Е. Деза_ М.М. Деза. Энциклопедический словарь расстояний (2008) (1185330), страница 53
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èÂÂÏ¢ÂÌË ·‡ ÔÂÂÒ˜ÂÌËÈ – ÔÂÓ·‡ÁÓ‚‡ÌË ·Â,ÒÛÚ¸ ÍÓÚÓÓ„Ó Á‡Íβ˜‡ÂÚÒfl ‚ ‰Ó·‡‚ÎÂÌËË ÌÂÍÓÚÓÓ„Ó Â·‡  ‚ T ∈ T S Ë ÛÌ˘ÚÓÊÂÌËË ÌÂÍÓÚÓÓ„Ó Â·‡ f ËÁ ÔÓÎÛ˜ÂÌÌÓ„Ó ˆËÍ·, Ú‡Í ˜ÚÓ·˚ e Ë f Ì ÔÂÂÒÂ͇ÎËÒ¸.åÂÚË͇ ÒÍÓθÊÂÌËfl ·‡ ·ÂÁ ÔÂÂÒ˜ÂÌËÈ ÂÒÚ¸ ÏÂÚË͇ ̇ ÏÌÓÊÂÒÚ‚Â T S ‚ÒÂıÓÒÚÓ‚Ì˚ı ‰Â‚¸Â‚ ·ÂÁ ÔÂÂÒ˜ÂÌËÈ ÏÌÓÊÂÒÚ‚‡ S, ÓÔ‰ÂÎÂÌ̇fl ‰Îfl β·˚ı T1, T 2 ∈∈ T S Í‡Í ÏËÌËχθÌÓ ˜ËÒÎÓ ÒÍÓθÊÂÌËÈ Â·Â ·ÂÁ ÔÂÂÒ˜ÂÌËÈ, ÌÂÓ·ıÓ‰ËÏ˚ı ‰ÎflÔÂÓ·‡ÁÓ‚‡ÌËfl T1 ‚ T 2 . ëÍÓθÊÂÌË ·‡ ·ÂÁ ÔÂÂÒ˜ÂÌËÈ ÂÒÚ¸ Ó‰ÌÓ ËÁ ÔÂÓ·‡ÁÓ‚‡ÌËÈ Â·Â, ‚ ıӉ ÍÓÚÓÓ„Ó ·ÂÂÚÒfl ÌÂÍÓÚÓÓÂ Â·Ó Â ‚ T ∈ TS Ë Ó‰Ì‡ ËÁ „ÓÍÓ̈‚˚ı ÚÓ˜ÂÍ ÔÂÂÏ¢‡ÂÚÒfl ‚‰Óθ ÌÂÍÓÚÓÓ„Ó ÒÏÂÊÌÓ„Ó Ò Â Â·‡ ‚ T Ú‡Í, ˜ÚÓ·˚Ì ‚ÓÁÌËÍÎÓ ÔÂÂÒ˜ÂÌËfl Â·Â Ë "Á‡ÏÂÚ‡ÌËfl" ÚÓ˜ÂÍ ËÁ S (˝ÚÓ ‰‡ÂÚ Ì‡Ï ‚ÏÂÒÚÓ ÂÌÓ‚ÓÂ Â·Ó f).
ëÍÓθÊÂÌË ·‡ fl‚ÎflÂÚÒfl ÓÒÓ·˚Ï ÒÎÛ˜‡ÂÏ ÔÂÂÏ¢ÂÌËfl ·‡·ÂÁ ÔÂÂÒ˜ÂÌËÈ: ÌÓ‚Ó ‰ÂÂ‚Ó Ó·‡ÁÛÂÚÒfl ‚ ÂÁÛθڇÚ Á‡Ï˚͇ÌËfl Ò ÔÓÏÓ˘¸˛ fˆËÍ· ë ‰ÎËÌ˚ 3 ‚ í Ë Û‰‡ÎÂÌËfl  ËÁ ë Ú‡ÍËÏ Ó·‡ÁÓÏ, ˜ÚÓ·˚ f Ì ÔÓÔ‡‰‡ÎÓ ‚ÌÛÚ¸ÚÂÛ„ÓθÌË͇ ë.ê‡ÒÒÚÓflÌËfl χ¯ÛÚÓ‚ ÍÓÏÏË‚ÓflʇèÓ·ÎÂχ ÍÓÏÏË‚Óflʇ ËÁ‚ÂÒÚ̇ Í‡Í Á‡‰‡˜‡ ̇ıÓʉÂÌËfl ͇ژ‡È¯Â„Ó Ï‡¯ÛÚ‡ ‰Îfl ÔÓÒ¢ÂÌËfl ÌÂÍÓÚÓÓ„Ó ÏÌÓÊÂÒÚ‚‡ „ÓÓ‰Ó‚.
