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Optimal Implementation of Watched Literals and More General Techniques

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The document describes three approaches to maintaining a pointer called "last" when searching a list in backtracking search. The simplest approach searches the list from the start each time and may check the same elements multiple times per branch. The second approach restores the value of last on backtracking so the list only needs to be scanned once per branch. The third, "circular" approach stores the value of last without backtracking, continuing the search from last+1 and looping back to the start of the list when it reaches the end. An example search tree demonstrates how the circular approach works and amortizes the cost of calls to an acceptability function over the nodes in the tree.

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Optimal Implementation of Watched Literals and More General Techniques Ian Gent University of St Andrews Scotland ian.gent@st-andrews.ac.uk Note: this paper will appear in JAIR, in press Monday, 16 September 13
A surprising result • Dangerous to use the word “surprising” • in fact deleted from final version of paper • But it surprised me • And it surprised Pete Nightingale • And it surprised Christian Bessiere • And there’s no hint of it in the literature • So I call it surprising Monday, 16 September 13
Some conventional wisdom is wrong • Searching a linear list for key element • e.g. watched literals in SAT • e.g. support lists in CSP • Don’t have to backtrack pointer (well known) • And can still retain optimality (result of this paper) • Was thought to be non-optimal • work can be repeated many times on a single branch • But a different partition of tree gives result Monday, 16 September 13
The Setup • We are in backtracking search (depth-first like) • Elements in LIST might be acceptable • e.g. domain value still valid • Acceptability is monotonic • if an element of LIST is unacceptable • it is unacceptable at all descendant nodes • When LIST has no acceptable elements • e.g. no valid values still exist • Search engine needs to be notified immediately • typically to perform some propagation Monday, 16 September 13
The Setup • We maintain a pointer last • With the following invariant • EITHER LIST[last] is acceptable • OR no element in LIST is acceptable • When LIST[last] becomes unacceptable • System notifies us (through some assumed mechanism) • Look for new value of last • If we can’t find one... • then no element in LIST is acceptable • We can notify search engine this information Monday, 16 September 13
Three ways to maintain last 1. We do not store state of last at all • Search LIST from start every time • Can check the same element multiple times per branch 2. Restore value of last on backtracking • Scan list once only per branch • But requires additional memory 3. Store value of last without backtracking • continue search from last+1 • but have to cycle round to start of LIST at end Monday, 16 September 13
Find-New-Element (FNE) • FNE: procedure to find new value of last • Called by our assumed mechanism • when LIST[last] becomes unacceptable • Three variants for three approaches 1. last not stored between calls 2. last stored and restored on backtracking 3. last stored but not restored • call this “circular” because of cycle at end of LIST Monday, 16 September 13
No State Stored (unacceptable), so we have to scan for a new watched literal propagations at the same node can make this new value unsatisfi This paper compares three implementations of findNewEle properties of the last one. The differences arise from how las invocations of findNewElement. The simplest variant is sh time it is called the previous value of last is discarded and beginning. Procedure 1: FNE-NoState(list) 1: last := -1 2: repeat 3: last := last + 1 4: if acceptable(list,last) then return true 5: until last = N 6: return false Procedure 1 is simple, certainly maintains Invariant 2, and has a significant disadvantage in that its worst case is to require Θ at every leaf node, stated here as Proposition 3 and proved in A omitted from the main text but is available online, see Appendi Monday, 16 September 13
