Foundations of Constraint Processing, Fall 2005 September 11, 2005Backtracking1 Foundations of...

September 11, 2005 Backtracking 1

Foundations of Constraint Processing, Fall 2005

Foundations of Constraint Processing

CSCE421/821, Fall 2005: www.cse.unl.edu/~choueiry/F05-421-821/

Berthe Y. Choueiry (Shu-we-ri)

Avery Hall, Room 123B

[email protected]

Tel: +1(402)472-5444

Intelligent Backtracking Algorithms



Reading• Required reading

Hybrid Algorithms for the Constraint Satisfaction Problem [Prosser, CI 93]

• Recommended reading– Chapters 5 and 6 of Dechter’s book– Tsang, Chapter 5

• Notes available upon demand– Notes of Fahiem Bacchus: Chapter 2, Section 2.4– Handout 4 and 5 of Pandu Nayak (Stanford)



Outline

• Review of terminology of search

• Hybrid backtracking algorithms

• Looking ahead

• Evaluation of (deterministic) BT search algorithms



Backtrack search (BT)

Var 1 v1 v2

S

• Variable/value ordering

• Variable instantiation

• (Current) path

• Current variable

• Past variables

• Future variables

• Shallow/deep levels /nodes

• Search space / search tree

• Back-checking

• Backtracking



Outline

• Review of terminology of search • Hybrid backtracking algorithms

– Vanilla: BT– Improving back steps: {BJ, CBJ} – Improving forward step: {BM, FC}

• Looking ahead• Evaluation of (deterministic) BT search

algorithms



Two main mechanisms in BT

1. Backtracking: • To recover from dead-ends • To go back

2. Consistency checking: • To expand consistent paths• To move forward



Backtracking

To recover from dead-ends

1. Chronological (BT)

2. Intelligent• Backjumping (BJ)• Conflict directed backjumping (CBJ)• With learning algorithms (Dechter Chapt 6.4)• Etc.



Consistency checking

To expand consistent paths1. Back-checking: against past variables

• Backmarking (BM)

2. Look-ahead: against future variables• Forward checking (FC) (partial look-ahead)• Directional Arc-Consistency (DAC) (partial

look-ahead)• Maintaining Arc-Consistency (MAC) (full

look-ahead)



Hybrid algorithms

Backtracking + checking = new hybrids

BT BJ CBJ

BM BMJ BM-CBJ

FC FC-BJ FC-CBJ

Evaluation:

• Empirical: Prosser 93. 450 instances of Zebra

• Theoretical: Kondrak & Van Beek 95



Notations (in Prosser’s paper)

• Variables: Vi, i in [1, n]• Domain: Di = {vi1, vi2, …,viMi}• Constraint between Vi and Vj: Ci,j

• Constraint graph: G• Arcs of G: Arc(G)• Instantiation order (static or dynamic)• Language primitives: list, push, pushnew,

remove, set-difference, union, max-list



Main data structures• v: a (1xn) array to store assignments

– v[i] gives the value assigned to ith variable – v[0]: pseudo variable (root of tree), backtracking to

v[0] indicates insolvability

• domain[i]: a (1xn) array to store the original domains of variables

• current-domain[i]: a (1xn) array to store the current domains of variables– Upon backtracking, current-domain[i] of future

variables must be refreshed

• check(i,j): a function that checks whether the values assigned to v[i] and v[j] are consistent



Generic search: bcssp1. Procedure bcssp (n, status)2. Begin3. consistent true4. status unknown5. i 16. While status = unknown

7. Do Begin8. If consistent

9. Then i label (i, consistent)10. Else i unlabel (i, consistent)

11. If i > n12. Then status “solution”13. Else If i=0 then status “impossible”14. End

15. End

•Forward move: x-label

•Backward move: x-unlabel

•Parameters: current variable, Boolean

•Return: new current variable



Chronological backtracking (BT)

• Uses bt-label and bt-unlabel• When v[i] is assigned a value from current-domain[i], we

perform back-checking against past variables (check(i,k))

• If back-checking succeeds, bt-label returns i+1• If back-checking fails, we remove the assigned value

from current-domain[i], assign the next value in current-domain[i], etc.

