Class 2 Recursion on Lists

Transcription

1 Class 2 Recursion on Lists The list data type Recursive methods on lists MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 1

2 Data structures Contain large amounts of data Allow for access to that data Many different data structures, allowing efficiency for various operations; more from your data structures module MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 2

3 Lists Lists represent a simple data structure in which data are stored in a row. We will talk first about lists of integers: x 0, x 1,..., x n-1 (n >= 0) Elements can be added and removed only at the beginning (x 0 ) MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 3

4 Lists (cont.) Terminology: First element (x 0 ) is head of the list List of remaining elements (x 1,..., x n-1 ) is tail of the list Adding a new element to the front is called consing (for constructing ) The empty list is the list with no elements (n=0) MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 4

5 List operations We assume a type List, for variables that contain such lists. The List type will be defined by a class that also provides the following static methods: List cons (int i, List L) - construct a list containing i at the front List empty() - the empty (zero-element) list MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 5

6 List operations (cont.) List objects have the following instance methods: int gethead() return the head of the list List gettail() - return the tail of the list boolean isempty()- is the list empty? MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 6

7 Examples of list operations Suppose L is the list 2, 3, 5, 7 cons(2,cons(3,cons(5,cons(7,empty())))) ) L.getHead() returns 2 L.getTail() returns the list 3, 5, 7 L.isEmpty() returns false cons(13, L) returns the list 13, 2, 3, 5, 7 cons(13, L.getTail()) returns the list 13, 3, 5, 7 L.getTail().getTail() returns the list 5, MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 7

8 Example Assume we have defined the class List as described above. public static void main (String[] args) { List L = List.empty(); L = List.cons(5, L); L = List.cons(10, L); System.out.println(L.getHead()); System.out.println(L.getTail().getHead());} To avoid having to write List.empty and List.cons, use import static List.empty; import static List.cons; MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 8

9 Recursion on lists Writing recursive methods on lists follows same principle as for integers: To compute f(l), assume f(l ) can be calculated for lists L smaller than L, and use f(l ) to calculate f(l). Some lists are small enough for f to be calculated directly MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 9

10 Example: printing lists Print all the elements in a list, one per line: Assume you can print the tail of L (i.e. printlist(l.gettail())), so how do you print L? MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 10

11 Example: printing lists (cont.) static void printlist (List L) { if (L.isEmpty()) return; else { System.out.println(L.getHead()); printlist(l.gettail()); } } MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 11

12 Example: printing lists (cont.) Print all the elements in a list, all on one line, separated by a comma and a space: Assume you can print the tail of L (i.e. printlist(l.gettail())), so how do you print L? Answer: print L.getHead(), followed by a comma and space, but only if the tail of L is nonempty MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 12

13 Example: printing lists (cont.) static void printlist (List L) { if (L.isEmpty()) return; else if (L.getTail().isEmpty())) System.out.print(L.getHead()); else { System.out.print(L.getHead()); System.out.print(, ); printlist(l.gettail()); } } MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 13

14 The method Example: addtoend List addtoend (List L, int i) can be defined recursively. It adds the integer i at the end of the list L. For example, if L is the list 3, 5, 7, then addtoend(l, 2) returns the list 3, 5, 7, 2. We will see how to define addtoend later, but we can use it for MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 14

15 Example: reading integers Given integer n, read n integers from the user and place them in a list. Assume you can read n-1 integers and place them in a list; how can you read n integers? Answer: read the n-1 integers, creating a list L, then read one more integer and place it at the end of L MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 15

16 Example: reading integers (cont.) static List readlist (int n) { if (n == 0) return List.empty(); else { List L = readlist(n-1); int i = Keyboard.readInt(); return addtoend(l, i); } } MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 16

17 Recursion on lists (review) Writing recursive methods on lists follows the same principle as for integers: To compute f(l), assume f(l ) can be calculated for lists L smaller than L, and use f(l ) to calculate f(l). Some lists are small enough for f to be calculated directly MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 17

18 Example: joining lists Given lists L and M, create a list consisting of the elements of L followed by the elements of M. Assume you can append the tail of L and M (i.e. append(l.gettail(), M)); how can you append L and M? MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 18

19 Example: joining lists (cont.) static List append (List L, List M) { if (L.isEmpty()) return M; else return cons(l.gethead(), append(l.gettail(), M)); } MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 19

20 Example: addtoend Given list L and integer i, construct a list that contains the elements of L with i at the end. static List addtoend(list L, int i) { return append(l, cons(i, empty())); } MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 20

21 Example: finding maximum element Find the maximum integer in a non-empty list. Assume you can find the maximum element of the tail (if the tail is non-empty) MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 21

22 Example: finding maximum element (cont.) static int max (List L) { if ((L.getTail().isEmpty())) return L.getHead(); else { int m = max(l.gettail()); return (L.getHead() > m)? L.getHead(): m; } } MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 22

23 Example: list reversal Given list L, construct the reversal of L. Assume you can reverse the tail of L. How can you place the head of L in the reversal of the tail of L so as to get the reversal of L itself? MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 23

24 Example: list reversal (cont.) static List reverse (List L) { if (L.isEmpty()) return L; else return addtoend(reverse(l.gettail()), L.getHead()); } MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 24

25 Analysis of list reversal reverse is rather slow. Consider how many method calls it makes in relation to the size of its argument. Note that addtoend(l) calls itself recursively n times, where n is the length of L. How many method calls does reverse(l) make? MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 25

26 Analysis of list reversal (cont.) After recursive call to itself, it calls addtoend, on a list of length n-1. So this makes n-1 calls to addtoend. Now consider the first recursive call. It in turn has a recursive call; after that inner recursive call, there is a call to addtoend, which entails n-2 calls to addtoend. Continuing in this way, there is a call to addtoend that involves n-3 calls, one involving n-4 calls, etc MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 26

27 Analysis of list reversal (cont.) Thus, the total number of calls to addtoend is (n-1) + (n-2) + + 1, which equals n(n-1)/2 n 2. This is a much larger number than n itself MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 27

28 Faster list reversal To avoid the quadratic cost of reversal, we can define another version, which we will call rev (just to distinguish it). The important trick is that rev uses an auxiliary (helper) method rev1 with type List rev1 (List L, List M) rev1(l,m) is defined to return the reversal of L appended to M MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 28

29 Faster list reversal (cont.) rev1 can be defined recursively. The key observation is this one: Placing the reversal of L onto the front of M is the same thing as placing the reversal of the tail of L onto cons(gethead(l), M). For example, if we want to place the reversal of 1,2,3 onto the front of list 4,5 - obtaining 3,2,1,4,5 - we can just as well place the reversal of 2,3 onto the front of 1,4, MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 29

30 Faster list reversal (cont.) This observation leads to: static List rev1 (List L, List M) { if (L.isEmpty()) return M; else return rev1(l.gettail(), List.cons(L.getHead(), M)); } Note that this requires only n method calls, where n is the length of L MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 30

31 Faster list reversal (cont.) We can easily program rev once we ve got rev1. static List rev (List L) { return rev1(l, List.empty); } MSc Workshop S.Kamin, U.S.Reddy 1B - Lists - 31