CS 410 Applied Algorithms Applied Algorithms Lecture #7 Counting

CS 410 Applied Algorithms

Applied Algorithms

Lecture #7

Counting


Exam Next Week

• Don’t Forget• We will use half of next weeks class time

for an exam.• The exam will be closed book. You may

bring one sheet (8.5 x 11.0 inch) of paper with notes on it.

• The exam will cover the reading, assignments and lectures upto the day of the exam.


Quiz

• We have a 10 minute quiz today on the reading – Chapter 5.

• We will continue to have quizzes until the class gets an average score high enough to convince me that people are doing the reading.


Pairsumonious

• [2, 4, 5, 6] -- Unknown original numbers

• [ 2+4, 2+5, 2+6, 4+5, 4+6, 5+6 ]

-- all pair wise sums

• [6,7,8,9,10,11] -- Given numbers


Computing pair-wise positions

• N = 4[[(1,2),(1,3),(1,4)], [(2,3),(2,4)], [(3,4)]]

• N = 5[[(1,2),(1,3),(1,4),(1,5)], [(2,3),(2,4),(2,5)], [(3,4),(3,5)], [(4,5)]]


Sort Assumptions

• Assume that both the unknown list is sorted, and that the given list is sorted. We can sort it, if it isn’t.

• [x1, x2, x3, x4 ] -- Unknowns

• where x1 x2, x2 x3, x3 x4

• [6,7,8,9,10] -- Given numbers sorted

• x1 + x2 = 6

• x1 + x3 = 7 -- Why?


xm + xn = 8

• What are the possible values for m and n?

• x1 + x4 = 8

• x2 + x3 = 8

• Why isn’t x2 + x4 = 8, a possiblility?


Enumerate the possibilities (X3+X4 = 11) (X2+X4 = 10) (X2+X3 = 9) (X1+X4 = 8)(X1+X2 = 6) (X1+X3 = 7) (X2+X3 = 8) (X1+X4 = 9) (X2+X4 = 10) (X3+X4 = 11)

Every path from root to leaf is a possible solution.


Given a Table choose the next nodes in the tree

[[(1,2),(1,3),(1,4)], [(2,3),(2,4)], [(3,4)]]

next2 :: Table -> [((Int,Int),Table)]next2 [] = []next2 [(x:xs)] = [(x,[xs])]next2 ([]:ys) = next2 ysnext2 ((x:xs):(y:ys):zs) = case order x y of Less -> [(x,xs:(y:ys):zs)] Greater -> [(y,(x:xs):ys:zs)] Incomp -> [(x,xs:(y:ys):zs),(y,(x:xs):ys:zs)]


Generate All Paths

paths :: [Int] -> Path ->

((Int,Int),Table) -> [Path]

paths (s:sums) path ((i,j),xs) =

case next2 xs of

[] -> [reverse ((i,j,s):path)]

zs -> concat(map (paths sums ((i,j,s):path)) zs)


Two Paths

• [(1,2,6),(1,3,7),(1,4,8),(2,3,9),(2,4,10),(3,4,11)]• [(1,2,6),(1,3,7),(2,3,8),(1,4,9),(2,4,10),(3,4,11)]


Example of size 5

[[(1,2),(1,3),(1,4),(1,5)], [(2,3),(2,4),(2,5)], [(3,4),(3,5)], [(4,5)]] • when n= 5 • sums = [-1,0,-1,-2,1,0,-1,1,0,1]• [-2,-1,-1,-1,0,0,0,1,1,1]


(X4+X5 = 1)

(X3+X5 = 1)

(X3+X4 = 1)

(X2+X5 = 0)

(X2+X3 = 0)

(X1+X5 = -1)

(X2+X4 = 0)

(X3+X4 = 0)

(X2+X5 = 1)

(X3+X5 = 1)

(X4+X5 = 1)

(X1+X4 = -1)

(X4+X5 = 1)

(X3+X5 = 1)

(X3+X4 = 1)

(X2+X5 = 0)

(X2+X4 = 0)

(X1+X5 = 0)

(X3+X4 = 0)

(X2+X5 = 1)

(X3+X5 = 1)

(X4+X5 = 1)

(X2+X3 = -1)

(X4+X5 = 1)

(X3+X5 = 1)

(X2+X5 = 1)

(X3+X4 = 0)

(X2+X4 = 0)

(X1+X5 = 0)

(X2+X5 = 0)

(X3+X4 = 1)

(X3+X5 = 1)

(X4+X5 = 1)

(X1+X2 = -2)

(X1+X3 = -1)

(X4+X5 = 1)

(X3+X5 = 1)

(X3+X4 = 1)

(X2+X5 = 0)

(X2+X4 = 0)

(X1+X5 = 0)

(X3+X4 = 0)

(X2+X5 = 1)

(X3+X5 = 1)

(X4+X5 = 1)

(X1+X4 = -1)

(X2+X3 = -1)

(X4+X5 = 1)

(X3+X5 = 1)

(X2+X5 = 1)

(X3+X4 = 0)

(X2+X4 = 0)

(X1+X5 = 0)

(X2+X5 = 0)

(X3+X4 = 1)

(X3+X5 = 1)

(X4+X5 = 1)


