1

Lecture 16:

Lists and vectorsBinary search,

Sorting

2

How can we implement a pair?

•Each pair consists of two consecutive memory locations.The first holds a pointer to the carThe second holds a pointer to the cdr

•Data is typed, so we know if we have a pointer or not.

3

Vectors – storing lists differently

We can directly access the k’th element of the list.

base

We have a pointer to the address base where the lists starts.We assume each element takes the same amount of space (say 1)

The first element is at address baseThe second element is at address base+1The k’th element is at address base+(k-1)

4

Lists vs. Vectors

On the one hand:We can directly access the k’th element of a vector. O(1) time.We need k operations to access the k’th element of a list.

On the other hand:Lists are more flexible.We can add/delete elements in between.We can reverse the order of pointers, shuffle them,…

Conclusion: Different things suit different purposes.

5

The vector interface.

Methods:(vector? obj) ; returns true if obj is a vector, false otherwise(vector-length vec) ; returns the length of the vector vec(vector-ref vec k) ; returns the element of vec with index k ; k should be between 0 and length-1

Constructors:(vector ... ) ; create a vector from 0 or more elements(make-vector k) ; create a vector of length k(make-vector k init) ; same, but each element is initialized to init

Mutator:(vector-set! vec k elt) ; stores elt in the element k of the ; vector vec. k should be between 0 and length-1

6

Examples

(define vec1 (vector 9 7 4 'anna))

vec1 ==> #4(9 7 4 anna)(vector-length vec1) ==> 4

(vector-ref vec1 1) ==> 7(vector-set! vec1 2 'foo)

vec1 ==> #(9 7 ‘foo ‘anna)

(define vec2 (make-vector 4 '()))

vec2 ==> #4(() () () ())(vector-set! vec2 2 (vector-ref vec1 2))

vec2 ==> #(() () foo ())

(define list_of_two_vecs (list vec1 vec2))

(vector? (car list_of_two_vecs)) ==> #t

7

Binary search

We are given:

• a sorted vector vec, and

•An element elm

We want to know whether elm is in vec.

Example: We want to check whether elm = גולית הפלישתי Appears in the Telephone book.

8

Searching for גולית

Alternatively:We can use the fact that the phone book is sorted.We first look for the letter גThen for the letter וAnd so on.

Impatient people don’t have time to think about the problem,They start reading the phone book from the beginningUntil they fall asleep.

But how do we find where does ג starts?

9

Binary search

Compare elm to the element x that is at the middle of the vector.

• If elm=x, we have found elm

• If elm<x, we recursively check in the first half of the vector.

• If elm>x, we recursively check in the second half of the vector.

We open the phone book in the middle,And we adaptively decide in which half to continue the search.Then solve recursively.

10

Complexity

With one comparison we reduce search space to half its size,

therefore, the run time is (log n)

We require:

• An order relation on the elements.

• The input vector must be sorted

• Random access to the vector.

11

Implementation

(define (bin-search vec elm) (define (search left right) (if (> left right) nil (let* ((mid (average left right)) (mid-item (vector-ref vec mid))) (cond ((= elm mid-item) mid) ((< elm mid-item) (search left (- mid 1))) (else (search (+ mid 1) right)))))) (search 0 (- (vector-length vec) 1)))

(define (average x y) (round (/ (+ x y) 2)))

12

Sorting

Input: A collection of data compound elements, each data element has a key from some ordered domain.

For our example there will be just keys.

Output: The same elements ordered according to the keys in increasing order.

As we saw, sorting can be sometimes useful, e.g., for searching in a phone book.

But how do we create the phone book?

13

Bubble sort

Basic step: The algorithm compares neighbors, and exchanges them when they are out of order.

A round: Go through the elements from last to first, performing this check on each pair of neighbors.

Run n rounds.

Correctness: After the first round, the smallest element is first. After the second round, the first two elements are correctAnd so on.

14

Bubble sort

3 1 5 2 4

3 1 2 5 4

1 3 2 5 4

1 3 2 4 5

1 2 3 4 5

First round

Second round

15

Implementation of Bubble sort

(define (bubble-sort vec) (define n (vector-length vec)) (define (iter i) (define (bubble j) (if (>= j i) (let ((prev (vector-ref vec (- j 1))) (cur (vector-ref vec j))) (cond ((> prev cur) (vector-set! vec (- j 1) cur) (vector-set! vec j prev))) (bubble (- j 1))))) (cond ((< i n) (bubble (- n 1)) (iter (+ i 1))))) (iter 1))

16

The complexity of Bubble sort

Number of comparisons is:

(n-1) + (n-2) + . . . . + 1 = n*(n-1)/2 = (n2 ).

To sort TA phone book, we need about(10^7)^2 = 10^14 operations.

