Approaches | Nbr and list of parameters | Functions | Output | Structure | Technics | Constraints | |
Greedy algorithm | 3 | $V=\left\{{v}_{1},{v}_{2},\cdots ,{v}_{n}\right\}$ Set of candidate view, $C=\left\{{c}_{1},{c}_{2},\cdots ,{c}_{n}\right\}$ where c_{i} as the associated size of each view candidate v_{i} K: number max of view to be materialized, | $\mathrm{max}\left(B\left(v,S\right)\right)=\mathrm{max}\left({\displaystyle {\sum}_{W\le U}{B}_{W}}\right)$ | S: Subset of materialized views with Max(B(v, S)) | Data cube lattice | Select a Subset S of k views v including a top view where the benefit B(v, S) is maximized. For each view u relative to S calculate $B\left(v,S\right)={\displaystyle {\sum}_{W\le U}{B}_{W}}$ . Where w is all descending of u. | K: number max of view to be materialized |
Greedy Genetic Algorithm | 6 | lattice of N Cubes $C=\left\{{c}_{1},{c}_{2},\cdots ,{c}_{i},\cdots ,{c}_{n}\right\}$ , A Set of Query $Q=\left\{{q}_{1},{q}_{2},\cdots ,{q}_{i},\cdots ,{q}_{k}\right\}$ , Frequency Values for each query $F=\left\{f{q}_{1},f{q}_{2},\cdots ,f{q}_{i},\cdots ,f{q}_{k}\right\}$ , Update Frequency of cubes $G=\left\{g{c}_{1},g{c}_{2},\cdots ,g{c}_{i},\cdots ,g{c}_{n}\right\}$ , Constraint Space S. | $\begin{array}{l}\mu (\xb7)={\displaystyle {\sum}_{i=1}^{k}f{q}_{i}E\left({q}_{i},M\right)}\\ +{\displaystyle {\sum}_{c\in M}gcU\left(c,M\right)}\end{array}$ | ῼ = (L, C, R, Q, F, G, S) | Data cube lattice | Genetic Greedy Method. | Storage Space, ${\sum}_{c\in Mc}\left|c\right|}\le S$ |
Genetic Algorithm | 4 | Lattice of Views with size of each view, P_{c}: Probability of crossover, P_{m}: Probability of mutation, G: Pre-defined number of generations. | $\begin{array}{l}TVEC={\displaystyle {\sum}_{i=1\wedge SMvi=1}^{N}Size\left(Vi\right)}\\ +{\displaystyle {\sum}_{i=1\wedge SMvi=0}^{N}SizeSMA\left(Vi\right)}\end{array}$ | Top-T Views. | Data cube lattice | Genetic method. | G: Number of generations |
A Distributed Clustering with Intelligent Multi Agents System | 1 | Queries workload. | $DIC=\frac{{\displaystyle {\sum}_{i=1}^{j=k}Dist\left({O}_{j},{O}_{i}\right)}}{K}$ $\begin{array}{l}Dist\left({O}_{j},{O}_{i}\right)\\ ={\displaystyle \sum _{k=1}^{m}\left|M\left({O}_{i},{q}_{k}\right)-M\left({O}_{j},{q}_{k}\right)\right|}\end{array}$ | Cluster with configuration provide minimal queries cost. | NONE | Generates a predicates usage matrix M with row presented by all queries and column defined by the restriction predicate and joint predicate. Use a distributed clustering based on k-means algorithm created and managed by COA. Each COA responsible of set of Cluster Agents (CAs). | Storage Space, MinDist. |
Clustering technique | 3 | Set of relation: $R\left({A}_{1},{A}_{2},\cdots ,{A}_{m}\right)$ ; The set of queries: $Q\text{}=\left\{{Q}_{1},{Q}_{2},\cdots ,{Q}_{n}\right\}$ Average Similarity: cut-off:. | $\begin{array}{l}TVEC={\displaystyle {\sum}_{i=1\wedge SMvi=1}^{N}Size\left(Vi\right)}\\ +{\displaystyle {\sum}_{i=1\wedge SMvi=0}^{N}SizeSMA\left(Vi\right)}\end{array}$ | Cluster of materialized views | NONE | Construct Attribute Usage Matrix M attributes * N queries. Generate Attribute Similarity Matrix M * M between attributes. Calculate similarity by $Jaccard\left(A,B\right)=\frac{\left|A\right|\cap \left|B\right|}{\left|A\right|\cup \left|B\right|}$ | J(A_{i}, A_{j}) = J(A_{j}, A_{k}) ≥ cut-off. So |