å˚ ‡ÒÒÏÓÚËÏ ÔÓ·ÎÂÏÛÍÓÏÏË‚Óflʇ ÚÓθÍÓ ‰Îfl ÌÂÓËÂÌÚËÓ‚‡ÌÌÓ„Ó ÒÎÛ˜‡fl. ÑÎfl ¯ÂÌËfl ÔÓ·ÎÂÏ˚ÍÓÏÏË‚Óflʇ ÔËÏÂÌËÚÂθÌÓ Í N „ÓÓ‰‡Ï ‡ÒÒÏÓÚËÏ ÔÓÒÚ‡ÌÒÚ‚Ó N( N − 1)!χ¯ÛÚÓ‚ Í‡Í ÏÌÓÊÂÒÚ‚Ó, ÒÓÒÚÓfl˘Â ËÁˆËÍ΢ÂÒÍËı ÔÂÂÒÚ‡ÌÓ‚ÓÍ2„ÓÓ‰Ó‚ 1, 2,…, N.åÂÚË͇ D ̇ N ÓÔ‰ÂÎflÂÚÒfl ‚ ÚÂÏË̇ı ‡Á΢Ëfl ÒÎÂ‰Û˛˘ËÏ Ó·‡ÁÓÏ: ÂÒÎËχ¯ÛÚ˚ T, T' ∈ N ‡Á΢‡˛ÚÒfl ‚ m ·‡ı, ÚÓ D(T, T') = m.k-OPT ÔÂÓ·‡ÁÓ‚‡ÌË χ¯ÛÚ‡ í ÔÓÎÛ˜‡˛Ú ÔÓÒ‰ÒÚ‚ÓÏ Û‰‡ÎÂÌËfl k · ËÁ íË ÔÓÒÚÓÂÌËfl ‰Û„Ëı ·Â.
凯ÛÚ T', ÔÓÎÛ˜‡ÂÏ˚È ËÁ í Ò ËÒÔÓθÁÓ‚‡ÌËÂÏ k-OPTÔÂÓ·‡ÁÓ‚‡ÌËfl, ̇Á˚‚‡ÂÚÒfl k-OPTÓÏ ‰Îfl í . ê‡ÒÒÚÓflÌË d ̇ ÏÌÓÊÂÒÚ‚Â NÓÔ‰ÂÎflÂÚÒfl ‚ ÚÂÏË̇ı 2-OPT ÔÂÓ·‡ÁÓ‚‡ÌËÈ: d(T, T') ÂÒÚ¸ ÏËÌËχθÌÓ ˜ËÒÎÓ i,238ó‡ÒÚ¸ IV. ê‡ÒÒÚÓflÌËfl ‚ ÔËÍ·‰ÌÓÈ Ï‡ÚÂχÚË͉Îfl ÍÓÚÓÓ„Ó ÒÛ˘ÂÒÚ‚ÛÂÚ ÔÓÒΉӂ‡ÚÂθÌÓÒÚ¸ ËÁ i 2-OPT ÔÂÓ·‡ÁÓ‚‡ÌËÈ, Ô‚Ӊfl˘‡fl í ‚ T'.ÑÎfl β·˚ı T, T' ∈ N ËÏÂÂÚ ÏÂÒÚÓ Ì‡‚ÂÌÒÚ‚Ó d(T, T') ≤ D(T, T') (ÒÏ., ̇ÔËÏÂ,[MaMo95]).ê‡ÒÒÚÓflÌËfl ÏÂÊ‰Û ÔÓ‰„‡Ù‡ÏËëڇ̉‡ÚÌÓ ‡ÒÒÚÓflÌË ̇ ÏÌÓÊÂÒÚ‚Â ‚ÒÂı ÔÓ‰„‡ÙÓ‚ Ò‚flÁÌÓ„Ó „‡Ù‡ G = (V, E)ÓÔ‰ÂÎflÂÚÒfl ͇Ímin{d path (u, v) : u ∈ V ( F ), v ∈ V ( H )}‰Îfl β·˚ı ÔÓ‰„‡ÙÓ‚ F, H „‡Ù‡ G.