State restored on backtracking The remaining two variants both use FNE-NoState to in calls thereafter. The second variant is based on state restorat from the most recent value of last found at the current node the value of last may be changed by descendent nodes, some search solver must be exploited to restore the value of last wh this method is will vary between solvers, and is not critical f method is shown as Procedure 2. Procedure 2: FNE-RestoreState(list) 1: repeat 2: last := last + 1 3: if acceptable(list,last) then return true 4: until last = N 5: return false Since acceptability is monotonic and last is always restored the invariant is guaranteed. We have the following, for which Proposition 4. The procedure FNE-RestoreState maint branch of the search tree from root to leaf node, at most N c Monday, 16 September 13
Circular: State Stored but not backtracked node. To deal with this correctly, we have to allow every elemen even if some may have already been checked higher on the curren same node. The focus of this paper is the “circular” method for c the end of the list we circle around from the end of the list to zer until we hit the value that last had when the call was made. For above procedure FNE-NoState. All subsequent calls are to the f Procedure 3: FNE-Circular(list) 1: last-cache := last 2: repeat 3: last := last + 1 4: if last = N then last := 0 5: if acceptable(list,last) then return true 6: until last = last-cache 7: return false No claim is made for originality of the circular method: it was and Miguel (2006b) and probably by earlier authors. Unlike th invariant does not hold for all search algorithms. Correctness wil below, in the context of ‘downwards-explored search trees’, as defi Monday, 16 September 13
Example Search Tree 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Say length of LIST = N = 20 Monday, 16 September 13
Right Hand Branch 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node At Node 0, we set last=14, total cost = 14 Monday, 16 September 13
2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Stuff happens on left hand subtree and value of last changes Right Hand Branch Monday, 16 September 13
2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Stuff happens on left hand subtree and value of last changes Right Hand Branch Monday, 16 September 13
2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node We loop round to set last=14 (again!) total cost = 17 Right Hand Branch Monday, 16 September 13
2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Stuff happens on left hand subtree and value of last changes Right Hand Branch Monday, 16 September 13
2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Stuff happens on left hand subtree and value of last changes Right Hand Branch Monday, 16 September 13
2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Right Hand Branch We loop round to set last=14 (again!) total cost = 18 Monday, 16 September 13
2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Stuff happens on left hand subtree and value of last changes Right Hand Branch Monday, 16 September 13
2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Stuff happens on left hand subtree and value of last changes Right Hand Branch Monday, 16 September 13
2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Right Hand Branch We loop round to set last=14 (again!) cost = 18 Monday, 16 September 13
2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Right Hand Branch At the same node we search again and fail, total cost at node = 38 Monday, 16 September 13
2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Right Hand Branch Total cost down branch = 17+16+15+38 = 87 Monday, 16 September 13
Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the Left Branch Segment is sequence of nodes from solid box down double lines Monday, 16 September 13
Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the Left Branch Segment is sequence of nodes where we always branch left Monday, 16 September 13
Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the In LBS sequence of nodes is consecutive in number without backtracking Monday, 16 September 13
Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the Each node is in exactly one LBS Monday, 16 September 13
Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the There are the same number of LBSs as branches/leaf nodes Monday, 16 September 13
Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the Never make as many as 2N = 40 calls to Acceptable in LBS Monday, 16 September 13
Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the All these statements are general. Last is crux of optimality result. Monday, 16 September 13
Key Lemma • Say length of LIST=N • In any Left Branch Segment • there are at most N-1 successful calls to Acceptable() • Proof is easy: • if there are N it has checked every element • after a successful call there is only another call if that element becomes unacceptable • if there are N every element has become unacceptable Monday, 16 September 13
