### Table 6: Backtracking search tree.

### Table 7: Performance of compressing backtracking search and set pruning trees #28with the DAG optimization#29 using one 5- dimensional trie, where BB and PB stand for basic backtracking and pruned backtracking #28described in Section 4.2#29. The numbers in the parentheses are the height of the compressed trie. Sometimes the height is smaller than the memory accesses of pruned backtracking, since some nodes can not be reached after pruning. For linear search, we assume each #0Clter #0Cts in a cache line, and a #0Clter rule takes 1 memory accesses.

2001

"... In PAGE 9: ... This is because the high compression ratio of the paths in the backtracking search trie o#0Bsets the cost of a small number of backtracks. More speci#0Ccally, as Table7 shows, the total number of memory accesses is usually less than twice the height of the compressed tries. On the other hand, the compression ratio for the backtracking trie is much higher: mostly above 3 or more.... In PAGE 9: ... As wewould expect, both backtracking search trie and linear search have low storage cost. On the other hand, the lookup time of linear searchis 3 - 5 times larger than that required by backtracking search or set pruning trees, as shown in Table7 . If we use multi-bit tries, the performance of backtracking search and set pruning trees can be even better.... ..."

Cited by 5

### Table 1 contains the five problems. The first problem we consider is GACSUPPORT. It is at the core of all the generic arc consistency algorithms. The second problem, ISITGAC, is not directly related with operations used in basic propagation algorithms. It is largely introduced for academic purposes. The third question, NOGACWIPEOUT, can be used to decide if we do not need to backtrack at a given node in the search tree. (Note that a12 a16a15a18a17 a12

2004

"... In PAGE 3: ...Table1 . The five problems related to genaralized arc consistency Problem Instance Question/Output GACSUPPORT(a0 ) a1a3a2a4a0 , a5 on a6a8a7a10a9a12a11a13a1a15a14 , a16a17a2 a6a8a7a10a9a12a11a13a1a15a14 , and a6a18a2a19a5a20a11a21a16a22a14 Does value a6 for a16 have a support on a1 in a5 ? ISITGAC(a0 ) a1a23a2a24a0 , a5 on a6a8a7a10a9a25a11a13a1a15a14 Does GACSUPPORTa26 a1a28a27a29a5a19a27a30a16a20a27a29a6a32a31 answer yes for each variable a16a33a2a34a6a8a7a10a9a25a11a13a1a15a14 and each value a6a35a2a20a5a20a11a21a16a36a14 ? NOGACWIPEOUT(a0 ) a1a37a2a38a0 , a5 on a6a8a7a10a9a12a11a13a1a15a14 Is there any non empty a5a18a39a41a40a42a5 on which ISITGACa26 a1a28a27a43a5 a39 a31 answers yes ? MAXGAC(a0 ) a1a44a2a45a0 , a5 on a6a8a7a12a9a12a11a13a1a15a14 , and a5a46a40a47a5a49a48 Is it the case that ISITGACa26 a1a28a27a29a5a49a31 answers yes and a50 a51a25a5 a39 , a5a53a52a34a5 a39 a40a47a5a49a48 , on which ISITGACa26 a1a28a27a43a5 a39 a31 answers yes ? GACDOMAIN(a0 ) a1a23a2a24a0 , a5 a48 on a6a8a7a10a9a12a11a13a1a15a14 The domain a5 such that MAXGACa26 a1a28a27a43a5 a48 a27a30a5a49a31 answers yes 3.... ..."

Cited by 7

### Table 2 gives the numbers of backtracks performed by the three solvers. BP-AC makes the same number of backtracks as BP-IC except for alpha and olympic, and GP makes fewer backtracks than BP-AC for cars. Basically, the three solvers explore the same search trees for most of the programs. Therefore, the comparison results shown in Table 1 reflect the real performance of the solvers. GP and BP are quite different. In GP constraints are compiled into indexicals

2005

"... In PAGE 20: ...F. Zhou Table2 . Comparison of numbers of backtracks.... ..."

### Table 4: Performance of backtracking search using one 5-dimensional trie, where BB, PB, and SB stands for basic backtracking, pruned backtracking, and pruned backtracking with the extended switch pointer optimization #28described in Section 4.2#29.

