📄 usaco 2_3_4 money systems 题解_leokan的blog.mht
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<DIV class=3Dtit>USACO 2.3.4 Money Systems =CC=E2=BD=E2</DIV>
<DIV class=3Ddate>2008=C4=EA01=D4=C230=C8=D5 =D0=C7=C6=DA=C8=FD =
11:20</DIV>
<TABLE style=3D"TABLE-LAYOUT: fixed">
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<TD>
<DIV class=3Dcnt>
<H2>USACO 2.3.4 Money Systems</H2>
<DIV class=3Dt_msgfont>Money Systems<BR><BR>The cows have not only =
created=20
their own government but they have chosen to create their own =
money=20
system. In their own rebellious way, they are curious about values =
of=20
coinage. Traditionally, coins come in values like 1, 5, 10, 20 or =
25, 50,=20
and 100 units, sometimes with a 2 unit coin thrown in for good =
measure.=20
<BR><BR>The cows want to know how many different ways it is =
possible to=20
dispense a certain amount of money using various coin systems. For =
instance, using a system of {1, 2, 5, 10, ...} it is possible to =
create 18=20
units several different ways, including: 18x1, 9x2, 8x2+2x1, =
3x5+2+1, and=20
many others. <BR><BR>Write a program to compute how many ways to =
construct=20
a given amount of money using supplied coinage. It is guaranteed =
that the=20
total will fit into both a signed long long (C/C++) and Int64 =
(Free=20
Pascal). <BR><BR>PROGRAM NAME: money<BR>INPUT FORMAT<BR>The number =
of=20
coins in the system is V (1 <=3D V <=3D 25). <BR><BR>The =
amount money to=20
construct is N (1 <=3D N <=3D 10,000). Line 1: =
Two integers,=20
V and N <BR>Lines 2..: V integers that =
represent=20
the available coins (no particular number of integers per line)=20
<BR><BR><BR>SAMPLE INPUT (file money.in) <BR>3 10<BR>1 2 =
5<BR><BR>OUTPUT=20
FORMAT<BR>A single line containing the total number of ways to =
construct N=20
money units using V coins. <BR>SAMPLE OUTPUT (file=20
money.out)<BR>10<BR><BR><BR><BR>Money =
Systems<BR><BR>=BB=F5=B1=D2=CF=B5=CD=B3<BR><BR>=D2=EB by=20
=
timgreen<BR><BR>=C4=B8=C5=A3=C3=C7=B2=BB=B5=AB=B4=B4=BD=A8=C1=CB=CB=FB=C3=
=C7=D7=D4=BC=BA=B5=C4=D5=FE=B8=AE=B6=F8=C7=D2=D1=A1=D4=F1=C1=CB=BD=A8=C1=A2=
=C1=CB=D7=D4=BC=BA=B5=C4=BB=F5=B1=D2=CF=B5=CD=B3=A1=A3<BR>[In their own=20
rebellious =
way],=A3=AC=CB=FB=C3=C7=B6=D4=BB=F5=B1=D2=B5=C4=CA=FD=D6=B5=B8=D0=B5=BD=BA=
=C3=C6=E6=A1=A3<BR>=B4=AB=CD=B3=B5=D8=A3=AC=D2=BB=B8=F6=BB=F5=B1=D2=CF=B5=
=CD=B3=CA=C7=D3=C91,5,10,20 =BB=F2 25,50, =BA=CD=20
=
100=B5=C4=B5=A5=CE=BB=C3=E6=D6=B5=D7=E9=B3=C9=B5=C4=A1=A3<BR>=C4=B8=C5=A3=
=CF=EB=D6=AA=B5=C0=D3=D0=B6=E0=C9=D9=D6=D6=B2=BB=CD=AC=B5=C4=B7=BD=B7=A8=C0=
=B4=D3=C3=BB=F5=B1=D2=CF=B5=CD=B3=D6=D0=B5=C4=BB=F5=B1=D2=C0=B4=B9=B9=D4=EC=
=D2=BB=B8=F6=C8=B7=B6=A8=B5=C4=CA=FD=D6=B5=A1=A3<BR>=BE=D9=C0=FD=C0=B4=CB=
