7.2 The General Backtrack Algorithm with Ordered Partitions

Section Backtrack in the reference manual describes the basic functions for a backtrack search. The purpose of this section is to document how the general backtrack algorithm is implemented in GAP and which parts you have to modify if you want to write your own backtrack routines.

Internal representation of ordered partitions. GAP represents an ordered partition as a record with the following components.

points: a list of all points contained in the partition, such that the points of each cell from lie consecutively,
cellno: a list whose ith entry is the number of the cell which contains the point i,
firsts: a list such that points[firsts[j]] is the first point in points which is in cell j,
lengths: a list of the cell lengths.

Some of the information is redundant, e.g., the lengths could also be read off the firsts list, but since this need not be increasing, it would require some searching. Similar for cellno, which could be replaced by a systematic search of points, keeping track of what cell is currently being traversed. With the above components, the mth cell of a partition P is expressed as P.points [ P.firsts[m

] ..

P.firsts[m] + P.lengths[m] - 1 ] . The most important operations, however, to be performed upon P are the splitting of a cell and the reuniting of the two parts. Following the strategy of J. Leon, this is done as follows:

(1) The points which make up the cell that is to be split are sorted so that the ones that remain inside occupy positions [ P.firsts[m] ..last ] in the list P.points (for a suitable value of last).

(2) The points at positions [ last + 1 .. P.firsts[m] +P.lengths[m] - 1 ] will form the additional cell. For this new cell requires additional entries are added to the lists P.firsts (namely, last+1) and P.lengths (namely, P.firsts[m] +P.lengths[m] - last - 1).

(3) The entries of the sublist P.cellno [ last+1 .. P.firsts[m] + P.lengths[m]-1 ] must be set to the number of the new cell.

(4) The entry P.lengths[m] must be reduced to last-P.firsts[m] + 1.

Then reuniting the two cells requires only the reversal of steps 2 to 4 above. The list P.points need not be rearranged.

Functions for setting up an R-base. This subsection explains some GAP functions which are local to the library file ``lib/stbcbckt.gi'' which contains the code for backtracking in permutation groups. They are mentioned here because you might find them helpful when you want to implement you own backtracking function based on the partition concept. An important argument to most of the functions is the R-base \R, which you should regard as a black box. We will tell you how to set it up, how to maintain it and where to pass it as argument, but it is not necessary for you to know its internal representation. However, if you insist to learn the whole story: Here are the record components from which an R-base is made up:

domain: the set W on which the group G operates
base: the sequence (a₁,Ľ,a_r) of base points
partition: an ordered partition, initially P₀, this will be refined to P₁,Ľ,P_r during the backtrack algorithm
where: a list such that a_i lies in cell number where[ i ] of P_i
rfm: a list whose ith entry is a list of refinements which take S_i to S_i+1; the structure of a refinement is described below
chain: a (copy of a) stabilizer chain for G (not if G is a symmetric group)
fix: only if G is a symmetric group: a list whose i entry contains Fixcells( P_i )
level: initially equal to chain, this will be changed to chains for the stabilizers G_{a₁... a_i} for i = 1,Ľ,r during the backtrack algorithm; if G is a symmetric group, only the number of moved points is stored for each stabilizer
lev: a list whose ith entry remembers the level entry for G_{a₁Ľa_i-1}
level2, lev2: a similar construction for a second group (used in intersection calculations), false otherwise. This second group H activated if the R-base is constructed as EmptyRBase( [ G, H ], W, P₀ ) (if G = H, GAP sets level2 = true instead).
nextLevel: this is described below

As our guiding example, we present code for the function Centralizer which calculates the centralizer of an element g in the group G. (The real code is more general and has a few more subtleties.)

P₀ := TrivialPartition( W );
\R := EmptyRBase( G, W, P₀ );

\R.nextLevel := function( P, rbase )
local fix, p, q, where;
NextRBasePoint( P, rbase );
fix := Fixcells( P );
for p in fix do
   q := p ^ g;
   where := IsolatePoint( P, q );
   if where <> false then
    Add( fix, q );
    ProcessFixpoint( \R, q );
    AddRefinement( \R, "Centralizer", [ P.cellno[ p ], q, where ] );
    if P.lengths[ where ] = 1 then
     p := FixpointCellNo( P, where );
     ProcessFixpoint( \R, p );
     AddRefinement( \R, "ProcessFixpoint", [ p, where ] );
    fi;
   fi;
od;
end;

return PartitionBacktrack( G,
   c -> g ^ c = g,
   false,
   \R,
   [ P₀, g ],
   L, R );

1. W is the set on which G acts and P₀ is the first member of the decreasing sequence of partitions mentioned in Backtrack searching in the reference manual. We set P₀ = (W), which is constructed as TrivialPartition( W )), but we could have started with a finer partition, e.g., into unions of g-cycles of the same length.

