## Resolution versus Search: Two Strategies for SAT (2000)

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Venue: | Journal of Automated Reasoning |

Citations: | 50 - 1 self |

### BibTeX

@ARTICLE{Rish00resolutionversus,

author = {Irina Rish and Rina Dechter},

title = {Resolution versus Search: Two Strategies for SAT},

journal = {Journal of Automated Reasoning},

year = {2000},

volume = {24},

pages = {215--259}

}

### Years of Citing Articles

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### Abstract

The paper compares two popular strategies for solving propositional satisfiability, backtracking search and resolution, and analyzes the complexity of a directional resolution algorithm (DR) as a function of the "width" (w) of the problem's graph.

### Citations

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Citation Context ...ional resolution belongs to a family of elimination algorithms rst analyzed for optimization tasks in dynamic programming [6] and later used in constraint satisfaction [57, 20] and in belief networks =-=[47]-=-. In fact, DR can be viewed as an adaptation of the constraint-satisfaction algorithm adaptive consistency to propositional satis ability where the project-join operation over relational constraints i... |

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Citation Context ... incompleteslocal search techniques [59, 58]. The two most widely used complete techniques are backtracking search (e.g., the Davis-Putnam Procedure [11]) and resolution (e.g., Directional Resolution =-=[12, 23]-=-). We compare both approaches theoretically and empirically, suggesting several ways of combining them into more effective hybrid algorithms. In 1960, Davis and Putnam presented a resolution algorithm... |

1000 |
Consistency in networks of relations
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Citation Context ...onstraints is replaced by resolution over clauses [20, 24]. Using the same analogy, bounded resolution can be related to bounded consistency-enforcing algorithms, such as arc- path- and i-consistency =-=[48, 30, 14]-=-, while bounded directional resolution, BDR(i), parallels directional i-consistency [20, 24]. InResolved Variables 40 30 20 10 0 -1 0 DCDR on uniform 3-cnfs 100 variables, 400 clauses 100 experiments ... |

944 | Temporal constraint networks
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Citation Context ...rdering (a), only one binary clause is generated along ordering (c). Although nding an ordering that yields the smallest induced width is NP-hard [1], good heuristic orderings are currently available =-=[6, 14, 55]-=- and continue to be explored [4]. Furthermore, there is a class of graphs, known as k-trees, that have w <kand can be recognized in O(n exp(k)) time [1]. De nition 2: (k-trees) 1. A clique of size k (... |

774 | The complexity of theorem-proving procedures
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Citation Context ...ported by NSF grant IRI-9157636. 1s1 Introduction Propositional satis ability (SAT) is a prototypical example of an NP-complete problem; any NP-complete problem is reducible to SAT in polynomial time =-=[8]-=-. Since many practical applications such as planning, scheduling, and diagnosis can be formulated as propositional satis ability, nding algorithms with good average performance has been a focus of ext... |

768 | Constraint Logic Programming: A Survey
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Citation Context ... recent success of constraint processing which can be attributed to techniques combining search with limited forms of constraint propagation (e.g., forward-checking, MAC, constraint logic programming =-=[41, 36, 56, 43]-=-) that motivated our hybrid algorithms. In the SAT community, a popular form of combining constraint propagation with search is unit-propagation in DP. Our work extends this idea. The hybrid algorithm... |

740 |
A machine program for theorem proving
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Citation Context ...at can always determine satis ability as opposed to incomplete local search techniques [59, 58]. The two most widely used complete techniques are backtracking search (e.g., the Davis-Putnam Procedure =-=[11]-=-) and resolution (e.g., Directional Resolution [12, 23]). We compare both approaches theoretically and empirically, suggesting several ways of combining them into more e ective hybrid algorithms. In 1... |

680 | A New Method for Solving Hard Satisfiability Problems
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- 1992
(Show Context)
Citation Context ...s such as planning, scheduling, and diagnosis can be formulated as propositional satisfiability, finding algorithms with good average performance has been a focus of extensive research for many years =-=[59, 10, 34, 45, 46, 3]-=-. In this paper, we consider complete SAT algorithms that can always determine satisfiability as opposed to incompleteslocal search techniques [59, 58]. The two most widely used complete techniques ar... |