=B5, =CA=B9=D3=C3=D2=BB=B8=F6=BB=F5=B1=D2=CF=B5=CD=B3=20
{1,2,5,10,...}=B2=FA=C9=FA =
18=B5=A5=CE=BB=C3=E6=D6=B5=B5=C4=D2=BB=D0=A9=BF=C9=C4=DC=B5=C4=B7=BD=B7=A8=
=CA=C7:18x1, 9x2, 8x2+2x1,=20
=
3x5+2+1,=B5=C8=B5=C8=C6=E4=CB=FC=A1=A3<BR>=D0=B4=D2=BB=B8=F6=B3=CC=D0=F2=C0=
=B4=BC=C6=CB=E3=D3=D0=B6=E0=C9=D9=D6=D6=B7=BD=B7=A8=D3=C3=B8=F8=B6=A8=B5=C4=
=BB=F5=B1=D2=CF=B5=CD=B3=C0=B4=B9=B9=D4=EC=D2=BB=B6=A8=CA=FD=C1=BF=B5=C4=C3=
=E6=D6=B5=A1=A3<BR>=B1=A3=D6=A4=D7=DC=CA=FD=BD=AB=BB=E1=CA=CA=BA=CFlong =
long=20
(C/C++) =BA=CD Int64 (Free Pascal)=A1=A3<BR><BR>PROGRAM NAME: =
money<BR><BR>INPUT=20
=
FORMAT<BR><BR>=BB=F5=B1=D2=CF=B5=CD=B3=D6=D0=BB=F5=B1=D2=B5=C4=D6=D6=C0=E0=
=CA=FD=C4=BF=CA=C7 V =A1=A3 (1<=3D =
V<=3D25)<BR>=D2=AA=B9=B9=D4=EC=B5=C4=CA=FD=C1=BF=C7=AE=CA=C7 N =A1=A3 =
(1<=3D=20
N<=3D10,000)<BR><BR>=B5=DA 1 =D0=D0: =
=B6=FE=D5=FB=CA=FD=A3=AC V =BA=CD N <BR>=B5=DA 2 ..V+1=D0=D0=A3=BA =
=BF=C9=D3=C3=B5=C4=BB=F5=B1=D2 V=20
=
=B8=F6=D5=FB=CA=FD=A1=A1(=C3=BF=D0=D0=D2=BB=B8=F6=A1=A1=C3=BF=D0=D0=C3=BB=
=D3=D0=C6=E4=CB=FC=B5=C4=CA=FD)=A1=A3 <BR><BR>SAMPLE INPUT (file =
money.in) <BR>3 10<BR>1 2=20
5<BR>OUTPUT =
FORMAT<BR>=B5=A5=B6=C0=B5=C4=D2=BB=D0=D0=B0=FC=BA=AC=C4=C7=B8=F6=BF=C9=C4=
=DC=B5=C4=B9=B9=D4=EC=B5=C4=B7=BD=B0=B8=CA=FD=A1=A3<BR>SAMPLE OUTPUT =
(file=20
money.out)<BR>10</DIV>
<P></P>
<P></P>
<HR>
<P></P>
<P><STRONG>USACO 2.2.4 Money Systems =
<BR>=CC=E1=BD=BB=B4=CE=CA=FD:2=B4=CE</STRONG></P>
=
<P><STRONG>=D5=E2=CC=E2=CE=D2=D2=BB=BF=AA=CA=BC=CF=EB=B5=BD=D3=C3=B5=C4=B6=
=FE=CE=AC=CA=FD=D7=E9,cash[i]=B1=ED=CA=BE=B5=DAi=BC=FE=CE=EF=C6=B7=B5=C4=BC=
=DB=C7=AE,f(i,j)=B1=ED=CA=BE=D3=C3=C7=B0i=BC=FE=CE=EF=C6=B7=B1=ED=CA=BEj=D5=
=E2=B8=F6=BC=DB=C7=AE=B5=C4=BF=C9=C4=DC=CA=FD.=C8=BB=BA=F3f(i,j)=3D=A1=C6=
f(i-k,j-cash</STRONG><STRONG>[i]).AC=D6=AE=BA=F3=BA=F6=C8=BB=CF=EB=B5=BD,=
=C6=E4=CA=B5=D2=BB=CE=AC=CA=FD=D7=E9=BE=CD=BF=C9=D2=D4=C1=CB,=D4=AD=CF=C8=
f(i,j)=B5=C4=C7=B0=D2=BB=B8=F6=B2=CE=CA=FD=C6=E4=CA=B5=CA=C7=B2=BB=B1=D8=D2=
=AA=B5=C4,f(i-k,j-cash[i])=BF=C9=D2=D4=D3=C3=D2=BB=CE=AC=B5=C4f(j-cash[i]=
)=BC=D3=D1=AD=BB=B7</STRONG><STRONG>=B1=ED=CA=BE,i-k=C4=C7=B8=F6=C1=BF=D2=
=D1=BE=AD=B0=B5=B2=D8=D4=DA=C0=EF=C3=E6=C1=CB.</STRONG></P>
=
<P><STRONG>=D5=E2=C1=EE=CE=D2=D1=A7=CF=B0=B5=BD,DP=BD=B5=CE=AC=CA=C7=BA=DC=
=D6=D8=D2=AA=B5=C4,=BC=D9=C8=E7=D5=E2=CC=E2=D3=D0250=B8=F6=CA=FD100000=B8=
=F6=BC=DB=D6=B5,=C4=C7=CE=D2=D4=AD=CF=C8=C4=C7=D6=D6=B7=BD=B7=A8=BE=CD=B9=
=FD=B2=BB=C1=CB=C1=CB.</STRONG></P>
=
<P><BR><STRONG>=CE=CA=CC=E2=D6=D8=CA=F6:<BR>=B8=F8=B3=F6=D7=D4=C8=BB=CA=FD=
n,=BC=B0v=B8=F6=BB=A5=B2=BB=CF=E0=CD=AC=B5=C4=CA=FD=D7=D6,=C7=F3=D3=C3v=B8=
=F6=BB=A5=B2=BB=CF=E0=CD=AC=B5=C4=CA=FD=D7=D6=C0=DB=BC=D3=B5=C3n=D5=E2=B8=