2. This statement sets up the R-base in the variable \R.

3--21. These lines define a function \R.nextLevel which is called whenever an additional member in the sequence P₀ ł P₁ ł Ľ of partitions is needed. If P_i does not yet contain enough base points in one-point cells, GAP will call \R.nextLevel( P_i, \R ), and this function will choose a new base point a_i+1, refine P_i to P_i+1 (thereby changing the first argument) and store all necessary information in \R.

5. This statement selects a new base point a_i+1, which is not yet in a one-point cell of P and still moved by the stabilizer G_{a₁Ľa_i} of the earlier base points. If certain points of W should are preferred as base point (e.g., because they belong to long cycles of g), a list of points starting with the most wanted ones, can be given as an optional third argument to NextRBasePoint (actually, this is done in the real code for Centralizer).

6. Fixcells( P ) returns the list of points in one-point cells of P (ordered as the cells are ordered in P).

7. For every point p Î fix, if we know the image p ^ g under c Î C_G(e), we also know ( p ^ g ) ^ c = ( p ^ c ) ^ g. We therefore want to isolate these extra points in P.

9. This statement puts point q in a cell of its own, returning in where the number of the cell of P from which q was taken. If q was already the only point in its cell, where = false instead.

12. This command does the necessary bookkeeping for the extra base point q: It prescribes q as next base in the stabilizer chain for G (needed, e.g., in line 5) and returns false if q was already fixed the stabilizer of the earlier base points (and true otherwise; this is not used here). Another call to ProcessFixpoint like this was implicitly made by the function NextRBasePoint to register the chosen base point. By contrast, the point q was not chosen this way, so ProcessFixpoint must be called explicitly for q.

13. This statement registers the function which will be used during the backtrack search to perform the corresponding refinements on the ``image partition'' S_i (to yield the refined S_i+1). After choosing an image b_i+1 for the base point a_i+1, GAP will compute S_i Ů({b_i+1},W-{b_i+1}) and store this partition in \I.partition, where \I is a black box similar to \R, but corresponding to the current ``image partition'' (hence it is an ``R-image'' in analogy to the R-base). Then GAP will call the function Refinements.Centralizer( \R, \I, P.cellno[ p ], p, where ), with the then current values of \R and \I, but where P.cellno[ p ], p, where still have the values they have at the time of this AddRefinement command. This function call will further refine \I.partition to yield S_i+1 as it is programmed in the function Refinements.Centralizer, which is described below. (The global variable Refinements is a record which contains all refinement functions for all backtracking procedures.)

14--18. If the cell from which q was taken out had only two points, we now have an additional one-point cell. This condition is checked in line 13 and if it is true, this extra fixpoint p is taken (line 15), processed like q before (line 16) and is then (line 17) passed to another refinement function Refinements.ProcessFixpoint( \R, \I, p, where ), which is also described below.

22--27. This command starts the backtrack search. Its result will be the centralizer as a subgroup of G. Its arguments are

22.: the group we want to run through,
23.: the property we want to test, as a GAP function,
24.: false if we are looking for a subgroup, true in the case of a representative search (when the result would be one representative),
25.: the R-base,
26.: a list of data, to be stored in \I.data, which has in position 1 the first member S₀ of the decreasing sequence of ``image partitions'' mentioned in Backtrack searching in the reference manual. In the centralizer example, position 2 contains the element that is to be centralized. In the case of a representative search, i.e., a conjugacy test g ^ c = h, we would have h instead of g here, and possibly a S₀ different from P₀ (e.g., a partition into unions of h=cycles of same length).
27.: two subgroups L Ł C_G(g) and R Ł C_G(h) known in advance (we have L = R in the centralizer case).

Refinement functions for the backtrack search. The last subsection showed how the refinement process leading from P_i to P_i+1 is coded in the function \R.nextLevel, this has to be executed once the base point a_i+1. The analogous refinement step from S_i to S_i+1 must be performed for each choice of an image b_i+1 for a_i+1, and it will depend on the corresponding value of S_iŮ({b_i+1}, W-{b_i+1}). But before we can continue our centralizer example, we must, for the interested reader, document the record components of the other black box \I, as we did above for the R-base black box \R. Most of the components change as GAP walks up and down the levels of the search tree.

