Эффективные алгоритмы, осень 2007: Нахождение максимального потока

  • Published on
    08-Feb-2017

  • View
    124

  • Download
    1

Embed Size (px)

Transcript

<ul><li><p>/ 18: </p><p>. </p><p>Computer Science http://logic.pdmi.ras.ru/infclub/</p><p>. (CS ) 18. 1 / 43</p><p>http://logic.pdmi.ras.ru/~infclub/</p></li><li><p>1 </p><p>2 -</p><p>3 </p><p>4 ---</p><p>. (CS ) 18. 2 / 43</p></li><li><p>1 </p><p>2 -</p><p>3 </p><p>4 ---</p><p>. (CS ) 18. 2 / 43</p></li><li><p>1 </p><p>2 -</p><p>3 </p><p>4 ---</p><p>. (CS ) 18. 2 / 43</p></li><li><p>1 </p><p>2 -</p><p>3 </p><p>4 ---</p><p>. (CS ) 18. 2 / 43</p></li><li><p>1 </p><p>2 -</p><p>3 </p><p>4 ---</p><p>. (CS ) 18. 3 / 43</p></li><li><p> (flow network) , (u, v) c(u, v)( (u, v) E , c(u, v) = 0), (capacity), , s (source) t (sink). (flow) G = (V ,E ) f : V V R, :</p><p>I : u, v V , f (u, v) c(u, v);I : u, v V , f (u, v) = f (v , u);I : u V {s, t},</p><p>vV f (u, v) = 0.</p><p> (value) </p><p>vV f (s, v). (maximum flow-problem) .</p><p>. (CS ) 18. 4 / 43</p></li><li><p> (flow network) , (u, v) c(u, v)( (u, v) E , c(u, v) = 0), (capacity), , s (source) t (sink).</p><p> (flow) G = (V ,E ) f : V V R, :</p><p>I : u, v V , f (u, v) c(u, v);I : u, v V , f (u, v) = f (v , u);I : u V {s, t},</p><p>vV f (u, v) = 0.</p><p> (value) </p><p>vV f (s, v). (maximum flow-problem) .</p><p>. (CS ) 18. 4 / 43</p></li><li><p> (flow network) , (u, v) c(u, v)( (u, v) E , c(u, v) = 0), (capacity), , s (source) t (sink). (flow) G = (V ,E ) f : V V R, :</p><p>I : u, v V , f (u, v) c(u, v);I : u, v V , f (u, v) = f (v , u);I : u V {s, t},</p><p>vV f (u, v) = 0.</p><p> (value) </p><p>vV f (s, v). (maximum flow-problem) .</p><p>. (CS ) 18. 4 / 43</p></li><li><p> (flow network) , (u, v) c(u, v)( (u, v) E , c(u, v) = 0), (capacity), , s (source) t (sink). (flow) G = (V ,E ) f : V V R, :</p><p>I : u, v V , f (u, v) c(u, v);</p><p>I : u, v V , f (u, v) = f (v , u);I : u V {s, t},</p><p>vV f (u, v) = 0.</p><p> (value) </p><p>vV f (s, v). (maximum flow-problem) .</p><p>. (CS ) 18. 4 / 43</p></li><li><p> (flow network) , (u, v) c(u, v)( (u, v) E , c(u, v) = 0), (capacity), , s (source) t (sink). (flow) G = (V ,E ) f : V V R, :</p><p>I : u, v V , f (u, v) c(u, v);I : u, v V , f (u, v) = f (v , u);</p><p>I : u V {s, t},</p><p>vV f (u, v) = 0.</p><p> (value) </p><p>vV f (s, v). (maximum flow-problem) .</p><p>. (CS ) 18. 4 / 43</p></li><li><p> (flow network) , (u, v) c(u, v)( (u, v) E , c(u, v) = 0), (capacity), , s (source) t (sink). (flow) G = (V ,E ) f : V V R, :</p><p>I : u, v V , f (u, v) c(u, v);I : u, v V , f (u, v) = f (v , u);I : u V {s, t},</p><p>vV f (u, v) = 0.