Key Corollary • In any Left Branch Segment • there are at most 2N-1 calls to Acceptable() • Proof is easy: • N-1 successful calls • then N including cycle for the last failed attempt Monday, 16 September 13
Optimality Theorem • In a search tree, FNE-Circular can make no more than 2N calls to Acceptable() per branch of the tree • Note this applies to every search tree • no average case or asymptotics • This is optimal because we must check O(N) per branch in any method • Worst case is only 2x worst case of state restoration • and only worse when failure on branch Monday, 16 September 13
Application: Constraints • Typically in GAC algorithm • (Generalised arc consistency) • Need support for each variable/value • Each variable/value has list of support tuples • acceptable means: • every assignment in tuple is still valid • Losing support = element unacceptable Monday, 16 September 13
From GAC to MGAC • MGAC = Maintaining GAC in search • Two approaches can both be improved • Restore State • e.g. Bessiere et al implementation of MGAC • Optimal per branch but uses additional space to store pointer • Circular • e.g. Gent et al, various papers • Used because of simplicity and effectiveness in practice • With warning in paper that approach is not optimal • but it was optimal! Monday, 16 September 13
Application: SAT • Watched literals in SAT • Need two acceptable elements • Generalisation is not hard • though did not find till the last minute • Fun Fact: • if you want w watched elements • number of calls to Acceptable() does not increase with w Monday, 16 September 13
Application: SAT • I thought that Circular was standard implementation • But Reviewer pointed out that was wrong • Actually most people do FNE-NoState • e.g. MiniSat, tiniSat • meaning can be O(N2) calls per branch • Complete surprise to me • So is Circular (optimal) better? Monday, 16 September 13
Is circular better than no-state in MiniSat?Gent 0.5 0.6 0.8 1 2 3 4 5 6 8 10 0.5 0.6 0.8 1 1.2 2 3 4 5 6 8 10 20 Circular:Stock 0.548x + 0.523 Yes! Monday, 16 September 13
Is circular better than no-state in MiniSat?Gent 0.5 0.6 0.8 1 2 3 4 5 6 8 10 0.5 0.6 0.8 1 1.2 2 3 4 5 6 8 10 20 Circular:Stock 0.548x + 0.523 Average of 29% faster propagation Monday, 16 September 13
Is circular better than no-state in MiniSat?Gent 0.5 0.6 0.8 1 2 3 4 5 6 8 10 0.5 0.6 0.8 1 1.2 2 3 4 5 6 8 10 20 Circular:Stock 0.548x + 0.523 Circular can propagate almost 10x faster Monday, 16 September 13
Is circular better than no-state in MiniSat?Gent 0.5 0.6 0.8 1 2 3 4 5 6 8 10 0.5 0.6 0.8 1 1.2 2 3 4 5 6 8 10 20 Circular:Stock 0.548x + 0.523 Circular is better at finding literals that are not falsified Monday, 16 September 13
Is circular better than no-state in MiniSat? No! Gent 0.01 0.1 1 10 100 0.01 0.1 1 10 100 1000 Circular ratio TwoPointer ratio Monday, 16 September 13
Is circular better than no-state in MiniSat? No significant improvement in overall search Gent 0.01 0.1 1 10 100 0.01 0.1 1 10 100 1000 Circular ratio TwoPointer ratio Monday, 16 September 13
Is circular better than no-state in MiniSat? Different propagations = different explanations = different heuristics = different search Gent 0.01 0.1 1 10 100 0.01 0.1 1 10 100 1000 Circular ratio TwoPointer ratio Monday, 16 September 13
Conclusions • The circular approach is optimal • and this is surprising • minimises space and very easy to implement • Some existing implementations are optimal • even though their authors said not • Generalises to finding multiple elements • e.g. watched literals in SAT • Circular can improve propagation in MiniSat • but there’s no evidence it improves overall solving speed Monday, 16 September 13
And finally... • JAIR paper gives me particular pleasure... • I have published papers in JAIR 20 years apart • Gent &Walsh, September 1993 • Gent, September, 2013 • JAIR was platinum open access from day one • Original publication by anon ftp and gopher • web page set up a few months later! Monday, 16 September 13

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Optimal Implementation of Watched Literals and More General Techniques

  • 1. Optimal Implementation of Watched Literals and More General Techniques Ian Gent University of St Andrews Scotland ian.gent@st-andrews.ac.uk Note: this paper will appear in JAIR, in press Monday, 16 September 13