577 | Where the really hard problems are
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Citation Context ...ques and with increasing number of clauses per clique. Further empirical studies are required to investigate the possible phase transition phenomenon in chains as it was done for uniform random 3cnfs =-=[7, 49, 9]-=-. In our experiments nearly all of the 3-cnf chain problems that were di cult for DP were unsatis able. One plausible explanation is that inconsistent chain theories mayhave an unsatis able subtheory ... |

521 | Pushing the envelope: planning, propositional logic and stochastic search
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Citation Context ...ions such as planning, scheduling, and diagnosis can be formulated as propositional satis ability, nding algorithms with good average performance has been a focus of extensive research for many years =-=[59, 10, 34, 45, 46, 3]-=-. In this paper, we consider complete SAT algorithms that can always determine satis ability as opposed to incomplete local search techniques [59, 58]. The two most widely used complete techniques are... |

457 |
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Citation Context ... "smart" variable- and value-ordering heuristics ([9], [33]). More efficient backtracking mechanisms, such as backjumping [36, 13, 50], constraint propagation (e.g., arc-consistency, forward=-= checking [41]-=-), or learning (recording constraints) [13, 31, 2] are available. The Davis-Putnam Procedure (DP) [11] shown in Figure 17b is a backtracking search algorithm for deciding propositional satisfiability ... |

389 |
Network-based heuristics for constraint-satisfaction problems
- Dechter, Pearl
- 1987
(Show Context)
Citation Context ...hybrids run DP on the cutset variables and DR on the remaining ones, thus combining the virtues of both approaches. Like DR, they 10 This is a generalization of the cycle-cutset algorithm proposed in =-=[20]-=- which transforms the interaction graph of a theory into a tree. 39 E B C D Esexploit low-w structure and produce an output theory that facilitates model generation, while using less space and allowin... |

361 | Noise strategies for improving local search
- Selman, Kautz, et al.
- 1994
(Show Context)
Citation Context ... extensive research for many years [59, 10, 34, 45, 46, 3]. In this paper, we consider complete SAT algorithms that can always determine satis ability as opposed to incomplete local search techniques =-=[59, 58]-=-. The two most widely used complete techniques are backtracking search (e.g., the Davis-Putnam Procedure [11]) and resolution (e.g., Directional Resolution [12, 23]). We compare both approaches theore... |

346 | Hybrid algorithms for the constraint satisfaction problem
- Prosser
- 1993
(Show Context)
Citation Context ...nced backtracking algorithms for solving CSPs that improve the basic scheme using \smart" variable- and value-ordering heuristics ([9], [33]). More e cient backtracking mechanisms, such asbackjum=-=ping [36, 13, 50]-=-, constraint propagation (e.g., arc-consistency, forward checking [41]), or learning (recording constraints) [13, 31, 2] are available. The Davis-Putnam Procedure (DP) [11] shown in Figure 17b is a ba... |

327 |
Complexity of finding embeddings in a k-tree
- Arnborg, Corneil, et al.
- 1987
(Show Context)
Citation Context ...the clique sizes are substantially smaller than the overall number of variables. In general, one can imagine a variety of real-life domains having such structure that is captured by k-tree-embeddings =-=[1]-=- used in our random problem generators. Our empirical studies of SAT algorithms confirm previous results: DR is very inefficient when dealing with unstructured uniform random problems. However, on str... |

288 | Bucket elimination: a unifying framework for probabilistic inference
- Dechter
- 1996
(Show Context)
Citation Context ...s the known relationship between variable elimination and the tree-clustering compilation scheme that was presented for constraint satisfaction in [21] and was extended to probabilistic frameworks in =-=[15]-=-. 11 DCDR(b) performs resolution on variables that are connected to at most b other variables; therefore, the size of resolvents is bounded b. It does not, however, resolve over the variables having d... |

283 |
and easy distributions of sat problems
- Hard
- 1992
(Show Context)
Citation Context ...explosion. However, this modi cation changed the nature of the algorithm and transformed it into a backtracking scheme. Most of the work on propositional satis ability quotes the backtracking version =-=[40, 49]-=-. We will refer to the original Davis-Putnam algorithm as DP-resolution, ordirectional resolution (DR) 1 , and to its later modi cation as DP-backtracking, orDP (also called DPLL in the SAT community)... |

