P. Prosser. Forward checking with backmarking. Constraint Processing (LNCS), 923:185-204, 1995.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....in x i s current session. 4. 4 Con ict directed backjumping The two ideas, jumping back to a variable that, as instantiated, is in con ict with the current variable, and jumping back at internal dead ends, can be integrated into a single algorithm, the con ict directed backjumping algorithm [67]. This algorithm uses the scheme we have outlined for graph based backjumping but, rather than relying on graph information, exploits information 20 gathered during search. For each variable, the algorithm maintains an induced jumpback set. Given a dead end tuple a i , we de ne next the ....

P. Prosser. Forward checking with backmarking. Technical Report AISL{48-93, University of Strathclyde, 1993.

....well for MAC. Nevertheless, the results (and many other similar parameter settings of the CT model) do serve to demonstrate that there exist classes of problems for which EFC is a significant assist to FC. One final point is that EFC is identical to the algorithm FC BM described by Prosser in [9]. In particular, Prosser identified that when backing up from a DWO the gains of Prune: Backup could be achieved. However, the algorithm he described uti 6 Prosser s FC D2C algorithm [4] uses a special test to recognize the case when a value can be pruned back to level 0. This special case is ....

P. Prosser. Forward checking with backmarking. In M. Meyer, editor, Constraint Processing, LNCS 923, pages 185--204. Springer-Verlag, New York, 1995.

....of each future variable which is compatible with the committed labels. This is the principle of introducing delayed evaluation in FC based algorithms. We refer to this class of algorithms as lazy algorithms. As mentioned in [4, 2] MFC can be extended with Prosser s type of partial backmarking [17] and CBJ. Two types of savings can be achieved in BM [14] a) if the current variable s label was found incompatible with a past variable s label, unless that past variable s label has been changed the compatibility check between this pair of labels can be saved as the check will fail again; b) ....

....In this paper, an algorithm with these benefits is proposed. It is called minimal forward checking with backmarking and conflict directed backjumping (MFC BM2 CBJ) 1 Albeit MFC BM2 CBJ can be regarded as Prosser s forward checking with backmarking and conflict directed backjumping (FC BMCBJ) [17] with delayed evaluation, it adopts a 2dimensional data structure for storing the minimum backup level (mbl) 10] while FC BM CBJ uses an one dimensional array for mbl. The more involved mbl allows more precise backmarking information to be kept during search than Gaschnig s backmarking and ....

[Article contains additional citation context not shown here]

Prosser, P. Forward Checking with Backmarking. In Meyer (ed.): Constraint Processing, LNCS 923, SpringerVerlag, 185-204, 1995.

....) For a proof see [BM96] 5. 4 Conflict directed backjumping The two ideas, jumping back to a variable that, as instantiated, is in conflict with the current variable, and jumping back at internal dead ends, can be integrated into a single algorithm, the conflict directed backjumping algorithm [Pro93a]. This algorithm uses the scheme we have outlined for graph based backjumping but, rather than using graph information, exploits information gathered during search. For each variable, the algorithm maintains an induced jumpback set. Given a dead end tuple a i , we define the jumpback set of a ....

P. Prosser. Forward checking with backmarking. Technical Report AISL--48--93, University of Strathclyde, 1993.

....bounds for backjumping. 4. 4 Conflict directed backjumping The two ideas, jumping back to a variable that, as instantiated, is in conflict with the current variable, and jumping back at internal dead ends, can be integrated into a single algorithm, the conflict directed backjumping algorithm [Pro93a]. This algorithm uses the scheme we have outlined for graph based backjumping but, rather than using graph information, exploits information gathered during search. For each variable, the algorithm maintains an induced jumpback set. Given a dead end tuple a i , we define next the jumpback set ....

P. Prosser. Forward checking with backmarking. Technical Report AISL--48--93, University of Strathclyde, 1993.

....that are generated outside of the phase transition. Both Smith and Prosser propose that Smith s model of hard binary CSPs be used to compare CSP algorithms empirically. This has already been done by many authors 1 [ Dechter and Meiri, 1994; Dent and Mercer, 1994; Frost and Dechter, 1994a; 1994b; Prosser, 1995 ] We believe that the underlying structure of a problem needs to be included into models of hard problems. Smith Dyer (1995) also conclude that local graph topology needs to be included in order to better model the transition point. With this in mind we generalize the above CSP ....

P. Prosser. Forward Checking with Backmarking, pages 185--204. in M. Meyer (ed); Constraint Processing, LNCS 923. Springer-Verlag, 1995.