• If no other value exists, v[i-1] is un-instantiated and we seek a new value for it… (notation: in general v[h])

• For all future variables j: current-domain[j] = domain[j]• For all past variables g: current-domain[g] domain[g]



BT-label1. Function bt-label(i,consistent): INTEGER2. BEGIN3. consistent false4. For v[i] each element of current-domain[i] while not consistent

5. Do Begin6. consistent true7. For h 1 to (i-1) While consistent8. Do consistent check(i,h)9. If not consistent10. Then current-domain[i] remove(v[i], current-domain[i])11. End

12. If consistent then return(i+1) ELSE return(i)13. END

Terminates:

• consistent=true, return i+1

• consistent=false, current-domain[i]=nil, returns i



BT-unlabel1. FUNCTION bt-unlabel(i,consistent):INTEGER

2. BEGIN

3. h i -1

4. current-domain[i] domain[i]

5. current-domain[h] remove(v[h],current-domain[h])

6. consistent current-domain[h] nil

7. return(h)

8. END • Is called when consistent=false and current-domain[i]=nil

• Selects vh to backtrack to

• Uninstantiates all variables between vh and vi

• Removes v[h] from current-domain [h]

• Sets consistent to true if current-domain[h] 0

• Returns h



Example: BT (the dumbest example ever)

{1,2,3,4,5}

{1,2,3,4,5}

{1,2,3,4,5}

{1,2,3,4,5}

{1,2,3,4,5}

V2

V1

V3

V4

V5

CV3,V4={(V3=1,V4=3)}

CV2,V5={(V2=5,V5=1),(V2=5,V5=4)}

-

v[1]

v[2]

v[3]

v[4]

v[5]

v[0]

1

1

1

1

21 3 4

2 3 4 5

etc…



Outline


– Vanilla: BT– Improving back steps: BJ, CBJ – Improving forward step: BM, FC


algorithms



Danger of BT: thrashing• BT assumes that the instantiation of v[i]

was prevented by a bad choice at (i-1). • It tries to change the assignment of v[i-1]• When this assumption is wrong, we suffer

from thrashing (exploring ‘barren’ parts of solution space)

• Backjumping (BT) tries to avoid that– Jumps to the reason of failure – Then proceeds as BT



Backjumping (BJ)• Tries to reduce thrashing by saving some

backtracking effort• When v[i] is instantiated, BJ remembers

v[h], the deepest node of past variables that v[i] has checked against.

• Uses: max-check[i], global, initialized to 0• At level i, when check(i,h) succeeds

max-check[i] max(max-check[i], h)

• If current-domain[h] is getting empty, simple chronological backtracking is performed from h– BJ jumps then steps!

12

3

0

23

1

i

h-1h-1

h

h

h-2

0

0

0

Current variablePast variable



BJ: label/unlabel• bj-label: same as bt-label, but updates max-

check[i]• bj-unlabel, same as bt-unlabel but

– Backtracks to h = max-check[i]– Resets max-check[j] 0 for j in [h+1,i]

Important: max-check is the deepest level we checked against, could have been success or could have been failure

12

3

0

23

1

i

h-1h-1

h

h

h-2

0

0

0



Example: BJ

2

{1,2,3,4,5}

{1,2,3,4,5}

{1,2,3,4,5}

{1,2,3,4,5}

{1,2,3,4,5}

V2

V1

V3

V4

V5

CV2,V4={(V2=1,V4=3)}

CV1,V5={(V1=1,V5=2)}

CV2,V5={(V2=5,V5=1)}

-

v[1]

v[2]

v[3]

v[4]

v[5]

v[0] = 0

1

1

1

1

21 3 4

2 3 4 5

Max-check[1] = 0

Max-check[2] = 1

max-check[4] = 3

max-check[5] = 2

V4=1, fails for V2, mc=2V4=2, fails for V2, mc=2 V4=3, succeeds

V5=1, fails for V1, mc=1V5=2, fails for V2, mc=2V5=3, fails for V1V5=4, fails for V1V5=5, fails for V1



Conflict-directed backjumping (CBJ)

• Backjumping– jumps from v[i] to v[h], – but then, it steps back from v[h] to v[h-1]

• CBJ improves on BJ– Jumps from v[i] to v[h]– And jumps back again, across conflicts

involving both v[i] and v[h]– To maintain completeness, we jump back to

the level of deepest conflict



CBJ: data structure

• Maintains a conflict set: conf-set• conf-set[i] are first initialized to {0}• At any point, conf-set[i] is a subset of

past variables that are in conflict with i

{0}

{0}

{0}

{0}{0}{0}

conf-set[g]

conf-set[h]

conf-set[i]

01

2

g

h-1

h

i

conf-set



CBJ: conflict-set

{x}

{3}

{1, g, h}

{0}{0}{0}

conf-set[g]

conf-set[h]

conf-set[i]

