doc/informe.rtf

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  61 <p style="Heading 1">Estrategia de resolución</p>
  62 <p style="Heading 2">Representación de números grandes</p>
  63 <p style="Normal">El número grande es representado por una cola de "doble-fin" (double ended queue o std::deque en C++), que permite operaciones de inserción al frente y al final en O(1), además de acceso aleatorio a cualquier posición del contenedor también O(1)<field footnote-id="1" type="footnote_ref" style="Footnote Reference"></field><foot footnote-id="1">
  64 <p footnote-id="1" style="Footnote Text" props="text-position:subscript; font-weight:normal; font-style:normal"><field footnote-id="1" type="footnote_anchor" style="Footnote Text" props="text-position:superscript"></field><c footnote-id="1"> </c><c footnote-id="1" props="font-family:Times New Roman; font-size:12pt; color:000000; text-decoration:none; text-position:normal; font-weight:normal; font-style:normal; lang:es-ES">The C++ Programming Language, Bjarne Stroustrup, 3ra edición, sección 17.1.2.</c></p>
  65 </foot>. Esto permite hacer shifts O(pos) siendo pos la cantidad de posiciones a desplazar. Dentro de esta deque se guardan valores de tipo nativo (parametrizable, por omisión se utiliza uin32_t), quedando como primer elemento (llamaremos a partir de ahora a cada elemento de este contenedor <c props="font-weight:bold">chunk</c>) del deque los bits menos significativos del número (podría decirse que es little endian).</p>
  66 <p style="Normal" props="font-weight:normal; font-style:normal">Supongamos un tipo nativo de 4 bits (para hacer el ejemplo bien simple), el número 159 (0x103 == 0001 0000 0011<c props="text-position:subscript">b</c>) quedaría representado por:</p>
  67 <p style="Normal" props="font-weight:normal; font-style:normal"></p>
  68 <table props="table-column-props:1.62cm/1.00cm/1.00cm/1.00cm/; table-row-heights:0.62cm/0.62cm/; table-column-leftpos:5.00cm; list-tag:1">
  69 <cell props="left-attach:0; right-attach:1; bot-style:1; bot-attach:1; top-attach:0; bot-color:000000; bg-style:1">
  70 <p style="Normal" props="font-weight:normal; font-style:normal">posición</p>
  71 </cell>
  72 <cell props="left-attach:1; right-attach:2; bot-style:1; bot-attach:1; top-attach:0; bot-color:000000; bg-style:1">
  73 <p props="font-weight:normal; font-style:normal">0</p>
  74 </cell>
  75 <cell props="left-attach:2; right-attach:3; bot-style:1; bot-attach:1; top-attach:0; bot-color:000000; bg-style:1">
  76 <p style="Normal" props="font-weight:normal; font-style:normal">1</p>
  77 </cell>
  78 <cell props="left-attach:3; right-attach:4; bot-style:1; bot-attach:1; top-attach:0; bot-color:000000; bg-style:1">
  79 <p style="Normal" props="font-weight:normal; font-style:normal">2</p>
  80 </cell>
  81 <cell props="bot-attach:2; left-attach:0; right-attach:1; top-attach:1">
  82 <p style="Normal" props="font-weight:normal; font-style:normal">valor</p>
  83 </cell>
  84 <cell props="bot-attach:2; left-attach:1; right-attach:2; top-attach:1">
  85 <p props="font-weight:normal; font-style:normal">0011</p>
  86 </cell>
  87 <cell props="bot-attach:2; left-attach:2; right-attach:3; top-attach:1">
  88 <p style="Normal" props="font-weight:normal; font-style:normal">0000</p>
  89 </cell>
  90 <cell props="bot-attach:2; left-attach:3; right-attach:4; top-attach:1">
  91 <p style="Normal" props="font-weight:normal; font-style:normal">0001</p>
  92 </cell>
  93 </table>
  94 <p style="Normal" props="font-weight:normal; font-style:normal"></p>
  95 <p style="Normal">Los número nativos usados son no signados, guardándose el signo por separado en una variable enumerada que puede tomar los valores <c props="font-weight:normal">positive</c> o <c props="font-weight:normal; font-style:normal">negative</c>.</p>
  96 <p style="Heading 2">Suma</p>
  97 <p style="Normal">La suma se divide en dos casos particulares:</p>
