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Problem potpunog prekrivanja razmatramo u kontekstu računalne konstrukcije (v, k, 1)-dizajna. Potrebno je pronaći k-člane blokove dizajna koji ujedno tvore egzaktno pokrivanje nad skupom parova svih točaka. Za traženje pokrivača koristimo algoritam X, algoritam kojeg je osmislio Donald Knuth. Knuthova implementacija oslanja se na metodu "dancing links". Program implementation of Donald Knuth's algorithm X is used to solve an exact cover problem for (v, k, 1)-designs. Algorithm finds k-sized subsets called blocks which cover a set of point pairs. Algorithm X is implemented using "dancing links" method.

Exact cover problem is a well-known NP-complete decision problem to determine if the exact cover really exists. In this paper, we show how to solve a modified version of the famous Hexomino puzzle (being a noteworthy example of an exact cover problem) using a Dancing-links based algorithm. In this modified problem, a limited number of gaps in the rectangular box may be left uncovered (this is a common scenario in a variety of spatial planning problems). Additionally, we present the benchmark generator which allows for elaborating very demanding yet solvable problem instances. These instances were used during the qualifying round of Deadline24—an international 24-h programming marathon. Finally, we confront our baseline solutions with those submitted by the contestants, and elaborated using our two solvers.

Multiple patterning lithography has been recognized as one of the most promising solutions, in addition to extreme ultraviolet lithography, directed self-assembly, nanoimprint lithography, and electron beam lithography, for advancing the resolution limit of conventional optical lithography. Multiple patterning layout decomposition (MPLD) becomes more challenging as advanced technology introduces complex coloring rules. Existing works model MPLD as a graph coloring problem; nevertheless, when complex coloring rules are considered, layout decomposition can no longer be modeled accurately by graph coloring. Therefore, in this paper, for capturing the essence of layout decomposition with complex coloring rules, we model the MPLD problem as an exact cover problem. We then propose a fast and exact MPLD framework based on augmented dancing links. Our method is flexible and general: It can consider the basic and complex coloring rules simultaneously, and it can handle quadruple patterning and beyond. Experimental results show that our approach outperforms state-of-the-art works on reported conflicts and stitches and is promising for handling complex coloring rules as well.

Many tasks in combinatorial computation admit a natural formulation as instances of the exact cover problem. We observe that the exact cover problem can in turn be compactly translated to the satisfiability (SAT) problem, allowing the basic Davis-Putnam-Logemann-Loveland procedure with no clause learning to linearly simulate backtrack search for exact cover. This SAT-based approach is empirically compared with a widely applied backtrack search algorithm, Knuth's "Dancing Links X" algorithm, on a set of benchmark problems arising in combinatorial enumeration. The experimental results indicate that the current model-counting SAT solvers are in general superior in pruning the search space, but still need to be optimized for running time.

We discuss weighted deductive parsing and consider the problem of finding the derivation with the lowest weight. We show that Knuth's generalization of Dijkstra's algorithm for the shortest-path problem offers a general method to solve this problem. Our approach is modular in the sense that Knuth's algorithm is formulated independently from the weighted deduction system.

In this paper we will take a detailed look at a larger example of program analysis by transformation. We will be considering Algorithm 2.3.3.A from Knuth's «Fundamental Algorithms» Knuth (1968) (p. 357) which is an algorithm for the addition of polynomials represented using four-directional links. Knuth (1974) describes this as having «a complicated structure with excessively unrestrained goto statements» and goes on to say «I hope someday to see the algorithm cleaned up without loss of its efficiency». Our aim is to manipulate the program, using semantics-preserving operations, into an equivalent, high-level specification. The transformations are carried out in the WSL language, a «wide spectrum language» which includes both low-level program operations and high level specifications, and which has been specifically designed to be easy to transform