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Table 2 Precomputed expressions used in the proof of Lemma 4.7

From: Consistency of the Neighbor-Net Algorithm

Case l = 1 r 1 d S ( x l , x l + 1 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaaiabdsgaKnaaBaaaleaacqWGtbWuaeqaaOGaeiikaGIaemiEaG3aaSbaaSqaaiabdYgaSbqabaGccqGGSaalcqWG4baEdaWgaaWcbaGaemiBaWMaey4kaSIaeGymaedabeaakiabcMcaPaWcbaGaemiBaWMaeyypa0JaeGymaedabaGaemOCaiNaeyOeI0IaeGymaedaniabggHiLdaaaa@4289@ d S (x1, x r ) l = 1 n d S ( x 1 , x l ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaaiabdsgaKnaaBaaaleaacqWGtbWuaeqaaOGaeiikaGIaemiEaG3aaSbaaSqaaiabigdaXaqabaGccqGGSaalcqWG4baEdaWgaaWcbaGaemiBaWgabeaakiabcMcaPaWcbaGaemiBaWMaeyypa0JaeGymaedabaGaemOBa4ganiabggHiLdaaaa@3E61@
(i) 1 1 n - j
(ii) 0 0 n - j
(iii) 2 0 j - i + 1
(iv) 1 1 j - i + 1
(v) 0 0 j - i + 1
Case l = 2 r 1 k = 1 n d S ( x l , x k ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaamaaqadabaGaemizaq2aaSbaaSqaaiabdofatbqabaGccqGGOaakcqWG4baEdaWgaaWcbaGaemiBaWgabeaakiabcYcaSiabdIha4naaBaaaleaacqWGRbWAaeqaaOGaeiykaKcaleaacqWGRbWAcqGH9aqpcqaIXaqmaeaacqWGUbGBa0GaeyyeIuoaaSqaaiabdYgaSjabg2da9iabikdaYaqaaiabdkhaYjabgkHiTiabigdaXaqdcqGHris5aaaa@4773@ l = 1 n d S ( x r , x l ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaaeWaqaaiabdsgaKnaaBaaaleaacqWGtbWuaeqaaOGaeiikaGIaemiEaG3aaSbaaSqaaiabdkhaYbqabaGccqGGSaalcqWG4baEdaWgaaWcbaGaemiBaWgabeaakiabcMcaPaWcbaGaemiBaWMaeyypa0JaeGymaedabaGaemOBa4ganiabggHiLdaaaa@3EDE@
(i) (j - 1)(n - j) + (r - j - 1)j j
(ii) (r - 2)(n - j) n - j
(iii) (j - i + 1)(n - 2j + 2i + r - 4) j - i + 1
(iv) (i - 2)(j - i + 1) + (r - i)(i - 1 + n - j) i - 1 + n - j
(v) (r - 2)(j - i + 1) j - i + 1