  98 <p level="1" listid="1000" parentid="0" style="Normal" props="start-value:1; text-indent:-0.2993in; list-style:Numbered List; field-font:NULL; margin-left:0.5000in"><field type="list_label" props="width:0in; font-family:Times New Roman; display:inline; font-style:normal; font-weight:normal; bgcolor:transparent; lang:es-ES; text-position:normal; homogeneous:1; color:000000; font-variant:normal; text-decoration:none; font-size:12pt; height:0in; list-style:Numbered List; font-stretch:normal"></field><c props="width:0in; font-family:Times New Roman; display:inline; font-style:normal; font-weight:normal; bgcolor:transparent; lang:es-ES; text-position:normal; homogeneous:1; color:000000; font-variant:normal; text-decoration:none; font-size:12pt; height:0in; list-style:Numbered List; font-stretch:normal">       Los operandos tienen signo distinto.</c></p>
  99 <p level="1" listid="1000" parentid="0" style="Normal" props="start-value:1; text-indent:-0.2993in; list-style:Numbered List; field-font:NULL; margin-left:0.5000in"><field type="list_label" props="width:0in; font-family:Times New Roman; display:inline; font-style:normal; font-weight:normal; bgcolor:transparent; lang:es-ES; text-position:normal; homogeneous:1; color:000000; font-variant:normal; text-decoration:none; font-size:12pt; height:0in; list-style:Numbered List; font-stretch:normal"></field><c props="width:0in; font-family:Times New Roman; display:inline; font-style:normal; font-weight:normal; bgcolor:transparent; lang:es-ES; text-position:normal; homogeneous:1; color:000000; font-variant:normal; text-decoration:none; font-size:12pt; height:0in; list-style:Numbered List; font-stretch:normal">       Los operandos tienen </c><c props="font-family:Times New Roman; font-size:12pt; color:000000; text-decoration:none; text-position:normal; font-weight:normal; font-style:normal; lang:es-ES">el mismo </c><c props="width:0in; font-family:Times New Roman; display:inline; font-style:normal; font-weight:normal; bgcolor:transparent; lang:es-ES; text-position:normal; homogeneous:1; color:000000; font-variant:normal; text-decoration:none; font-size:12pt; height:0in; list-style:Numbered List; font-stretch:normal">signo.</c></p>
 100 <p style="Normal"><c props="width:0in; font-family:Times New Roman; display:inline; color:000000; font-weight:normal; text-position:normal; lang:es-ES; font-style:normal; font-variant:normal; bgcolor:transparent; homogeneous:1; text-decoration:none; height:0in; font-size:12pt; font-stretch:normal">En el primer caso, directamente se utiliza el algoritmo de la resta. En el segundo caso, la suma se hace de forma lineal, sumando los dos chunks de la misma posición de cada número entre sí mientras la cantidad de chunks de cada número lo permita. Si un operando tiene más chunks que el otro, se continuará la suma simplemente propagando el carry o sumando los chunks restantes según corresponda. El peor caso es O(n).</c></p>
 101 <p style="Normal"></p>
 102 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">// suma haciendo los cambios en "numero", sería el operador +=</c></p>
 103 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">numero.sumar_inplace(otro)</c></p>
 104 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">{</c></p>
 105 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  fin = max(numero.size(), otro.size())</c></p>
 106 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  carry = 0;</c></p>
 107 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  // Procesa parte en común sumando</c></p>
 108 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  for (i = 0; i &lt; fin; i++)</c></p>
 109 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  {</c></p>
 110 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">          numero[i] = numero[i] + otro[i] + carry;</c></p>
 111 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">          if (hubo_carry)</c></p>
 112 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">                  carry = 1;</c></p>
 113 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">          else</c></p>
 114 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">                  carry = 0;</c></p>
 115 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  }</c></p>
 116 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  // Si "numero" tiene más chunks que "otro", sólo queda </c></p>
 117 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  // propagar el carry.</c></p>
 118 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  if (numero.size() &gt; otro.size())</c></p>