</p><p> (value) </p><p>vV f (s, v). (maximum flow-problem) .</p><p>. (CS ) 18. 4 / 43</p></li><li><p> (flow network) , (u, v) c(u, v)( (u, v) E , c(u, v) = 0), (capacity), , s (source) t (sink). (flow) G = (V ,E ) f : V V R, :</p><p>I : u, v V , f (u, v) c(u, v);I : u, v V , f (u, v) = f (v , u);I : u V {s, t},</p><p>vV f (u, v) = 0.</p><p> (value) </p><p>vV f (s, v).</p><p> (maximum flow-problem) .</p><p>. (CS ) 18. 4 / 43</p></li><li><p> (flow network) , (u, v) c(u, v)( (u, v) E , c(u, v) = 0), (capacity), , s (source) t (sink). (flow) G = (V ,E ) f : V V R, :</p><p>I : u, v V , f (u, v) c(u, v);I : u, v V , f (u, v) = f (v , u);I : u V {s, t},</p><p>vV f (u, v) = 0.</p><p> (value) </p><p>vV f (s, v). (maximum flow-problem) .</p><p>. (CS ) 18. 4 / 43</p></li><li><p>3 10</p><p>3</p><p>4</p><p>5</p><p>2</p><p>1</p><p>11</p><p>2</p><p>5</p><p>2 0</p><p>1</p><p>4</p><p>5</p><p>2</p><p>1</p><p>01</p><p>2</p><p>5</p><p>. (CS ) 18. 5 / 43</p></li><li><p> . ( , , , ). , , (multicommodity flow).</p><p>. (CS ) 18. 6 / 43</p></li><li><p> .</p><p> ( , , , ). , , (multicommodity flow).</p><p>. (CS ) 18. 6 / 43</p></li><li><p> . ( , , , ).</p><p> , , (multicommodity flow).</p><p>. (CS ) 18. 6 / 43</p></li><li><p> . ( , , , ). , , (multicommodity flow).</p><p>. (CS ) 18. 6 / 43</p></li><li><p>1 </p><p>2 -</p><p>3 </p><p>4 ---</p><p>. (CS ) 18. 7 / 43</p></li><li><p> -</p><p>Ford-Fulkerson-Method(G , s, t)1 f 02 while p3 do f p4 return f</p><p>. (CS ) 18. 8 / 43</p></li><li><p> G f .</p><p> u v (residual capacity of(u, v)) cf (u, v) = c(u, v) f (u, v). (residual network) G , f , Gf = (V ,Ef ), </p><p>Ef = {(u, v) V V : cf (u, v) &gt; 0}.</p><p>cf (u, v) c(u, v). .</p><p>. (CS ) 18. 9 / 43</p></li><li><p> G f .</p><p> u v (residual capacity of(u, v)) cf (u, v) = c(u, v) f (u, v).</p><p> (residual network) G , f , Gf = (V ,Ef ), </p><p>Ef = {(u, v) V V : cf (u, v) &gt; 0}.</p><p>cf (u, v) c(u, v). .</p><p>. (CS ) 18. 9 / 43</p></li><li><p> G f .</p><p> u v (residual capacity of(u, v)) cf (u, v) = c(u, v) f (u, v). (residual network) G , f , Gf = (V ,Ef ), </p><p>Ef = {(u, v) V V : cf (u, v) &gt; 0}.</p><p>cf (u, v) c(u, v). .</p><p>. (CS ) 18. 9 / 43</p></li><li><p> G f .</p><p> u v (residual capacity of(u, v)) cf (u, v) = c(u, v) f (u, v). (residual network) G , f , Gf = (V ,Ef ), </p><p>Ef = {(u, v) V V : cf (u, v) &gt; 0}.</p><p>cf (u, v) c(u, v). .</p><p>. (CS ) 18. 9 / 43</p></li><li><p> G f .</p><p> u v (residual capacity of(u, v)) cf (u, v) = c(u, v) f (u, v). (residual network) G , f , Gf = (V ,Ef ), </p><p>Ef = {(u, v) V V : cf (u, v) &gt; 0}.</p><p>cf (u, v) c(u, v).</p><p> .</p><p>. (CS ) 18. 