### Table 2: Number of consistency checks performed when finding one solutionto crossword puzzles. The absence of an entry indicates that the problem could not be solved within 5 10 8 consistency checks. For the dual problem, n is the number of variables and m is the number of constraints; for the non-binary problems, m is the number of variables and n is the number of constraints.

1998

"... In PAGE 5: ... This provides some evidence that the experimental predictions scale for larger n. The results for 20 crossword puzzles (Gin93) are shown in Table2 . In the non-binary formulation of cross- word puzzles there is a variable for each letter to be filled in and the constraints are the words in the Unix dictionary.... ..."

Cited by 76

### Table 2: Number of consistency checks performed when finding one solutionto crossword puzzles. The absence of an entry indicates that the problem could not be solved within 5 108 consistency checks. For the dual problem, n is the number of variables and m is the number of constraints; for the non-binary problems, m is the number of variables and n is the number of constraints.

1998

"... In PAGE 5: ... This provides some evidence that the experimental predictions scale for larger n. The results for 20 crossword puzzles (Gin93) are shown in Table2 . In the non-binary formulation of cross- word puzzles there is a variable for each letter to be filled in and the constraints are the words in the Unix dictionary.... ..."

Cited by 76

### Table 2: Number of consistency checks performed when finding one solutionto crossword puzzles. The absence of an entry indicates that the problem could not be solved within 5 10 8 consistency checks. For the dual problem, n is the number of variables and m is the number of constraints; for the non-binary problems, m is the number of variables and n is the number of constraints.

1998

"... In PAGE 5: ... This provides some evidence that the experimental predictions scale for larger n. The results for 20 crossword puzzles (Gin93) are shown in Table2 . In the non-binary formulation of cross- word puzzles there is a variable for each letter to be filled in and the constraints are the words in the Unix dictionary.... ..."

Cited by 76

### Table 1 Percentiles in branches searched to complete a quasigroup of order 10 using algorithms that maintain either arc-consistency on the binary decomposition (MAC) or generalized arc-consistency on the non-binary representation (MGAC). means that the instance was abandoned after 10,000 branches. 100 problems were solved at each data point p MAC MGAC

"... In PAGE 19: ... We therefore focus on the higher percentiles. Table1 gives branches explored to complete an order 10 quasigroup with p% of entries preassigned, maintaining either AC on the binary representation or GAC on the all-different constraints. We see a very significant advantage in enforcing GAC.... ..."

### Table 2: Speedups obtained using adaptive QE strategy for two problems from the MIPLIB benchmark suite, one with the maximum number of non-binary integer variables (bell3a) and the other with the maximum number of constraints (rentacar).

1998

"... In PAGE 4: ...35%), there are several instances where it yields considerably better performance, up to as much as almost 80%, and the average speedup improvement obtainedis 15%. Next, Table2... ..."

Cited by 1

### Table 1 compares three encodings of some standard crossword puzzles. The worst-time complexity of GAC on the non-binary encoding of the puzzles is in the order of O(mk), where m is the number of letters in the alphabet and k is the length of the words. In the puzzles we gener- ated, k was up to 10. This obviously makes the non-binary encoding completely impractical, so we did not consider it in the experiments. The small domain size of the original variables compared to the dual variables makes the hidden representation better. Note, that because of the large size of the dual variables, AC is expensive and it may be the case that forward checking is enough to solve these problems in reasonable time.

"... In PAGE 5: ... Table1 : Branches and CPU time when generating cross- word puzzles. Fail First was used for variable ordering.... ..."

### Table 2: Branches explored and CPU time (seconds) used to find an optimal golomb ruler or prove that none exists. The variables were ordered lexicographically. The numbers of branches in the hidden representation are not given because they are always equal to the corresponding numbers in the non-binary representation. A * means that there was a cut off after 1 hour of CPU.

"... In PAGE 5: ... This gives a0 a3 a0 a22 a38a26a13a9a5a39a40 ternary constraints and a clique of binary `not equals apos; constraints. Table2 compares MGAC on the ternary representation to MAC on the hid- den and double representations. As Theorem 1 predicted, MGAC in the non-binary encoding explores the same num- ber of branches as MAC in the hidden.... ..."

### Table 2: Branches explored and CPU time (seconds) used to find an optimal golomb ruler or prove that none exists. The variables were ordered lexicographically. The numbers of branches in the hidden representation are not given because they are always equal to the corresponding numbers in the non-binary representation. A * means that there was a cut off after 1 hour of CPU.

"... In PAGE 5: ... This gives n(n ? 1)=2 ternary constraints and a clique of binary `not equals apos; constraints. Table2 compares MGAC on the ternary representation to MAC on the hid- den and double representations. As Theorem 1 predicted, MGAC in the non-binary encoding explores the same num- ber of branches as MAC in the hidden.... ..."

### Table 3: The non-binary pseudo-Graycode(2,2)

"... In PAGE 4: ... start with the all zero code word corresponding to the source zero 2. form the next codeword bychanging the least signi cant group by the least amount that results in an unused code word;; the change can be either an increment (add 1) or decrement (subtract 1) of the value in the group;; if an increment or decrement does not generate a new code word, move to the next higher signi cance group A(2;; 2) non-binary pseudo-Graycodeisshown in Table3 along with a 4 bit binary Graycodeand the 4 bit weighted binary representation for comparison. Immediately noticeable is the fact that the... ..."

### Table 1: Performance of a discourse segmenter that uses a decision-tree, non-binary classifier.

1999

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