 119 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  {</c></p>
 120 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">          if (carry)</c></p>
 121 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">                  propagar_carry();</c></p>
 122 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  }</c></p>
 123 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  // Si "otro" tiene más chunks que "numero"</c></p>
 124 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  else if (numero.size() &lt; otro.size())</c></p>
 125 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">  {</c></p>
 126 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">          // Recorre los chunks de "otro" que faltan</c></p>
 127 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">          for (i = fin; i &lt; otro.size(); i++)</c></p>
 128 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">          {</c></p>
 129 <p style="Plain Text"><c props="height:0in; display:inline; width:0in; homogeneous:1">                  // Va haciendo crecer a "numero" con la suma</c></p>
 130 <p style="Plain Text" props="text-align:left; dom-dir:ltr"><c props="height:0in; display:inline; width:0in; homogeneous:1">                     numero.insertar_atras(otro[i] + c);</c></p>
 131 <p style="Plain Text" props="text-align:left; dom-dir:ltr"><c props="font-family:Courier New; font-size:12pt; lang:es-ES; text-position:normal; font-weight:normal; font-style:normal; text-decoration:none">                   if (hubo_carry)</c></p>
 132 <p style="Plain Text" props="margin-top:0.0000in; margin-left:0.0000in; text-indent:0.0000in; dom-dir:ltr; margin-bottom:0.0000in; line-height:1.000000; text-align:left; margin-right:0.0000in"><c props="font-family:Courier New; font-size:12pt; lang:es-ES; text-position:normal; font-weight:normal; font-style:normal; text-decoration:none">                             carry = 1;</c></p>
 133 <p style="Plain Text" props="margin-top:0.0000in; margin-left:0.0000in; text-indent:0.0000in; dom-dir:ltr; margin-bottom:0.0000in; line-height:1.000000; text-align:left; margin-right:0.0000in"><c props="font-family:Courier New; font-size:12pt; lang:es-ES; text-position:normal; font-weight:normal; font-style:normal; text-decoration:none">                     else</c></p>
 134 <p style="Plain Text" props="margin-top:0.0000in; margin-left:0.0000in; text-indent:0.0000in; dom-dir:ltr; margin-bottom:0.0000in; line-height:1.000000; text-align:left; margin-right:0.0000in"><c props="font-family:Courier New; font-size:12pt; lang:es-ES; text-position:normal; font-weight:normal; font-style:normal; text-decoration:none">                             carry = 0;</c></p>
 135 <p style="Plain Text" props="margin-top:0.0000in; margin-left:0.0000in; text-indent:0.0000in; dom-dir:ltr; margin-bottom:0.0000in; line-height:1.000000; text-align:left; margin-right:0.0000in"><c props="font-family:Courier New; font-size:12pt; lang:es-ES; text-position:normal; font-weight:normal; font-style:normal; text-decoration:none">             }</c></p>
 136 <p style="Plain Text" props="margin-top:0.0000in; margin-left:0.0000in; text-indent:0.0000in; dom-dir:ltr; margin-bottom:0.0000in; line-height:1.000000; text-align:left; margin-right:0.0000in"><c props="font-family:Courier New; font-size:12pt; lang:es-ES; text-position:normal; font-weight:normal; font-style:normal; text-decoration:none">             // Si quedó un carry lo propaga</c></p>
 137 <p style="Plain Text" props="margin-top:0.0000in; margin-left:0.0000in; text-indent:0.0000in; dom-dir:ltr; margin-bottom:0.0000in; line-height:1.000000; text-align:left; margin-right:0.0000in"><c props="font-family:Courier New; font-size:12pt; lang:es-ES; text-position:normal; font-weight:normal; font-style:normal; text-decoration:none">             if (carry)</c></p>
 138 <p style="Plain Text" props="margin-top:0.0000in; margin-left:0.0000in; text-indent:0.0000in; dom-dir:ltr; margin-bottom:0.0000in; line-height:1.000000; text-align:left; margin-right:0.0000in"><c props="font-family:Courier New; font-size:12pt; lang:es-ES; text-position:normal; font-weight:normal; font-style:normal; text-decoration:none">                     propagar_carry();</c></p>
