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      >> VC++,C,Perl,Asp...ѧϰ,㷨.  ҵռ (0) 
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               ̰㷨

               1   ̰㷨
              Ȼһõ㷨һǼȻһЩ֮Чܹڽ㷨ƷʹЩ㷨۲Щ㷨ιġһ£Ϊ˻ýϺõܣ㷨ϸµĵĳЩ£㷨֮޷ﵽҪʱͱѰķ⡣
              ŻĸȻһֱ۵̰ⷽ㷨Ӧø㷨װ⡢⡢⡢ָ⡢·⡢Сⷽ
              1.1 Ż
              ¼½еӶŻ⣨ optimization problemÿŻⶼһ c o 
              n s t r a i n tһŻ optimization 
              functionⷽΪн⣨ feasible 
              solutionʹŻȡֵĿнΪŽ⣨optimal solution
              1-1 [ Ӥ] 
              һǳʵġСӤܵõĶһˮһͰţ̡޲ͬĹ֭಻ͬװƿӻеմˮӤɵõn 
              ֲͬϡǰn 
              ϵĲͬ飬Ӥ֪ĳЩϸԼθڣˣӤȡ·Ϊÿһϸһֵ1˾i 
              ϣۣһֵsi Ϊȸi ϡ
              ͨӤᾡþֵ޶ȵʵҪǲҵǣֵʱû㹻ӤʵҪaiǵi 
              ϵ԰˾ΪλӤҪt ˾ʣôҪnֲͬϸӤʵأ
              ϵ֪xi ΪӤҪõĵi ϵҪǣ
              ҵһʵxi1inʹn &aring;i = 1si xi 󣬲㣺n &aring;i=1xi =t 0xiai 
              Ҫָǣn &aring;i = 1ai < t򲻿ҵⷽΪʹȹеҲʹӤʡ
              ⾫ȷѧȷָ˳ɵĹЩѧʽԶ/ ʽ
              룺ntsi ai1inn Ϊtsi ai Ϊʵ
              ʵxi1inʹn &aring;i= 1si xi n &aring;i=1xi =t0xiain &aring;i = 1ai 
              <tʵĴϢ
              Уn &aring;i= 1xi =t 0xiai1inŻn &aring;i= 1si xi 
              κһʵxi ǿн⣬ʹn &aring;i= 1si xi ĿнŽ⡣
              1-2 [װ] һҴ׼װػдװﶼװڻлĴСһͬi 
              Ϊwi1inΪcǵĿڻװĻ
              ΪŻһxi ȡֵΪ01xi Ϊ0i װϴxi 
              Ϊ1i װϴǵĿҵһxi ʹn &aring;i = 1wi xi c x i &Icirc; {0, 
              1}, 1 inӦŻn &aring;i= 1xi 
              ÿһxi һн⣬ʹn &aring;i= 1xi ȡֵķŽ⡣
              1-3 [СͨѶ] 
              м֮пܵͨӿɱһͼͼÿ߶һȨֵȨֵʾʾͨҪĴۡͼж㣨Уͨͼһн⡣еȨֵǸпܵĿнⶼɱʾͼһŽоС۵
              УҪѡһͼеı߼ϵӼӼеı߹һŻӼбߵȨֵ֮͡


              ----------------------------------------------

              plot(100+t+15*cos(3.05*t),t=0..200,coords=polar,axes=none,scaling=constrained); 


       2004-5-27 19:37:49      

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              1.2 㷨˼
              ̰㷨greedy 
              methodв𲽹Žķÿ׶Σһȥŵľߣһı׼£һͲٸġ̰ߵݳΪ̰׼greedy 
              criterion
              1-4 [Ǯ] 
              һС˼ֵ1Ԫǣ1ԪǮۻԱۻԱϣĿٵӲҸСṩĿ޵ֵΪ2 
              5֡1 
              0֡5֡1ֵӲҡۻԱֲҪҵǮÿμһӲҡѡӲʱõ̰׼£ÿһѡӦʹǮΪ֤ⷨĿԣǮҪҵǮѡӲҲӦʹǮĿ
              ҪҸС6 7֣ѡö2 5ֵӲңöѡĲ2 
              5ֵӲңӲҵѡ񽫲УǮ6 7֣öӦѡ1 
              0ֵӲңȻ5ֵģ1ֵӲҡ
              ̰㷨ֱǮʱֱӦʹҳӲĿ٣ǽӽٵĿ֤̰㷨ǮʱõӲĿȷ٣ϰ1
              1-5 [] n ޶̨ĻڻϵõÿĿʼʱΪsiʱΪfi si 
              < fi [si , fi ] Ϊi ʱ䷶Χij ָʱ䷶Χصָij 
              յغϡ磺[ 14 ][ 24 ]ص[ 47 
              ]صһеָڷûصͬһ̨ˣڿеķÿ̨κʱֻһŷָʹõĻٵĿз䷽
              n= 7񣬱Ϊa gǵĿʼʱͼ13-1a ʾaָM1b ָM2. . 
              .g 
              ָM7ַǿеķ䣬ʹ̨ŷ䣬Ϊ䷽ʹõĻĿ٣磺Խabdͬһ̨ĿΪ̨
              һֻŷ̰𲽷ÿһҰʼʱķǵݼз䡣Ѿһĳ̨̨ǾɵģǾɣµġѡʱ̰׼򣺸Ŀʼʱ䣬ʱоɵĻãָɵĻ򣬽һ̨µĻ
              еݣ̰㷨Ϊn = 7˳Ϊafbcgedһûоɻ˽a 
              һ̨»M1̨02ʱ̴æ״̬ڵڶfڵf ʱɻԴæ״̬˽f 
              һ̨»(ΪM2 )b, ھɻM1Sb = 
              3ʱѴ״̬˽bM1ִУM1һοʱ̱fb = 7M2Ŀʱ̱ff = 
              5ĲcûоɻSc = 4ʱ̿ã˽c һ̨»M3̨һοʱΪfc = 
              7岽gM2e M1, ڵ߲2M3ע⣺d 
              ҲɷM2
              ̰㷨ܵŻ֤ϰϰ7ɰ·ʽʵһΪO (nl o 
              gn)̰㷨ȲһΪO (nl o gn)㷨򣩰Si 
              ĵиȻʹһھɻʱСѡ
              1-6 [·] һ磬·ĳȶΪ·ĸߵĺķ֮͡Ҫһӳʼs ĿĶd 
              ·
              ̰㷨ֲ·ÿһ·мһ㡣赱ǰ·ѵﶥq
              Ҷq ĿĶdһõ̰׼Ϊѡq Ŀǰ·еĶ㡣
              ̰㷨һܻ·磬ͼ1 3 - 
              2ϣӶ15·̰㷨Ӷ1ʼѰĿǰ·е붥1Ķ㡣ﶥ3ȽΪ2λӶ3ԵΪ4Ӷ4ﶥ2󵽴ĿĶ5·Ϊ1 
              , 3 , 4 , 2 , 5䳤Ϊ1 
              0·ͼд15·ʵϣм̵·ڣ·145ĳΪ6
              ӣһǰһЩӦãм㷨Ҳ̰㷨磬㷨n- 
              1СȨⲿ·ĶÿһöϲΪһã㷨ʹõ̰׼ΪӿõĶѡȨСáL 
              P TȹҲһ̰㷨n n 
              ҵȽҵʱ䳤ȻÿһΪһһ̨ѡõ̰׼ΪʹĿǰĵʱ̡ҵȵɵĻϣȿеĻ
              ע⵽ڻУ̰㷨ܱ֤ţȻһֱһ½ǷǳӽֵõĹʵʻϣ˹õĹ㷨֤õŽͨýŽ޼㷨ҲΪʽ 
              h e u r i s t i c s )L P Tһʽȷ9 - 2L P 
              Tȵʱѵȵʱ֮ĹϵL P Tʽ޶
              ܣ bounded performance ޶ܵʽΪ㷨 a p p r o x i m a t i 
              o na l g o r i t h m
              µಿֽܼ̰㷨ӦáЩӦУ̰㷨ĽŵĽһЩӦãɵ㷨ֻһʽҲܲǽ㷨


