David Evanshttp://www.cs.virginia.edu/~evans

CS200: Computer ScienceUniversity of VirginiaComputer Science

Lecture 14: P = NP?

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Menu

• Golden Ages• Permuted Sorting• Complexity Classes

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The Real Golden Rule?Why do fields like astrophysics, medicine, biology and computer science (?) have “endless golden ages”, but fields like– music (1775-1825)– rock n’ roll (1962-1973, or whatever was popular when you

were 16)– philosophy (400BC-350BC?)– art (1875-1925?)– soccer (1950-1974)– baseball (1925-1950)– movies (1930-1940)

have short golden ages?

From Lecture 13:

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1

2

3

4

5

6Goal-den age

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rage

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ame,

FIF

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Changed goalkeeperpassback rule

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Endless Golden Age and “Grade Inflation”

• Average student gets twice as smart and well-prepared every 15 years– You had grade school teachers who

went to college!• If average GPA in 1970 is 2.00

what should it be today (if Prof’s didn’t change grading standards)?

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Grade Inflation or Scale Compression?

2.00 average GPA in 1970 (“gentleman’s C”?)* 2 better students 1970-1985* 2 better students 1985-2000* 2 admitting women (1971)* 1.54 population increase* 0.58 increase in enrollment

Virginia 1970 4,648,494

Virginia 2000 7,078,515

Average GPA today should be:14.3 CS200 has only the best of the best students, and only

the best half of them stayed in the course after PS1, so theaverage grade in CS200 should be 14.3*2*2*2 = 114.3

Students 1970 11,000Students 2002 18,848

(12,595 UG)

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Permuted Sorting

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Permuted Sorting

• A (possibly) really dumb way to sort:– Find all possible orderings of the list

(permutations)– Check each permutation in order, until you

find one that is sorted• Example: sort (3 1 2)

All permutations: (3 1 2) (3 2 1) (2 1 3) (2 3 1) (1 3 2) (1 2 3)

is-sorted? is-sorted?is-sorted? is-sorted? is-sorted? is-sorted?

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is-sorted?

(define (is-sorted? cf lst) (or (null? lst) (= 1 (length lst)) (and (cf (car lst) (cadr lst)) (is-sorted? cf (cdr lst)))))

Is or a procedure or a special form?

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all-permutations(define (all-permutations lst) (flat-one (map (lambda (n) (if (= (length lst) 1) (list lst) ; The permutations of (a) are ((a)) (map (lambda (oneperm) (cons (nth lst n) oneperm)) (all-permutations (exceptnth lst n))))) (intsto (length lst)))))

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permute-sort

(define (permute-sort cf lst) (car (filter (lambda (lst) (is-sorted? cf lst)) (all-permutations lst))))

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> (time (permute-sort <= (rand-int-list 5)))cpu time: 10 real time: 10 gc time: 0(4 14 14 45 51)> (time (permute-sort <= (rand-int-list 6)))cpu time: 40 real time: 40 gc time: 0(6 29 39 40 54 69)> (time (permute-sort <= (rand-int-list 7)))cpu time: 261 real time: 260 gc time: 0(6 7 35 47 79 82 84)> (time (permute-sort <= (rand-int-list 8)))cpu time: 3585 real time: 3586 gc time: 0(4 10 40 50 50 58 69 84)> (time (permute-sort <= (rand-int-list 9)))Crashes!

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How much work is permute-sort?

• We evaluated is-sorted? once for each permutation of lst.

• How much work is is-sorted??

(n)• How many permutations of the list are

there?

(define (permute-sort cf lst) (car (filter (lambda (lst) (is-sorted? cf lst)) (all-permutations lst))))

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Number of permutations

• There are n = (length lst) values in the first map, for each possible first element

• Then, we call all-permutations on the list without that element (length = n – 1)

• There are n * n – 1 * (n – 1) – 1 * … * 1 permutations

• Hence, there are n! lists to check: (n!)