 139 <p style="Plain Text" props="margin-top:0.0000in; margin-left:0.0000in; text-indent:0.0000in; dom-dir:ltr; margin-bottom:0.0000in; line-height:1.000000; text-align:left; margin-right:0.0000in"><c props="font-family:Courier New; font-size:12pt; lang:es-ES; text-position:normal; font-weight:normal; font-style:normal; text-decoration:none">     }</c></p>
 140 <p style="Plain Text" props="margin-top:0.0000in; margin-left:0.0000in; text-indent:0.0000in; dom-dir:ltr; margin-bottom:0.0000in; line-height:1.000000; text-align:left; margin-right:0.0000in"><c props="font-family:Courier New; font-size:12pt; lang:es-ES; text-position:normal; font-weight:normal; font-style:normal; text-decoration:none">}</c></p>
 141 <p style="Heading 2"><c props="height:0in; display:inline; width:0in; homogeneous:1">Resta</c></p>
 142 <p style="Normal"><c props="bgcolor:fffa7a; height:0in; display:inline; width:0in; homogeneous:1">TODO</c></p>
 143 <p style="Heading 2"><c props="height:0in; display:inline; width:0in; homogeneous:1">Multiplicación ingenua</c></p>
 144 <p style="Normal"><c props="font-family:Times New Roman; bgcolor:fffa7a; font-size:12pt; color:000000; text-decoration:none; text-position:normal; font-weight:normal; font-style:normal; lang:es-ES">TODO</c></p>
 145 <p style="Heading 2"><c props="height:0in; display:inline; width:0in; homogeneous:1">Multiplicación Karatsuba-Ofman</c></p>
 146 <p style="Normal"><c props="font-family:Times New Roman; bgcolor:fffa7a; font-size:12pt; color:000000; text-decoration:none; text-position:normal; font-weight:normal; font-style:normal; lang:es-ES">TODO</c></p>
 147 <p style="Heading 2"><c props="height:0in; display:inline; width:0in; homogeneous:1">Potenciación ingenua</c></p>
 148 <p style="Normal"><c props="font-family:Times New Roman; bgcolor:fffa7a; font-size:12pt; color:000000; text-decoration:none; text-position:normal; font-weight:normal; font-style:normal; lang:es-ES">TODO</c></p>
 149 <p style="Heading 2"><c props="height:0in; display:inline; width:0in; homogeneous:1">Potenciación por división y conquista</c></p>
 150 <p style="Normal"><c props="height:0in; display:inline; width:0in; homogeneous:1">Para diseñar el algoritmo de potenciación por división y conquista se pensó en la potenciación como un arbol ternario, cuyos nodos son el exponente, de manera tal que dos hijos de un exponente son la mitad del exponente (redondeando para abajo) y el tercer hijo, de estar presente, es 1 (y está presente si el padre es impar). Entonces para obtener la potencia indicada por el padre, hay que multiplicar los hijos. Un ejemplo del árbol sería:</c></p>
 151 <p style="Normal"><image dataid="3a0a2f62-3532-11da-8840-e92ea42174cc" props="height:5.01in; width:14.24in"/></p>
 152 <p style="Normal"></p>
 153 <p style="Normal">Donde X<c props="text-position:superscript">13</c> podría escribirse como: X<c props="text-position:superscript">6</c> * X * X<c props="text-position:superscript">6</c></p>
 154 <p style="Normal">X<c props="text-position:superscript">6</c> se escribiría como: X<c props="text-position:superscript">3</c> * X<c props="text-position:superscript">3</c></p>
 155 <p style="Normal">Y finalmente X3 sería: X * X * X</p>
 156 <p style="Normal">Siendo necesarias sólo 5 multiplicaciones.</p>
 157 <p style="Normal"></p>
 158 <p style="Normal">El pseudocódigo del algoritmo es el siguiente:</p>
 159 <p style="Normal"></p>
 160 <p style="Plain Text">potencia(x, n):</p>
 161 <p style="Plain Text">  if n == 1:</p>
 162 <p style="Plain Text">          return x</p>
 163 <p style="Plain Text">  result = potencia(x, n/2) // división entera</p>
 164 <p style="Plain Text">  result = result * result</p>
 165 <p style="Plain Text">  if n % 2: // si es impar</p>
 166 <p style="Plain Text">          result = result * x</p>
 167 <p style="Plain Text">  return result</p>
 168 <p style="Normal"></p>
 169 <p style="Normal">De donde podemos ver que la ecuación de recursividad sería (evaluando cantidad de multiplicaciones):</p>
 170 <p style="Normal"></p>
 171 <p style="Plain Text">T(N) = T(N/2) + O(1)</p>
 172 <p style="Normal"></p>
 173 <p style="Normal">por lo tanto el algortimo es O(log(n)).</p>
 174 </section>
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1213 </d>
1214 </data>
1215 </abiword>