▁the ▁tec ni ques ▁util ized ▁by ▁Hu - Sh ing ▁in ▁their ▁work ▁easier . ▁ ▁To ▁each ▁vertex ▁V ▁of ▁the ▁polygon ▁is ▁associated ▁a ▁weight ▁w . ▁Suppose ▁we ▁have ▁three ▁consecutive ▁vertices ▁, ▁and ▁that ▁ ▁is ▁the ▁vertex ▁with ▁minimum ▁weight ▁. ▁We ▁look ▁at ▁the ▁quad ril ater al ▁with ▁vertices ▁ ▁( in ▁clock wise ▁order ). ▁We ▁can ▁tri ang ulate ▁it ▁in ▁two ▁ways : ▁ ▁and ▁, ▁with ▁cost ▁▁ ▁and ▁ ▁with ▁cost ▁. ▁ ▁Therefore ▁if ▁▁ ▁or ▁equival ently ▁ ▁we ▁remove ▁the ▁vertex ▁ ▁from ▁the ▁polygon ▁and ▁add ▁the ▁side ▁ ▁to ▁the ▁tri ang ulation . ▁We ▁repeat ▁this ▁process ▁until ▁no ▁ ▁satisfies ▁the ▁condition ▁above . ▁For ▁all ▁the ▁remaining ▁vertices ▁, ▁we ▁add ▁the ▁side ▁ ▁to ▁the ▁tri ang ulation . ▁ ▁This ▁gives ▁us ▁a ▁nearly ▁optimal ▁tri ang ulation . ▁ ▁General izations ▁▁ ▁The ▁matrix ▁chain ▁multiplication ▁problem ▁general izes ▁to ▁solving ▁a ▁more ▁abstract ▁problem : ▁given ▁a ▁linear ▁sequence ▁of ▁objects , ▁an ▁associ ative ▁binary ▁operation ▁on ▁those ▁objects , ▁and ▁a ▁way ▁to ▁compute ▁the ▁cost ▁of ▁performing ▁that ▁operation ▁on ▁any ▁two ▁given ▁objects ▁( as ▁well ▁as ▁all ▁partial ▁results ), ▁compute ▁the ▁minimum ▁cost ▁way ▁to ▁group ▁the ▁objects ▁to ▁apply ▁the ▁operation ▁over ▁the ▁sequence . ▁One ▁somewhat ▁cont riv ed ▁special ▁case ▁of ▁this ▁is ▁string ▁concaten ation ▁of ▁a ▁list ▁of ▁strings . ▁In ▁C , ▁for ▁example , ▁the