              ----------------------------------------------

              plot(100+t+15*cos(3.05*t),t=0..200,coords=polar,axes=none,scaling=constrained); 


       2004-5-27 19:39:12       

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              ϰ
              1. ֤Ǯ⣨1 3 - 4̰㷨ܲӲǮ
              2. 1 3 - 4Ǯ⣬ۻԱֻ޵2 5֣ 1 0֣ 
              5ֺ1ֵӲңһǮ̰㷨ַҳӲǮ֤ۡ
              3. 1 3 - 4㷨ٶۻԱӲ⻹50, 20, 10, 5, 1Ԫֽң˿۸Ϊx Ԫy 
              ֵƷʱĿΪu Ԫv ֡㷨ҳֽӲĿǮ֤ۡ
              4. дһC + +ʵ1 3 - 4Ǯ㷨ۻԱֵΪ1 0 02 01 
              051ԪֽҺ͸Ӳҡɰģ飨Ʒļ۸񼰹˿Ǯģ飨ǮĿҪҵĸֻҵĿͼģ飨Ǯ
              5. ĳʹӲҵıֵΪ1 4 , 2 , 51֣1 3 - 
              4̰㷨ܲӲǮ֤ۡ
              6. 1) ֤1 3 - 5̰㷨ҵ䷽
              2) ʵ㷨ʹ临ΪO (nl o gn)ʾʱ佨Сѣ
              *7. 1 3 - 
              5Ļ⡣ٶһ̨ãôѡִ̨С磬ѡ񼯺Ϊ{a,b,e}ѡ̰㷨ɰѡÿѡһ̰׼£ʣµѡСʱҲص
              1) ֤̰㷨ܹѡ
              2) ʵָ㷨临ӦΪO(nl o gn)ʾһʱСѣ


              ----------------------------------------------

              plot(100+t+15*cos(3.05*t),t=0..200,coords=polar,axes=none,scaling=constrained); 


       2004-5-27 19:39:26       

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                         4 ¥ 



               

              1.3 Ӧ
              1.3.1 װ
              1  - 
              2Էֲװأÿװһ䣬Ҫװһ䡣˼̰׼򣺴ʣµĻУѡСĻ䡣ѡԱ֤ѡĻСӶװظĻ䡣̰ԣѡĻ䣬ȻѡĻ䣬ȥֱлװϴϲκһ䡣
              1-7 n =8, [w1 , ... w8 ]=[100,200,50,90,150,50,20,80], c= 4 0 
              0̰㷨ʱ˳Ϊ7 , 3 , 6 , 8 , 4 , 1 , 5 , 27 , 3 , 6 , 8 , 
              4 , 1Ϊ3 9 0λѱװأʣµװΪ1 0λСʣµκһ䡣̰㷨еõ[x1 
              , ..., x8 ] = [ 1 , 0 , 1 , 1 , 0 , 1 , 1 , 1 ]&aring;xi = 6
              1-1 ̰㷨ܲװء
              ֤Բ·ʽ֤̰㷨ԣx = [x1 , ..., xn ]Ϊ̰㷨õĽ⣬y =[ y1 , 
              ..., yn ]Ϊһн⣬ֻ֤n &aring;i= 1xi n &aring;i= 1yi 
              ʧһԣԼ䶼ź򣺼wiwi + 11inȻּy 
              תΪxתÿһһеyn &aring;i = 1yi ڵδתǰֵ֤n &aring;i = 1xi n 
              &aring;j = 1yi 
              ̰㷨Ḷ֪́[0, n] ķΧһkʹxi =1, ikxi =0, i>kѰ[ 1 
              ,n]ΧСjʹxjyj ûj ڣn &aring;i= 1xi =n &aring;i = 1yi j 
              ڣjky Ͳһн⣬Ϊxjyj xj = 1yj = 0yj = 1õy 
              ǿн⣬[ j+ 1 ,n]Χڱһl ʹyl = 1yl = 0wjwl õy 
              ǿеġңõy ԭy ͬĿ1
              תɽy תΪxÿתy ǰһy ͬĿ1x ʼy 
              ͬĿ1װ㷨C + +ʵּ1 3 - 1̰㷨˳װأ1 3 - 
              1üѰַI n d i r e c t S o r tԻ򣨼3 . 
              5ڼѰַĶ壩ɰ˳װءڼѰַʱΪO (nl o gn)Ҳ9 . 5 . 
              1ڵĶ򼰵2µĹ鲢򣩣㷨ಿʱΪO (n)˳1 3 - 1ܵĸΪO (nl o gn)
              13-1 װ
              template<class T>
              void ContainerLoading(int x[], T w[], T c, int n)
              {// װ̰㷨
              // x[i] = 1 ҽiװأ 1<=i<=n
              // cǴ, w ǻ
              // Ѱַʽ
              // t ǼѰַ
              int *t = new int [n+1];
              I n d i r e c t S o r t ( w, t, n);
              // ʱ, w[t[i]] <= w[t[i+1]], 1<=i<n
              // ʼx
              for (int i = 1; i <= n; i++)
              x[i] = 0;
              // ѡƷ
              for (i = 1; i <= n && w[t[i]] <= c; i++) {
              x[t[i]] = 1;
              c -= w[t[i]];} // ʣ
              delete [] t;
              }


