Intelligent Backtracking Algorithms Foundations of Constraint Processing CSCE

Intelligent Backtracking Algorithms Foundations of Constraint Processing CSCE 421/821, Fall 2015: www. cse. unl. edu/~choueiry/F 15 -421 -821 Berthe Y. Choueiry (Shu-we-ri) Avery Hall, Room 360 Foundations of Constraint Processing Intelligent Backtracking Algorithms 1

Reading • Required reading – Hybrid Algorithms for the Constraint Satisfaction Problem [Prosser, CI 93] • Recommended reading – Chapters 5 and 6 of Dechter’s textbook – Tsang, Chapter 5 Foundations of Constraint Processing Intelligent Backtracking Algorithms 2

Outline • Review of terminology of search • Hybrid backtracking algorithms Foundations of Constraint Processing Intelligent Backtracking Algorithms 3

Backtrack search (BT) • Variable/value ordering • Variable instantiation • (Current) path • Current variable • Past variables • Future variables • Shallow/deep levels /nodes • Search space / search tree • Back-checking • Backtracking Foundations of Constraint Processing Intelligent Backtracking Algorithms 4

Outline • Review of terminology of search • Hybrid backtracking algorithms – Vanilla: BT – Improving back steps: {BJ, CBJ} – Improving forward step: {BM, FC} Foundations of Constraint Processing Intelligent Backtracking Algorithms 5

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 Foundations of Constraint Processing Intelligent Backtracking Algorithms 6

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. Foundations of Constraint Processing Intelligent Backtracking Algorithms 7

Consistency checking To expand consistent paths 1. 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) Foundations of Constraint Processing Intelligent Backtracking Algorithms 8

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 Foundations of Constraint Processing Intelligent Backtracking Algorithms 9

Notations (in Prosser’s paper) • • Variables: Vi, i in [1, n] Domain: Di = {vi 1, vi 2, …, vi. Mi} Constraint between Vi and Vj: Ci, j Constraint graph: G Arcs of G: Arc(G) Instantiation order (static or dynamic) Language primitives: list, pushnew, remove, set-difference, union, max-list Foundations of Constraint Processing Intelligent Backtracking Algorithms 10

Main data structures • v: a (1 xn) 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 (1 xn) array to store the original domains of variables • current-domain[i]: a (1 xn) 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 Foundations of Constraint Processing Intelligent Backtracking Algorithms 11

Generic search: bcssp 1. 2. 3. 4. 5. 6. Procedure bcssp (n, status) • Forward move: x-label Begin consistent true • Backward move: x-unlabel status unknown • Parameters: i: current variable, i 1 consistent: Boolean While status = unknown • Return: i: new current variable 7. Do Begin 8. If consistent 9. Then i label (i, consistent) 10. Else i unlabel (i, consistent) 11. If i > n 12. Then status “solution” 13. Else If i=0 then status “impossible” 14. End 15. End Foundations of Constraint Processing Intelligent Backtracking Algorithms 12

Chronological backtracking (BT) • Uses bt-label and bt-unlabel • bt-label: – 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 currentdomain[i], assign the next value in current-domain[i], etc. – If no other value exists, consistent nil (bt-unlabel will be called) • bt-unlabel – – Current level is set to i-1 (notation for current variable: v[h]) For all future variables j: current-domain[j] If domain[h] is not empty, consistent true (bt-label will be called) Note: for all past variables g, current-domain[g] Foundations of Constraint Processing Intelligent Backtracking Algorithms 13

BT-label 1. 2. 3. 4. Function bt-label(i, consistent): INTEGER BEGIN consistent false For v[i] each element of current-domain[i] while not consistent 5. Do Begin Terminates: 6. consistent true • consistent=true, return i+1 7. For h 1 to (i-1) While consistent • consistent=false, current 8. Do consistent check(i, h) domain[i]=nil, returns i 9. If not consistent 10. Then current-domain[i] remove(v[i], current-domain[i]) 11. End 12. If consistent then return(i+1) ELSE return(i) 13. END Foundations of Constraint Processing Intelligent Backtracking Algorithms 14

BT-unlabel 1. 2. FUNCTION bt-unlabel(i, consistent): INTEGER BEGIN 3. h i -1 4. current-domain[i] 5. current-domain[h] remove(v[h], current-domain[h]) 6. consistent current-domain[h] nil 7. return(h) • Is called when consistent=false and current-domain[i]=nil 8. END • Selects vh to backtrack to • (Uninstantiates all variables between vh and vi) • Uninstantiates v[h]: removes v[h] from current-domain [h]: • Sets consistent to true if current-domain[h] 0 • Returns h Foundations of Constraint Processing Intelligent Backtracking Algorithms 15

Example: BT (the dumbest example ever) {1, 2, 3, 4, 5} V 1 V 2 V 3 v[0] - v[1] 1 v[2] 1 v[3] 1 CV 3, V 4={(V 3=1, V 4=3)} {1, 2, 3, 4, 5} V 4 v[4] 1 2 3 4 etc… 4 5 CV 2, V 5={(V 2=5, V 5=1), (V 2=5, V 5=4)} {1, 2, 3, 4, 5} V 5 v[5] 1 2 3 Foundations of Constraint Processing Intelligent Backtracking Algorithms 16

