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{TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT 307 19 "Matem\341tica Discreta
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 22 "ALG
ORITMOS DE B\332SQUEDA" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 9 "Problema:" }{TEXT 
-1 27 " dada una lista de n\372meros " }{TEXT 298 1 "a" }{TEXT -1 13 "
 y un n\372mero " }{TEXT 257 1 "b" }{TEXT -1 64 " determinar si dicho \+
n\372mero es uno de los elementos de la lista." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 19 "B\372squeda secue
ncial" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "
El primer algoritmo que presentamos toma una lista de n\372meros " }
{TEXT 257 2 "a " }{TEXT -1 12 "y un n\372mero " }{TEXT 258 0 "" }
{TEXT 299 1 "b" }{TEXT -1 105 " y devuelve un vector donde cada entrad
a es: un uno si la entrada de la lista correspondiente es igual a " }
{TEXT 259 0 "" }{TEXT -1 0 "" }{TEXT 260 2 "b " }{TEXT -1 28 "y un cer
o en caso contrario." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 144 "Vector:=proc(a::list,b)\nlocal c,j,i:\nc
:=[seq(0*j,j=1..nops(a))]: \nfor i from 1 to nops(a) do \n    if a[i]=
b then\n    c[i]:=1 fi: \nod:\nc; \nend: \n" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 22 "Vector([1,2,0,4,5],4);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT 297 7 "Ejemplo" }{TEXT -1 42 " de una aplicaci\363n a la teor
\355a de n\372meros:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 41 "\277Es 356 un cuadrado de un n\372mero entero? " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 169 "Como 10^
2 =100 y 20^2=400 entonces si 356 fuera un cuadrado, lo ser\355a de un
 n\372mero entre 11 y 19, tomando la lista de dichos cuadrados podemos
 ver si 356 es un cuadrado." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Vector([seq(x^2,x=11..19)],356);" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Como no
 est\341 en la lista (pues la salida del procedimiento " }{TEXT 300 6 
"Vector" }{TEXT -1 61 " esta formada \372nicamente por ceros) no puede
 ser un cuadrado." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 78 "Dado este algoritmo se pueden responder distintas pregunt
as, como por ejemplo:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 15 "a) el elemento " }{TEXT 261 2 "b " }{TEXT -1 17 "est
\341 en la lista " }{TEXT 262 2 "a " }{TEXT -1 44 "si el resultado de \+
aplicar el procedimiento " }{TEXT 265 6 "Vector" }{TEXT -1 12 " a la l
ista " }{TEXT 266 2 "a " }{TEXT -1 19 "contiene alg\372n uno;" }}
{PARA 0 "" 0 "" {TEXT -1 24 "b) el n\372mero de unos de " }{TEXT 268 
9 "Vector(a)" }{TEXT -1 25 " dice cu\341ntas veces est\341 " }{TEXT 
269 1 "b" }{TEXT -1 13 " en la lista " }{TEXT 270 1 "a" }{TEXT -1 1 ";
" }{TEXT 267 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "c) si " }{TEXT 301 1 "
b" }{TEXT -1 18 " est\341 en la lista " }{TEXT 302 3 "a, " }{TEXT -1 
82 "se puede construir un sencillo algoritmo que obtenga todos los lug
ares donde est\341 " }{TEXT 263 1 "b" }{TEXT -1 13 " en la lista " }
{TEXT 303 2 "a." }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 264 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 143 "preguntasayb:=proc(a::list,b)\nlocal i,l,c:\nl:
=add(Vector(a,b)[i],i=1..nops(Vector(a,b))): \nif l=0 then c:=[no,0] \+
\nelse c:=[si,l] \nfi: \nc; \nend:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 35 "Lo aplicamos a un ejemplo concreto:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "preguntasayb([1,2,1,3,0,1,0],1);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 31 "Para responder a la pregunta c)" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
188 "preguntac:=proc(a::list,b)\nlocal c,j,i:\nc:=[seq(0*j,j=1..pregun
