## Conjunctive Query Containment Revisited (1998)

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Citations: | 98 - 0 self |

### BibTeX

@INPROCEEDINGS{Chekuri98conjunctivequery,

author = {Chandra Chekuri and Anand Rajaraman},

title = {Conjunctive Query Containment Revisited},

booktitle = {},

year = {1998},

pages = {56--70},

publisher = {Springer-Verlag}

}

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### Abstract

We consider the problems of conjunctive query containment and minimization, which are known to be NP-complete, and show that these problems can be solved in polynomial time for the class of acyclic queries. We then generalize the notion of acyclicity and define a parameter called query width that captures the "degree of cyclicity" of a query: in particular, a query is acyclic if and only if its query width is 1. We give algorithms for containment and minimization that run in time polynomial in n k , where n is the input size and k is the query width. These algorithms naturally generalize those for acyclic queries, and are of practical significance because many queries have small query width compared to their sizes. We show that good bounds on the query width of Q can be obtained using the treewidth of the incidence graph of Q. We then consider the problem of finding an equivalent query to a given conjunctive query Q that has the least number of subgoals. We show that a polynomial tim...

### Citations

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Citation Context ...me a fixed set of predicates, called the database predicates, over which queries are posed and views are defined. All queries in this paper are conjunctive queries, defined in the conventional manner =-=[21]-=-. We say query Q 1 is contained in query Q 2 , denoted by Q 1 ` Q 2 , if for every state of the relations corresponding to the database predicates, the relation corresponding to the head of Q 1 is a s... |

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Citation Context ... 98101. E-mail: anand@amazon.com. Supported at Stanford University by an NSF grant IRI--92--23405, ARO grant DAAH04--95--1--0192, and USAF contract F33615--93--1--1339. 1 well-known to be NP-complete =-=[7]-=-. In view of its practical significance, considerable attention has been devoted to finding classes of conjunctive queries that admit polynomial-time algorithms for equivalence and minimization [1, 2,... |

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337 |
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248 |
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Citation Context ...clic queries that enables a polynomial-time query containment algorithm for such queries (Section 4). Our algorithms for acyclic queries assume the elimination tree as an input. Tarjan and Yannakakis =-=[20]-=- present a simple linear-time algorithm that tests whether Q is acyclic and if so, constructs an elimination tree for it. 3.2 The Width of a Query It is natural to ask whether we can generalize the no... |

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Citation Context ... studied formally by Levy et al. [15], who showed the problem to be NP-complete. Rajaraman et al. [19] extended the results of Levy et al. to queries and views with binding patterns. Chaudhuri et al. =-=[8]-=- considered the problem of finding rewritings for SPJ queries and views, when the queries and views use bag semantics instead of the usual set semantics. They also suggest a way to extend a traditiona... |

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Citation Context ...d therefore their algorithm can miss some rewritings. The problem of finding equivalent rewritings was studied formally by Levy et al. [15], who showed the problem to be NP-complete. Rajaraman et al. =-=[19]-=- extended the results of Levy et al. to queries and views with binding patterns. Chaudhuri et al. [8] considered the problem of finding rewritings for SPJ queries and views, when the queries and views... |

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Citation Context ...idth is an extensively studied concept in graph theory. Though computing the treewidth of a graph is NP-Complete in general [3], checking a graph for small treewidth is in polynomial time. Bodlaender =-=[6]-=- presents a polynomial time algorithm to determine, for a given k, whether the treewidth of a graph is bounded by k. The running time of the algorithm is exponential in k but linear in the size of the... |

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- 1998
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Citation Context ... in the query. We will establish this by showing that the reduction of Chandra and Merlin from graph coloring is in fact approximation preserving, and then using the recent result of Feige and Kilian =-=[9]-=- on the hardness of coloring (based on a long and remarkable line of work, most notably the result of Hastad [11] on hardness of approximating clique). Definition 1 Given a undirected graph G = (V; E)... |

141 | Query folding
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- 1996
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Citation Context ...per. The algorithms of Aho et al. , Johnson and Klug, and Biskup et al. also differ from ours in that they cannot be generalized to test query containment instead of query equivalence. Recently, Qian =-=[18]-=- independently showed that acyclic queries admit polynomial time algorithms for containment and minimization. Our work treats acyclic queries as a special case of queries with width 1, and so Qian's a... |

113 |
Algorithms for acyclic database schemes
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- 1981
(Show Context)
Citation Context ..., 12, 4]. Acyclic queries in particular have been extensively studied in the context of query optimization in distributed database systems, and are well-known to have desirable algorithmic properties =-=[23]-=-. In this paper, we first present polynomial-time algorithms to test containment of an arbitrary conjunctive query in an acyclic query, and to minimize an acyclic query. We then introduce a new parame... |