data: this will be mentioned below
depth: the level i in the search tree of the current node S_i
bimg: a list of images of the points in \R.base
partition: the partition S_i of the current node
level: the stabilizer chain \R.lev[ i ] at the current level
perm: a permutation mapping Fixcells( P_i ) to Fixcells( S_i ) (this implies mapping (a₁,Ľ,a_i) to (b₁,Ľ,b_i))
level2, perm2: a similar construction for the second stabilizer chain, false otherwise (and true if \R.level2 = true)

As declared in the above code for Centralizer, the refinement is performed by the function Refinement.Centralizer( \R, \I, P.cellno[ p ], p, where ). The functions in the record Refinement always take two additional arguments before the ones specified in the AddRefinement call (in line 13 above), namely the R-base \R and the current value \I of the ``R-image''. In our example, p is a fixpoint of P = P_i Ů({a_i+1}, W-{a_i+1}) such that where = P.cellno[ p ^ g ]. The Refinement functions must return false if the refinement is unsuccessful (e.g., because it leads to S_i+1 having different cell sizes from P_i+1) and true otherwise. Our particular function looks like this.

Refinements.Centralizer := function( \R, \I, cellno, p, where )
local S, q;
S := \I.partition;
q := FixpointCellNo( S, cellno ) ^ \I.data[ 2 ];
return IsolatePoint( S, q ) = where and ProcessFixpoint( \I, p, q );
end;

3. The current value of S_iŮ({b_i+1}, W-{b_i+1}) is always found in \I.partition.

4. The image of the only point in cell number cellno = P_i.cellno[ p ] in S under g = \I.data[ 2 ] is calculated.

5. The function returns true only if the image q has the same cell number in S as p had in P (i.e., where) and if q can be prescribed as an image for p under the coset of the stabilizer G_{a₁Ľa_i+1}.c where c Î G is an (already constructed) element mapping the earlier base points a₁,Ľ,a_i+1 to the already chosen images b₁,Ľ,b_i+1. This latter condition is tested by ProcessFixpoint( \I, p, q ) which, if successful, also does the necessary bookkeeping in \I. In analogy to the remark about line 12 in the program above, the chosen image b_i+1 for the base point a_i+1 has already been processed implicitly by the function PartitionBacktrack, and this processing includes the construction of an element c Î G which maps Fixcells( P_i ) to Fixcells( S_i ) and a_i+1 to b_i+1. By contrast, the extra fixpoints p and q in P_i+1 and S_i+1 were not chosen automatically, so they require an explicit call of ProcessFixpoint, which replaces the element c by some c˘.c (with c˘ Î G_{a₁Ľa_i+1}) which in addition maps p to q, or returns false if this is impossible.

You should now be able to guess what Refinements.ProcessFixpoint( \R, \I, p, where ) does: it simply returns ProcessFixpoint( \I, p, FixpointCellNo( \I.partition, where ) ).

Summary. When you write your own backtrack functions using the partition technique, you have to supply an R-base, including a component nextLevel, and the functions in the Refinements record which you need. Then you can start the backtrack by passing the R-base and the additional data (for the data component of the ``R-image'') to PartitionBacktrack.

Functions for meeting ordered partitions. A kind of refinement that occurs in particular in the normalizer calculation involves computing the meet of P (cf. lines 6ff. above) with an arbitrary other partition L, not just with one point. To do this efficiently, GAP uses the following two functions.

StratMeetPartition( \R, P, L [, g ] )

MeetPartitionStrat( \R, \I, L˘ [, g˘ ], strat )

Such a StratMeetPartition command would typically appear in the function call \R.nextLevel( P, \R ) (during the refinement of P_i to P_i+1). This command replaces P by PŮL (thereby changing the second argument) and retuns a ``meet strategy'' strat. This is (for us) a black box which serves two purposes: First, it allows GAP to calculate faster the corresponding meet SŮL˘, which must then appear in a Refinements function (during the refinement of S_i to S_i+1). It is faster to compute SŮL˘ with the ``meet strategy'' of PŮL because if the refinement of S is successful at all, the intersection of a cell from the left hand side of the Ů sign with a cell from the right hand side must have the same size in both cases (and strat records these sizes, so that only non-empty intersections must be calculated for SŮL˘). Second, if there is a discrepancy between the behaviour prescribed by strat and the behaviour observed when refining S, the refinement can immediately be abandoned.