tasayb(a,b)[2])]: \nj:=1: \nfor i from 1 to nops(a) do\n    if Vector(
a,b)[i]=1 then\n    c[j]:=i: j:=j+1: \n    fi: \nod:\nc; \nend:" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Lo aplica
mos a un ejemplo concreto:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "preguntac([1,2,1,3,0,1,0],1);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "En cualquier caso estos algorit
mos tienen " }{TEXT 271 19 "complejidad lineal " }{TEXT -1 40 "ya que \+
cuando tiene una lista de tama\361o " }{TEXT 272 2 "n " }{TEXT -1 23 "
realizan esencialmente " }{TEXT 273 3 "kn " }{TEXT -1 16 "operaciones \+
con " }{TEXT 274 1 "k" }{TEXT -1 47 " constante. Comprobamos esta \372
ltima afirmaci\363n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 24 "El procedimiento vector:" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 105 "Vector:=proc(a::list,b)\nlocal c,j,
i:\nc:=[seq(0*j,j=1..nops(a))]:                                se hace
n " }{TEXT 287 2 "n " }{TEXT -1 97 "asignaciones \nfor i from 1 to nop
s(a) do                                  un bucle que se repite " }
{TEXT 288 1 "n" }{TEXT -1 123 " veces     \nif a[i]=b then c[i]:=1 fi:
 od:                                        se hace una comparaci\363n
 y una asignaci\363n " }}{PARA 0 "" 0 "" {TEXT -1 7 "c; end:" }}{PARA 
0 "" 0 "" {TEXT -1 74 "                                               \+
                           " }{TEXT 289 7 "Total: " }{TEXT -1 0 "" }
{TEXT 290 14 "3n operaciones" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 30 
"El procedimiento preguntasayb:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 68 "preguntasayb:=proc(a::list,b)            \+
                           " }}{PARA 0 "" 0 "" {TEXT -1 104 "local i,l
,c:\nl:=add(Vector(a,b)[i],i=1..nops(Vector(a,b))):   se hace el proce
dimiento Vector por tanto " }{TEXT 291 3 "3n " }{TEXT -1 11 "operacion
es" }}{PARA 0 "" 0 "" {TEXT -1 99 "                                   \+
                                     luego se hacen como m\341ximo " }
{TEXT 292 2 "n " }{TEXT -1 6 "sumas " }}{PARA 0 "" 0 "" {TEXT -1 124 "
if l=0 then c:=[no,0]                                          una com
paraci\363n y una asignaci\363n \nelse c:=[si,l] fi: \nc; end:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "         \+
                                                                " }
{TEXT 256 5 "Total" }{TEXT 293 2 ": " }{TEXT -1 0 "" }{TEXT 257 16 "4n
+2 operaciones" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "El proced
imiento " }{TEXT 275 9 "preguntac" }{TEXT -1 1 ":" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 117 "preguntac:=proc(a::list,
b)\nc:=[seq(0*j,j=1..preguntasayb(a,b)[2])]:    se hace  procedimiento
 preguntasayb por tanto " }{TEXT 294 5 "4n+2 " }{TEXT -1 11 "operacion
es" }}{PARA 0 "" 0 "" {TEXT -1 92 "                                   \+
                              luego se hacen como m\341ximo " }{TEXT 
276 1 "n" }{TEXT -1 14 "  asignaciones" }}{PARA 0 "" 0 "" {TEXT -1 75 
"j:=1:                                                          1 asig
naci\363n" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
74 "for i from 1 to nops(a) do                         un bucle que se
 repite " }{TEXT 277 2 "n " }{TEXT -1 51 "veces y cada vez que se repi
te hace 3 operaciones  " }}{PARA 0 "" 0 "" {TEXT -1 42 "if Vector(a,b)
[i]=1 then c[j]:=i: j:=j+1: " }}{PARA 0 "" 0 "" {TEXT -1 69 "fi: od:\n
c; end:                                                      " }{TEXT 
286 5 "Total" }{TEXT -1 2 ": " }{TEXT 295 4 "8n+3" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 296 16 "Complejid
ad O(n)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 16 "B\372squeda binaria" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 114 "Se trata de un algoritmo que toma una li