              ----------------------------------------------

              plot(100+t+15*cos(3.05*t),t=0..200,coords=polar,axes=none,scaling=constrained); 


       2004-5-27 19:39:39       

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                         5 ¥ 



               

              1.3.2 0/1
              0 / 1УΪc ıװءn Ʒѡȡװ뱳ƷÿƷi Ϊwi ֵΪpi 
              ڿеıװأƷܳװָװƷֵߣn &aring;i=1pi xi 
              ȡֵԼΪn &aring;i =1wi xic xi&Icirc;[ 0 , 1 ] ( 1in)
              ʽУxt ֵxi = 1ʾƷi װ뱳Уxi =0 ʾƷi װ뱳0 / 
              1һһ㻯Ļװ⣬ÿõļֵͬװתΪʽΪΪΪװ뱳Ʒ
              1-8 ӻ˵һƷһӻn 
              ֲͬĻ涨ÿֻֻһӵΪcƷi ռwi Ŀռ䣬ֵΪpi 
              ĿʹװصƷֵ󡣵ȻװﲻܳͬһƷ߶ɷ0 / 
              1нģгӦڱӦƷ
              0 / 
              1кṵ̈ԣÿ̰Զöಽɱװ롣ÿһ̰׼ѡһƷװ뱳һ̰׼ΪʣƷУѡװ뱳ļֵƷֹ򣬼ֵƷȱװ루㹻ȻһֵƷ˼ȥֲԲܱ֤õŽ⡣磬n=2, 
              w=[100,10,10], p =[20,15,15], c = 1 0 5ṵֵ̈׼ʱõĽΪx= [ 1 , 0 
              , 0 ]ֵַܼΪ2 0ŽΪ[ 0 , 1 , 1 ]ֵܼΪ3 0
              һַ̰׼ǣʣµƷѡװ뱳СƷȻֹǰܲŽ⣬һһܵõŽ⡣n= 
              2 ,w=[10,20], p=[5,100], c= 2 5̰ʱõĽΪx =[1,0], Ž[ 0 
              , 1 ]Ҫ
              һֵܶpi /wi ̰㷨ѡ׼ΪʣƷѡ
              װpi /wi ֵƷֲҲܱ֤õŽ⡣ô˲Խn= 3 ,w=[20,15,15], 
              p=[40,25,25], c=30 ʱŽ⡣
              ǲļ̰㷨ܱ֤õŽɥ 0 / 1һN 
              P-⡣⣬ҲͲҵжʽʱ㷨Ȼpi /wi 
              ǵݣĴװƷܱ֤õŽ⣬һֱϽƵĽ⡣ϣһõʽ㷨Ҵʱܺܺõؽӽ㷨6 
              0 0ıУʽ̰㷨2 3 9ΪŽ⡣5 8 3Ž1 0 %6 0 
              0Ž֮ȫ2 5 %ڡ㷨O (nl o gn)ʱڻ˺õܡҲʣǷһx 
              (x<1 0 0 )ʹ̰Ľֵx%ڡǷ񶨵ġΪ˵һ㣬n =2, w = [ 1 
              ,y], p= [ 1 0 , 9y], c= y̰㷨Ϊx=[1,0], ֵַΪ1 0y1 0 / 
              9ŽֵΪ9 yˣ̰㷨ֵŽĲŽıΪ( ( 9y - 1 0)/9y* 1 0 0 ) 
              %ڴyֵ1 0 0 %ǿԽ̰ʽṩ⣬ʹĽŽֵֵ֮x% 
              (x<100) ֮ڡȽk Ʒ뱳k ƷcʣǽʣƷpi 
              /wi ݼ˳װ롣ͨĽⷨΪk ƷпܵӼõŽ⡣
              13-9 n =4, w=[2,4,6,7], p=[6,10,12,13], c = 11k= 
              0ʱƷֵܶȷǵݼ˳װ룬ȽƷ1뱳ȻƷ2ʣµΪ5ԪʣµƷûһʵģ˽Ϊx 
              = [ 1 , 1 , 0 , 0 ]˽õļֵΪ1 6
              ڿk = 1ʱ̰ӼΪ{ 1 } , { 2 } , { 3 } , { 4 }Ӽ{ 1 } , { 2 
              }k= 0ʱͬĽӼ{ 3 }x3 
              Ϊ1ʱʣ5λֵܶȷǵ˳5λȿƷ1ʺϣȡx1 
              Ϊ1ʱʣ3λˣʣƷûܹ뱳еƷͨӼ{ 3 }ʼýΪx = [ 1 , 0 , 1 
              , 0 ]õļֵΪ1 8Ӽ{ 4 }ʼĽΪx = [ 1 , 0 , 0 , 1 ]õļֵΪ1 
              9ӼСΪ01ʱõŽΪ[ 1 , 0 , 0 , 1 ]ͨk= 1̰ʽ㷨õġ
              k= 2˿k< 2Ӽ迼Ӽ{ 1 , 2 } , { 1 , 3 } , { 1 , 4 } , { 2 , 
              3 } , { 2 , 4 }{ 3 , 4 
              }ȴһӼʼǲеģʽʣµӼֱõ½[ 1 , 1 , 0 , 0 ] , [ 
              1 , 0 , 1 , 0 ] , [ 1 , 0 , 0 , 1 ] , [ 0 , 1 , 1 , 0 ][ 0 , 1 , 
              0 , 1 ]ЩһֵΪ2 3ֵk= 0k= 1ʱõĽҪߣ𰸼ΪʽĽ
              ޸ĺ̰ΪkŻk - o p t i m a lҲǣӴȡk Ʒk 
              õĽԭĺãַʽõֵֵ( 1 0 0 / (k + 1 ) ) %ڡk= 
              1ʱ֤սֵ5 0 %ڣk= 2ʱ3 3 . 3 3 %ڵȵȣʽִʱk 
              ӣҪԵӼĿΪO (nk )ÿһӼʱΪO (n)˵k >0ʱܵʱ俪ΪO (nk+1 
              )ʵʹ۲쵽Ҫõöࡣ


              ----------------------------------------------

              plot(100+t+15*cos(3.05*t),t=0..200,coords=polar,axes=none,scaling=constrained); 


       2004-5-27 19:39:53       

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                         6 ¥ 



               