Outline • Review of terminology of search • Hybrid backtracking algorithms – Vanilla: BT – Improving back steps: BJ, CBJ – Improving forward step: BM, FC Foundations of Constraint Processing Intelligent Backtracking Algorithms 17

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 Foundations of Constraint Processing Intelligent Backtracking Algorithms 18

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 1 2 3 0 1 2 3 h-1 h h-2 i h 0 0 h-1 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! Past variable Current variable 0 Foundations of Constraint Processing Intelligent Backtracking Algorithms 19

BJ: label/unlabel • bj-label: same as bt-label, but updates 1 2 max-check[i] 3 • 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 0 1 2 3 h-1 h h-2 i h 0 0 h-1 0 Foundations of Constraint Processing Intelligent Backtracking Algorithms 20

Example: BJ v[0] = 0 {1, 2, 3, 4, 5} V 1 V 2 - v[1] 1 v[2] 1 v[3] 1 Max-check[1] = 0 2 Max-check[2] = 1 CV 2, V 5={(V 2=5, V 5=1)} {1, 2, 3, 4, 5} V 3 CV 2, V 4={(V 2=1, V 4=3)} {1, 2, 3, 4, 5} V 4=1, fails for V 2, mc=2 V 4=2, fails for V 2, mc=2 V 4=3, succeeds max-check[4] = 3 v[4] V 4 V 5=1, fails for V 1, mc=1 V 5=2, fails for V 2, mc=2 V 5=3, fails for V 1 V 5=4, fails for V 1 v[5] V 5=5, fails for V 1 max-check[5] = 2 CV 1, V 5={(V 1=1, V 5=2)} {1, 2, 3, 4, 5} V 5 1 1 2 2 3 3 4 4 5 Foundations of Constraint Processing Intelligent Backtracking Algorithms 21

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 Foundations of Constraint Processing Backtracking Intelligent Backtracking Algorithms 22

CBJ: data structure 0 1 conf-set 2 • Maintains a conflict set: conf-set g • conf-set[i] are first initialized to {0} h-1 • At any point, conf-set[i] is a subset of h past variables that are in conflict with i i conf-set[g] {0} conf-set[h] {0} conf-set[i] {0} {0} Foundations of Constraint Processing Intelligent Backtracking Algorithms 23

CBJ: conflict-set 1 2 3 conf-set[i] {h} • When current-domain[i] empty 1. Jumps to deepest past variable h in conf-set[i] Past variables • When a check(i, h) fails g conf-set[g] {x} {x, 3, 1} h-1 h conf-set[h] {3} {3, 1, g} Current variable i conf-set[i] {1, g, h} 2. Updates conf-set[h] (conf-set[i] {h}) • Primitive form of learning (while searching) {0} {0} Foundations of Constraint Processing Intelligent Backtracking Algorithms 24

Example CBJ V 1 V 2 V 3 v[0] = 0 {1, 2, 3, 4, 5} {(V 2=1, V 4=3), (V 2=4, V 4=5)} V 4 V 5 {1, 2, 3, 4, 5} {(V 4=5, V 6=3)} V 6 {1, 2, 3, 4, 5} - v[1] 1 conf-set[1] = {0} v[2] 1 conf-set[2] = {0} v[3] 1 conf-set[3] = {0} v[4] 1 2 conf-set[4] = {2} {(V 1=1, V 5=3)} v[5] 1 conf-set[5] = {1} v[6] 1 2 2 3 conf-set[4] = {1, 2} 3 3 4 5 conf-set[6] = {1} conf-set[6] = {1, 4} {(V 1=1, V 6=3)} conf-set[6] = {1, 4} Foundations of Constraint Processing Intelligent Backtracking Algorithms 25

CBJ for finding all solutions • After finding a solution, if we jump from this last variable, then we may miss some solutions and lose completeness • Two solutions, proposed by Chris Thiel (S 08) 1. Using conflict sets 2. Using cbf of Kondrak, a clear pseudo-code • Rationale by Rahul Purandare (S 08) – – We cannot skip any variable without chronologically backtracking to it at least once In fact, exactly once Foundations of Constraint Processing Intelligent Backtracking Algorithms 26

CBJ/All solutions without cbf • When a solution is found, force the last variable, N, to conflict with everything before it – conf-set[N] {1, 2, . . . , N-1}. • This operation, in turn, forces some chronological backtracking as the conf-sets are propagated backward Foundations of Constraint Processing Intelligent Backtracking Algorithms 27

CBJ/All solutions with cbf • Kondrak proposed to fix the problem using cbf (flag), a 1 xn vector – i, cbf[i] 0 – When you find a solution, i, cbf[i] 1 • In unlabel – if (cbf[i]=1) • Then h i-1; cbf[i] 0 • Else h max-list (conf-set[i]) Foundations of Constraint Processing Intelligent Backtracking Algorithms 28