sta ordenada de menor a mayor de n\372meros y realiza la b\372squeda c
omparando " }{TEXT 278 1 "b" }{TEXT -1 320 " con el elemento que est
\341 en el medio de la lista: si es igual, el algoritmo termina; si es
 menor, trabaja con una nueva lista (la de los elementos menores que e
l elemento intermedio) e itera el algoritmo; si es mayor, trabaja con \+
una nueva lista (la de los elementos mayores que el elemento intermedi
o) y hace lo mismo." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 288 "Binaria:=
proc(a::list,b)\nlocal c,i,j,k,s:\ni:=1: j:=nops(a): c:=0; \nfor s fro
m 1 to nops(a) while j-i>-1 do\n    k:=floor((i+j)/2):\n    if b=a[k] \+
then c:=1: s:=nops(a)+1: fi: \n    if b>a[k] then i:=k+1: fi:\n    if \+
b<a[k] then j:=k-1: fi:\nod:\nif c=1 then RETURN(si) \nelse RETURN(no)
 fi:\nend:" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 45 "Comprobemos el algoritmo en algunos ejemplos:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "Binaria([3
,4,5,6,7,7.5,8,9],7.25);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 
"Binaria([3,4,5,6,7,7.5,8,9],9);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "Binaria([3,4,5,6,7,7.5,8,9],4);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 33 "Binaria([seq(x^2,x=10..20)],169);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 56 "En el \372ltimo ejemplo la respuesta es \+
s\355 porque 169=13^2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 126 "Para estudiar la co
mplejidad es relevante saber cu\341ntas veces como m\341ximo se repite
 el bucle, ya que cada vez que el contador " }{TEXT 306 2 "s " }{TEXT 
-1 221 "toma un valor fijado, el n\372mero de operaciones que se efect
\372an es constante. Como la lista se va dividiendo en una lista de ta
ma\361o la mitad, hasta que queda una lista con un \372nico elemento, \+
el bucle se repite como m\341ximo " }{TEXT 304 0 "" }{TEXT 305 0 "" }
{TEXT 279 7 "log(n) " }{TEXT -1 7 " donde " }{TEXT 280 4 "log " }
{TEXT -1 26 "es el logaritmo en base 2." }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 129 "Binaria:=proc(a::list,b)\nlocal c,i
,j,k,s:\ni:=1: j:=nops(a): c:=0;                                      \+
         tres asignaciones" }}{PARA 257 "" 0 "" {TEXT -1 79 "         \+
                                                                      \+
" }{TEXT 281 18 "el bucle se repite" }{TEXT -1 7 " log(n)" }{TEXT 282 
25 " veces y cada vez se hace" }}{PARA 0 "" 0 "" {TEXT -1 230 "for s f
rom 1 to nops(a) while j-i>-1 do                                   una
 comparaci\363n\nk:=floor((i+j)/2):                                   \+
                                 una asignaci\363n\nif b=a[k] then c:=
1: s:=nops(a)+1: fi: " }}{PARA 0 "" 0 "" {TEXT -1 642 "               \+
                                                                      \+
       una comparaci\363n y una asignaci\363n  (en el peor de los caso
s el                                                                  \+
                                                                      \+
                                                 n\372mero no est\341 \+
en la lista)                                            \nif b>a[k] th
en i:=k+1: fi:\nif b<a[k] then j:=k-1: fi: od:           \nif c=1 then
 RETURN(si) else RETURN(no) fi:          una comparaci\363n y una asig
naci\363n\nend:                                                       \+
                   " }{TEXT 283 7 "Total: " }{TEXT 284 9 "4log(n)+5" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 21 "Este algoritmo tiene " }{TEXT 285 33 "com
plejidad logar\355tmica O(log(n))" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
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