95 |
Query transformation for PSJ-queries
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- 1987
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Citation Context ...for containment and minimization. Our work treats acyclic queries as a special case of queries with width 1, and so Qian's algorithm for query containment falls out as a special case. Yang and Larson =-=[14, 22]-=- considered the problem of finding rewritings for SPJ queries using SPJ views. In their analysis, they considered what amounts to 1-1 mappings from the views to the query, and therefore their algorith... |

81 |
Computing queries from derived relations
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(Show Context)
Citation Context ...for containment and minimization. Our work treats acyclic queries as a special case of queries with width 1, and so Qian's algorithm for query containment falls out as a special case. Yang and Larson =-=[14, 22]-=- considered the problem of finding rewritings for SPJ queries using SPJ views. In their analysis, they considered what amounts to 1-1 mappings from the views to the query, and therefore their algorith... |

67 |
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Citation Context ... in fact approximation preserving, and then using the recent result of Feige and Kilian [9] on the hardness of coloring (based on a long and remarkable line of work, most notably the result of Hastad =-=[11]-=- on hardness of approximating clique). Definition 1 Given a undirected graph G = (V; E), we define the conjunctive query QG corresponding to G as the following query. QG : q() : \Gamma (u;v)2E R(u; v)... |

58 |
Efficient Optimization of a Class of Relational Expressions
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- 1979
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Citation Context ...te [7]. In view of its practical significance, considerable attention has been devoted to finding classes of conjunctive queries that admit polynomial-time algorithms for equivalence and minimization =-=[1, 2, 12, 4]-=-. Acyclic queries in particular have been extensively studied in the context of query optimization in distributed database systems, and are well-known to have desirable algorithmic properties [23]. In... |

58 | Approximating treewidth, pathwidth, frontsize, and shortest elimination tree
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Citation Context ...graph is hard to approximate within a factor larger than 2, it would imply NP-hardness of computing query width. The best known approximation algorithm for computing treewidth has a ratio of O(log n) =-=[5]-=-, so the above possibility is not ruled out. In addition, even if computing query width is hard, we believe that finding a query decomposition of size k, if it exists, can be computed in polynomial ti... |

45 |
Equivalence of relational expressions
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Citation Context ...te [7]. In view of its practical significance, considerable attention has been devoted to finding classes of conjunctive queries that admit polynomial-time algorithms for equivalence and minimization =-=[1, 2, 12, 4]-=-. Acyclic queries in particular have been extensively studied in the context of query optimization in distributed database systems, and are well-known to have desirable algorithmic properties [23]. In... |

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Citation Context ... polynomial-time algorithm that, given an acyclic query Q and a set of views V, determines whether there is a rewriting using V that is contained in (but not necessarily equivalent to) Q. Levy et al. =-=[17]-=- study the problem of finding an equivalent rewriting when the set of views is possibly infinite, albeit encoded in some finite fashion. The Information Manifold system [16] implements heuristics that... |

23 |
Optimizing conjunctive queries that contain untyped variables
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Citation Context ...te [7]. In view of its practical significance, considerable attention has been devoted to finding classes of conjunctive queries that admit polynomial-time algorithms for equivalence and minimization =-=[1, 2, 12, 4]-=-. Acyclic queries in particular have been extensively studied in the context of query optimization in distributed database systems, and are well-known to have desirable algorithmic properties [23]. In... |

22 |
Ozsoyoglu, An algorithm for tree-query membership of a distributed query
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Citation Context ...unique to E; that is, they appear in no other hyperedge of G. Then we call E an ear, the removal of E from G ear removal, and say that "E is removed in favor of F ." The GYO-reduction of a h=-=ypergraph [10, 24]-=- is obtained by removing ears until no further ear removals are possible. A hypergraph is acyclic if its GYO-reduction is the empty hypergraph; otherwise it is cyclic. A query is cyclic (acyclic) if i... |

21 |
On the universal relation
- Graham
- 1980
(Show Context)
Citation Context ...unique to E; that is, they appear in no other hyperedge of G. Then we call E an ear, the removal of E from G ear removal, and say that "E is removed in favor of F ." The GYO-reduction of a h=-=ypergraph [10, 24]-=- is obtained by removing ears until no further ear removals are possible. A hypergraph is acyclic if its GYO-reduction is the empty hypergraph; otherwise it is cyclic. A query is cyclic (acyclic) if i... |

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Optimization of a subclass of conjunctive queries
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Citation Context |