Possible paths from root to leaf[(1,2,-2),(1,3,-1),(1,4,-1),(1,5,-1),(2,3,0),(2,4,0),(2,5,0),(3,4,1),(3,5,1),(4,5,1)]

[(1,2,-2),(1,3,-1),(1,4,-1),(1,5,-1),(2,3,0),(2,4,0),(3,4,0),(2,5,1),(3,5,1),(4,5,1)]

[(1,2,-2),(1,3,-1),(1,4,-1),(2,3,-1),(1,5,0),(2,4,0),(2,5,0),(3,4,1),(3,5,1),(4,5,1)]

[(1,2,-2),(1,3,-1),(1,4,-1),(2,3,-1),(1,5,0),(2,4,0),(3,4,0),(2,5,1),(3,5,1),(4,5,1)]

[(1,2,-2),(1,3,-1),(1,4,-1),(2,3,-1),(2,4,0),(1,5,0),(2,5,0),(3,4,1),(3,5,1),(4,5,1)]

[(1,2,-2),(1,3,-1),(1,4,-1),(2,3,-1),(2,4,0),(1,5,0),(3,4,0),(2,5,1),(3,5,1),(4,5,1)]

[(1,2,-2),(1,3,-1),(2,3,-1),(1,4,-1),(1,5,0),(2,4,0),(2,5,0),(3,4,1),(3,5,1),(4,5,1)]

[(1,2,-2),(1,3,-1),(2,3,-1),(1,4,-1),(1,5,0),(2,4,0),(3,4,0),(2,5,1),(3,5,1),(4,5,1)]

[(1,2,-2),(1,3,-1),(2,3,-1),(1,4,-1),(2,4,0),(1,5,0),(2,5,0),(3,4,1),(3,5,1),(4,5,1)]

[(1,2,-2),(1,3,-1),(2,3,-1),(1,4,-1),(2,4,0),(1,5,0),(3,4,0),(2,5,1),(3,5,1),(4,5,1)]


Is a path feasible?

• [(1,2,6),(1,3,7),(1,4,8),(2,3,9),(2,4,10),(3,4,11)]

• [(x1+x2=6),(x1+x3=7),(x1+x4=8),

(x2+x3=9),(x2+x4=10),(x3+x4=11)]

How can we test this?


Every path contains all pair-wise sums

• So they must contain the following (x1+x2=A) (x1+x3=B) (x2+x3=C)

• N=4 examples– [(1,2,6),(1,3,7),(1,4,8),(2,3,9),(2,4,10),(3,4,11)]– [(1,2,6),(1,3,7),(2,3,8),(1,4,9),(2,4,10),(3,4,11)]

• N=5 example[(1,2,-2),(1,3,-1),(1,4,-1),(2,3,-1),(2,4,0),(1,5,0),(2,5,0),(3,4,1),(3,5,1),(4,5,1)]


• (x1+x2=A) (x1+x3=B) (x2+x3=C)

(x1+x2) + (x1+x3) - (x2+x3) = A + B – C

2 * x1 = A + B – C

Solving for x1 we get x1 = (A+B-C)/2

Algorithm find the 3 pairs in the path and solve


CodesolveX1 :: Int -> Int -> Int -> Path -> Maybe IntsolveX1 x12 x13 x23 ((1,2,w):ys) = solveX1 w x13 x23 yssolveX1 x12 x13 x23 ((1,3,w):ys) = solveX1 x12 w x23 yssolveX1 x12 x13 x23 ((2,3,w):ys) = solveX1 x12 x13 w yssolveX1 x12 x13 x23 (y:ys) = solveX1 x12 x13 x23 yssolveX1 x12 x13 x23 [] = if even sum then Just(sum `div` 2) else Nothing where sum = x12+x13-x23


Every path also contains

• N=4 examples– [(1,2,6),(1,3,7),(1,4,8),(2,3,9),(2,4,10),(3,4,11)]– [(1,2,6),(1,3,7),(2,3,8),(1,4,9),(2,4,10),(3,4,11)]

• N=5 example[(1,2,-2),(1,3,-1),(1,4,-1),(2,3,1),(2,4,0),(1,5,0),(2,5,0),(3,4,1),(3,5,1),(4,5,1)]


CodesolveRest :: Int -> Path -> [(Int,Int)] -> [(Int,Int)]solveRest x1 [] pairs = pairssolveRest x1 ((1,xn,sum):xs) pairs = solveRest x1 xs ((xn,sum - x1):pairs)solveRest x1 (x:xs) pairs = solveRest x1 xs pairs

solve [] = Nothingsolve (p:paths) = case solveX1 0 0 0 p of Just x1 -> Just(map snd (sort(solveRest x1 p [(1,x1)]))) Nothing -> solve paths


In Class Problems

• How Many Fibs? 6.6.1– Page 137 of the text– We will write this together as a class

• Expressions 6.6.4– Page 140 of the text–


Today’s Assignments

Read for next time

Chapter 7 of the text. pp 147-166 Be prepared to answer questions in class next Friday from

the reading.

Programming assignment

• 6.6.8 Steps• Page 145• Write a solutionSubmit your solution (until you get it right)Hand in both your program, and the judge output.Those who volunteer to discuss their program get class participation points.

Email me solutions before noon on Friday, May 6.

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CS 410 Applied Algorithms Applied Algorithms Lecture #7 Counting