ÑÎfl β·˚ı ÔÓ‰„‡ÙÓ‚ F, H ÒËθÌÓ Ò‚flÁÌÓ„ÓÓ„‡Ù‡ D = (V, E) Òڇ̉‡ÚÌÓ ͂‡ÁˇÒÒÚÓflÌË ÓÔ‰ÂÎflÂÚÒfl ͇Ímin{ddpath (u, v) : u ∈V ( F ), v ∈V ( H )}.ê‡ÒÒÚÓflÌË ‚‡˘ÂÌËfl ·‡ ̇ ÏÌÓÊÂÒÚ‚Â Sk(G) ‚ÒÂı ·ÂÌÓ-Ë̉ۈËÓ‚‡ÌÌ˚ıÔÓ‰„‡ÙÓ‚ Ò k ·‡ÏË ‚ Ò‚flÁÌÓÏ „‡Ù G ÓÔ‰ÂÎflÂÚÒfl Í‡Í ÏËÌËχθÌÓ ˜ËÒÎÓ‚‡˘ÂÌËÈ Â·‡, ÌÂÓ·ıÓ‰ËÏ˚ı ‰Îfl ÔÂÓ·‡ÁÓ‚‡ÌËfl F ∈ Sk(G) ‚ H ∈ Sk(G). ÉÓ‚ÓflÚ,˜ÚÓ H ÔÓÎÛ˜‡ÂÚÒfl ËÁ F ‚‡˘ÂÌËÂÏ Â·‡, ÂÒÎË ÒÛ˘ÂÒÚ‚Û˛Ú ‡Á΢Ì˚ ‚¯ËÌ˚ u, vË w ‚ G, Ú‡ÍË ˜ÚÓ uv ∈ E(F), uw ∈ E(G) Ë H = F – uv + uw.ê‡ÒÒÚÓflÌË ÒÏ¢ÂÌËfl ·‡ ̇ ÏÌÓÊÂÒÚ‚Â S k(G) ‚ÒÂı ·ÂÌÓ-Ë̉ۈËÓ‚‡ÌÌ˚ıÔÓ‰„‡ÙÓ‚ Ò k ·‡ÏË ‚ Ò‚flÁÌÓÏ „‡Ù G ÓÔ‰ÂÎflÂÚÒfl Í‡Í ÏËÌËχθÌÓ ˜ËÒÎÓÒÏ¢ÂÌËÈ Â·‡, ÌÂÓ·ıÓ‰ËÏ˚ı ‰Îfl ÔÂÓ·‡ÁÓ‚‡ÌËfl F ∈ Sk(G) ‚ H ∈ Sk(G). ÉÓ‚ÓflÚ,˜ÚÓ ç ÔÓÎÛ˜‡ÂÚÒfl ËÁ F ÒÏ¢ÂÌËÂÏ Â·‡, ÂÒÎË ÒÛ˘ÂÒÚ‚Û˛Ú ‡Á΢Ì˚ ‚¯ËÌ˚ u, vË w ‚ G, Ú‡ÍË ˜ÚÓ uv ∈ E(F), uw ∈ E(G)\E(F) Ë H = F – uv + uw.ê‡ÒÒÚÓflÌË ÔÂÂÏ¢ÂÌËfl ·‡ ̇ ÏÌÓÊÂÒÚ‚Â Sk(G) ‚ÒÂı ·ÂÌÓ-Ë̉ۈËÓ‚‡ÌÌ˚ı ÔÓ‰„‡ÙÓ‚ Ò k ·‡ÏË ‚ „‡Ù G (Ì ӷflÁ‡ÚÂθÌÓ Ò‚flÁÌÓÏ) ÓÔ‰ÂÎflÂÚÒflÍ‡Í ÏËÌËχθÌÓ ˜ËÒÎÓ ÔÂÂÏ¢ÂÌËÈ Â·‡, ÌÂÓ·ıÓ‰ËÏ˚ı ‰Îfl ÔÂÓ·‡ÁÓ‚‡ÌËfl F ∈∈ S k(G) ‚ H ∈ S k(G).
ÉÓ‚ÓflÚ, ˜ÚÓ ç ÔÓÎÛ˜‡ÂÚÒfl ËÁ F ÔÂÂÏ¢ÂÌËflÏ Â·‡, ÂÒÎËÒÛ˘ÂÒÚ‚Û˛Ú (Ì ӷflÁ‡ÚÂθÌÓ ‡Á΢Ì˚Â) ‚¯ËÌ˚ u, v, w Ë x ‚ G, Ú‡ÍË ˜ÚÓuv ∈ E(F), wx ∈ E(G)\E(F) Ë H = F – uv + w x. ê‡ÒÒÚÓflÌË ÔÂÂÏ¢ÂÌËfl ·‡ –ÏÂÚË͇ ̇ S k(G). ÖÒÎË F Ë H ËÏÂ˛Ú s Ó·˘Ëı e·Â, ÚÓ ÓÌÓ ‡‚ÌÓ k – s.ê‡ÒÒÚÓflÌË Ò͇˜Í‡ ·‡ (ÍÓÚÓÓ ‚ Ó·˘ÂÏ ÒÎÛ˜‡Â ÏÓÊÂÚ ÔËÌËχڸ Á̇˜ÂÌËÂ∞) ̇ ÏÌÓÊÂÒÚ‚Â S k(G) ‚ÒÂı ·ÂÌÓ-Ë̉ۈËÓ‚‡ÌÌ˚ı ÔÓ‰„‡ÙÓ‚ Ò k e·‡ÏË „‡Ù‡G (Ì ӷflÁ‡ÚÂθÌÓ Ò‚flÁÌÓ„Ó) ÓÔ‰ÂÎflÂÚÒfl Í‡Í ÏËÌËχθÌÓ ˜ËÒÎÓ Ò͇˜ÍÓ‚ ·‡,ÌÂÓ·ıÓ‰ËÏ˚ı ‰Îfl ÔÂÓ·‡ÁÓ‚‡ÌËfl F ∈ S k(G) ‚ H ∈ S k(G). ÉÓ‚ÓflÚ, ˜ÚÓ H ÔÓÎÛ˜‡ÂÚÒflËÁ F Ò͇˜ÍÓÏ Â·‡, ÂÒÎË ÒÛ˘ÂÒÚ‚Û˛Ú Ú‡ÍË ˜ÂÚ˚ ‡Á΢Ì˚ ‚¯ËÌ˚ u, v, w Ë x‚ G, ˜ÚÓ uv ∈ E(F), wx ∈ E(G)\E(F) Ë H = F – uv + wx.15.4.
