1. Interchangeability: An algorithm for computing interchangeability in non-binary CSPs.
2. Dynamic bundling: Integration of the above with backtrack search for solving non-binary CSPs.
3. Experiments demonstrating the benefits of dynamic bundling
·Finding multiple, robust solutions.
·Decreasing computational cost of search

Lal, Anagh

Choueiry, Berthe Y.

Freuder, Eugene C.

English

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DigitalCommons@University of Nebraska - Lincoln 
CSE Conference and Workshop Papers Computer Science and Engineering, Department of 
2005 
Neighborhood Interchangeability and Dynamic Bundling for Non-
binary CSPs 
Anagh Lal 
University of Nebraska-Lincoln, alal@cse.unl.edu 
Berthe Y. Choueiry 
University of Nebraska-Lincoln, choueiry@cse.unl.edu 
Eugene C. Freuder 
University College Cork, e.freuder@cs.ucc.ie 
Follow this and additional works at: https://digitalcommons.unl.edu/cseconfwork 
 Part of the Computer Sciences Commons 
Lal, Anagh; Choueiry, Berthe Y.; and Freuder, Eugene C., "Neighborhood Interchangeability and Dynamic 
Bundling for Non-binary CSPs" (2005). CSE Conference and Workshop Papers. 168. 
https://digitalcommons.unl.edu/cseconfwork/168 
This Article is brought to you for free and open access by the Computer Science and Engineering, Department of at 
DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in CSE Conference and 
Workshop Papers by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 
A Constraint Satisfaction Problem (CSP) is a combinatorial decision 
problem defined by a set of variables, a set of domain values for 
these variables, and a set of constraints restricting the allowable 
combinations of values for variables, where the task is to find a 
solution (i.e., an assignment of a value to each variable satisfying all 
constraints), or to find all such solutions. 
Neighborhood Interchangeability and Dynamic Bundling for Non-binary CSPs 
S 
c d, e, f 
d V1 
V2 
Dynamic bundling [1]: 
Interchangeability sets are 
updated during search 
Static bundling [3]: 
Interchangeability sets are 
computed before search 
c e, f d 
d V1 
V2 
S 
Search without bundling 
c e f d 
d V1 
V2 
S 
1. Interchangeability: An algorithm for computing interchangeability in 
non-binary CSPs. 
2. Dynamic bundling: Integration of the above with backtrack search for 
solving non-binary CSPs. 
3. Experiments demonstrating the benefits of dynamic bundling 
 Finding multiple, robust solutions. 
 Decreasing computational cost of search. 
Contributions 
C4 
{1, 2, 3,  
4, 5, 6} 
{1, 2, 3} 
{1, 2, 3} 
{1, 2, 3} 
{1, 2, 3} 
C2 
C1 
C3 
Constraint 
Variable 
Non-binary CSP (nb-CSP) 
  
SELECT R1.A,R1.B,R1.C 
FROM   R1,R2 
WHERE  R1.A=R2.A  
AND    R1.B=R2.B 
AND    R1.C=R2.C 
Result:  10 tuples 
in 3 nested tuples 
 Use new join algorithm when materializing join queries. 
 Exploit bundled results in data-analysis/data-mining packages. 
 Assist in query size estimation 
 Improve accuracy of sampling operators 
Interchangeability identifies values equivalent in all solutions of a 
CSP [2]. 
Full Interchangeability (FI): d, e, and f can be 
swapped for V2 in any solution. 
Neighborhood Interchangeability (NI): finds e 
and f but misses d. It efficiently approximates FI. 
Interchangeability & Bundling 
Modeling the join query as a CSP 
Attributes            variables 
Attribute values  domains 
Relations             relational constraints 
Join conditions   join-condition constraints 
Dynamic Bundling in Databases [4] 
Experiments: FC versus DynBndl 
Relational constraint 
Join-condition constraint 
R1.A R1.B R1.C 
R2.A R2.B R2.C 
R1 R2 
 
