1. Isomorphism Checking in GROOVE
Kaminski, Seidl et al. Muscholl
Arend Rensink, University of Twente
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

2. Isomorphism Checking in GROOVE
Basics
Universe of labels Lab
Graphs: tuples G = <N,E,src,tgt,lab>
Finite set of nodes N
Finite set of edges E
Source and target mappings src,tgt: E  N
Edge labelling lab: E  Lab
Isomorphism between G and H (G  H):
Mapping f: G  H
Bijective node and edge components fN, fE
Structure-preserving: fNsrcG = srcHfE etc
G and H are “essentially the same”
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

3. Isomorphism Checking in GROOVE
Examples 1 2 3 4 5 a b a b a b a b a b
Which of these graphs are isomorphic?
1  3; all others are distinct
Which of these graphs are bisimilar? (roughly: can mutually simulate one another)
1  3  5; all others are distinct
Which have non-trivial symmetries?
1, 3, 4 and 5
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

4. Isomorphism Checking in GROOVE
Complexity
Traditional isomorphism question
For given graphs G and H, is G  H?
In NP; unknown to be either in P or NP-hard
Essential concept: graph certificate
Existing algorithms use partition refinement
Extended isomorphism question
For given graph G and set of graphs S, find H  S such that G  H, if it exists
More general: above case given by S = {H}
Graph certificates also help in this problem
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

5. Isomorphism Checking in GROOVE
Graph certificates (1)
Graph cerficate mapping
Universe Cert of certificates; e.g., Nat
Mapping c: Graph  Cert
Invariance: G  H implies c(G) = c(H)
Example: c: G  |EG|
Use 1: conservative approximation of iso
c(G)  c(H) implies (G  H)
Use 2: hash value in set of graphs S a b a b a b a b a b
c=4
c=4
c=4
c=4
c=4
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

6. Isomorphism Checking in GROOVE
Graph certificates (2)
Refinement: element certificate mapping
Mappings cN: Graph  Node  Cert cE: Graph  Edge  Cert
Invariance: isomorphism f: G  H implies cN(G) = cN(H)  fN and cE(G) = cE(H)  fE
Example: cN(G): n  |tgtG-1(n)|, cE(G): e  hash(labG(e)) a b 1 2 1 2 a b 1 2 a b a b 1 2 a b 1 2
hash: a  1, b  2
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

7. Isomorphism Checking in GROOVE
Graph Certificates (3)
Use 1: Predict isomorphism
f: G  H implies fN  cN-1(H)  cN(G) fE  cE-1(H)  cE(G)
Use 2: Define graph certificate mapping
c1(G) = nN cV(G,n) + eE cE(G,e)
c2(G) = <{cV(G,n)|n N},{cE(G,e)|e E}> 1 2 1 2 1 2 1 2 1 2
<{1},{1,2}>
c1=12
<{0,1,2},{1,2}>
c1=12
<{1},{1,2}>
c1=12
<{0,2},{1,2}>
c1=12
<{1},{1,2}>
c1=12
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

8. Inspiration: bisimilarity
Relation G over nodes + edges
Largest symmetrical relation such that
src(e1)  n2 implies e1  e2 for some e2 with src(e2)=n2 and lab(e2)=lab(e1)
vice versa for target nodes
Weaker than isomorphism:
f: G  H an iso implies n GH fN(n) and e GH fE(e) a b a b
Bisimilar graphs
Equally coloured nodes
Equally labelled edges
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

9. Bisimilarity co-inductively
Algorithm: G = i i given
Sequence of relations with i+1  i
0 = (NGNG)  { (e1,e2)  EGEG | lab(e1)=lab(e2) }
e1 i+1 e2 if src(e1) i src(e2) and tgt(e1) i tgt(e2)
n1 i+1 n2 if i+1 is a total relation between src-1(n1) and src-1(n2) as well as between tgt-1(n1) and tgt-1(n2)
Sequence converges for finite graphs
Average complexity: |G| log |G|
First iteration:
All nodes, equally labelled edges
Second iteration:
Equally coloured nodes
Equally labelled edges
Third iteration: convergence a b a b
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

10. Isomorphism Checking in GROOVE
The algorithm (1)
Iterated construction of certificates
Inspired by bisimilarity characterisation
Sequence of mappings (cNi,cEi)i
Starting point: simple case like before
Next steps: use end nodes/incident edges
n i n2 implies cNi(n1)=cNi(n2), also for edges
Not vice versa: certificates not precise
Convergence: use partition size |N/i|
Partition size may grow due to imprecision
Stop when |N/i|  |N/i-1|
Derived graph certificate as hash key
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