êÄëëíéüçàü çÄ ÑÖêÖÇúüïèÛÒÚ¸ í – ÍÓ̂Ӡ‰Â‚Ó, Ú.Â. ‰Â‚Ó, Û ÍÓÚÓÓ„Ó Ó‰Ì‡ ËÁ Â„Ó ‚¯ËÌ ‚˚·‡Ì‡ ‚͇˜ÂÒÚ‚Â ÍÓÌfl. ÉÎÛ·Ë̇ ‚¯ËÌ˚ v, depth(v) – ˝ÚÓ ˜ËÒÎÓ e·Â ̇ ÔÛÚË ÓÚ v ÍÍÓÌ˛. ǯË̇ v ̇Á˚‚‡ÂÚÒfl Ó‰ËÚÂθÒÍÓÈ ‰Îfl ‚¯ËÌ˚ u, v = par(u), ÂÒÎË ÓÌËÒÏÂÊÌ˚Â Ë ËÏÂÂÚ ÏÂÒÚÓ ‡‚ÂÌÒÚ‚Ó depth(u) = depth(v) + 1; ‚ ˝ÚÓÏ ÒÎÛ˜‡Â u ̇Á˚‚‡ÂÚÒfl ‰Ó˜ÂÌÂÈ ‰Îfl v. т ‚¯ËÌ˚ ̇Á˚‚‡˛ÚÒfl ÒÂÒÚ‡ÏË, ÂÒÎË ËÏÂ˛Ú Ó‰ÌÓ„Ó ËÚÓ„Ó Ê ӉËÚÂÎfl. ëÚÂÔÂ̸ ‚˚ıÓ‰‡ ‚¯ËÌ˚ – ˝ÚÓ ÍÓ΢ÂÒÚ‚Ó Â ‰Ó˜ÂÌËı ‚¯ËÌ. T(v) ÂÒÚ¸ ÔÓ‰‰ÂÂ‚Ó ‰Â‚‡ í Ò ÍÓÌÂÏ ‚ ‚¯ËÌ v ∈ V(T). ÖÒÎË w ∈ V(T(v)),ÚÓ v fl‚ÎflÂÚÒfl Ô‰ÍÓÏ ‰Îfl w, ‡ w – ÔÓÚÓÏÍÓÏ ‰Îfl v; nca(u, v) – ·ÎËʇȯËÈ Ó·˘ËÈÔ‰ÓÍ ‰Îfl ‚¯ËÌ u Ë v. ÑÂÂ‚Ó Ì‡Á˚‚‡ÂÚÒfl ÔÓϘÂÌÌ˚Ï ‰Â‚ÓÏ, ÂÒÎË Í‡Ê‰‡fl ËÁ239É·‚‡ 15.
ê‡ÒÒÚÓflÌËfl ‚ ÚÂÓËË „‡ÙÓ‚Â„Ó ‚¯ËÌ Ó·ÓÁ̇˜Â̇ ÒËÏ‚ÓÎÓÏ Á‡‰‡ÌÌÓ„Ó ÍÓ̘ÌÓ„Ó ‡ÎÙ‡‚ËÚ‡ . ÑÂÂ‚Ó í̇Á˚‚‡ÂÚÒfl ÛÔÓfl‰Ó˜ÂÌÌ˚Ï ‰Â‚ÓÏ, ÂÒÎË Á‡‰‡Ì ÔÓfl‰ÓÍ (Ò΂‡ ̇ԇ‚Ó) ̇‚¯Ë̇ı-ÒÂÒÚ‡ı.ç‡ ÏÌÓÊÂÒÚ‚Â rlo ‚ÒÂı ÍÓÌ‚˚ı ÔÓϘÂÌÌ˚ı ÛÔÓfl‰Ó˜ÂÌÌ˚ı ‰Â‚¸Â‚ ‰ÓÔÛÒ͇˛ÚÒfl ÒÎÂ‰Û˛˘Ë ÓÔ‡ˆËË Â‰‡ÍÚËÓ‚‡ÌËfl:èÂÂË̉ÂÍÒ‡ˆËfl – ËÁÏÂÌÂÌË ÏÂÚÍË ‚¯ËÌ˚ v.쉇ÎÂÌË – Û‰‡ÎÂÌË ÌÂÍÓÌ‚ÓÈ ‚¯ËÌ˚ v Ò Ó‰ËÚÂÎÂÏ v', Ú‡Í ˜ÚÓ ‰Ó˜ÂÌ˽ÎÂÏÂÌÚ˚ v ÒÚ‡ÌÓ‚flÚÒfl ‰Ó˜ÂÌËÏË ˝ÎÂÏÂÌÚ‡ÏË v'; ˝ÚË ‰Ó˜ÂÌË ˝ÎÂÏÂÌÚ˚ ‚ÒÚ‡‚Îfl˛ÚÒfl ‚ÏÂÒÚÓ v Í‡Í ÛÔÓfl‰Ó˜ÂÌ̇fl Ò΂‡ ̇ԇ‚Ó ÔÓ‰ÔÓÒΉӂÚÂθÌÓÒÚ¸ ‰Ó˜ÂÌËı˝ÎÂÏÂÌÚÓ‚ v'.