R1       R2 (Compacted) 
A B C 
{1, 5} {12, 13, 14} {23} 
{2, 4} {10} {25} 
{6} {13, 14} {27} 
Sorting-based bundling algorithm 
Sort-merge join algorithm based on dynamic bundling 
 Reduces number of tuples compared in the main memory. 
 Is memory efficient and produces compacted results, saving  
– I/O for the next operator and 
– disk space (and network bandwidth in distributed 
databases). 
Experiments 
 Compaction rate achieved in a real-world problem: 2.26. 
 Compaction rate achieved on a random data-set: 1.48 (10’000 
tuples; memory size: 4’000 tuples; page size 200 tuples). 
Task: Join query 
V3 
{d} 
{a, b, d} {a, b, c} 
{c, d, e, f} 
V4 
V2 V1 
Binary CSP 
1. Choueiry, B.Y., Davis, A.M.: Dynamic bundling: Less Effort for More Solutions. SARA 02. 
2. Freuder, E.C.: Eliminating Interchangeable Values in Constraint Satisfaction Problems. AAAI 
91. 
3. Haselböck, A: Exploiting Interchangeabilities in Constraint Satisfaction Problems. IJCAI 93. 
4. Lal, A., Choueiry, B.Y.: Constraint Processing Techniques for Improving Join Computation. 
CDB 04. 
{1, 2} 
{3} 
{3} 
{1, 2} 
{1, 3} {3} 
{3} {3} 
No-good 
bundle 
V 
D 
C 
A 
B 
Solution  
bundle 
Constraint Satisfaction Problems 
{c, d, e, f  } 
V3 
{d} 
{a, b, d} {a, b, c} 
V4 
V2 V1 
This research was supported by a Maude Hammond Fling Faculty Research Fellowship, National Science Foundation CAREER Award 
#0133568, and Science Foundation Ireland Grant 00/PI.I/C075. 
Experiments were conducted utilizing the Research Computing Facility of the University of Nebraska-Lincoln. 
Dynamic Bundling for Non-Binary CSPs 
Interchangeability in non-binary CSPs 
We show how to compute NI for non-binary constraints by: 
1. Building the non-binary discrimination tree, nb-DT(V, C), a data-
structure that determines the NI sets of a variable V given a constraint 
C defined on V. 
2. Intersecting the NI sets from the nb-DTs of a set of constraints, which 
yields the domain partition of the variable V given the constraints.  
 Tested random CSPs, Model B, 1000 instances per sample 
 Criteria: FBS (First Bundle Size), CPU time, number of nodes visited 
(NV), and number of constraint checks. 
 Statistical tools: ANOVA and t-distribution for confidence intervals. 
 partitions domains in a memory 
efficient manner. 
 fits into the iterator model of 
databases and produces one 
bundle at a time. 
Future Research Directions 
References 
Anagh Lal and Berthe Y. Choueiry  
Constraint Systems Laboratory • University of Nebraska-Lincoln 
Eugene C. Freuder 
Constraint Computation Center • University College Cork 
Dynamic bundling (DynBndl) was thought to be 
an overkill.  We show DynBndl is worthwhile for: 
 Finding all solutions: theoretically best 
 Finding first solutions: empirical evidence 
 
Because DynBndl: 
 Bundles solutions 
 Bundles no-goods (i.e., bundles of 
inconsistent partial solutions). 
Constraint Ratio  p2 c3  c4 
CR1 0.25 3 2 
CR2 0.25 6 5 
CR3 0.40 3 2 
CR4 0.40 6 5 
High tightness 
 Problems mostly 
unsolvable. 
 Minimal bundling 
overhead. 
Phase transition 
 Multiple solutions exist. 
 Maximum no-good 
bundling yields max 
savings in CPU time, 
NV,  & CC. 
Low tightness 
 Large FBS: 33 at t = 0.35 
(2254 in Dataset #13). 
 Small bundling overhead. 
Varying tightness 
Increasing domain size 
Tightness 
n=20 
a=15 
CR=CR3 
0 
2 
4 
6 
8 
10 
12 
14 
16 
18 
20 
0.325 0.35 0.375 0.4 0.425 0.45 0.475 0.5 0.525 0.55 0.575 0.6 
T
im
e
 [
s
e
c
] 
#
N
V
, 
h
u
n
d
re
d
s
 
    t       FBS 
0.350   33.44 
0.400   10.91  
0.425     7.13 
0.437     6.38  
0.450     5.62 
0.462     2.37 
0.475     0.66 
0.500     0.03 
0.550     0.00  
#NV 
CPU time 
DynBndl 
FC 
DynBndl 
FC 
CR FBS CPU improvement 
%  
a=10 a=15 a=10 a=15 
CR1 5.5 11.9 33.3 34.3 
CR2 5.0 5.5 28.6 33.0 
CR3 3.6 5.0 29.8 31.7 
CR4 1.2 1.4 28.4 31.6 
Bundling = Search + neighborhood interchangeability [3] 
Dynamic bundling = Search + dynamic NI  [1] 
Increasing a, we note in the phase-
transition area: 
 FBS increases (more chances for 
symmetry). 
 CPU time decreases (better no-good 
bundling). 
The benefits of DynBndl increase with 
increasing domain size: Use DynBndl in 
database applications where large 
domains are typical. 
A B C
1 12 23
1 13 23
1 14 23
2 10 25
3 16 30
3 16 24
4 10 25
5 12 23
5 13 23
5 14 23
6 13 27
6 14 27
7 14 28
7 19 20
R1
A B C
1 12 23
1 13 23
1 14 23
1 15 23
2 10 25
3 17 20
3 18 22
4 10 25
5 12 23
5 13 23
5 14 23
5 15 23
6 13 27
6 14 27
8 14 28
R2
Partition 
Unequal 
partitions 
Symmetric 
partitions 
Bundle for 
R1.A: {1, 5} 
R1   
A  B  C  
1   12   23   
1  13   23   
1  14   23   
2  10   25   
      
5  12   23   
5  13   23   
5  14   23   
  