12

3

g

h-1h

Current variable i

Pas

t var

iabl

es

{3,1, g}

{x, 3,1}

• When a check(i,h) failsconf-set[i] conf-set[i] {h}

• When current-domain[i] empty

1. Jumps to deepest past variable

h in conf-set[i]

2. Updates

conf-set[h] conf-set[h] (conf-set[i] \{h})

• Primitive form of learning (while searching)



Example CBJ{1,2,3,4,5}

{1,2,3,4,5}

{1,2,3,4,5}

{1,2,3,4,5}

{1,2,3,4,5}

V2

V1

V3

V4

V5

{(V1=1, V6=3)}

-

v[1]

v[2]

v[3]

v[4]

v[6]

v[0] = 0

1

1

1

1

21 3

2 3 4 5

conf-set[1] = {0}

conf-set[2] = {0}

conf-set[3] = {0}

{(V4=5, V6=3)}

{(V2=1, V4=3), (V2=4, V4=5)}

conf-set[6] = {1}

{1,2,3,4,5}V6

{(V1=1, V5=3)}

conf-set[4] = {2}

v[5] 21 3

conf-set[6] = {1}conf-set[6] = {1,4}

conf-set[6] = {1,4}conf-set[6] = {1,4}

conf-set[4] = {1, 2}

conf-set[5] = {1}



Backtracking: summary• Chronological backtracking

– Steps back to previous level– No extra data structures required

• Backjumping– Jumps to deepest checked-against variable, then

steps back– Uses array of integers: max-check[i]

• Conflict-directed backjumping– Jumps across deepest conflicting variables– Uses array of sets: conf-set[i]



Outline


– Vanilla: BT– Improving back steps: BJ, CBJ – Improving forward step: BM, FC


algorithms



Backmarking: goal

• Tries to reduce amount of consistency checking

• Situation:– v[i] about to be re-assigned k– v[i] k was checked against v[h]g

– v[h] has not been modified

v[h] = g

v[i] kk



BM: motivation• Two situations

1. Either (v[i]=k,v[h]=g) has failed it will fail again2. Or, (v[i]=k,v[h]=g) was founded consistent it will remain consistent

v[h] = g

v[i] kk

v[h] = g

v[i] kk

• In either case, back-checking effort against v[h] can be saved!



Data structures for BM: 2 arrays

0 0 0 0 0 0 0 0 0

0

0

0

0

Num

ber

of v

aria

bles

n

max domain size m

Num

ber

of v

aria

bles

n

• maximum checking level: mcl (n x m)• Minimum backup level: mbl (n x 1)



Maximum checking level

0 0 0 0 0 0 0 0 0

00

0

0

Num

ber

of v

aria

bles

n

max domain size m

• mcl[i,k] stores the deepest variable that v[i]k checked against

• mcl[i,k] is a finer version of max-check[i]



Minimum backup level

Num

ber

of v

aria

bles

n

• mbl[i] gives the shallowest past variable whose value has changed since v[i] was the current variable

• BM (and all its hybrid) do not allow dynamic variable ordering



When mcl[i,k]=mbl[i]=j

v[i] kk

v[j]

mbl[i] = j

BM is aware that• The deepest variable that (v[i] k)

checked against is v[j]• Values of variables in the past of

v[j] (h<j) have not changed

So• We do need to check (v[i] k)

against the values of the variables between v[j] and v[i]

• We do not need to check (v[i] k) against the values of the variables in the past of v[j]



Type a savings

v[h]

v[i] kk

v[j]

mcl[i,k]=h mcl[i,k] < mbl[i]=j

When mcl[i,k] < mbl[i], do not check v[i] k because it will fail



Type b savings

h

v[i] kk

v[j]

v[g]

mcl[i,k]=g

mbl[i] = j

mcl[i,k]mbl[i]

When mcl[i,k] mbl[i], do not check (i,h<j) because they will succeed



Hybrids of BM

• mcl can be used to allow backjumping in BJ

• Mixing BJ & BM yields BMJ– avoids redundant consistency checking (types

a+b savings) and – reduces the number of nodes visited during

search (by jumping)

• Mixing BM & CBJ yields BM-CBJ



Problem of BM and its hybrids: warning

v[m]v[m]

v[g]

v[h]

v[i]

v[g]

v[h]

v[i]

v[m]

v[g]

v[h]

v[i]

v[h]

v[f]

• Backjumping from v[i]:– v[i] backjumps up to v[g]