9 / 43</p></li><li><p> G f .</p><p> u v (residual capacity of(u, v)) cf (u, v) = c(u, v) f (u, v). (residual network) G , f , Gf = (V ,Ef ), </p><p>Ef = {(u, v) V V : cf (u, v) &gt; 0}.</p><p>cf (u, v) c(u, v). .</p><p>. (CS ) 18. 9 / 43</p></li><li><p> (augmenting path) Gf . p(residual capacity of f ) , p: cf (p) = min{cf (u, v) : (u, v) p}.</p><p> , .</p><p>. (CS ) 18. 10 / 43</p></li><li><p> (augmenting path) Gf .</p><p> p(residual capacity of f ) , p: cf (p) = min{cf (u, v) : (u, v) p}.</p><p> , .</p><p>. (CS ) 18. 10 / 43</p></li><li><p> (augmenting path) Gf . p(residual capacity of f ) , p: cf (p) = min{cf (u, v) : (u, v) p}.</p><p> , .</p><p>. (CS ) 18. 10 / 43</p></li><li><p> (augmenting path) Gf . p(residual capacity of f ) , p: cf (p) = min{cf (u, v) : (u, v) p}.</p><p> , .</p><p>. (CS ) 18. 10 / 43</p></li><li><p> (cut) S T = V S , s S t T . (cut capacity) S T . (cut flow) , . (minimum cut) .</p><p> , .</p><p>. (CS ) 18. 11 / 43</p></li><li><p> (cut) S T = V S , s S t T .</p><p> (cut capacity) S T . (cut flow) , . (minimum cut) .</p><p> , .</p><p>. (CS ) 18. 11 / 43</p></li><li><p> (cut) S T = V S , s S t T . (cut capacity) S T .</p><p> (cut flow) , . (minimum cut) .</p><p> , .</p><p>. (CS ) 18. 11 / 43</p></li><li><p> (cut) S T = V S , s S t T . (cut capacity) S T . (cut flow) , .</p><p> (minimum cut) .</p><p> , .</p><p>. (CS ) 18. 11 / 43</p></li><li><p> (cut) S T = V S , s S t T . (cut capacity) S T . (cut flow) , . (minimum cut) .</p><p> , .</p><p>. (CS ) 18. 11 / 43</p></li><li><p> (cut) S T = V S , s S t T . (cut capacity) S T . (cut flow) , . (minimum cut) .</p><p> , .</p><p>. (CS ) 18. 11 / 43</p></li><li><p>3 10</p><p>3</p><p>4</p><p>5</p><p>2</p><p>1</p><p>11</p><p>2</p><p>5</p><p>. (CS ) 18. 12 / 43</p></li><li><p> f G . :</p><p>1 f .2 Gf .3 (S ,T ), |f | = c(S ,T ).</p><p> :</p><p>1 2: , .2 3: , S </p><p> , s .3 1: </p><p> .</p><p>. (CS ) 18. 13 / 43</p></li><li><p> f G . :</p><p>1 f .</p><p>2 Gf .3 (S ,T ), |f | = c(S ,T ).</p><p> :</p><p>1 2: , .2 3: , S </p><p> , s .3 1: </p><p> .</p><p>. (CS ) 18. 13 / 43</p></li><li><p> f G . :</p><p>1 f .2 Gf .</p><p>3 (S ,T ), |f | = c(S ,T ).</p><p> :</p><p>1 2: , .2 3: , S </p><p> , s .3 1: </p><p> .</p><p>. (CS ) 18. 13 / 43</p></li><li><p> f G . :</p><p>1 f .2 Gf .3 (S ,T ), |f | = c(S ,T ).</p><p> :</p><p>1 2: , .2 3: , S </p><p> , s .3 1: </p><p> .</p><p>. (CS ) 18. 13 / 43</p></li><li><p> f G . :</p><p>1 f .2 Gf .3 (S ,T ), |f | = c(S ,T ).</p><p> :</p><p>1 2: , .2 3: , S </p><p> , s .3 1: </p><p> .</p><p>. (CS ) 18. 13 / 43</p></li><li><p> f G . :</p><p>1 f .2 Gf .3 (S ,T ), |f | = c(S ,T ).</p><p> :</p><p>1 2: , .