ÇÒÚ‡‚͇ – ‰ÓÔÓÎÌÂÌËÂ Í Û‰‡ÎÂÌ˲; ‚ÒÚ‡‚͇ ‚¯ËÌ˚ v ‚ ͇˜ÂÒÚ‚Â ‰Ó˜ÂÌ„ӽÎÂÏÂÌÚ‡ v', ˜ÚÓ ‰Â·ÂÚ v Ó‰ËÚÂÎÂÏ ÔÓÒÎÂ‰Û˛˘ÂÈ ÔÓ‰ÔÓÒΉӂ‡ÚÂθÌÓÒÚË ‰Ó˜ÂÌËı ˝ÎÂÏÂÌÚÓ‚ v'.ÑÎfl ÌÂÛÔÓfl‰Ó˜ÂÌÌ˚ı ‰Â‚¸Â‚ ÓÔ‡ˆËË Â‰‡ÍÚËÓ‚‡ÌËfl ÓÔ‰ÂÎfl˛ÚÒfl ‡Ì‡Îӄ˘Ì˚Ï Ó·‡ÁÓÏ, ÌÓ ÓÔ‡ˆËË ‚ÒÚ‡‚ÍË Ë Û‰‡ÎÂÌËfl ‰ÂÈÒÚ‚Û˛Ú Ì‡ ÔÓ‰ÏÌÓÊÂÒÚ‚Â, ‡ ÌÂ̇ ÔÓ‰ÔÓÒΉӂ‡ÚÂθÌÓÒÚË.è‰ÔÓ·„‡ÂÚÒfl, ˜ÚÓ ÒÛ˘ÂÒÚ‚ÛÂÚ ÙÛÌ͈Ëfl ˆÂÌ˚, ÓÔ‰ÂÎflÂχfl ‰Îfl ͇ʉÓÈ ÓÔ‡ˆËË Â‰‡ÍÚËÓ‚‡ÌËfl, ‡ ˆÂ̇ ÔÓÒΉӂ‡ÚÂθÌÓÒÚË ÓÔ‡ˆËÈ Â‰‡ÍÚËÓ‚‡ÌËflÓÔ‰ÂÎflÂÚÒfl Í‡Í ÒÛÏχ ˆÂÌ ˝ÚËı ÓÔ‡ˆËÈ.ìÔÓfl‰Ó˜ÂÌÌÓ ÓÚÓ·‡ÊÂÌË ‡ÒÒÚÓflÌËfl ‰‡ÍÚËÓ‚‡ÌËfl – ÒÔˆˇθ̇flËÌÚÂÔÂÚ‡ˆËfl ÓÔ‡ˆËÈ Â‰‡ÍÚËÓ‚‡ÌËfl.
îÓχθÌÓ, ̇ÁÓ‚ÂÏ ÚÓÈÍÛ (M, T1, T2)Í‡Í ÛÔÓfl‰Ó˜ÂÌÌ˚Ï ÓÚÓ·‡ÊÂÌËÂÏ ‡ÒÒÚÓflÌËfl ‰‡ÍÚËÓ‚‡ÌËfl ‰Â‚‡ í1 ‚‰ÂÂ‚Ó í2, T 1 , T 2 ∈ rlo, ÂÒÎË M ⊂ V(T 1 ) × V(T 2 ) Ë, ‰Îfl β·˚ı (v1, w 1 ), (v2 , w 2 ) ∈ M‚˚ÔÓÎÌflÂÚÒfl ÒÎÂ‰Û˛˘Â ÛÒÎÓ‚ËÂ: v1 = v2 ÚÓ„‰‡ Ë ÚÓθÍÓ ÚÓ„‰‡, ÍÓ„‰‡ w 1 = w 2(ÛÒÎÓ‚Ë ‚Á‡ËÏÌÓÈ Ó‰ÌÓÁ̇˜ÌÓÒÚË), v1 fl‚ÎflÂÚÒfl Ô‰ÍÓÏ ‰Îfl v2 ÚÓ„‰‡ Ë ÚÓθÍÓÚÓ„‰‡, ÍÓ„‰‡ w1 fl‚ÎflÂÚÒfl Ô‰ÍÓÏ w2 (ÛÒÎÓ‚Ë Ô‰ÍÓ‚), v 1 ̇ıÓ‰ËÚÒfl Ò΂‡ ÓÚ v2ÚÓ„‰‡ Ë ÚÓθÍÓ ÚÓ„‰‡, ÍÓ„‰‡ w 1 ̇ıÓ‰ËÚÒfl Ò΂‡ ÓÚ w2 (ÛÒÎÓ‚Ë ÒÂÒÚÂ).ÉÓ‚ÓflÚ, ˜ÚÓ ‚¯Ë̇ v ‚ T 1 Ë T2 ÚÓÌÛÚ‡ ÎËÌËÂÈ ‚ å , ÂÒÎË v ÔÓfl‚ÎflÂÚÒfl‚ ÌÂÍÓÚÓÓÈ Ô‡Â ËÁ å.
