Dyn Prog Change

8/10/2019 Dyn Prog Change

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The Change Problem

"The Change Store" was an old SNL skit (a pretty dumb

one...) where they would say things like, "ou need !hange ora #$% &e'll gie you two tens, or a ten and two ies, or our

ies, et!."

you are a dorky minded CS # student, you might ask

yoursel (ater you ask yoursel why those writers get paid so

mu!h or writing the !rap that they do), "*ien a !ertain

amount o money, how many dierent ways are there to make

!hange or that amount o money%"

Let us simpliy the problem as ollows+

*ien a positie integer n, how many ways !an we make

!hange or n !ents using pennies, ni!kels, dimes and uarters%

-e!ursiely, we !ould break down the problem as ollows+

To make !hange or n !ents we !ould+) *ie the !ustomer a uarter. Then we hae to make !hange

or n/#0 !ents

#) *ie the !ustomer a dime. Then we hae to make !hange or

n/$ !ents

1) *ie the !ustomer a ni!kel. Then we hae to make !hange

or n/0 !ents

2) *ie the !ustomer a penny. Then we hae to make !hange

or n/ !ents.

we let T(n) 3 number o ways to make !hange or n !ents, we

get the ormula

T(n) 3 T(n/#0)4T(n/$)4T(n/0)4T(n/)


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2) d is , we !an simply return sin!e i you are only allowed

to gie pennies, you !an only make !hange in one way.

;lthough this seems uite a bit more !omple< than beore, the

!ode itsel isn't so long. Let's take a look at it+

publi! stati! int makeChange(int n, int d) @

i (n 5 $)

return $A

else i (n33$)

return A

else @ int sum 3 $A

swit!h (d) @

!ase #0+ sum43makeChange(n/#0,#0)A

!ase $+ sum43makeChange(n/$,$)A

!ase 0+ sum 43 makeChange(n/0,0)A

!ase + sum44A

B

return sumA B

B

There's a whole bun!h o stu going on here, but one o the

things you'll noti!e is that the larger n gets, the slower and

slower this will run, or maybe your !omputer will run out o

sta!k spa!e. urther analysis will show that many, many

method !alls get repeated in the !ourse o a single initial

method !all.

n dynami! programming, we want to ;87= these reo!!uring

!alls. To do this, rather than making those three re!ursie !alls

aboe, we !ould store the alues o ea!h o those in a two

dimensional array.


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7ur array !ould look like this

# 1 2 0 6 D E F $ # 1 2 0

0 # # # # # 1 1 1 1 1 2

$ # # # # # 2 2 2 2 2 6

#0 # # # # # 2 2 2 2 2 6

9ssentially, ea!h row label stands or the number o !ents we

are making !hange or and ea!h !olumn label stands or the

largest !oin alue allowed to make !hange.

(Note+ The lightly !olored suares with , # and are added to

!al!ulate the lightly !olored suare with 2, based on the

re!ursie algorithm.)

Now, let us try to write some !ode that would emulate building

this table by hand, rom let to right.

publi! stati! int makeChangedyn(int n, int d) @

GG Take !are o simple !ases.

i (n 5 $)

return $A

else i ((nH3$) II (n 5 0))

return A

GG Juild table here.

else @

intK denominations 3 @, 0, $, #0BA

intKK table 3 new intK2Kn4A


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GG nitialiMe table

or (int i3$A i5n4Ai44)

tableK$Ki 3 A

or (int i3$A i50A i44) @ tableKKi 3 A

tableK#Ki 3 A

tableK1Ki 3 A

B

or (int i30Ai5n4Ai44) @

tableKKi 3 $A

tableK#Ki 3 $A

tableK1Ki 3 $A B

GG ill in table, row by row.

or (int i3A i52A i44) @

or (int ?30A ?5n4A ?44) @

or (int k3$A k53iA k44) @

i ( ? H3 denominationsKk)

tableKiK? 43 tableKkK? / denominationsKkA B

B

B

return tableKlookup(d)KnA

B

B

;n alternate way to !ode this up is to realiMe that we =7N'T

need to add many dierent !ases up together. nstead, we note

that the number o ways to make !hange or n !ents using

denomination d !an be split up into !ounting two groups+


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) The number o ways to make !hange or n !ents using

denominations L9SS than d

#) The number o ways to make !hange or n !ents using atleast 7N9 !oin o denomination d.

The ormer is simply the alue in the table that is dire!tly

aboe the one we are trying to ill.

The latter is the alue on the table that is on the same row, by d

spots to the let.

8isually, !onsider ?ust adding two alues rom our preious

e

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Dyn Prog Change