• Backmarking of v[h]:– When reconsidering v[h], v[h] will

be checked against all f in [m,g( – effort could be saved

• Phenomenon will worsen with CBJ

• Problem fixed by Kondrak & van Beek 95

BMJ enjoys only some of the advantages of BM

Assume: mbl[h] = m and max-check[i]=max(mcl[i,x])=g



Forward checking (FC)• Looking ahead: from current variable, consider all future

variables and clear from their domains the values that are not consistent with current partial solution

• FC makes more work at every instantiation, but will expand fewer nodes

• When FC moves forward, the values in current-domain of future variables are all compatible with past assignment, thus saving backchecking

• FC may “wipe out” the domain of a future variable (aka, domain annihilation) and thus discover conflicts early on. FC then backtracks chronologically

• Goal of FC is to fail early (avoid expanding fruitless subtrees)



FC: data structures

v[i]

v[k]

v[l]

v[n]

v[m]

v[j]

• When v[i] is instantiated, current-domain[j] are filtered for all j connected to i and i<j<n

• reduction[j] store sets of values remove from current-domain[j] by some variable before v[j]

reductions[j] = {{a, b}, {c, d, e}, {f, g, h}}

• future-fc[i]: subset of the future variables that v[i] checks against (redundant)

future-fc[i] = {k, j, n}

• past-fc[i]: past variables that checked against v[j]

• All these sets are treated like stacks



Forward Checking: functions

• check-forward

• undo-reductions

• update-current-domain

• fc-label

• fc-unlabel



FC: functions• check-forward(i,j) is called when instantiating v[i]

– It performs REVISE(j,i)– Returns false if current-domain[j] is empty, true

otherwise– Values removed from current-domain[j] are pushed,

as a set, into reductions[j]

• These values will be popped back if we have to backtrack over v[i] (undo-reductions)



FC: functions• update-current-domain

– current-domain[i] domain[i] \ reductions[i] – actually, we have to iterate over reductions=set of

sets

• fc-label– Attempts to instantiate current-variable– Then filters domains of all future variables (push into

reductions)– Whenever current-domain of a future variable is

wiped-out: • v[i] is un-instantiated and • domain filtering is undone (pop reductions)



Hybrids of FC• FC suffers from thrashing: it is based on BT• FC-BJ:

– max-check is integrated in fc-bj-label and fc-bj-unlabel– Enjoys advantages of FC and BJ… but suffers

malady of BJ (jump the step)

• FC-CBJ: – Best algorithm for far (assuming static variable

ordering)– fc-cbj-label and fc-cbj-unlabel



Consistency checking: summary

• Chronological backtracking– Uses back-checking– No extra data structures

• Backmarking– Uses mcl and mbl– Two types of consistency-checking savings

• Forward-checking– Works more at every instantiation, but expands fewer

subtrees– Uses: reductions[i], future-fc[i], past-fc[i]



Experiments• Empirical evaluations on Zebra

– Representative of design/scheduling problems– 25 variables, 122 binary constraints– Permutation of variable ordering yields new search

spaces– Variable ordering: different bandwidth/induced width

of graph

• 450 problem instances were generated• Each algorithm was applied to each instance

Experiments were carried out under static variable ordering



Analysis of experiments

Algorithms compared with respect to:

1. Number of consistency checks (average)FC-CBJ < FC-BJ < BM-CBJ < FC < CBJ < BMJ < BM < BJ < BT

2. Number of nodes visited (average)FC-CBJ < FC-BJ < FC < BM-CBJ < BMJ =BJ < BM = BT

3. CPU time (average)FC-CBJ < FC-BJ < FC < BM-CBJ < CBJ < BMJ < BJ < BT < BM

FC-CBJ apparently the champion



Additional developments• Other backtracking algorithms exist:

– Graph-based backjumping (GBJ), etc. [Dechter]

– Pseudo-trees [Freuder 85]

• Other look-ahead techniques exist:– DAC, MAC, etc.