              1.3.3 
              һӵĹͨԷֽһСļϣЩСζ̵ɡ磬װ乤̿ɷֽΪ񣺽̷װߣװᣬλװڵϣᣬװɲװŵȵȡ֮Ⱥϵװ֮ǰȽװװߡȺ˳ͼʾΪ 
              Activity On Vertex, AOV硣ͼĶ(i, j) ʾȺϵj ʼǰi 
              ɡͼ1  - 4ʾḶ́ߣ 1 , 4ʾ14ʼǰɣͬߣ 4 , 
              6ʾ46ʼǰɣߣ1 , 4루4 , 
              6֪16ʼǰɣǰϵǴݵġɴ˿֪ߣ1 , 4ǶģΪߣ1 , 3ͣ3 , 
              4Ѱʾֹϵ
              ںܶ£ִеģװƽʱıСҪװ䡱ƷгСǧװãݻȵȣǿɸ˳װ䡣ͼУߣ 
              i, jʾװi j ǰ棬ʵгΪУtopological 
              orderstopological sequences)ͼеĹ̳Ϊtopological 
              sortingͼ1 - 4ͼжУеΪ1 2 3 4 5 61 3 2 4 5 62 1 5 3 
              4 61 4 2 3 5 6ͲУΪ43ǰ棬ͼеıΪ 3 , 
              4ߣ 3 , 
              4ָʾìܡ̰㷨С㷨ҵĲ蹹Уÿһźõмһ㡣̰׼ѡ񶥵㣺ʣµĶУѡ񶥵wʹw 
              ߣ v,wжv źõнṹг֡ע⵽ĶwΥ׼򣨼ͼдڱߣ 
              v,wv ѹУ޷Ϊv ڶw ̰֮㷨αͼ1 3 - 
              5ʾwhile ѭÿε̰㷨һ衣
              ̰㷨ͼ1 - 
              4ͼȴһVʼһѡVĵһ㡣ʱͼѡ12ѡ񶥵2V = 
              2һɡڶѡVĵڶ㣬̰׼֪ѡΪ15ѡ5V = 2 
              5һ1ΨһĺѡV = 2 5 1Ĳ3Ψһĺѡ˰Ѷ3V
              õV = 2 5 1 3ֱ붥46 V = 2 5 1 3 4 6
              1. ̰㷨ȷ
              Ϊ֤̰㷨ȷԣҪ֤ 1) 㷨ʧʱͼûУ 2) 
              㷨ûʧܣVС2) ̰׼ѡȡһֱӽ 1) ֤1 3 - 
              2֤㷨ʧܣͼл·ͼаqj qj + 1.qk qj , ûУΪаʾqj 
              һҪqj ʼǰɡ
              1-2 ͼ1 3 - 5㷨ʧܣͼл·
              ֤ע⵽ʧʱ| V |<n, ûкѡܼVУһq1 VУͼбذߣ q2 , 
              q1q2 VУ q1 ǿɼVĺѡ㡣ͬбߣq3 , q2ʹq3 VУq3 = q1 
              q1 q2 q3 ͼеһ·q3 q1شq4 ʹq4 , q3ͼıq4 VУq3 
              Vһѡ㡣q4 Ϊq1 , q2 , q3 
              еκһֿ֪ͼлΪͼ޸nҵһ·
              2. ݽṹѡ
              Ϊͼ1 - 5C + 
              +ʵ֣뿼VԼҳɼVĺѡ㡣һָЧʵַǽVһάv 
              ģһջɼVĺѡ㡣һһάI n D e g r e eI n D e g r e e[ j 
              ]ʾ붥jĽڵi ĿжiVеĳԱ֮ͼı߱ʾΪ i, jI n D e g r e 
              e[ j ]Ϊ0ʱʾj Ϊһѡڵ㡣VʼʱΪաI n D e g r e e[ j ]Ϊj 
              ȡÿVмһʱ¼붥ڽӵĶjI n D e g r e e[ j ]1ͼ1 - 
              4ʼʱI n D e g r e e [ 1 : 6 ] = [ 0 , 0 , 1 , 3 , 1 , 3 
              ]ڶ12I n D e g r e 
              eֵΪ0ǿɼVĺѡ㣬ɴˣ12ջÿһջȡһ㽫VͬʱȥڽӵĶI 
              n D e g r e eֵڵһʱջȡ2Võv [ 0 ] = 2I n D e g r e 
              e [ 1 : 6 ] = [ 0 , 0 , 1 , 2 , 0 , 3 ]I n D e g r e e [ 5 
              ]ոձΪ0˽5ջ
              1 3 - 2ӦC + +룬뱻ΪN e t w o r 
              kһԱң޼Ȩͼáͼ޼ȨõĽΪͼġΪ⣬ͬģԱAdjacencyGraph, 
              AdjacencyWGraphL i n k e d G r a p hL i n k e d W G r a p 
              hЩN e t w o r kеĺϢҵУTopological t r u 
              eͼ򷵻f a l s eҵʱ䷵صv [ 0 :n- 1 ]С
              3. Network:Topological ĸ
              һ͵f o rѭʱ俪Ϊ(n )ʹãķѣڽӾ,ڶfor ѭõʱΪ(n2 
              )ʹڽ,ʱΪ(n+e)Ƕ׵while ѭУѭִnΣÿνw 뵽v 
              Уʼڲwhile ѭʹڽӾʱڲw h i l eѭÿw 
              軨(n)ʱ䣻ڽѭ軨dwout ʱ䣬ˣڲwhile ѭʱ俪Ϊ(n2 
              )(n+e)ԣڽӾ󣬳1 3 - 2ʱ临Ϊ(n2 )ڽΪ(n+e)
              13-2 
              bool Network::Topological(int v[])
              {// ͼж˴
              // ҵһ˴򷵻t r u eʱv [ 0 : n - 1 ]м¼˴
              // ˴򷵻f a l s e
              int n = Ve r t i c e s ( ) ;
              // 
              int *InDegree = new int [n+1];
              InitializePos(); // ͼ
              for (int i = 1; i <= n; i++) // ʼ
              InDegree[i] = 0;
              for (i = 1; i <= n; i++) {// i ı
              int u = Begin(i);
              while (u) {
              I n D e g r e e [ u ] + + ;
              u = NextVe r t e x ( i ) ; }
              }
              // ΪĶѹջ
              LinkedStack<int> S;
              for (i = 1; i <= n; i++)
              if (!InDegree[i]) S.Add(i);
              // ˴
              i = 0; // v α
              while (!S.IsEmpty()) {// Ӷջѡ
              int w; // һ
              S . D e l e t e ( w ) ;
              v[i++] = w;
              int u = Begin(w);
              while (u) {// ޸
              I n D e g r e e [ u ] - - ;
              if (!InDegree[u]) S.Add(u);
              u = NextVe r t e x ( w ) ; }
              }
              D e a c t i v a t e P o s ( ) ;
              delete [] InDegree;
              return (i == n);
              }