252 |
Graph minors XIII. The disjoint paths problem
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- 1995
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Citation Context ...rdering (a), only one binary clause is generated along ordering (c). Although nding an ordering that yields the smallest induced width is NP-hard [1], good heuristic orderings are currently available =-=[6, 14, 55]-=- and continue to be explored [4]. Furthermore, there is a class of graphs, known as k-trees, that have w <kand can be recognized in O(n exp(k)) time [1]. De nition 2: (k-trees) 1. A clique of size k (... |

251 |
Tree Clustering for Constraint Networks
- Dechter, Pearl
- 1989
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Citation Context ...lgorithm max-cardinality. 5 Directional Resolution and Tree-Clustering In this section we further discuss the knowledge-compilation aspects of directional resolution, and relate it to tree-clustering =-=[21]-=-, a general preprocessing technique commonly used in constraint and belief networks. As stated in Theorem 2, given an input theory and a variable ordering, algorithm DR produces a directional extensio... |

245 |
Synthesizing Constraint Expressions
- Freuder
- 1978
(Show Context)
Citation Context ...onstraints is replaced by resolution over clauses [20, 24]. Using the same analogy, bounded resolution can be related to bounded consistency-enforcing algorithms, such as arc- path- and i-consistency =-=[48, 30, 14]-=-, while bounded directional resolution, BDR(i), parallels directional i-consistency [20, 24]. InResolved Variables 40 30 20 10 0 -1 0 DCDR on uniform 3-cnfs 100 variables, 400 clauses 100 experiments ... |

244 |
M.: Simple linear-time algorithms to test chordality of graphs, test acyclity of hypergraphs, and selectively reduce acyclic hypergraphs
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- 1984
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Citation Context ...ming it into a chordal graph (a graph is chordal if every cycle of length at least four has a chord, i.e. an edge between two non-sequential nodes in that cycle). This procedure, called triangulation =-=[61]-=-, processes the nodes along some order of the variables o, going from the last node to the rst, connecting edges between the earlier neighbors of each node. The result is the induced graph along o, wh... |

204 | Using CSP look-back techniques to solve real-world SAT instances
- Bayardo, Schrag
- 1997
(Show Context)
Citation Context ...ions such as planning, scheduling, and diagnosis can be formulated as propositional satis ability, nding algorithms with good average performance has been a focus of extensive research for many years =-=[59, 10, 34, 45, 46, 3]-=-. In this paper, we consider complete SAT algorithms that can always determine satis ability as opposed to incomplete local search techniques [59, 58]. The two most widely used complete techniques are... |

203 | E.: Contradicting conventional wisdom in constraint satisfaction
- Sabin, Freuder
- 1994
(Show Context)
Citation Context ... recent success of constraint processing which can be attributed to techniques combining search with limited forms of constraint propagation (e.g., forward-checking, MAC, constraint logic programming =-=[41, 36, 56, 43]-=-) that motivated our hybrid algorithms. In the SAT community, a popular form of combining constraint propagation with search is unit-propagation in DP. Our work extends this idea. The hybrid algorithm... |

201 | Experimental results on the cross-over point in satisfiability problems
- Crawford, Auton
- 1993
(Show Context)
Citation Context ...ssed out. Theory '5 is obviously inconsistent. There are various advanced backtracking algorithms for solving CSPs that improve the basic scheme using \smart" variable- and value-ordering heurist=-=ics ([9]-=-, [33]). More e cient backtracking mechanisms, such asbackjumping [36, 13, 50], constraint propagation (e.g., arc-consistency, forward checking [41]), or learning (recording constraints) [13, 31, 2] a... |

179 |
Nonserial Dynamic Programming
- BertelĂ¨, Brioschi
- 1972
(Show Context)
Citation Context ...rdering (a), only one binary clause is generated along ordering (c). Although nding an ordering that yields the smallest induced width is NP-hard [1], good heuristic orderings are currently available =-=[6, 14, 55]-=- and continue to be explored [4]. Furthermore, there is a class of graphs, known as k-trees, that have w <kand can be recognized in O(n exp(k)) time [1]. De nition 2: (k-trees) 1. A clique of size k (... |