èÛÒÚ¸ N1 Ë N2 – ÏÌÓÊÂÒÚ‚‡ ‚¯ËÌ ‰Â‚¸Â‚ T 1 Ë T2 ÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓ, ÍÓÚÓ˚ Ì ÚÓÌÛÚ˚ ÎËÌËflÏË ‚ å.ñÂ̇ å Á‡‰‡ÂÚÒfl ͇Íγ (M) =γ ( v → w) +γ (v → λ) +γ (λ → w ), „‰Â γ ( a → b) = γ ( a, b) – ˆÂ̇ ÓÔÂ-∑( v , w ) ∈M∑v ∈N1∑w ∈N 2‡ˆËË Â‰‡ÍÚËÓ‚‡ÌËfl a → b, ÍÓÚÓ‡fl fl‚ÎflÂÚÒfl ÔÂÂË̉ÂÍÒ‡ˆËÂÈ, ÂÒÎË a, b ∈ ,Û‰‡ÎÂÌËÂÏ, ÂÒÎË b = λ, Ë ‚ÒÚ‡‚ÍÓÈ, ÂÒÎË a = λ. á‰ÂÒ¸ ÒËÏ‚ÓÎ λ ∉ ‚˚ÒÚÛÔ‡ÂÚ Í‡ÍÒÔˆˇθÌ˚È ÒËÏ‚ÓÎ Ôӷ·, Ë γ fl‚ÎflÂÚÒfl ÏÂÚËÍÓÈ Ì‡ ÏÌÓÊÂÒÚ‚Â ∪ λ (ËÒÍβ˜‡fl Á̇˜ÂÌË γ(λ, λ)).ê‡ÒÒÚÓflÌË ‰‡ÍÚËÓ‚‡ÌËfl ‰Â‚‡ê‡ÒÒÚÓflÌË ‰‡ÍÚËÓ‚‡ÌËfl ‰Â‚‡ ([Tai79]) ̇ ÏÌÓÊÂÒÚ‚Â rlo ‚ÒÂı ÍÓÌ‚˚ıÔÓϘÂÌÌ˚ı ÛÔÓfl‰Ó˜ÂÌÌ˚ı ‰Â‚¸Â‚ ÓÔ‰ÂÎflÂÚÒfl ‰Îfl β·˚ı T1, T 2 ∈ rlo ͇ÍÏËÌËχθ̇fl ˆÂ̇ ÔÓÒΉӂ‡ÚÂθÌÓÒÚË ÓÔ‡ˆËÈ Â‰‡ÍÚËÓ‚‡ÌËfl (ÔÂÂË̉ÂÍÒ‡ˆËÈ,‚ÒÚ‡‚ÓÍ Ë Û‰‡ÎÂÌËÈ), Ô‚Ӊfl˘ÂÈ T 1 ‚ T2.Ç ÚÂÏË̇ı ÛÔÓfl‰Ó˜ÂÌÌÓ„Ó ÓÚÓ·‡ÊÂÌËfl ‡ÒÒÚÓflÌËfl ‰‡ÍÚËÓ‚‡ÌËfl ˝ÚÓ‡ÒÒÚÓflÌË ‡‚ÌÓ min ( M , T1 , T2 ) γ ( M ), „‰Â ÏËÌËÏÛÏ ·ÂÂÚÒfl ÔÓ ‚ÒÂÏ ÛÔÓfl‰Ó˜ÂÌÌ˚ÏÓÚÓ·‡ÊÂÌËflÏ ‡ÒÒÚÓflÌËfl ‰‡ÍÚËÓ‚‡ÌËfl (M, T1 , T 2 ).ê‡ÒÒÚÓflÌË ‰‡ÍÚËÓ‚‡ÌËfl ‰Â‚‡ ÏÓÊÌÓ ÓÔ‰ÂÎËÚ¸ ‡Ì‡Îӄ˘Ì˚Ï Ó·‡ÁÓÏ Ì‡ÏÌÓÊÂÒÚ‚Â ‚ÒÂı ÍÓÌ‚˚ı ÌÂÛÔÓfl‰Ó˜ÂÌÌ˚ı ‰Â‚¸Â‚.240ó‡ÒÚ¸ IV.