              ----------------------------------------------

              plot(100+t+15*cos(3.05*t),t=0..200,coords=polar,axes=none,scaling=constrained); 


       2004-5-27 19:40:10       

              b  
        
        
        ȼְҵ 
        £470
        ֣956
        ɣڿ͵۹ 
        ע᣺2003-8-28
                         7 ¥ 



               

              1.3.4 ָ
              ͼһͼn 
              ɶΪAͼBͬһеͼޱκһ߶һڼAУһڼBУҽBеÿAһʱAһӼA' 
              ǼB򵥵˵A' һǣA' ĴСΪA' еĶĿҽA' ǸBӼСʱA' 
              ΪСǡ
              1-10 ͼ1 - 6ʾľ1 7ĶͼA={1, 2, 3, 16, 17}B={4, 5, 6, 7, 
              8, 9,10, 11, 12, 13, 14, 15}ӼA' = { 1 , 1 6 , 1 7 
              }BСǡڶͼѰСǵΪָǣ b i p a r t i t e - c o v e r⡣1 
              2 - 3˵СǺõģΪܽڻʹٵķԱз롱һ⡣
              ָڼϸǣ s e t - c o v e r⡣ڼϸиk S= {S1 , S2 
              ,., Sk }ÿSi еԪؾȫUеĳԱҽ&Egrave;i S'Si =UʱSӼS' US 
              'еļĿΪǵĴСҽûܸUĸСļʱS' 
              ΪСǡԽϸתΪָ⣨֮ȻAĶʾS1 , ., Sk 
              BеĶUеԪءҽSӦаUеĶӦԪʱABĶ֮һߡ
              1 - 11 S= {S1. . .S5 }, U= { 45. . .15}, S1 = { 467891 
              3 }S2 = { 4568 }S3 = { 81 01 21 41 5 }S4 = { 5681 21 
              41 5 }S5 = { 491 011 }S ' = {S1S4S5 }һСΪ3ĸǣûиСĸǣ 
              S' ΪСǡϸӳΪͼ1-6Ķͼö1231 61 7ֱʾS1S2S3S4 
              S5j ʾеԪj4j1 5
              ϸΪN P-⡣ڼϸָͬһ⣬ָҲN 
              P-⡣˿޷ҵһٵ㷨ǿ̰㷨ѰһֿʽһֿǷֲA' 
              ÿһѡAеһ븲ǡѡ̰׼򣺴AѡȡܸBлδǵԪĿĶ㡣
              1-12 ͼ1 - 6ʾĶͼʼA' = Bûж㱻ǣ11 
              6ܸBе㣬321 7ֱ𸲸ĸˣڵһA' м붥11 6붥1 
              6ǵĶΪ{ 5 , 6 , 8 , 1 2 , 1 4 , 1 5 }δǵĶΪ{ 4 , 7 , 9 , 1 0 
              , 11 , 1 3 }1ܸĸ㣨 { 4 , 7 , 9 , 1 3 }2 һ( { 4 } 
              )3һ{ 1 0 }1 61 7ĸ{ 4 , 9 , 1 0 , 11 
              }һѡ11 7A' ѡ񶥵1򶥵{ 1 0 , 11} Ȼδǣʱ121 
              6һ3һ1 7ѡ񶥵1 7жѱǣA' = { 1 6 , 1 
              , 1 7 }
              ͼ1 - 7̰ʽα룬֤ 1) ҽʼĶͼûиʱ㷨Ҳǣ2) 
              ʽҲͼСǡ
              1. ݽṹѡȡԷ
              Ϊʵͼ13 - 7㷨ҪѡA' μ¼AнڵܸǵBδǽڵĿڶԼA' 
              ʹüӷ㣬һάCA 'm ¼A' ԪظA' еĳԱ¼C[ 0 :m-1] 
              СAжiN e wi Ϊi ܸǵBδǵĶĿѡN e wi 
              ֵĶ㡣һЩԭδǵĶڱˣ˻Ҫ޸ĸN e wi 
              ֵָУBһαVǵĶ㣬j Ϊһ㣬Aиj ĶN e wi ֵ1
              1-13 ͼ1 - 6ʼʱ(N e w1 , N e w2 , N e w3 , N e w16 , N e w17 ) 
              = ( 6 , 4 , 5 , 6 , 4 )1 - 1 2Уһѡ񶥵1 6ΪN e wi 
              ֵBǵĶ㣬ЩΪ5 , 6 , 8 , 1 2 , 1 41 5鶥5ʱ21 6N 
              e wi ֱֵ1Ϊ5Ǳ21 6ǵδǽڵ㣻鶥6ʱ1 , 2 ,1 
              6Ӧֱֵ1ͬ鶥8ʱ1231 6ֱֵ1ǵĶ㣬õN e wi 
              ֵΪ4104һѡ񶥵1±ǵĶΪ4791 3鶥4ʱN e w1 , N e w2, N 
              e w1 7 ֵ1鶥7ʱN e w1 ֵ1Ϊ1Ǹ7Ψһ㡣
              ΪʵֶѡȡḶ́Ҫ֪N e wi ֵѱǵĶ㡣һάﵽĿģN e 
              wһ飬New[i] N e wic o vΪһ顣iδc o v [ i ]f a 
              l s ec o v [ i ]Ϊt r u eֽͼ1 - 7αϸõͼ1 - 8
              m=0; //ǰǵĴС
              AеiNew[i]=Degree[i]
              BеiC o v [ i ] = f a l s e
              while (AеĳЩi,New[i]>0) {
              vǾN e w [ i ]Ķ㣻
              C [ m + + ] = v ;
              for ( ڽvĶj) {
              if (!Cov[j]) {
              Cov[j]= true;
              ڽjĶ㣬ʹN e w [ k ]1
              } } }
              if (Щδ) ʧ
              else ҵһ
              ͼ1-8  ͼ1-7ϸ
              N e wʱΪO (e)e ΪͼбߵĿʹڽӾ軨(n2 ) 
              ʱѰͼеıߣڽ(n+e) ʱ䡣ʵʸʱĲͬΪO (n2 ) O 
              (n+e)ѡ񶥵ʱΪ(S i z e O f A)S i z e O f A=| A 
              |ΪAж㶼пܱѡ貽ΪO ( S i z e O f A )㷨ܵĸΪO ( S i z 
              e O f A 2+n2) = O ( n2)O (S i z e Of A2+n + e)
              2. ͸
              ͨʹN e wiѻѡmax selection treeɽÿѡȡvĸԽΪ( 1 
              )飬ÿN e wi ֵʹʱΪ(S i z e O f B 
              ) ( S i z e O fB =|B| ) 3 . 8 . 1򣩡һS i z e O f BS i z 
              e O f Aö࣬鲢ܡ
              ѣÿһҪؽ¼N e wֵı仯ÿN e wֵ1ʱؽּ𱻼N e 
              wֵڶһ㣬ؽÿN e wֵ1( 1 )ʱ䣬ܹļĿΪO 
              (e)㷨вУάѽO (e)ʱ䣬ʱ㷨ܸΪO (n2 )O (n+e)
              ѡÿθN e wֵʱҪؽѡʱΪ(log S i z e O f 
              A)ؽʱÿʱÿN e wֵ1ʱҪؽĴΪO (e)ܵؽʱΪO (e log 
              S i z e OfA)ʱѵؽʱ䳤һЩȻͨά־ͬN e 
              wֵĶӣҲɻúʱͬʱơN e wȡֵΧΪ0S i z e O f BҪS i z e 
              O f B+ 1ӣi һ˫N e wֵΪi Ķ㡣ĳһʱN e w [ 6 ]1 
              2䵽4Ҫӵ1 2Ƶ4ӡģָ뼰һڵn o d en o d e [ i 
              ]in o d e [ i ] . l e f tn o d e [ i ] . r i g h 
              tΪ˫ָ룩ɽ6ӵ1 2Ƶ4ӣӵ1 2ɾn o d e [ 0 
              ]4ӡģʽɵøʽ㷨ĸΪO (n2 
              )O(n+e)ȡڽӾԱͼ
              3. ˫ӵʵ
              Ϊʵ˫ӣͼ1 - 9U n d i r e c t e d˽гԱN o d e Ty p 
              eһ˽ͳԱl e f tr i g h t࣬˫ڵ㣬1 3 - 3U n d i 
              r e c t e d˽гԱĴ롣
              void CreateBins (int b, int n)
              bӺnڵ
              void DestroyBins() { delete [] node;
              delete [] bin;}
              void InsertBins(int b, int v)
              bӶv
              void MoveBins(int bMax, int ToBin, int v)
              ӵǰƶvTo B i n
              int *bin;
              b i n [ i ]ָӵ˫׽ڵ
              N o d e Type *node;
              n o d e [ i ]洢iĽڵ
              ͼ1-9 ʵ˫U n d i r e c t e d˽гԱ