=F6=CA=FD=D7=D6(=CA=FD=D7=D6=BF=C9=D2=D4=D6=D8=B8=B4=CA=B9=D3=C3)=B5=C4=B7=
=BD=B7=A8=D7=DC=CA=FD.</STRONG></P>
=
<P><STRONG>=CE=CA=CC=E2=B7=D6=CE=F6:<BR>=D2=C0=D5=D5=CC=E2=D2=E2=B5=C3=B5=
=BD,=C0=DB=BC=D3=D2=BB=B8=F6x=D7=E9=B3=C9=CA=FD=D7=D6y=B5=C4=B7=BD=B7=A8=BC=
=AF=BA=CFA,=BA=CD=B2=BB=C0=DB=BC=D3=D5=E2=B8=F6x=D7=E9=B3=C9=CA=FD=D7=D6y=
-x=B5=C4=B7=BD=B7=A8=BC=AF=BA=CFB=D3=D0=D2=BB=D6=D6=D3=B3=C9=E4=B9=D8=CF=B5=
,g:A=A1=FAB.g=BE=CD=CA=C7=D4=DAA=D6=D0=C4=B3=D2=BB=D6=D6=C0=DB=BC=D3</STR=
ONG><STRONG>=B7=BD=B7=A8=BC=F5=C8=A5x.=BC=C8=C8=BB=D3=D0=D5=E2=D6=D6=B9=D8=
=CF=B5,=C4=C7=C3=B4=CE=D2=C3=C7=B6=A8=D2=E5f(m)=CE=AA=D3=C3=D5=E2=D0=A9=CA=
=FD=C0=DB=BC=D3=B5=C3m=B5=C4=B7=BD=B7=A8=D7=DC=CA=FD,cash[i]=B1=ED=CA=BE=B5=
=DAi=B8=F6=CA=FD,=CE=D2=C3=C7=BF=C9=D2=D4=CB=B5f(m)=3D=A1=C6f(m-cash[i])(=
i=A1=CA</STRONG><STRONG>{1,2...v}.=BF=C9=D2=D4=D3=C3=B6=AF=CC=AC=B9=E6=BB=
=AE=C7=F3=BD=E2.</STRONG></P>
=
<P><STRONG>=B1=E0=B3=CC=BD=E2=BE=F6:<BR>=B1=DF=BD=E7=CC=F5=BC=FE=C9=E8f[0=
]=B5=C8=D3=DA1,=CA=B2=C3=B4=C0=DB=BC=D3=B5=C4=B7=BD=B7=A8=CA=FD=CA=C7=D2=BB=
,=C0=DB=BC=D3=B5=C30.<BR> =20
f[0]=A1=FB1;<BR> for i=A1=FB1 to v=20
do<BR> for =
j=A1=FBcash[i] to n=20
=
do<BR> =
f[j]=A1=FBf[j]+f[j-cash[i]]</STRONG></P>
<P><STRONG>code:</STRONG></P>
<P><STRONG>{<BR>TASK:money<BR>LANG:PASCAL<BR>}<BR>program=20
money;<BR>var<BR> =
n,v:integer;<BR> =20
cash:array[1.. 25] of integer;<BR> =
f:array[0..10000] of=20
qword;<BR>procedure init;<BR>var<BR> =20
i:integer;<BR>begin<BR> =20
assign(input,'money.in');reset(input);<BR> =20
readln(v,n);<BR> for i:=3D1 to v do=20
read(cash[i]);<BR> readln;<BR> =
close(input);<BR>end;<BR>procedure =
dp;<BR>var<BR> =20
i,j,k,l:integer;<BR>begin<BR> =20
assign(output,'money.out');rewrite(output);<BR> =20
fillchar(f,sizeof(f),0);<BR> =20
f[0]:=3D1;<BR> for i:=3D1 to v=20
do<BR> for j:=3Dcash[i] =
to n=20
=
do<BR> =
inc(f[j],f[j-cash[i]]);<BR> =20
writeln(f[n]);<BR> =20
close(output);<BR>end;<BR>begin<BR> =20
init;<BR> dp;<BR>end.<BR></STRONG></P><STRONG>
<HR>
</STRONG>
=
<P><STRONG>USACO=B5=C4=B7=D6=CE=F6,=C3=BB=D3=D0=CE=D2=B5=C4=B7=D6=CE=F6=CF=
=EA=BE=A1=B0=A1,o(=A1=C9_=A1=C9)o...=B9=FE=B9=FE</STRONG></P>
<P><STRONG>We use dynamic programming to count the number of ways =
to make=20
n cents with the given coins. If we denote the value of the kth =
coin by=20
c_k, then the recurrence is: </STRONG></P><PRE><STRONG> nway(n, =
k) =3D no. of ways to make n cents with the first k types of coins
nway(n, k) =3D nway(n, k-1) + nway(n-c_k, k)
</STRONG></PRE>
<P><STRONG>This just says the number of ways to make n cents with =
the=20
first k coins is the number of ways to make n cents using the =
first k-1=20
coins (i.e., without using the kth coin) plus the number of ways =