  • 2. A surprising result • Dangerous to use the word “surprising” • in fact deleted from final version of paper • But it surprised me • And it surprised Pete Nightingale • And it surprised Christian Bessiere • And there’s no hint of it in the literature • So I call it surprising Monday, 16 September 13
  • 3. Some conventional wisdom is wrong • Searching a linear list for key element • e.g. watched literals in SAT • e.g. support lists in CSP • Don’t have to backtrack pointer (well known) • And can still retain optimality (result of this paper) • Was thought to be non-optimal • work can be repeated many times on a single branch • But a different partition of tree gives result Monday, 16 September 13
  • 4. The Setup • We are in backtracking search (depth-first like) • Elements in LIST might be acceptable • e.g. domain value still valid • Acceptability is monotonic • if an element of LIST is unacceptable • it is unacceptable at all descendant nodes • When LIST has no acceptable elements • e.g. no valid values still exist • Search engine needs to be notified immediately • typically to perform some propagation Monday, 16 September 13
  • 5. The Setup • We maintain a pointer last • With the following invariant • EITHER LIST[last] is acceptable • OR no element in LIST is acceptable • When LIST[last] becomes unacceptable • System notifies us (through some assumed mechanism) • Look for new value of last • If we can’t find one... • then no element in LIST is acceptable • We can notify search engine this information Monday, 16 September 13
  • 6. Three ways to maintain last 1. We do not store state of last at all • Search LIST from start every time • Can check the same element multiple times per branch 2. Restore value of last on backtracking • Scan list once only per branch • But requires additional memory 3. Store value of last without backtracking • continue search from last+1 • but have to cycle round to start of LIST at end Monday, 16 September 13
  • 7. Find-New-Element (FNE) • FNE: procedure to find new value of last • Called by our assumed mechanism • when LIST[last] becomes unacceptable • Three variants for three approaches 1. last not stored between calls 2. last stored and restored on backtracking 3. last stored but not restored • call this “circular” because of cycle at end of LIST Monday, 16 September 13
  • 8. No State Stored (unacceptable), so we have to scan for a new watched literal propagations at the same node can make this new value unsatisfi This paper compares three implementations of findNewEle properties of the last one. The differences arise from how las invocations of findNewElement. The simplest variant is sh time it is called the previous value of last is discarded and beginning. Procedure 1: FNE-NoState(list) 1: last := -1 2: repeat 3: last := last + 1 4: if acceptable(list,last) then return true 5: until last = N 6: return false Procedure 1 is simple, certainly maintains Invariant 2, and has a significant disadvantage in that its worst case is to require Θ at every leaf node, stated here as Proposition 3 and proved in A omitted from the main text but is available online, see Appendi Monday, 16 September 13
  • 9. State restored on backtracking The remaining two variants both use FNE-NoState to in calls thereafter. The second variant is based on state restorat from the most recent value of last found at the current node the value of last may be changed by descendent nodes, some search solver must be exploited to restore the value of last wh this method is will vary between solvers, and is not critical f method is shown as Procedure 2. Procedure 2: FNE-RestoreState(list) 1: repeat 2: last := last + 1 3: if acceptable(list,last) then return true 4: until last = N 5: return false Since acceptability is monotonic and last is always restored the invariant is guaranteed. We have the following, for which Proposition 4. The procedure FNE-RestoreState maint branch of the search tree from root to leaf node, at most N c Monday, 16 September 13
  • 10. Circular: State Stored but not backtracked node. To deal with this correctly, we have to allow every elemen even if some may have already been checked higher on the curren same node. The focus of this paper is the “circular” method for c the end of the list we circle around from the end of the list to zer until we hit the value that last had when the call was made. For above procedure FNE-NoState. All subsequent calls are to the f Procedure 3: FNE-Circular(list) 1: last-cache := last 2: repeat 3: last := last + 1 4: if last = N then last := 0 5: if acceptable(list,last) then return true 6: until last = last-cache 7: return false No claim is made for originality of the circular method: it was and Miguel (2006b) and probably by earlier authors. Unlike th invariant does not hold for all search algorithms. Correctness wil below, in the context of ‘downwards-explored search trees’, as defi Monday, 16 September 13