By the time the phone book is ready we might need a new one.

17

Another approach

Start with an empty list. Always maintain a sorted (sub)-list.Each time add another element to the right location in the list.

If each insert takes O(log n)To sort a list would take O(n log n)

Problem:• With vectors we can find the right place in O(log n),

but we can not insert in the middle.• With lists we can insert in the middle, but we can not find the right location fast.

The approach can be made to work using more sophisticated data structures.

18

Divide and Conquer

Facts of life: Solving a 5,000 piece puzzle is difficult. Solving a 1,000 piece puzzle is doable. Solving a 500 piece puzzle is easy. Solving a 1 piece puzzle is trivial.

One way to solve a 5,000 piece puzzle• Find a way to divide the 5,000 pieces to five groups that are solved separately.• Solve each separately, and combine the five solutions to get a solution for the whole puzzle.

For example:• all the pieces on the border have a straight line, Take them alone and build the border.• all the pieces of the windmill have brown in them.

19

Divide and conquer

If we:1. Know how to break a problem into smaller problems, such that2. Given a solution to the smaller problems we can easily find

A solution to the bigger problem

Then we get an algorithm:1. Divide and solve the smaller pieces recursively2. Solve the big problem using the solutions to the

smaller problems.

20

Quicksort

The algorithm:

1. chooses some element, called pivot.

2. Splits the elements into three groups:

• smaller the pivot

• equal to pivot

• larger than pivot.

3. Recursively sort the smaller and the larger groups independently

21

Quicksort (Cont.)

(define (quicksort lst) (if (null? lst) nil (let ((pivot (car lst))) (let ((low (filter (lambda (x) (< x pivot)) lst)) (high (filter (lambda (x) (> x pivot)) lst)) (same (filter (lambda (x) (= x pivot)) lst))) (append (append (quicksort low) same) (quicksort high))))))

22

Quicksort -- brief analysis

T(n) <= cn + T(m) + T(n-m-1)

where m is the number of elements smaller than the pivot.

n

m n-m-1

cn

c(n-1) c(n-2)

So the running time is O(d n)Where d is the depth of the tree.

23

Quicksort -- brief analysis

Best case: when at each stage m is about n/2, i.e., the algorithm partitions the elements into two sets of about equal size.

This gives depth O(log n) and time T(n)=O(n log n)

Worst case: when m=0 (or m=n-1),

giving depth n and run time T(n) = (n2)

Average case: Average on what? How do we pick the pivot?Picking it “correctly”, “on average” it’s fast.

24

Merge Sort

Another divide and conquer algorithm.

Input: A list with n elements.

Algorithm: 1. Divide the list into the first half and the second half.2. Recursively Sort each half.3. Merge the two sorted lists.

25

Merge

Input: Two sorted lists (of may be different lengths)Output: One sorted list, with the same set of elements as the union of the two lists.Algorithm: Put a pointer at the beginning of each list. At each step pick the smallest element, and advance that pointer to the next element in its list.

26

Implementing merge

(define (merge list1 list2) (cond ((null? list1) list2) ((null? list2) list1) ((< (car list1) (car list2)) (cons (car list1) (merge (cdr list1) list2))) (else (cons (car list2) (merge list1 (cdr list2))))))

Complexity: linear in the size of the lists.

27

Running time

T(n) = 2 ( 2 T(n/4) + cn/2) + cn = 4 T(n/4) + cn + cn = 8 T(n/8) + cn + cn + cn = 2k T(n/2k) + k cn

= c n log(n)

T(n) = 2T(n/2) + cnT(2) = c

Worst case.

28

Can we do better?

Theorem: Any sorting algorithms that is based on comparisons requires (n log n) Comparisons.

Proof idea: There are n! possible orderings. We need to distinguish them all. Thus we need about log(n!) comparisons and that’s (n log n).

Making it precise: well, not now.

29

Q: We want to sort 52 cards,We don’t like the log(52) factor.Can we sort cards better?

Yes! but not based on comparisons.

Radix sort: Input: n elements between 1 to s. Output: The elements sorted. Algorithm: Create a vector of length s, initialized with 0 everywhere.

Go over the input, and for each element, if its value is k, increase vec[k] by 1.

Complexity: O(n+s)

30

Implementing Radix Sort.

(define (radix-sort lst m) ; sorts a list of elements between 0..m (define vec (make-vector (+ m 1) 0)) (define (help-func sub-list) (cond ((null? sub-list) 'done) (else (let* ((elm (car sub-list)) (val (vector-ref vec elm))) (vector-set! vec elm (+ val 1)) (help-func (cdr sub-list)))))) (help-func lst) vec) (radix-sort (list 1 5 3 6 8 1 3 6 9) 10) #11(0 2 0 2 0 1 2 0 1 1 0)