</p><p>2 3: , S , s .</p><p>3 1: .</p><p>. (CS ) 18. 13 / 43</p></li><li><p> f G . :</p><p>1 f .2 Gf .3 (S ,T ), |f | = c(S ,T ).</p><p> :</p><p>1 2: , .2 3: , S </p><p> , s .</p><p>3 1: .</p><p>. (CS ) 18. 13 / 43</p></li><li><p> f G . :</p><p>1 f .2 Gf .3 (S ,T ), |f | = c(S ,T ).</p><p> :</p><p>1 2: , .2 3: , S </p><p> , s .3 1: </p><p> .</p><p>. (CS ) 18. 13 / 43</p></li><li><p> -</p><p>Ford-Fulkerson(G , s, t)1 for (u, v) E2 do f [u, v ] 03 f [v , u] 04 while Gf p s t5 do cf (p) min{cf (u, v) : (u, v) p}6 for (u, v) p7 do f [u, v ] f [u, v ] + cf (p)8 f [v , u] f [u, v ]9 return f</p><p>. (CS ) 18. 14 / 43</p></li><li><p>10</p><p>3</p><p>4</p><p>1</p><p>1</p><p>23</p><p>2</p><p>1</p><p>5</p><p>5</p><p>. (CS ) 18. 15 / 43</p></li><li><p>1</p><p>1</p><p>1</p><p>1</p><p>1</p><p>1</p><p>1</p><p>1</p><p>41</p><p>1</p><p>10</p><p>3</p><p>4</p><p>1</p><p>1</p><p>22</p><p>1</p><p>1</p><p>4</p><p>. (CS ) 18. 15 / 43</p></li><li><p>2</p><p>2</p><p>1</p><p>1</p><p>2</p><p>1</p><p>21</p><p>2</p><p>1</p><p>41</p><p>2</p><p>10</p><p>3</p><p>4</p><p>1</p><p>1</p><p>2</p><p>4</p><p>4</p><p>3</p><p>. (CS ) 18. 15 / 43</p></li><li><p>2</p><p>3</p><p>2</p><p>1</p><p>4</p><p>5</p><p>1</p><p>3</p><p>21</p><p>2</p><p>1</p><p>4</p><p>5</p><p>10</p><p>3</p><p>4</p><p>1</p><p>1</p><p>2</p><p>1</p><p>1</p><p>1</p><p>2</p><p>. (CS ) 18. 15 / 43</p></li><li><p>2</p><p>4</p><p>2</p><p>1</p><p>5</p><p>5</p><p>1</p><p>4</p><p>21</p><p>2</p><p>1</p><p>5</p><p>1</p><p>1</p><p>5</p><p>101 2</p><p>3</p><p>1 1</p><p>. (CS ) 18. 15 / 43</p></li><li><p>2</p><p>4</p><p>1</p><p>2</p><p>1</p><p>1</p><p>5</p><p>5</p><p>2 12</p><p>21</p><p>2</p><p>110</p><p>4</p><p>1</p><p>5</p><p>1</p><p>2</p><p>5</p><p>. (CS ) 18. 15 / 43</p></li><li><p>, . , Ford-Fulkerson O(|E | |f *|), f * , 1., , , ., .</p><p>. (CS ) 18. 16 / 43</p></li><li><p> , .</p><p> , Ford-Fulkerson O(|E | |f *|), f * , 1., , , ., .</p><p>. (CS ) 18. 16 / 43</p></li><li><p> , . , Ford-Fulkerson O(|E | |f *|), f * , 1.</p><p>, , , ., .</p><p>. (CS ) 18. 16 / 43</p></li><li><p> , . , Ford-Fulkerson O(|E | |f *|), f * , 1., , , .</p><p>, .</p><p>. (CS ) 18. 16 / 43</p></li><li><p> , . , Ford-Fulkerson O(|E | |f *|), f * , 1., , , ., .</p><p>. (CS ) 18. 16 / 43</p></li><li><p> -</p><p> - - -, ( 1).</p><p> - O(|V | |E |2).</p><p>. (CS ) 18. 17 / 43</p></li><li><p> -</p><p> - - -, ( 1).</p><p> - O(|V | |E |2).</p><p>. (CS ) 18. 17 / 43</p></li><li><p> :</p><p> . , . O(|V |) . O(|E |) . O(|E |), O(|V ||E |2).</p><p>. (CS ) 18. 18 / 43</p></li><li><p> :</p><p> .</p><p> , . O(|V |) . O(|E |) . O(|E |), O(|V ||E |2).</p><p>. (CS ) 18. 18 / 43</p></li><li><p> :</p><p> . , .