On the other hand, if you only want to meet a partition P with L for a one-time use, without recording a strategy, you can simply type StratMeetPartition( P, L ) as in the following example, which also demonstrates some other partition-related commands.

    gap> P := Partition( [[1,2],[3,4,5],[6]] );;  Cells( P );
    [ [ 1, 2 ], [ 3, 4, 5 ], [ 6 ] ]
    gap> Q := OnPartitions( P, (1,3,6) );;  Cells( Q );
    [ [ 3, 2 ], [ 6, 4, 5 ], [ 1 ] ]
    gap> StratMeetPartition( P, Q );
    [  ]  # the ``meet strategy'' was not recorded, ignore this result
    gap> Cells( P );
    [ [ 1 ], [ 5, 4 ], [ 6 ], [ 2 ], [ 3 ] ]

You can even say StratMeetPartition( P, D ) where D is simple a subset of W, it will then be interpreted as the partition (D,W-D).

GAP makes use of the advantages of a ``meet strategy'' if the refinement function in Refinements contains a MeetPartitionStrat command where strat is the ``meet strategy'' calculated by StratMeetPartition before. Such a command replaces \I.partition by its meet with L˘, again changing the argument \I. The necessary reversal of these changes when backtracking from a node (and prescribing the next possible image for a base point) is automatically done by the function PartitionBacktrack.

In all cases, an additional argument g means that the meet is to be taken not with L, but instead with L.g^-1, where operation on ordered partitions is meant cellwise (and setwise on each cell). (Analogously for the primed arguments.)

    gap> P := Partition( [[1,2],[3,4,5],[6]] );;
    gap> StratMeetPartition( P, P, (1,6,3) );;  Cells( P );
    [ [ 1 ], [ 5, 4 ], [ 6 ], [ 2 ], [ 3 ] ]  # |$P.(1,3,6) = Q|$

Avoiding multiplication of permutations. In the description of the last subsections, the backtrack algorithm constructs an element c Î G mapping the base points to the prescribed images and finally tests the property in question for that element. During the construction, c is obtained as a product of transversal elements from the stabilizer chain for G, and so multiplications of permutations are required for every c submitted to the test, even if the test fails (i.e., in our centralizer example, if g ^ c <> g). Even if the construction of c stops before images for all base points have been chosen, because a refinement was unsuccessful, several multiplications will already have been performed by (explicit or implicit) calls of ProcessFixpoint, and, actually, the general backtrack procedure implemented in GAP avoids this.

For this purpose, GAP does not actually multiply the permutations but rather stores all the factors of the product in a list. Specifically, instead of carrying out the multiplication in c® c˘.c mentioned in the comment to line 5 of the above program --- where c˘ Î G_{a₁Ľa_i+1} is a product of factorized inverse transversal elements, see Stabilizer chain records --- GAP appends the list of these factorized inverse transversal elements (giving c˘) to the list of factors already collected for c. Here c˘ is multiplied from the left and is itself a product of inverses of strong generators of G, but GAP simply spares itself all the work of inverting permutations and stores only a ``list of inverses'', whose product is then (c˘.c)^-1 (which is the new value of c^-1). The ``list of inverses'' is extended this way whenever ProcessFixpoint is called to improve c.

The product has to be multiplied out only when the property is finally tested for the element c. But it is often possible to delay the multiplication even further, namely until after the test, so that no multiplication is required in the case of an unsuccessful test. Then the test itself must be carried out with the factorized version of the element c. For this purpose, PartitionBacktrack can be passed its second argument (the property in question) in a different way, not as a single GAP function, but as a list like in lines 2--4 of the following alternative excerpt from the code for Centralizer.

return PartitionBacktrack( G,
[ g, g,
OnPoints,
c -> c!.lftObj = c!.rgtObj ],
false, \R, [ P₀, g ], L, R );

The test for c to have the property in question is of the form opr( left, c ) = right where opr is an operation function as explained in External sets in the reference manual. In other words, c passes the test if and only if it maps a ``left object'' to a ``right object'' under a certain operation. In the centralizer example, we have opr = OnPoints and left = right = g, but in a conjugacy test, we would have right = h.

2. Two first two entries (here g and g) are the values of left and right.

3. The third entry (here OnPoints) is the operation opr.

4. The fourth entry is the test to be performed upon the mapped left object left and preimage of the right object opr( right, c^-1 ). Here GAP operates with the inverse of c because this is the product of the permutations stored in the ``list of inverses''. The preimage of right under c is then calculated by mapping right with the factors of c^-1 one by one, without the need to multiply these factors. This mapping of right is automatically done by the ProcessFixpoint function whenever c is extended, the current value of right is always stored in c!.rgtObj. When the test given by the fourth entry is finally performed, the element c has two components c!.lftObj = left and c!.rgtObj = opr( right, c^-1 ), which must be used to express the desired relation as a function of c. In our centralizer example, we simply have to test whether they are equal.

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GAP 4 manual
February 2000