• More empirical evaluations: – over randomly generated problems

• Theoretical evaluations: – Based on approach of Kondrak & Van Beek IJCAI’95



Outline

• Review of terminology of search • Hybrid backtracking algorithms• Looking ahead

– Forward checking (FC)– Directional Arc Consistency (DAC)– Maintaining Arc Consistency (a.k.a. full arc-

consistency)

• Evaluation of (deterministic) BT search algorithms



Looking ahead• Rationale:

– As decisions are made (conditioning), • Revise the domain of future variables to propagate the

effects of decisions • I.e., eliminate inconsistent choices in future sub-problem

– Domain annihilation of a future variable avoids expansion of useless portions of the tree

• Techniques– Partial: forward-checking (FC), directional arc-

consistency (DAC)– Full: Maintaining arc-consistency (MAC)– Use: Revise(Vf, Vc), Vf future variable, Vc current

variable



Revising the domain of Vi

• Revising the domain of Vi given a constraint C on Vi (i.e., Vi Scope(C))

• General notation: Revise(Vi,C)

• In a binary CSP:

Revise(Vi,C)=

Revise(Vi,CVi,Vj)=

Revise(Vi, Vj)



Revise(Vi, Vj)

• Revise(Vi, Vj)

1. x DVi

2. y DVj

3. If y DVj with (x,y) C return()

4. ElseIf DVi DVi \ {x}

NOTE: only DVi may be updated



Domain filtering in lookahead

• Vc current variable

• Vf future variable

• {Vf} all future variables

• Revise(Vf, Vc)

• FC(Vc):

1. Vf {Vf} connected to Vc

2. Revise(Vf,Vc)

3. If DVf ={} then return(failure)



• FC:

• DAC:– assumes a fixed variable ordering d

• MAC:

– does more pruning at the cost of more consistency checks

Look-ahead techniques: FC, DAC, MAC

FC(Vc); Vf {Vf}, FC(Vf) in the ordering d

FC(Vc); AC({Vf})

FC(Vc)

FC(Vc);

Repeat until quiescence or failure

Vf1,Vf2 {Vf}, Revise(Vf1,Vf2)



Terminology overload alert: FC

• FC is used to denote any of the following:– a partial look-ahead schema – a specific chronological backtrack search

algorithm that uses the partial look-ahead schema

• Meaning is inferred from context• Not a healthy situation, but a reality• Advice: state upfront the meaning of your terms

and stick to them throughout your paper



Outline

• Review of terminology of search • Hybrid backtracking algorithms• Looking ahead• Evaluation of (deterministic) BT search

algorithms [Dechter, 6.6.2]

– CSP parameters– Comparison criteria – Theoretical evaluations– Empirical evaluations



CSP parameters

• Number of variables: n

• Domain size: a, d

• Constraint tightness:

t = |forbidden tuples| / | all tuples |

• Proportion of constraints (a.k.a., constraint density, constraint probability):

p1 = e / emax, e is nbr of constraints



Comparison criteria1. Number of nodes visited (NV)

• Every time you call label

2. Number of constraint check (CC)• Every time you call check(i,j)

3. CPU time• Be as honest and consistent as possible

4. Some specific criterion for assessing the quality of the improvement proposed

Presentation of values:• Average or median of criterion• (qualified) run-time distribution• Solution-quality distribution



Theoretical evaluations

• Comparing NV and/or CC

• Common assumptions: – for finding all solutions

– static orderings



Empirical evaluation: data sets

• Use real-data sets (anecdotal evidence)

• Use benchmarks (csp library)

• Use randomly generated problems



Empirical evaluations: random problems

• Various models exist (use Model B)– Models A, B, C, E, F, etc.

• Vary parameters: <n, a, t, p>– Number of variables: n– Domain size: a, d– Constraint tightness: t = |forbidden tuples| / | all tuples |

– Proportion of constraints (a.k.a., constraint density, constraint probability): p1 = e / emax

• Issues: – Uniformity– Difficulty (phase transition)– Solvability of instances (for incomplete search techniques)



Model B1. Input: n, a, t, p12. Generate n nodes3. Generate a list of n2 tuples of all combinations of 2

nodes4. Choose e elements from above list as constraints to

between the n nodes5. If the graph is not connected, throw away, go back to

step 4, else proceed6. Generate a list of a2 tuples of all combinations of 2

values7. For each constraint, choose randomly a number of

tuples from the list to guarantee tightness t for the constraint



Example: MAC vs. FC

• Reference: [Sabin & Freuder, ECAI94], [Bessière & Régin, CP97], [Sabin & Freuder, CP97], [Gent & Prosser, APES-20-2000], [Experiments by Lin XU, 2001], [Yang, MS thesis 2003]

• Results: (sketchy)Low tightness High tightness

Low density (sparse) FC MAC

High density (dense) FC FC

Note: results depend on

• Variable ordering (static vs. dynamic)

• Problem difficulty (positive relative to crossover point)

Foundations of Constraint Processing, Fall 2005 September 11, 2005Backtracking1 Foundations of...

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