</p><p> O(|V |) . O(|E |) . O(|E |), O(|V ||E |2).</p><p>. (CS ) 18. 18 / 43</p></li><li><p> :</p><p> . , . O(|V |) .</p><p> O(|E |) . O(|E |), O(|V ||E |2).</p><p>. (CS ) 18. 18 / 43</p></li><li><p> :</p><p> . , . O(|V |) . O(|E |) .</p><p> O(|E |), O(|V ||E |2).</p><p>. (CS ) 18. 18 / 43</p></li><li><p> :</p><p> . , . O(|V |) . O(|E |) . O(|E |), O(|V ||E |2).</p><p>. (CS ) 18. 18 / 43</p></li><li><p>1 </p><p>2 -</p><p>3 </p><p>4 ---</p><p>. (CS ) 18. 19 / 43</p></li><li><p> (preflow) f : V V R, , , :u V {s}, f (V , u) =</p><p>vV f (v , u) 0</p><p> e(u) = f (V , u) (excess flow) u., , (overflowing), .</p><p>. (CS ) 18. 20 / 43</p></li><li><p> (preflow) f : V V R, , , :u V {s}, f (V , u) =</p><p>vV f (v , u) 0</p><p> e(u) = f (V , u) (excess flow) u., , (overflowing), .</p><p>. (CS ) 18. 20 / 43</p></li><li><p> (preflow) f : V V R, , , :u V {s}, f (V , u) =</p><p>vV f (v , u) 0</p><p> e(u) = f (V , u) (excess flow) u.</p><p>, , (overflowing), .</p><p>. (CS ) 18. 20 / 43</p></li><li><p> (preflow) f : V V R, , , :u V {s}, f (V , u) =</p><p>vV f (v , u) 0</p><p> e(u) = f (V , u) (excess flow) u., , (overflowing), .</p><p>. (CS ) 18. 20 / 43</p></li><li><p>, . . . , .</p><p>. (CS ) 18. 21 / 43</p></li><li><p> , .</p><p> . . , .</p><p>. (CS ) 18. 21 / 43</p></li><li><p> , . .</p><p> . , .</p><p>. (CS ) 18. 21 / 43</p></li><li><p> , . . .</p><p> , .</p><p>. (CS ) 18. 21 / 43</p></li><li><p> , . . . , .</p><p>. (CS ) 18. 21 / 43</p></li><li><p> f G . h : V N0 (height function) f , h(s) = |V |, h(t) = 0 h(u) h(v) + 1 (u, v) Ef .</p><p> , , : h(u) &gt; h(v) + 1, Ef (u, v).</p><p>. (CS ) 18. 22 / 43</p></li><li><p> f G . h : V N0 (height function) f , h(s) = |V |, h(t) = 0 h(u) h(v) + 1 (u, v) Ef .</p><p> , , : h(u) &gt; h(v) + 1, Ef (u, v).</p><p>. (CS ) 18. 22 / 43</p></li><li><p>Push(u, v)1 : u , cf (u, v) &gt; 0, h[u] = h[v ] + 12 df (u, v) min{e[u], cf (u, v)}3 f [u, v ] f [u, v ] + df (u, v)4 f [v , u] f [u, v ]5 e[u] e[u] df (u, v)6 e[v ] e[v ] + df (u, v)</p><p> u v . (saturating), .</p><p>. (CS ) 18. 23 / 43</p></li><li><p>Push(u, v)1 : u , cf (u, v) &gt; 0, h[u] = h[v ] + 12 df (u, v) min{e[u], cf (u, v)}3 f [u, v ] f [u, v ] + df (u, v)4 f [v , u] f [u, v ]5 e[u] e[u] df (u, v)6 e[v ] e[v ] + df (u, v)</p><p> u v . (saturating), .</p><p>. (CS ) 18. 23 / 43</p></li><li><p>Push(u, v)1 : u , cf (u, v) &gt; 0, h[u] = h[v ] + 12 df (u, v) min{e[u], cf (u, v)}3 f [u, v ] f [u, v ] + df (u, v)4 f [v , u] f [u, v ]5 e[u] e[u] df (u, v)6 e[v ] e[v ] + df (u, v)</p><p> u v .</p><p> (saturating), .</p><p>. (CS ) 18. 