dynprog knapsack

8/20/2019 DynProg Knapsack

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Dynamic Programming …Dynamic Programming …

ContinuedContinued0-1 Knapsack Problem


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Last ClassLast Class

Longest Common Subsequence (LCS◦ !ssigned problem " #ind t$e LCS bet%een

&')!*+ and &C',)P!*..+ #ill out t$e dynamic programming table /race back %$at t$e LCS is

.dit Distance◦ !ssigned problem " #ind t$e edit distance

required to transorm &K,L/+ into &K,//.*+ #ill out t$e dynamic programming table

/race back t$e edit pat$

◦ !ssigned problem " utline t$e recursi2e calls%$en using t$e recursi2e met$od or 3ndingt$e edit distance bet%een &*+ and &##+4


3/40

Last ClassLast ClassCan I have volunteers to do the following on

the board?1 #ill out t$e dynamic programming table or t$e LCS

bet%een &')!*+ and &C',)P!*..+

5 /race back %$at t$e LCS string is bet%een &')!*+

and &C',)P!*..+

6 #ill out t$e dynamic programming table to 3nd t$e editdistance required to transorm &K,L/+ into &K,//.*+

7 /race back t$e edit pat$ t$at transormed &K,L/+ into&K,//.*+

8 utline t$e recursi2e calls %$en using t$e recursi2emet$od or 3nding t$e edit distance bet%een &*+ and&##+4


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!nnouncements!nnouncements

!ssignment 97 D*! DistanceProblem


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Knapsack 0-1 ProblemKnapsack 0-1 Problem

/$e goal is tomaximize the valueof a knapsack t$atcan $old at most :units (i4e4 lbs or kg

%ort$ o goods rom alist o items I0, I1, … In-1.

◦ .ac$ item $as 5attributes;

1) Value – let this be vi foritem Ii

2) Weight – let this be wi foritem Ii


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/$e di


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>rute #orce◦ /$e na?2e %ay to sol2e t$is problem is

to cycle t$roug$ all 5n subsets o t$e n

items and pick t$e subset %it$ a legal%eig$t t$at ma@imiAes t$e 2alue o t$eknapsack4

◦ :e can come up %it$ a dynamicprogramming algorit$m t$at %illS!LL= do better t$an t$is brute orcetec$nique4


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!s %e did beore %e are going to sol2e t$eproblem in terms o sub-problems4◦ So letBs try to do t$at…

ur 3rst attempt mig$t be to c$aracteriAea sub-problem as ollo%s;◦ Let ! be the o"timal subset of elements from #I0, I1, …, I! $.

:$at %e 3nd is t$at t$e optimal subset rom t$e

elements #I0, I1, …, I!%1$ may not correspond to t$eoptimal subset o elements rom #I0, I1, …, I! $ in anyregular pattern4

◦ >asically t$e solution to t$e optimiAation

problem or !%1 mig$t */ contain t$e optimalsolution rom roblem 4


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Knapsack 0-1 ProblemKnapsack 0-1 ProblemLetBs illustrate t$at point %it$ an e@ample;

Item Weight Value

I0 3 0

I ! "

I# $ $

I3 !

%he maximum weight the knapsack can hold is #0&

/$e best set o items rom ,0 ,1 ,5E is ,0 ,1 ,5E

>/ t$e best set o items rom ,0 ,1 ,5 ,6E is ,0

,5 ,6E4◦ ,n t$is e@ample note t$at t$is optimal solution ,0 ,5 ,6E

does */ build upon t$e pre2ious optimal solution ,0 ,1,5E4

(,nstead it buildFs upon t$e solution ,0 ,5E %$ic$ is really t$e

optimal subset o ,0 ,1 ,5E %it$ %eig$t 15 or less4


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Knapsack 0-1 problemKnapsack 0-1 problemSo no% %e must re-%ork t$e %ay %e build upon

pre2ious sub-problems…

◦ Let '(k) w* represent t$e ma@imum total 2alue o a subset Sk

%it$ %eig$t %4

◦ ur goal is to 3nd '(n) W*) %$ere n is t$e total number oitems and : is t$e ma@imal %eig$t t$e knapsack can carry4

So our recursi2e ormula or subproblems;

B[k, w] = B[k - 1,w], if wk > w

= max { B[k - 1,w], B[k - 1,w - wk ] + vk }, otherwise

,n .nglis$ t$is means t$at t$e best subset o Sk t$at

$as total %eig$t % is;

1 /$e best subset o Sk-1 t$at $as total %eig$t % or

5 /$e best subset o Sk-1 t$at $as total %eig$t %-%k plus t$e item

k


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Knapsack 0-1 Problem "Knapsack 0-1 Problem "

