77通信网理论基础-第3章-通信网结构-最短路径问题

3.1.3.2.3.2.1.−=Gndijij12n≡vivjdij∞=n1−=PrimP≡≡∆00∆•vj1G1vj1•G1GG1vj2•2111minjjjjGGjdd=−∈•G2vj1,vj2•cj1j2=cj2j1=1∆•Grr•GrGGr1+rjv•rjrssjijGGjGiGGvGvddrrrr−∈∈=++−∈∈11,min•{}{}1211,,......,1++=∪=+rrrjjjjjrrvvvvvGG•111==++sjsjrrcc∆•r1n•r1n•GnC=P≡∆PPv1,v2,…,vn••rirdr=2,3,…,n,i2=1∆Q∆dijQP∆Vr-1={v1,v2,…,vr-1}V-Vr-1={vr,vr+1,…,vn}∆PdirrQ•Qvirvr•dirr•dirr•Vr-1V-Vr-1•dirrP•dirrQ•Q•dirrQ•r=2,3,…,nQP∆dijPQ∆=Prim≡4.532v1v25v34v42.5v5110≡G1v1∆GG1v2,v3,v4,v5∆{}{}12151413125,5.4,,1minmin1ddddddijGGj===∞===−∈∆v2∆c12c211≡G2v1,v2∆GG2v3,v4,v5∆{}252524231514132,,35,5.4,minmin22ddddddddijGGjGi==∞====∞==−∈∈∆v5∆c25c521≡G3v1,v2,v5∆GG3v3,v4∆{}545453242314135.2,4,35.4,minmin33ddddddddijGGjGi===∞===∞==−∈∈∆v4∆c54c451≡G4v1,v2,v5,v4∆GG4v3∆{}{}23534323134,10,3,minmin44ddddddijGGjGi====∞==−∈∈∆v3∆c23c321≡G5v1,v2,v5,v4,v3∆32v1v2v3v42.5v51∆54321010101000000010101010001054321vvvvvvvvvvC=∆l=d12+d25+d54+d231+2+2.5+38.5∆P≡QV4∆v4v5v2v1v3∆PP∆PQ∆QP=Prim≡n1≡GrrGGrnr≡rrnr1≡P[][])3)(2)(1(61623)1(61)1()12)(1(612)1()1(1)(221121111+−−⋅=−+−−⋅=−−−−⋅−−⋅=−−−=−−∑∑∑−=−=−=nnnnnnnnnnnnnnnrrnrnrnrnrnr≡Pn3∆P∆n2∆Pn2n3∆NP∆nnNP•NPNonPolynomial=KruskalK≡≡∆00•∆•di1j1vi1vj1•G1{vi1,vj1}•ci1j1cj1i11•∆•Grrr+12r••Gr••di(r+1)j(r+1)vi(r+1)vj(r+1)•Gr+1Gr{vi(r+1),vj(r+1)}•ci(r+1)j(r+1)cjrir1•∆•rn1•rn1•C=≡4.532v1v25v34v42.5v5110≡0∆v1v2v2v5v4v5v2v3v3v5v1v4v1v5v3v4122.5344.5510≡d121v1,v2∆G1v1,v2∆c12c211∆∆22.5344.5510≡d252∆v2v5∆G1v2v5∆G2G1{v2,v5}v1,v2,v5∆c25c521∆∆2.5344.5510≡d452.5∆v4,v5∆G2v4v5∆G3G2{v4,v5}v1,v2,v5,v4∆c45c541∆∆344.5510≡d233∆v2v3∆G3v2v3∆G4G3{v2,v3}v1,v2,v5,v4,v3∆c23c321∆G4r4n14∆54321010101000000010101010001054321vvvvvvvvvvC==K∆K∆≡∆KK∆Kdij∆K∆drs,∆•drs≤dij,•drsdij,dijdrsdrsdijdrs∆∆K=K≡K≡mm!∆log2(m!)≡nCn2n(n1)/2∆n2log2n−≡n∆∆∆≡≡∆∆=∆∆∆∆∆∆nnn-2NPHard∆≡----∆v1v2v3v4e1e2e3e4e531321∆0•43212110131111310112432100vvvvTvvvvAA−−−−−−−−−−=•‹‹0101•A0A0Tv382200012210131012=−−−++=−−−−=TAA‹‹‹0v4∆1t0•t0e1,e2,e4∆2t0•Se1(t0)e1,e5•Se2(t0)e2,e3•Se4(t0)e5,e3,e4∆3••t01‹t0‹tt0=1•t01‹t1e5,e2,e4e5t0e1‹t2e1,e3,e4e3t0e2‹t3e5,e2,e4e3t0e4‹t4e5,e2,e4e5t0e4•T1t1,t2,t3,t4∆4t02T2•T1•t0•t1‹Se2(t1)e2,e3‹Se4(t1)e4,e1,e3‹Se5(t1)e1,e5•‹t0‹t0Se2(t1)∩Se2(t0)e2,e3Se4(t1)∩Se4(t0)e4,e3e3e2e4‹t0T1‹t0t0T1‹e3t1e2t5e5,e3,e4‹e3t1e4t6e5,e2,e3•t2‹Se1(t2)e1,e5‹Se3(t2)e3,e2‹Se4(t2)e4,e2,e5•‹t0Se1(t2)∩Se1(t0){e1,e5}Se4(t2)∩Se4(t0){e4,e5}e5e1e4‹e5t2e4t7{e1,e3,e5}‹e5t2e1t8{e3,e4,e5}t5‹t5t8•t3t4‹•t02T2t5,t6,t7∆5t03T3•t03∆6t0t1t7•dt06dt18dt24dt35dt47dt56dt67dt75•t2PK•‹t0