              ----------------------------------------------

              plot(100+t+15*cos(3.05*t),t=0..200,coords=polar,axes=none,scaling=constrained); 


       2004-5-27 19:40:45       

              b  
        
        
        ȼְҵ 
        £470
        ֣956
        ɣڿ͵۹ 
        ע᣺2003-8-28
                         8 ¥ 



               

              13-3 ӺĶ
              void Undirected::CreateBins(int b, int n)
              {// bӺnڵ
              node = new NodeType [n+1];
              bin = new int [b+1];
              // ÿ
              for (int i = 1; i <= b; i++)
              bin[i] = 0;
              }
              void Undirected::InsertBins(int b, int v)
              {// bΪv b
              if (!b) return; // bΪ0
              node[v].left = b; // 
              if (bin[b]) node[bin[b]].left = v;
              node[v].right = bin[b];
              bin[b] = v;
              }
              void Undirected::MoveBins(int bMax, int ToBin, int v)
              {// v 䵱ǰƶTo B i n .
              // vҽڵ
              int l = node[v].left;
              int r = node[v].right;
              // ӵǰɾ
              if (r) node[r].left = node[v].left;
              if (l > bMax || bin[l] != v) // ڵ
              node[l].right = r;
              else bin[l] = r; // l
              // ӵTo B i n
              I n s e r t B i n s ( ToBin, v);
              }
              C r e a t e B i n s̬飺 n o d eb i nn o d e [i ]ʾi, 
              bin[i ]ָN e wֵΪi˫Ķ, f o rѭ˫Ϊաb0InsertBins 
              v b СΪb Ƕv New ֵb = 0ζŶv 
              ܸBеǰδǵκζ㣬ԣڽʱûôʿԽȥb0ʱn New ֵΪb 
              ˫ӵǰ棬ּ뷽ʽҪnode[v] bin[b] 
              еһڵߡڱڵӦָӣ˽node[v].left 
              ΪbӲգǰһڵleft ָ뱻Ϊָ½ڵ㡣node[v] ָ뱻Ϊb i n [ b 
              ]ֵΪ0ָһ׽ڵָ롣 b i n [ b ]Ϊָµĵһڵ㡣MoveBins v 
              ˫еĵǰλƵNew ֵΪToBin λϡдbMa xʹöеb i n[ j 
              ]Уj>bMa xb i n [ j ]Ϊաȷn o d e [ v 
              ]ڵǰ˫еҽڵ㣬Ŵ˫ȡn o d e [ v ]I n s e r t B i n 
              s²뵽b i n [ To B i n ]С
              4. Undirected::BipartiteCoverʵ
              LڷͼеĶ㣨䵽ABL [i ] = 1ʾiڼAУL[ i ] = 
              2ʾBС CmmΪĸǵĴС C [ 0 , m - 1 
              ]AγɸǵĶ㡣ͼûиǣf a l s e򷵻t r u eĴ1 3 - 4
              13-4 ̰
              bool Undirected::BipartiteCover(int L[], int C[], int& m)
              {// Ѱһͼ
              // L 붥ı, L[i] = 1 ҽi ڣ
              // C һ¼ǵ
              // ͼвڸǣ򷵻f a l s e
              // ͼһǣ򷵻t r u e ;
              // mзظǵĴС; C [ 0 : m - 1 ]зظ
              int n = Ve r t i c e s ( ) ;
              // ṹ
              int SizeOfA = 0;
              for (int i = 1; i <= n; i++) // ȷAĴС
              if (L[i] == 1) SizeOfA++;
              int SizeOfB = n - SizeOfA;
              CreateBins(SizeOfB, n);
              int *New = new int [n+1]; / /iˣN e w [ i ]δǵĶ
              bool *Change = new bool [n+1]; // Change[i]Ϊt r u eҽNew[i] Ѹı
              bool *Cov = new bool [n+1]; // Cov[i] Ϊtrue ҽi 
              I n i t i a l i z e P o s ( ) ;
              LinkedStack<int> S;
              // ʼ
              for (i = 1; i <= n; i++) {
              Cov[i] = Change[i] = false;
              if (L[i] == 1) {// i A
              New[i] = Degree(i); // i ô
              InsertBins(New[i], i);}}
              // 츲
              int covered = 0, // ǵĶ
              MaxBin = SizeOfB; // ܷǿյ
              m = 0; // Cα
              while (MaxBin > 0) { // 
              // ѡһ
              if (bin[MaxBin]) { // Ӳ
              int v = bin[MaxBin]; // һ
              C[m++] = v; // v 븲
              // ¸ǵĶ
              int j = Begin(v), k;
              while (j) {
              if (!Cov[j]) {// jδ
              Cov[j] = true;
              c o v e r e d + + ;
              // ޸N e w
              k = Begin(j);
              while (k) {
              New[k]--; // j 
              if (!Change[k]) {
              S.Add(k); // ջһ
              Change[k] = true;}
              k = NextVe r t e x ( j ) ; }
              }
              j = NextVe r t e x ( v ) ; }
              // 
              while (!S.IsEmpty()) {
              S . D e l e t e ( k ) ;
              Change[k] = false;
              MoveBins(SizeOfB, New[k], k);}
              }
              else MaxBin--;
              }
              D e a c t i v a t e P o s ( ) ;
              D e s t r o y B i n s ( ) ;
              delete [] New;
              delete [] Change;
              delete [] Cov;
              return (covered == SizeOfB);
              }
              1 3 - 4ȼABĴСʼҪ˫ṹ顢ʼͼһջȻC o 
              vC h a n g eʼΪf a l s eAеĶǸBжĿ뵽Ӧ˫С
              Ϊ˹츲ǣȰSizeOfB ݼ1˳˫һǿյıʱͽһv 
              뵽УֲԼΪѡNew ֵĶ㡣ѡĶ븲CBڽӵĶ㡣j v 
              ڽһδǣC o v [ j ]Ϊt r u eʾj ѱǣͬʱѱǵBеĶĿ1j 
              ǵģAj ڽӵĶNew ֵ1һwhile ѭЩNew ֵNew 
              ֵ͵Ķ㱣һջС붥vڽӵĶCov ֵϺN e 
              wֵӳAÿܸǵµĶȻAеĶNew ֵ£ڴ˫Уһwhile 
              ѭЩƵȷıС