174 | Performance measurement and analysis of certain search algorithms
- Gaschnig
- 1979
(Show Context)
Citation Context ...ories with an unsatis able theory close to the end of the ordering. Min-diversity and min-width orderings do not preclude such a situation. There are enhanced backtracking schemes, such asbackjumping =-=[36, 37, 13, 51]-=-, that are capable of exploiting the structure and preventing useless re-instantiations. Experiments with backjumping con rm that it 28 CPU time (log scale) 590 640 690sCPU-time (log-scale) 100 10 1 .... |

114 | Experimental results on the application of satisfiability algorithms to scheduling problems - Crawford, Baker - 1990 |

113 |
Solving propositional satisfiability problems
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- 1990
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Citation Context ... backtracking search algorithm for deciding propositional satisfiability combined with unit propagation. Various branching heuristics augmenting this basic version of DP have been proposed since 1962 =-=[44, 9, 42, 38]-=-. The worst-case time complexity of all backtracking algorithms is exponential in the 23 Frequency Nodes in Search Space 0 1,000 3,000 6,000 .005 .010 .015 .020 Figure 18: An empirical distribution of... |

112 | Locating the phase transition in binary constraint satisfaction problems
- Smith, Dyer
- 1996
(Show Context)
Citation Context ...d resolution. available { is often much more e cient than its worst-case bound. It demonstrates \great discrepancies in execution time" (D.E. Knuth), encountering rare but exceptionally hard prob=-=lems [60]-=-. Recent studies suggest that the empirical performance of backtracking algorithms can be modeled by long-tail exponential-family distributions, such as lognormal and Weibull [32, 54]. The average com... |

100 | Directional resolution: The davis-putnam procedure, revisited
- Dechter, Rish
- 1994
(Show Context)
Citation Context ... incomplete local search techniques [59, 58]. The two most widely used complete techniques are backtracking search (e.g., the Davis-Putnam Procedure [11]) and resolution (e.g., Directional Resolution =-=[12, 23]-=-). We compare both approaches theoretically and empirically, suggesting several ways of combining them into more e ective hybrid algorithms. In 1960, Davis and Putnam presented a resolution algorithm ... |

78 | Branching rules for satisfiability
- Hooker, Vinay
- 1995
(Show Context)
Citation Context ... backtracking search algorithm for deciding propositional satisfiability combined with unit propagation. Various branching heuristics augmenting this basic version of DP have been proposed since 1962 =-=[44, 9, 42, 38]-=-. The worst-case time complexity of all backtracking algorithms is exponential in the 23 Frequency Nodes in Search Space 0 1,000 3,000 6,000 .005 .010 .015 .020 Figure 18: An empirical distribution of... |

77 | D.P.: A complexity analysis of space-bounded learning algorithms for the constraint satisfaction problem
- Bayardo, Miranker
- 1996
(Show Context)
Citation Context ...uristics ([9], [33]). More e cient backtracking mechanisms, such asbackjumping [36, 13, 50], constraint propagation (e.g., arc-consistency, forward checking [41]), or learning (recording constraints) =-=[13, 31, 2]-=- are available. The Davis-Putnam Procedure (DP) [11] shown in Figure 17b is a backtracking search algorithm for deciding propositional satis ability combined with unit propagation. Various branching h... |

75 | R.: Dead-end driven learning
- Frost, Dechter
- 1994
(Show Context)
Citation Context ...uristics ([9], [33]). More e cient backtracking mechanisms, such asbackjumping [36, 13, 50], constraint propagation (e.g., arc-consistency, forward checking [41]), or learning (recording constraints) =-=[13, 31, 2]-=- are available. The Davis-Putnam Procedure (DP) [11] shown in Figure 17b is a backtracking search algorithm for deciding propositional satis ability combined with unit propagation. Various branching h... |

74 | A sufficiently fast algorithm for finding close to optimal junction trees
- Becker, Geiger
- 1996
(Show Context)
Citation Context ...erated along ordering (c). Although finding an ordering that yields the smallest induced width is NP-hard [1], good heuristic orderings are currently available [6, 14, 55] and continue to be explored =-=[4]-=-. Furthermore, there is a class of graphs, known as k-trees, that have w ! k and can be recognized in O(n \Delta exp(k)) time [1]. Definition 2: (k-trees) 1. A clique of size k (complete graph with k ... |