  • 11. Example Search Tree 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Say length of LIST = N = 20 Monday, 16 September 13
  • 12. Right Hand Branch 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node At Node 0, we set last=14, total cost = 14 Monday, 16 September 13
  • 13. 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Stuff happens on left hand subtree and value of last changes Right Hand Branch Monday, 16 September 13
  • 14. 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Stuff happens on left hand subtree and value of last changes Right Hand Branch Monday, 16 September 13
  • 15. 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node We loop round to set last=14 (again!) total cost = 17 Right Hand Branch Monday, 16 September 13
  • 16. 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Stuff happens on left hand subtree and value of last changes Right Hand Branch Monday, 16 September 13
  • 17. 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Stuff happens on left hand subtree and value of last changes Right Hand Branch Monday, 16 September 13
  • 18. 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Right Hand Branch We loop round to set last=14 (again!) total cost = 18 Monday, 16 September 13
  • 19. 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Stuff happens on left hand subtree and value of last changes Right Hand Branch Monday, 16 September 13
  • 20. 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Stuff happens on left hand subtree and value of last changes Right Hand Branch Monday, 16 September 13
  • 21. 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Right Hand Branch We loop round to set last=14 (again!) cost = 18 Monday, 16 September 13
  • 22. 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Right Hand Branch At the same node we search again and fail, total cost at node = 38 Monday, 16 September 13
  • 23. 2.1 Worked Example I present a worked example of Procedure 3, showing how we can amortize the count of calls to acceptable in a way which will form the basis of the optimality results of this paper. Figure 1 shows an example search using the circular approach. In the example we assume node id = 0:last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 0,1,2,4 = 19 0,1,5,6 = 52 0,1,5,7 = 51 0,8,9,10 = 52 0,8,9,11 = 52 0,8,12,13 = 50 0,8,12,14 = 87 Figure 1: A search using the circular approach. See main text for description. that the list being searched has 20 elements, indexed from 0 to 19. and that initialisation sets last = 0. The box at a node indicates the node number (italics), the value of last after any calls to FNE-Circular (bold), and the cost of those calls in total at the node Right Hand Branch Total cost down branch = 17+16+15+38 = 87 Monday, 16 September 13
  • 24. Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the Left Branch Segment is sequence of nodes from solid box down double lines Monday, 16 September 13
  • 25. Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the Left Branch Segment is sequence of nodes where we always branch left Monday, 16 September 13
  • 26. Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the In LBS sequence of nodes is consecutive in number without backtracking Monday, 16 September 13
  • 27. Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the Each node is in exactly one LBS Monday, 16 September 13
  • 28. Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the There are the same number of LBSs as branches/leaf nodes Monday, 16 September 13
  • 29. Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the Never make as many as 2N = 40 calls to Acceptable in LBS Monday, 16 September 13
  • 30. Left Branch Segment (LBS)Optimal Implementation of Watched Literals node id = 0: last=14, cost = 14 last=17 1:14, 0 last=19 8:14, 17 last=16 2:17, 3 5:17, 18 last=18 9:16, 21 × 12:14, 18 last=15 3:17, 19× 4:19, 2 6:18, 20 × 7:17, 19 10:16, 0 11:16, 0 13:15, 1 14:14, 38 × 0,1,2,3 = 36 4 = 2 5,6 = 38 7 = 19 8,9,10 = 38 11 = 0 12,13 = 19 14 = 38 Figure 2: The example from Figure 1 with additional annotations. A double line indicates a left hand branch, while a wavy line the right hand branch. A sequence of double lines indicates a left branch segment. A solid box around a node indicates the start of a left branch segment. Finally, the sums of costs are made only over the All these statements are general. Last is crux of optimality result. Monday, 16 September 13