              ----------------------------------------------

              plot(100+t+15*cos(3.05*t),t=0..200,coords=polar,axes=none,scaling=constrained); 


       2004-5-27 19:40:53       

              b  
        
        
        ȼְҵ 
        £470
        ֣956
        ɣڿ͵۹ 
        ע᣺2003-8-28
                         9 ¥ 



               

              1.3.5 Դ·
              УͼGÿ߶һǸĳȣķѣ a [i ][ j 
              ]·ĳȼΪ·ıߵĳ֮͡ڸԴsҳͼⶥ㣨ΪĿģ·ͼ13-10a 
              һͼϵΪȡԴs Ϊ1Ӷ1··˳ͼ13-10b 
              Уÿ·ǰΪ·ĳȡ
              E. 
              Dijkstra̰㷨Խ·⣬ֲͨ·ÿһһµĿĶ·һܴﵽĿĶ̰ͨ׼ѡȡڻδ·ĶУѡ·̵ĿĶ㡣Ҳ˵ 
              D i j k s t r aķ·˳·
              s 
              ··ûбߣ䳤Ϊ0̰㷨ÿһУһ·һַĿǰѲ·мһе̵ıߣ·ԭȲ·һߡֲԲáһַĿǰÿһ·УǼһ̵ıߣٴЩѡ̵ģֲԼD 
              i j k s t r a㷨
              ֤˳·ʱһ·һѲ·һγɡʵϣһ·Ѳ·һ̵ıߵõģ·Ķ·δͼ1 
              3 - 1 0Уb 
              еڶ·ǵһ·һγɵģ·ǵڶ·һߣ·ǵһ·һߣ·ǵ·һߡ
              ͨ۲һּķ洢·pp [ i ]s i·жi 
              ǰǸ㡣ڱp [ 1 : 5 ] = [ 0 , 1 , 1 , 3 , 4 ]s i 
              ·ɷ򴴽i p[i],p[p[i]],p[p[p[i]]], .˳ֱﶥs 0ڱУi 
              = 5ʼ򶥵Ϊp[i]=4, p[4]=3, p[3]=1=s·Ϊ1 , 3 , 4 , 5
              Ϊܷذȵ˳·d [ i 
              ]ΪѲ·мһ̱ߵĳȣӶʹ·ﶥiдs s 
              һΪ0·ʱÿid [ i ]a [ s ] [ i ]a 
              ͼĳڽӾ󣩡Ϊһ·Ҫѡδ·һڵ㣬ЩڵdֵСļΪһ·յ㡣һµ·µ·ܻСdֵЩdֵܻᷢ仯
              Եõͼ1 3 - 11ʾα룬 1) s ڽӵжp 
              ʼΪsʼڼ¼ǰõϢҲ˵s i 
              ·s·һߵõҵ̵·ʱ p [ i 
              ]ֵ¡һ·Ҫ·d ֵ
              1) ʼd[i ] =a[s] [i ]1in
              ڽsжip[i ] =s, Ķp[i ] = 0
              p[i]0ж㽨L
              2) LΪգֹת3 )
              3) LɾdֵСĶ㡣
              4) i ڽӵлδĶjd[ j ]ֵΪm i n{d[ j ], d[i ] +a[i ][ j ] 
              }d[ j ]˱仯j δ
              LУp[ j ] = 1j Lת2
              ͼ1 - 11 ·㷨
              1. ݽṹѡ
              ҪΪδĶбLѡһݽṹLпѡd ֵСĶ㡣LСѣ9 . 
              3ڣάѡȡڶʱɡ3) ִдΪO ( n )ʱΪO ( n l o g n 
              )һ߲µ·ʱʹδĶСd ֵ4) пҪıһЩd 
              ֵȻ㷨еļǱ׼СѲڶʱɡִмܴΪ O(ͼеı)= O ( 
              n2 )ִмʱΪO ( n2 l o g n )
              Lά3) 4) ѵʱΪO ( n2 )3) ÿִO(|L | ) =O( n 
              )ʱ䣬ÿμ( 1 )ʱ䣨Ҫȥd[j] ֵøı䣩ͼ1 - 11αϸΪ1 
              3 - 5ʹC h a i n (3 - 8 )C h a i n I t e r a t o rࣨ3 - 
              1 8
              13-5 ·
              template<class T>
              void AdjacencyWDigraph<T>::ShortestPaths(int s, T d[], int p[])
              {// ѰҴӶs·, dз̾
              // pзǰ̶
              if (s < 1 || s > n) throw OutOfBounds();
              Chain<int> L; // ·ɵﶥб
              ChainIterator<int> I;
              // ʼd, p, L
              for (int i = 1; i <= n; i++){
              d[i] = a[s][i];
              if (d[i] == NoEdge) p[i] = 0;
              else {p[i] = s;
              L . I n s e r t ( 0 , i ) ; }
              }
              // d, p
              while (!L.IsEmpty()) {// ѰҾСdĶv
              int *v = I.Initialize(L);
              int *w = I.Next();
              while (w) {
              if (d[*w] < d[*v]) v = w;
              w = I.Next();}
              // Lɾͨ򶥵vһ·d
              int i = *v;
              L . D e l e t e ( * v ) ;
              for (int j = 1; j <= n; j++) {
              if (a[i][j] != NoEdge && (!p[j] ||
              d[j] > d[i] + a[i][j])) {
              // Сd [ j ]
              d[j] = d[i] + a[i][j];
              // jL
              if (!p[j]) L.Insert(0,j);
              p[j] = i;}
              }
              }
              }
              N o E d g e㹻ʹû·ĳȴڻN o E d g eһfor ѭi 
              fɼΪif (d[j] > d[i] + a[i][j]))  NoEdge ֵӦʹd[j]+a[i][j] 
              ķΧڡ
              2. Է
              1 3 - 5ĸO ( n2 
              )κ·㷨ٶÿ߼һΣΪκһ߶п·С㷨СʱΪO ( e 
              )ʹúķڽӾͼͼоO ( n2 )ʱ䡣ˣ㷨軨O ( n2 
              )ʱ䡣1 3 - 5ŻӼʹıڽӱҲֻʹһf o rѭʱ併ΪO ( e 
              )Ϊֻi ڽӵĶd ֵı䣩LѡɾСĶʱȻO( n2 )


