▁ ▁The ▁valid ity ▁of ▁the ▁tree ▁structure ▁is ▁ens ured ▁by ▁creating ▁a ▁certificate ▁at ▁each ▁edge ▁that ▁enfor ces ▁the ▁heap ▁property ▁on ▁that ▁edge . ▁The ▁main ▁oper ational ▁difference ▁between ▁a ▁kin etic ▁heap ▁and ▁a ▁kin etic ▁he ater ▁is ▁in ▁how ▁they ▁respond ▁to ▁certificate ▁fail ures . ▁When ▁a ▁certificate ▁on ▁an ▁edge ▁fails , ▁a ▁kin etic ▁he ater ▁will ▁perform ▁a ▁rotation ▁around ▁the ▁nodes ▁that ▁failed ▁( inst ead ▁of ▁the ▁swap ▁that ▁a ▁kin etic ▁heap ▁would ▁perform ). ▁ ▁For ▁example , ▁consider ▁the ▁elements ▁B ▁( with ▁parent ▁F ) ▁and ▁its ▁left ▁child ▁D ▁( with ▁right ▁child ▁C ). ▁When ▁the ▁certificate ▁[ B > D ] ▁on ▁the ▁edge ▁B D ▁fails , ▁the ▁tree ▁will ▁be ▁rot ated ▁around ▁this ▁edge . ▁Thus ▁in ▁this ▁case ▁the ▁resulting ▁structure ▁has ▁D ▁in ▁place ▁of ▁B , ▁C ▁becomes ▁a ▁child ▁of ▁B ▁instead ▁of D , ▁and ▁there ▁are ▁three ▁certificate ▁changes ▁[ B > D ] ▁replaced ▁with ▁[ D > B ], ▁[ D > C ] ▁replaced ▁with ▁[ B > C ] ▁and ▁[ F > B ] ▁replaced ▁with ▁[ F > D ]. ▁Everything ▁else ▁stays ▁the ▁same . ▁ ▁Analysis ▁ ▁This ▁kin etic ▁data ▁structure ▁is : ▁ ▁Res p ons ive : ▁There ▁are ▁O ( 1 ) ▁certificate ▁updates ▁that ▁need ▁to ▁be ▁done ▁when ▁a ▁certificate ▁fails , ▁which ▁takes ▁O ( log ▁n )