  • 31. Key Lemma • Say length of LIST=N • In any Left Branch Segment • there are at most N-1 successful calls to Acceptable() • Proof is easy: • if there are N it has checked every element • after a successful call there is only another call if that element becomes unacceptable • if there are N every element has become unacceptable Monday, 16 September 13
  • 32. Key Corollary • In any Left Branch Segment • there are at most 2N-1 calls to Acceptable() • Proof is easy: • N-1 successful calls • then N including cycle for the last failed attempt Monday, 16 September 13
  • 33. Optimality Theorem • In a search tree, FNE-Circular can make no more than 2N calls to Acceptable() per branch of the tree • Note this applies to every search tree • no average case or asymptotics • This is optimal because we must check O(N) per branch in any method • Worst case is only 2x worst case of state restoration • and only worse when failure on branch Monday, 16 September 13
  • 34. Application: Constraints • Typically in GAC algorithm • (Generalised arc consistency) • Need support for each variable/value • Each variable/value has list of support tuples • acceptable means: • every assignment in tuple is still valid • Losing support = element unacceptable Monday, 16 September 13
  • 35. From GAC to MGAC • MGAC = Maintaining GAC in search • Two approaches can both be improved • Restore State • e.g. Bessiere et al implementation of MGAC • Optimal per branch but uses additional space to store pointer • Circular • e.g. Gent et al, various papers • Used because of simplicity and effectiveness in practice • With warning in paper that approach is not optimal • but it was optimal! Monday, 16 September 13
  • 36. Application: SAT • Watched literals in SAT • Need two acceptable elements • Generalisation is not hard • though did not find till the last minute • Fun Fact: • if you want w watched elements • number of calls to Acceptable() does not increase with w Monday, 16 September 13
  • 37. Application: SAT • I thought that Circular was standard implementation • But Reviewer pointed out that was wrong • Actually most people do FNE-NoState • e.g. MiniSat, tiniSat • meaning can be O(N2) calls per branch • Complete surprise to me • So is Circular (optimal) better? Monday, 16 September 13
  • 38. Is circular better than no-state in MiniSat?Gent 0.5 0.6 0.8 1 2 3 4 5 6 8 10 0.5 0.6 0.8 1 1.2 2 3 4 5 6 8 10 20 Circular:Stock 0.548x + 0.523 Yes! Monday, 16 September 13
  • 39. Is circular better than no-state in MiniSat?Gent 0.5 0.6 0.8 1 2 3 4 5 6 8 10 0.5 0.6 0.8 1 1.2 2 3 4 5 6 8 10 20 Circular:Stock 0.548x + 0.523 Average of 29% faster propagation Monday, 16 September 13
  • 40. Is circular better than no-state in MiniSat?Gent 0.5 0.6 0.8 1 2 3 4 5 6 8 10 0.5 0.6 0.8 1 1.2 2 3 4 5 6 8 10 20 Circular:Stock 0.548x + 0.523 Circular can propagate almost 10x faster Monday, 16 September 13
  • 41. Is circular better than no-state in MiniSat?Gent 0.5 0.6 0.8 1 2 3 4 5 6 8 10 0.5 0.6 0.8 1 1.2 2 3 4 5 6 8 10 20 Circular:Stock 0.548x + 0.523 Circular is better at finding literals that are not falsified Monday, 16 September 13
  • 42. Is circular better than no-state in MiniSat? No! Gent 0.01 0.1 1 10 100 0.01 0.1 1 10 100 1000 Circular ratio TwoPointer ratio Monday, 16 September 13
  • 43. Is circular better than no-state in MiniSat? No significant improvement in overall search Gent 0.01 0.1 1 10 100 0.01 0.1 1 10 100 1000 Circular ratio TwoPointer ratio Monday, 16 September 13
  • 44. Is circular better than no-state in MiniSat? Different propagations = different explanations = different heuristics = different search Gent 0.01 0.1 1 10 100 0.01 0.1 1 10 100 1000 Circular ratio TwoPointer ratio Monday, 16 September 13
  • 45. Conclusions • The circular approach is optimal • and this is surprising • minimises space and very easy to implement • Some existing implementations are optimal • even though their authors said not • Generalises to finding multiple elements • e.g. watched literals in SAT • Circular can improve propagation in MiniSat • but there’s no evidence it improves overall solving speed Monday, 16 September 13
  • 46. And finally... • JAIR paper gives me particular pleasure... • I have published papers in JAIR 20 years apart • Gent &Walsh, September 1993 • Gent, September, 2013 • JAIR was platinum open access from day one • Original publication by anon ftp and gopher • web page set up a few months later! Monday, 16 September 13