65 |
Probabilistic analysis of the Davisâ€“Putnam procedure for solving the satisfiability problem
- Franco, Paull
(Show Context)
Citation Context ...3 (the fraction of prohibited value pairs in a constraint). number of variables while their space complexity is linear. Yet, the average time complexity of DP depends on the distribution of instances =-=[29]-=- and is often much lower then its worst-case bound. Usually, its average performance is a ected by rare, but exceptionally hard instances. Exponential-family empirical distributions (e.g., lognormal, ... |

60 | Beek: Local and Global Relational Consistency
- Dechter, van
- 1997
(Show Context)
Citation Context ...tation of the constraint-satisfaction algorithm adaptive consistency to propositional satis ability where the project-join operation over relational constraints is replaced by resolution over clauses =-=[20, 24]-=-. Using the same analogy, bounded resolution can be related to bounded consistency-enforcing algorithms, such as arc- path- and i-consistency [48, 30, 14], while bounded directional resolution, BDR(i)... |

58 |
A new method for solving hard satis ability problems
- Selman, Levesque, et al.
- 1992
(Show Context)
Citation Context ...ions such as planning, scheduling, and diagnosis can be formulated as propositional satis ability, nding algorithms with good average performance has been a focus of extensive research for many years =-=[59, 10, 34, 45, 46, 3]-=-. In this paper, we consider complete SAT algorithms that can always determine satis ability as opposed to incomplete local search techniques [59, 58]. The two most widely used complete techniques are... |

57 |
Enhancement Schemes for Constraint Processing
- Dechter
- 1990
(Show Context)
Citation Context ...nced backtracking algorithms for solving CSPs that improve the basic scheme using \smart" variable- and value-ordering heuristics ([9], [33]). More e cient backtracking mechanisms, such asbackjum=-=ping [36, 13, 50]-=-, constraint propagation (e.g., arc-consistency, forward checking [41]), or learning (recording constraints) [13, 31, 2] are available. The Davis-Putnam Procedure (DP) [11] shown in Figure 17b is a ba... |

55 | Fattah, Y.: Topological parameters for time-space tradeoff
- Dechter, El
- 2001
(Show Context)
Citation Context ...coincides with DR if w osb. If b ! 0, the algorithm coincides with DP. Intermediate values of b allow trading space for time. As b increases, the algorithm requires more space and less time (see also =-=[16]-=-). However, there is no guaranteed worst-case time improvement over DR. It was shown [6] that the size of the smallest cycle-cutset C (a set of nodes that breaks all cycles in the interaction graph, l... |

53 | Experimental Evaluation of Preprocessing Techniques in ConstraintSatisfaction Problems
- Dechter, Meiri
- 1989
(Show Context)
Citation Context ... algorithms can be modeled by long-tail exponential-family distributions, such as lognormal and Weibull [32, 54]. The average complexity of algorithm DR, on the other hand, is close to its worst-case =-=[18]-=-. It is important to note that the space complexity ofDPisO(n), while DR is space-exponential in w . Another di erence is that in addition to deciding satis ability and nding a solution (a model), dir... |

53 | A General Backtrack Algorithm That Eliminates Most Redundant Tests
- Gaschnig
- 1977
(Show Context)
Citation Context ...nced backtracking algorithms for solving CSPs that improve the basic scheme using \smart" variable- and value-ordering heuristics ([9], [33]). More e cient backtracking mechanisms, such asbackjum=-=ping [36, 13, 50]-=-, constraint propagation (e.g., arc-consistency, forward checking [41]), or learning (recording constraints) [13, 31, 2] are available. The Davis-Putnam Procedure (DP) [11] shown in Figure 17b is a ba... |

51 | Satisfiability testing with more reasoning and less guessing
- Gelder, Tsuji
(Show Context)
Citation Context ... backtracking search algorithm for deciding propositional satisfiability combined with unit propagation. Various branching heuristics augmenting this basic version of DP have been proposed since 1962 =-=[44, 9, 42, 38]-=-. The worst-case time complexity of all backtracking algorithms is exponential in the 23 Frequency Nodes in Search Space 0 1,000 3,000 6,000 .005 .010 .015 .020 Figure 18: An empirical distribution of... |