2001

"... In PAGE 7: ...#29 Again our results are based on searching one bit at a time. Compared to several forms of backtracking search as shown in Table4 , set pruning trees provide an optimal number of memory accesses at the cost of large storage requirement. In particular, the storage requirement increases to 7 - 28 times as large as what is minimally required by the corresponding backtracking search trie.... In PAGE 9: ...ection 4.3.1 and Section 5.3#29. Compared with the perfor- mance results before compression, as shown in Table4 and Table 6, it is evident that compression improves performance signi#0Ccantly. More speci#0Ccally, compression reduces memory accesses and storage cost by a factor of 2-5for backtrack- ing search with and without cost based pruning.... ..."

Cited by 5

### Table 1. Backtracking (BT) and forward checking (FC) algorithms on the production planning problem from Section 3. CPU times are for a Common Lisp implementation running under Linux on an ancient 133MHz Pentium, whilst nodes are the number of nodes visited in the and/or search tree.

"... In PAGE 4: ... It will also ensure that the worlds in which we fail to satisfy the demand constraints are those where demand is much higher than average. Results are given in Table1 with the threshold for satisfi-... In PAGE 4: ... The performance advantage of the FC algorithm over the BT algo- rithm increases as the stochastic CSP increases in size. On the largest problem in Table1 , the FC algorithm visits approximately 1/6th the search nodes in roughly 1/6th the CPU time. This is in line with our results on random problems, where the FC algorithm is often an order of magnitude faster than the BT algorithm.... ..."

### Table 1. Backtracking (BT) and forward checking (FC) algorithms on the production planning problem from Section 3. CPU times are for a Common Lisp implementation running under Linux on an ancient 133MHz Pentium, whilst nodes are the number of nodes visited in the and/or search tree.

"... In PAGE 4: ... It will also ensure that the worlds in which we fail to satisfy the demand constraints are those where demand is much higher than average. Results are given in Table1 with the threshold for satisfi- ability AI set to 0.8.... In PAGE 4: ... The performance advantage of the FC algorithm over the BT algo- rithm increases as the stochastic CSP increases in size. On the largest problem in Table1 , the FC algorithm visits approximately 1/6th the search nodes in roughly 1/6th the CPU time. This is in line with our results on random problems, where the FCalgorithm isoften an order of magnitude faster than the BT algorithm.... ..."

### Table 3. Algorithm comparison that branch-and-bound algorithms are augmented with the ability for backtracking non-chronologically in the presence of conflicts that re- sult from bound conditions. In addition, we have established con- ditions for reducing the size of bound conflict explanations, which further elicits non-chronological backtracking. Preliminary results obtained on several instances of the Binate Covering Problem indicate that the proposed techniques are indeed effective and can be significant for specific classes of instances, in particular for instances of covering problems with sets of constraints that are hard to satisfy.

2000

"... In PAGE 5: ... A more efective pruning can also be observed, with an increase in the number of non-chronological backtracks and larger jumps in the search tree. Finally, in Table3 we can observe the results of several other al- gorithms on the same set of instances. Clearly, lp solve [3] (a generic Integer Linear Programming solver) is unable to solve almost all in- stances given the time limit.... ..."

Cited by 1

### Table 5: The State of the Search and Backtracking Stacks

1997

"... In PAGE 17: ...ransitions t1 and t2. Let t1have one local event d : w. Let t2have one local event d : z. Let all components be in the source state of these transitions. The states of the search and backtracking stacks at each mark#28#29 point of the algorithm are given in Table5 . The #0Crst column is used as an index, the second column shows the backtracking stack and the third column shows the search stack.... ..."

### Table 5: The State of the Search and Backtracking Stacks

"... In PAGE 16: ...ransitions t1 and t2. Let t1 have one local event d : w. Let t2 have one local event d : z. Let all components be in the source state of these transitions. The states of the search and backtracking stacks at each mark() point of the algorithm are given in Table5 . The rst column is used as an index, the second column shows the backtracking stack and the third column shows the search stack.... ..."