| To enhance storage systems performance,popularities of the data items must be taken into account to make frequencies of accesses to storage units as uniform as possible.To this end,Dau and Milenkovic in 2018 proposed a combinatorial model that ranks data items by popularity called a point-labeled Steiner system,the points are labled by 0,1,…,v-1,the data items and the storage units are associated with points and blocks in the Steiner system,respectively,so that the difference DiffSum between the largest and smallest block-sums is as small as possible,therefore the corresponding frequencies of accesses to storage units are as uniform as possible.They also established some upper and lower bounds on DiffSum for point-labeled Steiner systems.Chee et al.further improved the upper and lower bounds on DiffSum for point-labeled Steiner triple systems(STSs)in 2020.The sufficient and necessary condition for the existence of an STS(v)is r≡1,3(mod 6).When v(?)1,3(mod 6),we generalize a pointlabeled STS(v)to a point-labeled packing triple system.Since the number of blocks corresponds to the number of storage units in the storage system,we consider only the point-labeled optimal packing triple systems(OPTSs),and make DiffSum as small as possible.This thesis mainly studies the upper and lower bounds on DiffSum for point-labeled OPTSs and is organized as follows.In Chapter 1,we describe the background and main results of this thesis.In Chapter 2,we give the following lower bound on DiffSum for a point-labeled OPTS(v)by calculating the average value of block-sum and analyzing the leave structure of an OPTS(v).(?)In Chapter 3,we give the following upper bound on DiffSum for a point-labeled OPTS(v)by generalizing the constructions of point-labeled STSs.(?)If v≡0,2(mod 6)and for every nonzero x ∈Zv-1,the cycle for-2 containing x has even size;v≡4(mod 6)and for every x ∈Zv-1\(0,(v-1)/3,2(v-1)/3},the cycle for-2 containing x has even size;or r≡5(mod 6)and for every x∈Zv-2\{0,(v-2)/3,2(v-2)/3},the cycle for-2 containing x has even size,the upper bound on DiffSum for a point-labeled OPTS(v)is further improved as follows.(?)In Chapter 4,we give a result on DiffSum for point-labeled Steiner quadruple systems by using the doubling construction,and make a brief summary on this thesis. |