(map (lambda (n) (if (= (length lst) 1) (list lst) (map (lambda (oneperm) (cons (nth lst n) oneperm)) (all-permutations (exceptnth lst n))))) (intsto (length lst)))

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Big-O Notation• O(x) – it is no more than x work

(upper bound)(x) – work scales as x (tight bound)(x) – it is at least x work

(lower bound)If we can find the same x where O(x) and (x) are true, then (x) also.

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Procedures and Problems• So far we have been talking about

procedures (how much work is permute-sort?)

• We can also talk about problems: how much work is sorting?

• A problem defines a desired output for a given input. A solution to a problem is a procedure for finding the correct output for all possible inputs.

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The Sorting Problem

• Input: a list and a comparison function• Output: a list such that the elements

are the same elements as the input list, but in order so that the comparison function evaluates to true for any adjacent pair of elements

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O, , for problems• The sorting problem is O(n log2 n) since we

know a procedure (quicksort) that solves it in (n log2 n)

• Does this mean the sorting problem is

(n log2 n)?No, we would need to prove there is no better procedure. For some comparison functions (e.g., (lambda (a b) #t)), there are faster procedures. For <=, this is true, but we haven’t proven it (and won’t in this class).

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The Minimize Difference Problem

• Input: two same-length lists, lsta and lstb; a difference function, diff, that operates on pairs of elements of the lists

• Output: a list which is a permutation of lstb, such that the sum of diff applied to corresponding elements of lsta and the output list is the minimum possible for any permutation of lstb

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Example

> (minimize-difference (lambda (a b) (square (- a b))) (list 0 1 2 8) (list 6 4 5 3)) (3 4 5 6)

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minimize-difference(define (minimize-difference diff lsta lstb) (cdr (find-most (lambda (p1 p2) (< (car p1) (car p2))) (make-difference-lists diff (all-permutations lstb) lsta))))

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make-difference-lists(define (make-difference-lists diff plists tlist) (map (lambda (plist) (cons (sumlist (map (lambda (a b) (diff a b)) plist tlist)) plist)) plists))

Note: (map f lista listb) applies (f (car lista) (car listb)) …

(define (sumlist lst) (if (null? lst) 0 (+ (car lst) (sumlist (cdr lst)))))

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How much work?• How much work is our minimize-difference

procedure?

There are n! lists to check, each takes n work: (n!)

• If diff is a constant time ((1)) procedure, and we don’t know anything else about it, how much work is the minimize difference problem?

(n!)

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How much work?

• How much work is the minimize difference problem with – as the diff function?

O(n!) we know we can to it with no more than n! work

using minimize-differenceBut, is there a faster procedure to solve this problem?

Yes! It is O(1): (define (minimize-sub-difference lsta lstb) lstb)

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Have you seen anything like this?

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Perfect Photomosaic Problem

• Input: a set of tile images, a master image, a color difference function

• Output: an ordering of a subset of the tile images that minimizes the total color difference function for each image tile and the corresponding square in the master image

• How much work is finding a perfect photomosaic? O(n!) n is the number of tiles

(assumes same as number to cover master)

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Complexity Class P

Class P: problems that can be solved in polynomial time

O(nk) for some constant k.

Easy problems like sorting, making a photomosaic using duplicate tiles, understanding the universe.

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Complexity Class NPClass NP: problems that can be solved in nondeterministic polynomial time

If we could try all possible solutions at once, we could identify the solution in polynomial time.

Hard problems like minimize-difference, … (more examples Weds).

Making the best photomosaic without using duplicate tiles (PS5) may be even harder! (Weds)

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P = NP?• Is there a polynomial-time solution to the

“hardest” problems in NP?• No one knows the answer!• The most famous unsolved problem in

computer science and math• Listed first on Millennium Prize Problems

– win $1M if you can solve it – (also an automatic A+ in this course)

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Charge

• PS4 Due Friday– Lab Hours: Mon 7-9pm, Thu 7-9pm

• Wednesday: – What are the “hardest” problems in NP?

• Friday: review for exam– Covers through lecture 13 (not today)