Particle Swarm Optimization Algorithm (PSO) | 6 | lattice of N Cubes $C=\left\{{c}_{1},{c}_{2},\cdots ,{c}_{i},\cdots ,{c}_{n}\right\}$ , A Set of Query $Q=\left\{{q}_{1},{q}_{2},\cdots ,{q}_{i},\cdots ,{q}_{k}\right\}$ , Frequency Values for each query $F=\left\{f{q}_{1},f{q}_{2},\cdots ,f{q}_{i},\cdots ,f{q}_{k}\right\}$ , Update Frequency of cubes, cube invoking frequency $CF=\left(f{c}_{1},f{c}_{2},\cdots ,f{c}_{n}\right)$ , Constraint Space S. | $fm=\mathrm{min}\left({\displaystyle {\sum}_{i=1}^{k}f{q}_{i}E\left({q}_{i},M\right)}\right)$ | Binary string of length n bits [0, 1] witch is set of Cubes M to minimizing fm | Data cube lattice | Genetic Algorithm. Particle Swarm Optimization Algorithm | Storage Space ${\sum}_{c\in Mc}\left|c\right|}\le S$ |
coral reefs optimization algorithm (CROMVS) | 8 | MVPP, K = Number of generating populations, M = Number of Queries, N = Number of Relations, H = threshold times, F_{a} = Fraction of asexual reproduction, F_{b} = Ratio of number of selected solutions, F_{d} = Fraction of the worst solutions | $\begin{array}{l}{f}_{x}={\displaystyle {\sum}_{i\in M}\left({\displaystyle {\sum}_{q\in Q}e{f}_{q}\ast {C}_{i}^{q}}\right)}\\ -({\displaystyle {\sum}_{u\in Prej\cap x}{\displaystyle {\sum}_{q\in Q}e{f}_{q}\ast {C}_{u}^{q}}}\\ +{\displaystyle {\sum}_{r\in R}u{f}_{r}\ast {C}_{i}^{r}})\end{array}$ | MV with MAX(f_{x}) | MVPP | Coral reefs algorithm, is Meta-heuristic algorithm based on coral reproduction and coral reefs formation which performed using: external sexual reproduction, internal sexual reproduction, and asexual reproduction | Population List with Max(f_{x}) |
Multi- Objective MONPGA. | 5 | L: Size of each view, P_{c}: probability of crossover, P_{m}: probability of mutation, K: number of views to be selected and G: the maximum number of generations | $TVEC\left({V}_{Top\text{-}k}\right)={C}_{MV}+{C}_{NMV}$ | Top-K views TKV | Data cube lattice | Genetic Algorithm. | K: number of views to be selected and G: the maximum number of generations |
A game theory-based framework (GTMVS) | 4 | $R=\left\{{R}_{1},{R}_{2},\cdots ,{R}_{r}\right\}$ : Set of base relations uf_{i}: update frequency for R_{i}. $Q=\left\{{Q}_{1},{Q}_{2},\cdots ,{Q}_{q}\right\}$ : Set of queries, ef_{i} execution frequency for Q_{i}. | $\begin{array}{l}{g}_{x}={\displaystyle {\sum}_{i\in M}({\displaystyle {\sum}_{r\in R}u{f}_{r}\ast {C}_{i}^{r}}}\\ +{\displaystyle {\sum}_{q\in Q}e{f}_{q}\ast {C}_{i}^{q}})\end{array}$ | select the optimal set $M=\left\{{M}_{1},{M}_{2},\cdots ,{M}_{m}\right\}$ with lowest cost represented by both sets of player1 and player2. | MVPP | Game theory who the Players of game are: Query processing cost and view maintenance cost. Player strategy used TSGV. [35] | M: list of nodes in MVPP-Player 1 union Player 2. materialized view List |
Map-Reduce model (MR-MVPP) | 7 | Q: workload of queries f_{q}: list of query frequencies R: base relations f_{u}: list of update frequencies b: number of categories A: a number between 0 and 1 for hash function t: threshold of similarity | $Q{C}_{i}={f}_{qi}\ast {\displaystyle {\sum}_{\left\{j|{Q}_{i}\in Query\left({V}_{j}\right)\right\}}A\text{\hspace{0.05em}}Cost\left({V}_{j}\right)}$ | MVPP that has the least total cost. | MVPP | The map-reduce programming Model; The hashing technique; Use 4 algorithms. are: MR-MVPP, SSJoin, map, and reduce. Calculate similarity by Jaccard Function. $Jaccard\left(A,B\right)=\frac{\left|A\right|\cap \left|B\right|}{\left|A\right|\cup \left|B\right|}$ | MVPP with least total cost = Min(QC_{i}) |