41 |
A Computing Procedure for Quanti cation Theory
- Davis, Putnam
- 1960
(Show Context)
Citation Context ... incomplete local search techniques [59, 58]. The two most widely used complete techniques are backtracking search (e.g., the Davis-Putnam Procedure [11]) and resolution (e.g., Directional Resolution =-=[12, 23]-=-). We compare both approaches theoretically and empirically, suggesting several ways of combining them into more e ective hybrid algorithms. In 1960, Davis and Putnam presented a resolution algorithm ... |

38 |
A new method for solving constraint satisfaction problems
- Seidel
- 1981
(Show Context)
Citation Context ... problems. 9 Related Work Directional resolution belongs to a family of elimination algorithms rst analyzed for optimization tasks in dynamic programming [6] and later used in constraint satisfaction =-=[57, 20]-=- and in belief networks [47]. In fact, DR can be viewed as an adaptation of the constraint-satisfaction algorithm adaptive consistency to propositional satis ability where the project-join operation o... |

37 |
Experimental Evaluation of Preprocessing Algorithms for Constraints Satisfaction Problems
- Dechter, Meiri
- 1994
(Show Context)
Citation Context ...lly proposed in [23], corresponds to applying directional i-consistency prior to backtracking search for constraint processing. This approach was empirically evaluated for some constraint problems in =-=[19]-=-. However, those experiments were restricted to small and relatively easy problems, for which only a very limited amount of preprocessing was cost-e ective. The presented experiments with BDRDP(i) sug... |

34 |
Complexity of nding embedding in a k-tree
- Arnborg, Corneil, et al.
- 1987
(Show Context)
Citation Context ...the clique sizes are substantially smaller than the overall number of variables. In general, one can imagine a variety of real-life domains having such structure that is captured by k-tree-embeddings =-=[1]-=- used in our random problem generators. Our empirical studies of SAT algorithms con rm previous results: DR is very ine - cient when dealing with unstructured uniform random problems. However, on stru... |

34 |
On the complexity of regular resolution and the davisputnam procedure
- Galil
- 1977
(Show Context)
Citation Context ...propositional theory is su cient for deciding satis ability. However, this algorithm has received limited attention and analyses of its performance have emphasized its worst-case exponential behavior =-=[35, 39]-=-, while overlooking its virtues. It was quickly overshadowed by the Davis-Putnam Procedure, introduced in 1962 by Davis, Logemann, and Loveland [11]. They proposed a minor syntactic modi cation of the... |

31 |
Increasing tree search e ciency for constraint satisfaction problems
- Haralick, Elliot
- 1980
(Show Context)
Citation Context ...ing \smart" variable- and value-ordering heuristics ([9], [33]). More e cient backtracking mechanisms, such asbackjumping [36, 13, 50], constraint propagation (e.g., arc-consistency, forward chec=-=king [41]-=-), or learning (recording constraints) [13, 31, 2] are available. The Davis-Putnam Procedure (DP) [11] shown in Figure 17b is a backtracking search algorithm for deciding propositional satis ability c... |

29 | Directed constraint networks: A relational framework for causal modeling
- Dechter, Pearl
- 1991
(Show Context)
Citation Context .... 2 The proof follows immediately from Theorem 8 (see subsection 4.3). Zero-diversity theories generalize the notion of causal theories de ned for general constraint networks of multivalued relations =-=[22]-=-. According to this de nition, theories are causal if there is an ordering of the propositional variables such that each bucket contains a single clause. Consequently, the ordering has zero diversity.... |

29 | Summarizing CSP hardness with continuous probability distributions
- Frost, Rish, et al.
- 1997
(Show Context)
Citation Context ...onally hard problems [60]. Recent studies suggest that the empirical performance of backtracking algorithms can be modeled by long-tail exponential-family distributions, such as lognormal and Weibull =-=[32, 54]-=-. The average complexity of algorithm DR, on the other hand, is close to its worst-case [18]. It is important to note that the space complexity ofDPisO(n), while DR is space-exponential in w . Another... |