = 2T(n-1)+3 for every n>1 19

20 Recursively definitions from Nat to non-numerical objects CBT(n) : the complete binary tree on n nodes 20

21 Recursively definitions from Nat to non-numerical objects Hypercubes Q 0 = Graph with single node with no edges Q n+1 = Graph that has two copies of Q n with edges joining corresponding nodes (the copies ) in the two graphs. 21

22 Factorial 22

23 Fibonacci series 23

24 E.g.: proving properties of rec def functions For every natural number n, Fib(3n) is even. 24

25 E.g.: proving properties of rec def functions 25

26 Lists and (finite) sequences List/finite sequences over A Represented also as 26

27 Operations on lists () or null is the empty list cons(a, l): list obtained by inserting a into the list l as the first element first(l): the first element of the list, if the list is nonempty, otherwise undefined rest(l): the list obtained by removing the first element of l, if l is nonempty, otherwise undefined 27

28 More operations on lists null : denotes the null list null? (l) : returns true/false, checks if l is the null list or not list (a): returns the list containing the single element a. append(l1, l2): list obtained by concatenating the two lists l1 and l2, l2 after l1. Defined for all pairs of lists take (n, l) : If l has at least n elements, returns a list containing the first n elements, otherwise undefined drop (n, l) : If l has at least n elements, returns a list with all but the first n elements; otherwise undefined. 28

29 Let s write recursive functions on lists Key parts of definition Define on some base cases Define the function on length n lists using definition of function on lists of length strictly smaller than n. In fact, there is a programming language (many: lisp, ML, Scheme, Racket*) that can turn (syntactically wellformed) recursive definitions into programs. 29

30 Length of a list (define (my-length lst) (cond [(null? lst) 0] [else (+ 1 (my-length (rest lst)))])) 30

31 Reverse a list (define (rev-it lst) (cond [(empty? lst) null] [else (append (rev-it (rest lst)) (cons (first lst) null))])) 31

32 Merge two sorted lists into a single sorted list 32

33 Merge two sorted lists into a single sorted list (define (sortedmerge lst1 lst2) (cond [(empty? lst1) lst2] [(empty? lst2) lst1] [(< (first lst1) (first lst2)) (cons (first lst1) (sortedmerge (rest lst1) lst2))] [else (cons (first lst2) (sortedmerge lst1 (rest lst2)))] )) 33

34 Racket --- extra credit problems Install DrRacket (easy; see Play with it; there re some tutorials, youtube videos, etc. Concentrate on the parts that deal with lists and recursion only! (there s lots more to Racket and to functional prog.) Try doing some exercises on it We ll hand out one set of exercises that involve writing math recursive definitions, and for extra-credit to program them in Racket. 34

35 Next Tue Review of graphs, trees, and induction. And on doing induction problems, in preparation for Thursday s examlet. 35