              ----------------------------------------------

              plot(100+t+15*cos(3.05*t),t=0..200,coords=polar,axes=none,scaling=constrained); 


       2004-5-27 19:41:07       

              b  
        
        
        ȼְҵ 
        £470
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        ע᣺2003-8-28
                         10 ¥ 



               

              1.3.6 Сķ
              1 - 21 - 3ѿ⡣Ϊn 
              Gÿպþn-1ߣĳַѡn-1ʹγGСٿԲֲ̰ͬѡn-1ߡС̰㷨ǣ 
              K r u s k a l㷨P r i m㷨S o l l i n㷨
              1. Kruskal㷨
              (1) 㷨˼
              K r u s k a l㷨ÿѡn- 
              1ߣʹõ̰׼ǣʣµıѡһ·ľСķѵı߼ѡıߵļСע⵽ѡȡı·򲻿γһK 
              r u s k a l㷨e e бߵĿķѵ˳e 
              ߣÿοһߡĳʱ뵽ѡߵļлֻ·򣬽ѡ롣
              ͼ1-12a е硣ʼʱûκα߱ѡͼ13-12b ʾ˸ڵĵǰ״̬ߣ 1 , 
              6ѡıߣ뵽Уõͼ1 3 - 1 2 cһѡߣ 34Уͼ1 3 - 
              1 2 dʾȻǱ( 27 )в·Ǳõͼ1 3 - 1 2 eһǱߣ 
              23Уͼ1 3 - 1 2 
              fʾ໹δǵıУ74Сķѣȿڴл·Խ䶪˺󽫱ߣ 
              54Уõͼ13-12g ʾһǱߣ 75ڻ·䶪Ǳߣ 
              65УһķΪ9 9ͼ1 - 1 3K r u s k a l㷨α롣
              / /һn ҵһС
              TΪѡߵļϣʼT=
              E Ϊбߵļ
              w h i l e(E )&&(| T |n- 1 ) {
              (u,v)ΪEдСı
              E=E- { (u,v) } / /Eɾ
              i f( (u,v)Tв·) u,vT
              }
              i f(| T | = =n-1) TСķ
              e l s e 粻ǻģҵ
              ͼ13-13 Kruskao㷨α
              (2) ȷ֤
              ǰװõת֤ͼ1 3 - 1 3̰㷨ܽһСķҪ֤㣺 1) 
              ֻҪK r u s k a l㷨ܲһ 2) 
              СķѡGΪȨͼGһ磩1 2 . 11 . 
              3ڿ֪ҽһͼͨʱKruskal 
              㷨бܾıЩ·ıߡɾͨͼ·еһγɵͼͨͼGڿʼʱͨģTEеıγһͨͼҲGʼʱͨģ㷨ֹE= 
              | T |< n- 1
              ֤TСķѡG޿پһСUΪһС 
              TUպn- 1ߡT=U, T;Сķѣô֤ȥ˼TUk(k >0) 
              ΪTжUеıߵĸȻk ҲUжTеıߵĿ
              ͨU任ΪT֤UTͬĺķѣתk 
              ɡÿһʹTUеıߵĿպü1UĺķѲΪתı䡣k 
              תõUԭUͬĺķѣתUеı߾Tеıߡɴ˿֪ TСķѡÿתTһe 
              UУUƳһfe f ѡȡ·ʽУ
              1) e TжUеľСķѵıߡk >0߿϶ڡ
              2) e UʱγΨһһ·f Ϊ·ϲTеһߡ
              Tв·γɵĻ·һ߲TС
              e f ѡ񷽷пԿ V=U+ {e} -{ f } һTǡk- 
              1߲Vг֤֡VĺķUͬȻVĺķѵUĺķѼϱe ĺķټȥf ĺķѡe ĺķѱf 
              СVĺķѱUĺķСǲܵġe ĺķѸfK r u s k a l㷨f e 
              ֮ǰǡf TУKruskal 㷨ڿf ܷTʱѽf f TкķСڻf 
              ı߹ͬγɻ·ͨѡeЩ߾UУU϶л·ʵⲻܣΪUһe Ĵ۸f 
              ļ轫ᵼìܡʣµΨһĿe f ͬĺķѣɴ˿֪VUĺķͬ
              (3) ݽṹѡ񼰸Է


              ----------------------------------------------

              plot(100+t+15*cos(3.05*t),t=0..200,coords=polar,axes=none,scaling=constrained); 


       2004-5-27 19:41:42       

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