t5‹‹=∆•PK∆•∆∆•∆EW≡EWEsouWillian∆∆•Gnv1•dij(i,j=1,2,…,n)•Fii=2,3,…,n∆•v1KK1(K∈Z+)•MM∈R∆∆•EWnn••1•‹K•••dijtijdijD1i‹iGvidDii11min∈=‹Givi∆•EW00‹nn‹•EW11‹V1D1i‹tij•EW22‹tijijjijitt,**min=‹vi*vj*vj*v1v1KMvj*v1vi*vj*KM‹vi*vj*ci*j*1‹2EW2•EW33‹11EW1‹1≡∆30010v1v2v3v4v5134252312671040∆•dij•Fi•v1∆•K3K12•M5050∆∆EW0•n5G1={v1},G2={v2},……,G5={v5}•0∆EW1•{}5432105732501267103123230426124054321vvvvvijvvvvvd=•{}[]5432126124011vvvvvivd=•{}{}{}2,6,12,4,0min111===∈iiGviddDii•[]543211210021222vvvvvjjvDdt−−−=−=t22=0•[]5432151109031333vvvvvjjvDdt−−−=−=t33=0•[]543211054041444vvvvvjjvDdt−−−=−=t44=0•[]543210351051555vvvvvjjvDdt=−=t55=0•{}543210351010540511090121005432vvvvvijvvvvt−−−−−−−−−=∆EW2•tij11min34,**−===tttijjiji•v3v4‹v4v1‹v4v3‹1K‹401050≤M‹•v3v4‹v4v3G'3={v3,v4}‹c34c431∆EW3•4G1={v1},G2={v2},G'3={v3,v4},G5={v5}•EW1∆EW1•{}5432105732501267103123230426124054321vvvvvijvvvvvd=•{}[]5432126640min111vvvvviGvivdDii==∈v4v3,v16•{}54321035101054015036121005432vvvvvijvvvvt−−−−−−−−=∆EW2•tijt34t435‹•t424‹v2v410+40+30=80M‹•t323‹v2v310403080M‹•t242‹v2•t23t25t451‹v2v3‹v2v5K1301040M‹v4v510401060M•v2v5‹v2v5G'2={v2,v5}‹c25c521∆EW3•3G1={v1},G'2={v2,v5},G'3={v3,v4}•EW1∆EW1•{}5432105732501267103123230426124054321vvvvvijvvvvvd=•{}[]5432126620min111vvvvviGvivdDii==∈v2v5,v12•{}54321035101054015036101025432vvvvvijvvvvt−−−−−=∆EW2•tij‹t34t435‹t424‹t323‹t451•‹t240‹t410K2104050≤M‹t510K2301040M•v4v1v5v1‹G2v1G3v1‹c41c141‹v1v2v3v4v51362∆EW3•1••54321000110010101000100001100054321vvvvvvvvvvC=•d(T)=12≡EW∆∆∆3.2.2.−=DijkstraD≡G∆dij≡vs≡≡∆Gp∆GGp≡∆vs∆•≡vsGpvs•wj(vj∈GGp)≡sjGGvsddpj−∈=min1∆v1Gpvs,v1∆w1ds1∆v1vs∆w1≡v1∆),(min11*jjGGvjd=−∈≡wjw2v2Gpvs,v1,v2∆∆∆=D≡D0∆vsGpvs∆ws0wsvsvs∆wj(vj∈GGp)wjvsvj≡D1∆),(min*ijijGvGGvjd=∈−∈∆wivi∆wjvivj∆wjvivj∆wjwj≡D2∆jGGviwwpj−∈=min∆viGp∆Gpn∆Gpn≡vs∆∆=≡v1v2v3v4vs8342331≡D0∆vsGpvs∆ws0wj(j1,2,3,4)≡D1∆w1min(w1ws+ds1)min(0+8)8∆w2min(w2ws+ds2)min(0+4)4∆w3min(w3ws+ds3)min(0+2)2∆w4min(w4ws+ds4)min(0+)•wiwi•wiwi≡D2∆w3∆v3v3Gpvs,v3∆Gp2n=5D1≡D1∆w1min(w1w3+d31)min(82+)8∆w2min(w2w3+d32)min(42+1)3∆w4min(w4w3+d34)min(2+3)5≡D2∆w2∆v2v2GpGpvs,v3,v2∆vsv2vsv3v2∆Gp3n=5D1≡D1∆w1min(w1w2+d21)min(83+3)6∆w4min(w4w2+d24)min(53+3)5≡D2∆w4∆v4v4GpGpvs,v3,v2,v4∆vsv4vsv3v4•w4v2•w4w4v3∆Gp4n=5D1≡D1∆w1min(w1w4+d41)min(65+)6≡D2∆v1v1GpGpvs,v3,v2,v4,v1∆vsv1vsv3v2v1•w1v4•w1•w1v2•v2vsv3v2∆Gp5n≡wjvsv1v2v3v40∞∞∞∞vs842∞v3835v265v46v1ws0w32w23w45w16vsv3vsv3v2vsv3v4vsv3v2v1=D≡k∆nk•nk∆nk1∆3(nk)1≡)1(23)(31−⋅=−⋅∑=nnknnk≡n2∆−=Dn≡=≡FloydF=F≡D≡≡≡nGdij∆nnW∆nnR=F≡F0∆W(0)