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] Foundations of Constraint Processing Intelligent Backtracking Algorithms 29

Outline • Review of terminology of search • Hybrid backtracking algorithms – Vanilla: BT – Improving back steps: BJ, CBJ – Improving forward step: BM, FC Foundations of Constraint Processing Intelligent Backtracking Algorithms 30

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] = g – v[h] has not been modified v[i] k k Foundations of Constraint Processing Intelligent Backtracking Algorithms 31

BM: motivation • Two situations 1. Either (v[i]=k, v[h]=g) has failed it will fail again 2. Or, (v[i]=k, v[h]=g) was founded consistent it will remain consistent v[h] = g v[i] • v[h] = g k k v[i] k k In either case, back-checking effort against v[h] can be saved! Foundations of Constraint Processing Intelligent Backtracking Algorithms 32

Data structures for BM: 2 arrays • • maximum checking level: mcl (n x m) Minimum backup level: mbl (n x 1) 0 0 0 0 Number of variables n max domain size m Foundations of Constraint Processing 33

Maximum checking level • • mcl[i, k] stores the deepest variable that v[i] k checked against mcl[i, k] is a finer version of max-check[i] Number of variables n max domain size m 0 0 0 0 Foundations of Constraint Processing 34

Minimum backup level 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 Number of variables n • Foundations of Constraint Processing 35

When mcl[i, k]=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] v[i] k k mbl[i] = j Foundations of Constraint Processing Intelligent Backtracking Algorithms 36

Type a savings When mcl[i, k] < mbl[i], do not check v[i] k because it will fail v[h] v[j] v[i] k k mcl[i, k]=h mcl[i, k] < mbl[i]=j Foundations of Constraint Processing Intelligent Backtracking Algorithms 37

Type b savings When mcl[i, k] mbl[i], do not check (i, h<j) because they will succeed h v[j] v[g] v[i] k mcl[i, k]=g k mcl[i, k] mbl[i] = j Foundations of Constraint Processing Intelligent Backtracking Algorithms 38

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 Foundations of Constraint Processing Intelligent Backtracking Algorithms 39

Problem of BM and its hybrids: warning BMJ enjoys only some of the advantages of BM Assume: mbl[h] = m and max-check[i]=max(mcl[i, x])=g • Backjumping from v[i]: – v[i] backjumps up to v[g] v[m] v[f] • Backmarking of v[h]: – When reconsidering v[h], v[h] will be checked against all f [m, g) – effort could be saved • Phenomenon will worsen with CBJ • Problem fixed by Kondrak & van Beek 95 v[g] v[h] v[i] v[h] Foundations of Constraint Processing Intelligent Backtracking Algorithms 40

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) Foundations of Constraint Processing Intelligent Backtracking Algorithms 41

FC: data structures • 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[i] • All these sets are treated like stacks v[i] v[k] v[m] v[j] v[l] v[n] Foundations of Constraint Processing Intelligent Backtracking Algorithms 42

Forward Checking: functions • • • check-forward undo-reductions update-current-domain fc-label fc-unlabel Foundations of Constraint Processing Intelligent Backtracking Algorithms 43

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) Foundations of Constraint Processing Intelligent Backtracking Algorithms 44

FC: functions • update-current-domain – current-domain[i] reductions[i] – actually, we have to iterate over reductions, which is a 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) Foundations of Constraint Processing Intelligent Backtracking Algorithms 45

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 (first jumps, then steps back) • FC-CBJ: – Best algorithm so far – fc-cbj-label and fc-cbj-unlabel Foundations of Constraint Processing Intelligent Backtracking Algorithms 46

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] Foundations of Constraint Processing Intelligent Backtracking Algorithms 47

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 Foundations of Constraint Processing Intelligent Backtracking Algorithms 48

Analysis of experiments Algorithms compared with respect to: 1. Number of consistency checks (average) FC-CBJ ≼ FC-BJ ≼BM-CBJ ≼ FC ≼ CBJ ≼ 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 ≼ BMJ ≼ BT ≼ BM FC-CBJ apparently the champion Foundations of Constraint Processing Intelligent Backtracking Algorithms 49

Additional developments • Other backtracking algorithms exist: – Graph-based backjumping (GBJ), etc. – Pseudo-trees [Freuder 85] [Dechter] • Other look-ahead techniques exist – DAC, MAC, etc. • More empirical evaluations – over randomly generated problems • Theoretical comparisons [Kondrak & van Beek IJCAI’ 95] Foundations of Constraint Processing Intelligent Backtracking Algorithms 50

Implementing BT-based algorithms • Preprocessing – Enforce NC, do not include in #CC (e. g. , Zebra) – Normalize all constraints (fapp 01 -0200 -0) – Check for empty relations (bqwh-15 -106 -0_ext) • Interrupt as soon as you detect domain wipe out • Dynamic variable ordering – Apply domino effect Foundations of Constraint Processing Intelligent Backtracking Algorithms 51