ê‡ÒÒÚÓflÌËfl ‚ ÔËÍ·‰ÌÓÈ Ï‡ÚÂχÚËÍÂê‡ÒÒÚÓflÌË ëÂÎÍÓÛê‡ÒÒÚÓflÌË ëÂÎÍÓÛ (ËÎË ‡ÒÒÚÓflÌË ÌËÒıÓ‰fl˘Â„Ó Â‰‡ÍÚËÓ‚‡ÌËfl, ‡ÒÒÚÓflÌËfl‰‡ÍÚËÓ‚‡ÌËfl 1-ÒÚÂÔÂÌË) ÂÒÚ¸ ‡ÒÒÚÓflÌË ̇ ÏÌÓÊÂÒÚ‚Â rlo ‚ÒÂı ÍÓÌ‚˚ıÔÓϘÂÌÌ˚ı ÛÔÓfl‰Ó˜ÂÌÌ˚ı ‰Â‚¸Â‚, ÓÔ‰ÂÎÂÌÌÓ ‰Îfl β·˚ı T1, T2 ∈ rloÍ‡Í ÏËÌËχθ̇fl ˆÂ̇ ÔÓÒΉӂ‡ÚÂθÌÓÒÚË ÓÔ‡ˆËÈ Â‰‡ÍÚËÓ‚‡ÌËfl (ÔÂÂË̉ÂÍÒ‡ˆËÈ, ‚ÒÚ‡‚ÓÍ Ë Û‰‡ÎÂÌËÈ), Ô‚Ӊfl˘ÂÈ T 1 ‚ T2, ÂÒÎË ‚ÒÚ‡‚ÍË Ë Û‰‡ÎÂÌËfl‡ÒÔÓÒÚ‡Ìfl˛ÚÒfl ÚÓθÍÓ Ì‡ ÎËÒÚ¸fl ‰Â‚¸Â‚ ([Selk77]). äÓÂ̸ ‰Â‚‡ T1 ‰ÓÎÊÂÌÓÚÓ·‡Ê‡Ú¸Òfl ‚ ÍÓÂ̸ ‰Â‚‡ T 2 Ë, ÂÒÎË ‚¯Ë̇ v ÔÓ‰ÎÂÊËÚ Û‰‡ÎÂÌ˲ (‚ÒÚ‡‚ÍÂ),ÚÓ ÔÓ‰‰ÂÂ‚Ó Ò ÍÓÌÂÏ ‚ v, ÂÒÎË Ú‡ÍÓ‚Ó ËÏÂÂÚÒfl, ÔÓ‰ÎÂÊËÚ Û‰‡ÎÂÌ˲ (‚ÒÚ‡‚ÍÂ).Ç ÚÂÏË̇ı ÛÔÓfl‰Ó˜ÂÌÌÓ„Ó ÓÚÓ·‡ÊÂÌËfl ‡ÒÒÚÓflÌËfl ‰‡ÍÚËÓ‚‡ÌËfl ‡ÒÒÚÓflÌË ëÂÎÍÓÛ ‡‚ÌÓ min ( M , T1 , T2 ) γ ( M ), „‰Â ÏËÌËÏÛÏ ·ÂÂÚÒfl ÔÓ ‚ÒÂÏ ÓÚÓ·‡ÊÂÌËflχÒÒÚÓflÌËfl ÛÔÓfl‰Ó˜ÂÌÌÓ„Ó Â‰‡ÍÚËÓ‚‡ÌËfl (M, T1, T2 ), Û‰Ó‚ÎÂÚ‚Ófl˛˘ËÏ ÒÎÂ‰Û˛˘ÂÏÛ ÛÒÎӂ˲: ÂÒÎË (v, w) ∈ M , „‰Â ÌË v, ÌË w Ì fl‚Îfl˛ÚÒfl ÍÓÌflÏË, ÚÓ (par(v),par(w)) ∈ M.ê‡ÒÒÚÓflÌË ‰‡ÍÚËÓ‚‡ÌËfl Ò Ó„‡Ì˘ÂÌËÂÏê‡ÒÒÚÓflÌË ‰‡ÍÚËÓ‚‡ÌËfl Ò Ó„‡Ì˘ÂÌËÂÏ (ËÎË ‡ÒÒÚÓflÌË „·ÏÂÌÚËÓ‚‡ÌÌÓ„Ó Â‰‡ÍÚËÓ‚‡ÌËfl) ÂÒÚ¸ ‡ÒÒÚÓflÌË ̇ ÏÌÓÊÂÒÚ‚Â rlo ‚ÒÂı ÍÓÌ‚˚ı ÔÓϘÂÌÌ˚ı ÛÔÓfl‰Ó˜ÂÌÌ˚ı ‰Â‚¸Â‚, ÓÔ‰ÂÎÂÌÌÓ ‰Îfl β·˚ı T1, T2 ∈ rlo ͇ÍÏËÌËχθ̇fl ˆÂ̇ ÔÓÒΉӂ‡ÚÂθÌÓÒÚË ÓÔ‡ˆËÈ Â‰‡ÍÚËÓ‚‡ÌËfl (ÔÂÂË̉ÂÍÒ‡ˆËÈ,‚ÒÚ‡‚ÓÍ Ë Û‰‡ÎÂÌËÈ), Ô‚Ӊfl˘ÂÈ T 1 ‚ T 2 , Ò ÚÂÏ Ó„‡Ì˘ÂÌËÂÏ, ˜ÚÓ ÌÂÔÂÂÒÂ͇˛˘ËÂÒfl ÔÓ‰‰Â‚¸fl ‰ÓÎÊÌ˚ ÓÚÓ·‡Ê‡Ú¸Òfl ̇ ÌÂÔÂÂÒÂ͇˛˘ËÂÒfl ÔÓ‰‰Â‚¸fl.