to make=20
n-c_k cents using the first k coins. For the second set of ways, =
we then=20
add the kth coin to arrive at a total of n cents. </STRONG></P>
<P><STRONG>We keep track of the number of ways to total "n" cents =
in=20
"nway", updating the array as we read the value of each coin.=20
</STRONG></P><PRE><STRONG>/*
PROG: money
ID: rsc001
*/
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <assert.h>
#define MAXTOTAL 10000
long long nway[MAXTOTAL+1];
void
main(void)
{
FILE *fin, *fout;
int i, j, n, v, c;
fin =3D fopen("money.in", "r");
fout =3D fopen("money.out", "w");
assert(fin !=3D NULL && fout !=3D NULL);
fscanf(fin, "%d %d", &v, &n);
nway[0] =3D 1;
for(i=3D0; i<v; i++) {
fscanf(fin, "%d", &c);
for(j=3Dc; j<=3Dn; j++)
nway[j] +=3D nway[j-c];
}
fprintf(fout, "%lld\n", nway[n]);
}</STRONG></PRE></DIV></TD></TR></TBODY></TABLE><BR>
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>•</a></td><td><a href=3D"http://hi.baidu.com/' + arg[i][3] + =
'/blog/item/' + arg[i][2] + '.html" target=3D"_blank" title=3D"' + =
arg[i][0] + '">' + arg[i][1] + '</a>');
D(html, new Array(10).join('\u3000'));
D(html, '</td>');
if(arg[i + 1][0] !=3D "")
D(html, '<td width=3D"15px"><a style=3D"font-size:25px" =
>•</a></td><td><a href=3D"http://hi.baidu.com/' + arg[i + 1][3] + =
'/blog/item/' + arg[i + 1][2] + '.html" target=3D"_blank" title=3D"' + =
arg[i + 1][0] + '">' + arg[i + 1][1] + '</a></td>');
else
D(html, '<td> </td><td> </td>');
D(html, '</tr>');
}
if(hasMore) D(html, '<tr><td colspan=3D"4"><a target=3D"_blank" =
href=3D"/sys/search?pageno=3D1&type=3D7&sort=3D1&word=3DUSACO%202%2E3%2E4=
%20Money%20Systems%20%CC%E2%BD%E2&item=3D97e902b376e020a3d8335ae4">=B8=FC=
=B6=E0>></a></td></tr>');
D(html, '</table></div><div class=3D"line"> </div>');
var div =3D document.getElementById('in_related_tmp');
if(div){
div.innerHTML =3D html.join('');
while(div.firstChild){
div.parentNode.insertBefore(div.firstChild, div);
}
div.parentNode.removeChild(div);
}
}
if(RelatedDocData =3D=3D -1){ // not supported xhr
var script =3D document.createElement('script');
script.type =3D 'text/javascript';
script.src =3D =
'/sys/search?type=3D8&word=3DUSACO%202%2E3%2E4%20Money%20Systems%20%CC%E2=
%BD%E2&item=3D97e902b376e020a3d8335ae4&t=3D' + new Date().getTime();
document.getElementsByTagName('HEAD')[0].appendChild(script);
}else if(RelatedDocData =3D=3D null){
GetAndEval =3D true;
}else{
eval(RelatedDocData);
}
/*]]>*/
</SCRIPT>
<DIV id=3Din_reader>
<DIV class=3Dtit>=D7=EE=BD=FC=B6=C1=D5=DF=A3=BA</DIV>
<SCRIPT>
var g_spAnnony=3Dfalse;
var g_read=3D[
{}
];
g_read.length=3Dg_read.length-1;
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