Gecursi2e #ormulaGecursi2e #ormula

/$e best subset o Sk t$at $as t$e total%eig$t % eit$er contains item k or not4

+irst case, wk > w◦ ,tem k canBt be part o t$e solutionH , it %as

t$e total %eig$t %ould be I % %$ic$ isunacceptable4

-econd case, wk ≤ w◦

/$en t$e itemk can be in t$e solution and %ec$oose t$e case %it$ greater 2alue4


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Knapsack 0-1 !lgorit$mKnapsack 0-1 !lgorit$mfor w = 0 to W { // Initialize 1st row to 0’s

B[0,w] = 0

}

for i = 1 to n { // Initialize 1st column to 0’s

B[i,0] = 0

}

for i = 1 to n {

for w = 0 to W {

if wi


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LetBs run our algorit$m on t$e ollo%ingdata;

◦ n J 7 (9 o elements

◦ : J 8 (ma@ %eig$t

◦ .lements (%eig$t 2alue;

(56 (67 (78 (8


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Knapsack 0-1 .@ampleKnapsack 0-1 .@ample

i / w 0 1 2 3 4 0 0 0 0 0 0 0

1 0

2 0

3 0

4 0

&& Initiali'e the base (ases

for w 0 to W

*+0,w 0

for i 1 to n

*+i,0 0


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Knapsack 0-1 .@ampleKnapsack 0-1 .@ample,tems;

1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0

2 0

3 0

4 0

if wi w &&item i (an be in the solution

if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

*+i,w *+i-1,w

else B[i,w] = B[i-1,w] && wi w

i 1

vi /

wi 2

w 1w-wi -1

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0

2 0

3 0

4 0


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1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0

2 0

3 0

4 0


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

*+i,w *+i-1,w

else *+i,w *+i-1,w && wi w

i 1

vi /

wi 2

w = 2w-wi 0

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 /

2 0

3 0

4 0


if vi % *+i-1,w-wi *+i-1,w

B[i,w] = vi + B[i-1,w- wi]

else

*+i,w *+i-1,w


,


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1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 /

2 0

3 0

4 0


if vi % *+i-1,w-wi *+i-1,w

B[i,w] = vi + B[i-1,w- wi]

else

*+i,w *+i-1,w


i 1

vi /

wi 2

w = 3w-wi 1

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / /

2 0

3 0

4 0

,t


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1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / /

2 0

3 0

4 0


if vi % *+i-1,w-wi *+i-1,w

B[i,w] = vi + B[i-1,w- wi]

else

*+i,w *+i-1,w


i 1

vi /

wi 2

w = 4w-wi 2

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / /

2 0

3 0

4 0

,t


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1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / /

2 0

3 0

4 0


if vi % *+i-1,w-wi *+i-1,w

B[i,w] = vi + B[i-1,w- wi]

else

*+i,w *+i-1,w


i 1

vi /

wi 2

w = w-wi /

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0

3 0

4 0

,t


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1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0

3 0

4 0


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

*+i,w *+i-1,w


i 2

vi

wi /

w = 1w-wi -2

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0

3 0

4 0


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

*+i,w *+i-1,w


,t


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1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0

3 0

4 0


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

*+i,w *+i-1,w


i 2

vi

wi /

w = 2w-wi -1

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0

4 0

,tems


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1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0

4 0


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

*+i,w *+i-1,w


i 2

vi

wi /

w = 3w-wi 0

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0

4 0


if vi % *+i-1,w-wi *+i-1,w

B[i,w] = vi + B[i-1,w- wi]

else

*+i,w *+i-1,w


,tems;


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1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0

4 0


if vi % *+i-1,w-wi *+i-1,w

B[i,w] = vi + B[i-1,w- wi]

else

*+i,w *+i-1,w


i 2

vi

wi /

w = 4w-wi 1

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0

4 0

,tems;


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1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0

4 0


if vi % *+i-1,w-wi *+i-1,w

B[i,w] = vi + B[i-1,w- wi]

else

*+i,w *+i-1,w


i 2

vi

wi /

w = w-wi 2

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0

4 0

,tems;


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1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0

4 0


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

*+i,w *+i-1,w


i /

vi

wi

w = 1!!3w-wi -/..-1

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 /

4 0


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

*+i,w *+i-1,w


,tems;


26/40


1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 /

4 0


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

*+i,w *+i-1,w


i /

vi

wi

w = 4

w-wi 0

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 /

4 0


if vi % *+i-1,w-wi *+i-1,w

B[i,w] = vi + B[i-1,w- wi]

else

*+i,w *+i-1,w


,tems;


27/40


1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 /

4 0


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi else

*+i,w *+i-1,w


i /

vi

wi

w =

w-wi 1

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 /

4 0


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

B[i,w] = B[i-1,w]


,tems;


28/40


1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 /

4 0


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

*+i,w *+i-1,w


i

vi 3

wi

w = 1!!4

w-wi -..-1

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 /

4 0 0 /


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

*+i,w *+i-1,w


,tems;