Ç ÚÂÏË̇ı ÛÔÓfl‰Ó˜ÂÌÌÓ„Ó ÓÚÓ·‡ÊÂÌËfl ‡ÒÒÚÓflÌËfl ‰‡ÍÚËÓ‚‡ÌËfl ‡ÒÒÚÓflÌË ‰‡ÍÚËÓ‚‡ÌËfl Ò Ó„‡Ì˘ÂÌËÂÏ ‡‚ÌÓ min ( M , T1 , T2 ) γ ( M ), „‰Â ÏËÌËÏÛÏ ·ÂÂÚÒflÔÓ ‚ÒÂÏ ÛÔÓfl‰Ó˜ÂÌÌ˚Ï ÓÚÓ·‡ÊÂÌËflÏ ‡ÒÒÚÓflÌËfl ‰‡ÍÚËÓ‚‡ÌËfl (M, T1 , T 2 ),Û‰Ó‚ÎÂÚ‚Ófl˛˘ËÏ ÒÎÂ‰Û˛˘ÂÏÛ ÛÒÎӂ˲: ‰Îfl ‚ÒÂı (v1 , w 1 ), (v2, w 2 ), (v3, w 3 ) ∈ M,nca(v1 , v2 ) fl‚ÎflÂÚÒfl ÒÓ·ÒÚ‚ÂÌÌ˚Ï Ô‰ÍÓÏ v3 ÚÓ„‰‡ Ë ÚÓθÍÓ ÚÓ„‰‡, ÍÓ„‰‡ nca(w 1 , w2 )fl‚ÎflÂÚÒfl ÒÓ·ÒÚ‚ÂÌÌ˚Ï Ô‰ÍÓÏ w 3 .ùÚÓ ‡ÒÒÚÓflÌË ˝Í‚Ë‚‡ÎÂÌÚÌÓ ‡ÒÒÚÓflÌ˲ ‰‡ÍÚËÓ‚‡ÌËfl ÒÓÓÚ‚ÂÚÒÚ‚ÂÌÌÓÒÚÛÍÚÛÂ, ÓÔ‰ÂÎÂÌÌÓÏÛ Í‡Í min ( M , T1 , T2 ) γ ( M ), „‰Â ÏËÌËÏÛÏ ·ÂÂÚÒfl ÔÓ ‚ÒÂÏÛÔÓfl‰Ó˜ÂÌÌ˚Ï ÓÚÓ·‡ÊÂÌËflÏ ‡ÒÒÚÓflÌËfl ‰‡ÍÚËÓ‚‡ÌËfl (M, T1 , T 2 ), Û‰Ó‚ÎÂÚ‚Ófl˛˘ËÏ ÒÎÂ‰Û˛˘ÂÏÛ ÛÒÎӂ˲: ‰Îfl ‚ÒÂı (v1 , w1), (v2 , w2), (v3 , w3) ∈ M, Ú‡ÍËı ˜ÚÓ ÌËӉ̇ ËÁ v1 , v2 Ë v3 Ì fl‚ÎflÂÚÒfl Ô‰ÍÓÏ ‰Îfl ‰Û„Ëı, nca(v1, v2 ) = nca(v1 , v3 ) ÚÓ„‰‡ ËÚÓθÍÓ ÚÓ„‰‡, ÍÓ„‰‡ nca(w1, w 2 ) = nca(w 1 , w 3 )ê‡ÒÒÚÓflÌË ‰‡ÍÚËÓ‚‡ÌËfl ‰ËÌ˘ÌÓÈ ˆÂÌ˚ê‡ÒÒÚÓflÌË ‰‡ÍÚËÓ‚‡ÌËfl ‰ËÌ˘ÌÓÈ ˆÂÌ˚ – ‡ÒÒÚÓflÌË ̇ ÏÌÓÊÂÒÚ‚Â rlo ‚ÒÂıÍÓÌ‚˚ı ÔÓϘÂÌÌ˚ı ÛÔÓfl‰Ó˜ÂÌÌ˚ı ‰Â‚¸Â‚, ÓÔ‰ÂÎÂÌÌÓ ‰Îfl β·˚ıT 1 , T 2 ∈ rlo Í‡Í ÏËÌËχθÌÓ ˜ËÒÎÓ ÓÔ‡ˆËÈ Â‰‡ÍÚËÓ‚‡ÌËfl (ÔÂÂË̉ÂÍÒ‡ˆËÂÈ,‚ÒÚ‡‚ÓÍ Ë Û‰‡ÎÂÌËÈ), Ô‚Ӊfl˘Ëı T 1 ‚ T2.ê‡ÒÒÚÓflÌË ‚˚‡‚ÌË‚‡ÌËflê‡ÒÒÚÓflÌË ‚˚‡‚ÌË‚‡ÌËfl ([JWZ94]) ÂÒÚ¸ ‡ÒÒÚÓflÌË ̇ ÏÌÓÊÂÒÚ‚Â rlo ‚ÒÂıÍÓÌ‚˚ı ÔÓϘÂÌÌ˚ı ÛÔÓfl‰Ó˜ÂÌÌ˚ı ‰Â‚¸Â‚, ÓÔ‰ÂÎÂÌÌÓ ‰Îfl β·˚ıT 1 , T2 ∈ rlo Í‡Í ÏËÌËχθ̇fl ˆÂ̇ ‚˚‡‚ÌË‚‡ÌËfl T1 Ë T 2 .