....on each problem to find the first solution or exhaust the search space otherwise. These included backmarking and conflict directed backjumping (BM CBJ) Prosser 93] with minimal width heuristic (MWO) Freuder 82] forward checking with backmarking and conflict directed backjumping (FC BM CBJ) Prosser 94] with a heuristic called BZ which is derived from Br laz heuristic [Br laz 79] 4 , minimal forward checking with backmarking and conflict directed backjumping (MFC BM CBJ) Kwan Tsang 95] with either BZ or MWO. The backmarking element in all these algorithms employed a 2dimensional data ....

Prosser, P. 1994. Forward Checking with Backmarking. In Proceedings ECAI-94 Workshop on Constraint Processing, 59-68.

....to MAC by utilizing the minimal version of arc consistency developed by Schiex et al. SRGV96] Second, we can use the insight that FC s domain pruning is accomplishing the same thing as BM to develop more efficient versions and implementations of other CSP algorithms. In particular, Prosser [Pro95] has developed an algorithm he called forward checking with backmarking. However, we have shown that FC already achieves all BM savings. In examining Prosser s proposal it becomes clear that he has identified some additional improvements to FC. He develops a method of taking advantage of these ....

P. Prosser. Forward checking with backmarking. In M. Meyer, editor, Constraint Processing, LNCS 923, pages 185--204. Springer-Verlag, New York, 1995.

....50 problems were generated at each of the following data points. Note that asterisks mean all values within the range as mentioned above were used. 10, 5, 20, 5, 30, 5, 40, 5, 60, 5, 5 Note that FC BM2 CBJ is a variant of FC BM CBJ as described in [Prosser 94a] except the former used a 2 dimensional data structure to store the minimum backup level whereas the latter uses an onedimensional array. FC BM2 CBJ should be able to keep more accurate backmarking information than FC BM CBJ. CSM 241 8 To predict where phase transition happens using Equation ....

Prosser, P. 1994. Forward Checking with Backmarking. In Proceedings ECAI Workshop on Constraint Processing, 59-68.

....it as inconsistent with the deepest variable that can change d j . This would ensure that MFC never instantiates v i to that value as long as that value would empty the future connected domain d j . This optimization of FC can be seen as a form of partial Backmarking and is described in detail in [11]. A second optimization missing in MFC is the addition of a intelligent backtracking component. If the search jumps back to the source of a failure instead of the previously instantiated variable the delayed forward checks for the variables between will be avoided. Finally we would like to improve ....

P. Prosser. Forward Checking with Backmarking. Technical Report AISL-48-93, University of Strathclyde, 1993.

....new backtracking algorithms have been proposed. The basic ones include Backjumping [7] ConflictDirected Backjumping [18] Graph Based Backjumping [4] Backmarking [6] and Forward Checking [10] Several hybrid algorithms, which combine two or more basic algorithms, have also been developed [18, 19]. A question arises as to which of the known backtracking algorithms is the best one. There is no straightforward answer. First, the performance of backtracking algorithms depends heavily on the problem being solved. Often, it is possible to construct examples of constraint networks on which an ....

P. Prosser. Forward checking with backmarking. Technical Report AISL--48--93, University of Strathclyde, June 1993.

....may be overdone in FC which in turn alleviates the problem of using FC for CSPs with large domain size. In this paper, we suggest two new algorithms which achieve similar contributions. The essence of these algorithms is to incorporate laziness into forward checking with backmarking (FC BM) Prosser 95] One of them is further enhanced with the capability of conflict directed backjumping (CBJ) Prosser 93] They can be considered as extensions of MFC despite the new algorithms employ simpler data structures and thus have a smaller space complexity than MFC. In the next section, we present a ....

....are two major differences. Firstly, we adopt a 2 dimensional data structure for storing the minimum backup level as suggested in [Kondrak van Beek 95] while FC BM uses an one dimensional array for the same data structure. Secondly, unlike FC BM which does not have efficient type (b) saving [Prosser 95] MFC BM2 benefits from both type (a) and type (b) saving. It is the laziness of the new algorithm that makes type (b) saving feasible. 7 The 2 in MFC BM2 emphasises the fact that a 2 dimensional minimum backup level is used. CSM 260 7 Two 2 dimensional arrays are used for storing ....

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Prosser, P. Forward Checking with Backmarking. In Meyer (ed.): Constraint Processing, LNCS 923, Springer-Verlag, 185-204, 1995.

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P. Prosser. Forward checking with backmarking. Constraint Processing (LNCS), 923:185-204, 1995.

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