29/40


1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 /

4 0 0 /


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

*+i,w *+i-1,w


i

vi 3

wi

w =

w-wi 0

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 /

4 0 0 /


if vi % *+i-1,w-wi *+i-1,w

*+i,w vi % *+i-1,w- wi

else

B[i,w] = B[i-1,w]


,tems;


30/40


1;(56

5;

(67

6;(78

7;

(8

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 /

4 0 0 / "

We4re 567899

:he ma; "ossible value that (an be (arrie< in this !na"sa(! is $7


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Knapsack 0-1 !lgorit$mKnapsack 0-1 !lgorit$m

/$is algorit$m only 3nds t$e ma@possible 2alue t$at can be carriedin t$e knapsack

◦ /$e 2alue in >n:M

/o kno% t$e items t$at make

t$is ma@imum 2alue %e need totrace back t$roug$ t$e table4

K k 0 1 !l it$K k 0 1 !l it$


32/40

Knapsack 0-1 !lgorit$mKnapsack 0-1 !lgorit$m

#inding t$e ,tems#inding t$e ,tems

Let i n an< ! W

if *+i, ! = *+i-1, ! then

mar! the ith item as in the !na"sa(!

i i-1, ! !-wi

else

i i-1 && >ssume the ith item is not in the !na"sa(!

&& ?oul< it be in the o"timall@ "a(!e< !na"sa(!A

K k 0 1K k 0 1,tems; Knapsac


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Knapsack 0-1Knapsack 0-1

!lgorit$m!lgorit$m


,tems;

1;(56

5;

(67

6;(78

7;

(8

i

!

vi 3

wi

B[i,k] = "

*+i-1,!

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 / 4 0 0 /

i n , ! W

while i, ! 0

if B[i, k] ≠ B[i-1, k] thenmark the ith item as in the knapsack

i = i-1, k = k-wielse

i = i-1

Knapsack;

K k 0 1Knapsack 0 1,tems; Knapsac


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!lgorit$m!lgorit$m


,tems;

1;(56

5;

(67

6;(78

7;

(8

i /

!

vi

wi

B[i,k] = "

*+i-1,!

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 / 4 0 0 /

i n , ! W

while i, ! 0



i = i-1

Knapsack;



35/40


!lgorit$m!lgorit$m


,tems;

1;(56

5;

(67

6;(78

7;

(8

i 2

!

vi

wi /

B[i,k] = "

*+i-1,! /

! – wi 2

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 / 4 0 0 /

i n , ! W

while i, ! 0



i = i-1

Knapsack;Item 2



36/40


!lgorit$m!lgorit$m


,tems;

1;(56

5;

(67

6;(78

7;

(8

i 1

! 2

vi /

wi 2

B[i,k] = 3

*+i-1,! 0

! – wi 0

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

30 0 / 4 0 0 /

i n , ! W

while i, ! 0



i = i-1

Knapsack;

Item 1

Item 2

Knapsack 0 1Knapsack 0 1,tems; Knapsac


37/40


!lgorit$m!lgorit$m


,tems;

1;(56

5;

(67

6;(78

7;

(8

i 1

! 2

vi /

wi 2

B[i,k] = 3

*+i-1,! 0

! – wi 0

i / w 0 1 2 3 4

0 0 0 0 0 0 0

1 0 0 / / / /

2 0 0 /

3 0 0 /

4 0 0 /

k = 0, so we#re $%&'(

)he o*tima ka*sak sho. otai

Item 1 and Item 2

Knapsack;

Item 1

Item 2


38/40

Knapsack 0-1 Problem " GunKnapsack 0-1 Problem " Gun

/ime /imefor w 0 to W

*+0,w 0

for i 1 to n

*+i,0 0

for i 1 to n

for w 0 to W

the rest of the (o


39/40

Knapsack ProblemKnapsack Problem

1 #ill out t$e

dynamicprogrammingtable or t$e

knapsackproblem tot$e rig$t4

5 /race backt$roug$ t$etable to 3ndt$e items in

t$e knapsack4


40/40

GeerencesGeerences

Slides adapted rom !rup Nu$aBsComputer Science ,, Lecture notes;$ttp;OO%%%4cs4uc4eduOdmarinoOucOcop6806OlecturesO

!dditional material rom t$e te@tbook;

Data Structures and !lgorit$m !nalysis in Qa2a (Second .dition by )ark !llen :eiss

!dditional images;%%%4%ikipedia4com

@kcd4com
http://www.cs.ucf.edu/~dmarino/ucf/cop3503/lectures/http://www.cs.ucf.edu/~dmarino/ucf/cop3503/lectures/http://www.wikipedia.com/http://xkcd.com/http://xkcd.com/http://www.wikipedia.com/http://www.cs.ucf.edu/~dmarino/ucf/cop3503/lectures/http://www.cs.ucf.edu/~dmarino/ucf/cop3503/lectures/

dynprog knapsack

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