23 / 43</p></li><li><p>Push(u, v)1 : u , cf (u, v) &gt; 0, h[u] = h[v ] + 12 df (u, v) min{e[u], cf (u, v)}3 f [u, v ] f [u, v ] + df (u, v)4 f [v , u] f [u, v ]5 e[u] e[u] df (u, v)6 e[v ] e[v ] + df (u, v)</p><p> u v . (saturating), .</p><p>. (CS ) 18. 23 / 43</p></li><li><p>Lift(u)1 : u , (u, v) Ef , h[u] h[v ]2 h[u] 1+min{h[v ] | (u, v) Ef }</p><p>e[u] = f (V , u) &gt; 0, , v , f (v , u) &gt; 0. cf (u, v) = c(u, v) f (u, v) = c(u, v) + f (v , u) &gt; 0, , (u, v) Ef ., .</p><p>. (CS ) 18. 24 / 43</p></li><li><p>Lift(u)1 : u , (u, v) Ef , h[u] h[v ]2 h[u] 1+min{h[v ] | (u, v) Ef }</p><p>e[u] = f (V , u) &gt; 0, , v , f (v , u) &gt; 0. cf (u, v) = c(u, v) f (u, v) = c(u, v) + f (v , u) &gt; 0, , (u, v) Ef ., .</p><p>. (CS ) 18. 24 / 43</p></li><li><p>Lift(u)1 : u , (u, v) Ef , h[u] h[v ]2 h[u] 1+min{h[v ] | (u, v) Ef }</p><p>e[u] = f (V , u) &gt; 0, , v , f (v , u) &gt; 0.</p><p> cf (u, v) = c(u, v) f (u, v) = c(u, v) + f (v , u) &gt; 0, , (u, v) Ef ., .</p><p>. (CS ) 18. 24 / 43</p></li><li><p>Lift(u)1 : u , (u, v) Ef , h[u] h[v ]2 h[u] 1+min{h[v ] | (u, v) Ef }</p><p>e[u] = f (V , u) &gt; 0, , v , f (v , u) &gt; 0. cf (u, v) = c(u, v) f (u, v) = c(u, v) + f (v , u) &gt; 0, , (u, v) Ef .</p><p>, .</p><p>. (CS ) 18. 24 / 43</p></li><li><p>Lift(u)1 : u , (u, v) Ef , h[u] h[v ]2 h[u] 1+min{h[v ] | (u, v) Ef }</p><p>e[u] = f (V , u) &gt; 0, , v , f (v , u) &gt; 0. cf (u, v) = c(u, v) f (u, v) = c(u, v) + f (v , u) &gt; 0, , (u, v) Ef ., .</p><p>. (CS ) 18. 24 / 43</p></li><li><p>Generic-Preflow-Push(G )1 </p><p> , </p><p>2 |V |, </p><p>3 while 4 do </p><p>. (CS ) 18. 25 / 43</p></li><li><p> u , , . . : , |V |, |V | , 2. , . , .</p><p>. (CS ) 18. 26 / 43</p></li><li><p> u , , .</p><p> . : , |V |, |V | , 2. , . , .</p><p>. (CS ) 18. 26 / 43</p></li><li><p> u , , . .</p><p> : , |V |, |V | , 2. , . , .</p><p>. (CS ) 18. 26 / 43</p></li><li><p> u , , . . : , |V |, |V | , 2.</p><p> , . , .</p><p>. (CS ) 18. 26 / 43</p></li><li><p> u , , . . : , |V |, |V | , 2. , . , .</p><p>. (CS ) 18. 26 / 43</p></li><li><p> 2|V | 1.</p><p> u , Gf u s( G s u, ). 1., u 2|V | 1.</p><p>. (CS ) 18. 27 / 43</p></li><li><p> 2|V | 1.</p><p> u , Gf u s( G s u, ). 1., u 2|V | 1.</p><p>. (CS ) 18. 27 / 43</p></li><li><p> 2|V | 1.</p><p> u , Gf u s( G s u, ).</p><p> 1., u 2|V | 1.</p><p>. (CS ) 18. 27 / 43</p></li><li><p> 2|V | 1.</p><p> u , Gf u s( G s u, ). 1.</p><p>, u 2|V | 1.</p><p>. (CS ) 18. 27 / 43</p></li><li><p> 2|V | 1.</p><p> u , Gf u s( G s u, ). 1., u 2|V | 1.</p><p>. (CS ) 18. 27 / 43</p></li><li><p> 2|V |2.</p><p> 2|V | 1 , , , |V | 2.</p><p>. (CS ) 18. 28 / 43</p></li><li><p> 2|V |2.</p><p> 2|V | 1 , , , |V | 2.</p><p>. (CS ) 18. 28 / 43</p></li><li><p>...</p></li></ul>

Recommended

View more >