11. Isomorphism Checking in GROOVE
The algorithm (2)
Recall “extended isomorphism question”
For given graph G and set of graphs S, find H  S such that G  H, if it exists
If c(G) is fresh, no such H exists
If c(H) = c(G), test for G  H
Test for equality (equal node and edge sets)
If cNi is injective (|N/i|=|N|), G  H iff { (n,cNi(n)) | nNG } yields an iso
Otherwise, more complex extended iso test
If  G  H, it is a false positive: add G to S
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

12. Definition of element certificates
Node mapping: cNi(n) =
1 if i = 0
cNi-1(v) + 2 n=src(e) ciE(e) + n=tgt(e) ciE(e)
Edge mapping: cEi(e) =
hash(lab(e)) if i = 0
newCert(cEi-1(e),lab(e),cNi(src(e)),cNi(tgt(e)))
newCert determines “quality”:
Should reduce imprecision/confusion
Should be “maximally injective”
Derived graph certificate
c(G) = nN cVi(n) + eE cEi(e)
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

13. Isomorphism Checking in GROOVE
Example
newCert(old,lab,src,tgt) := old + 2 src + tgt a b a b a b a b a b 1 2 1 2 1 2 1 2 1 2 17 18 5 6 17 19 18 21 5 6 4 7 17 18 5 6 19 20 7 4 17 18 5 6
190
193 57 59
190
193 57 59
232
233 85 43
190
193 57 59
|N/| = 2
|N/| = 2 c = 998
|N/| = 1
|N/| = 1
|N/| = 4 c = 97
|N/| = 2 c = 998
|N/| = 2
|N/| = 1
|N/| = 2 c = 1186
|N/| = 1
|N/| = 2
|N/| = 2 c = 998
|N/| = 2
|N/| = 1
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

14. Implementation new edge certification function
int[] tmp = new int[vSize]; // temporary node certificates
int partSize = 1, oldPartSize; // partition size
// initialise the certificates
for (int e = 0; e < eSize; e++) eCert[e] = hash(lab[e]);
for (int v = 0; v < vSize; v++) vCert[v] = 1;
do { // calculate the new edge certificates
for (int e = 0; e < eSize; e++) {
eCert[e] = newCert(eCert[e], lab[e], vCert[src[e]], vCert[tgt[e]]);
// propagate to the endpoints
tmp[src[e]] += eCert[e]; tmp[tgt[e]] += 2*eCert[e]; }
// calculate new node certificates and determine partition size
IntSet certSet = new IntSet();
oldPartSize = partSize;
for (int v = 0; v < vSize; v++) {
// copy the temporary node certificates to the real ones
vCert[v] = tmp[v]; tmp[v] = 0;
certSet.add(vCert[v]);
partSize = certSet.size();
} while (partSize > oldPartSize); // continue while the number of cells still grows
new edge certification function
special set to count ints
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

15. Isomorphism Checking in GROOVE
Example (2) 1 2 3 4 5
190
193 57 59 17 19 18 21 5 6 4 7
190
193 57 59
232
233 85 43
190
193 57 59
c1 = 998
c2 = 97
c3 = 998
c4 = 1186
c5 = 998
S0 = empty: c1 = 998 not in c(S0), 1 is fresh
S1 = {1}: c2=97 not in c(S1), 2 is fresh
S2 = {1,2}: c3 = 998 = c1 in c(s2), |N/|  |N|; 3  1 after extended iso test, 3 not fresh
S3 = {1,2}: c4 = 1186 not in c(S3), 4 is fresh
S4 = {1,2,4}: c5 = 998 = c1 in c(s4), |N/|  |N|;  (5  1) after extended iso test, 5 is fresh
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

16. Isomorphism Checking in GROOVE
Experiments
Mutex algorithm [Heckel]
Small states (graphs up to 6 nodes)
“Unpredictable” symmetries
Dining philosophers
n-fold symmetry; here n = 12
No node creation/deletion
Concurrent append
Symmetry mainly due to confluence
List length 8, 4 concurrent methods
Gossiping girls
Huge amount of symmetry; here for 8 girls
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

17. Isomorphism Checking in GROOVE
Results
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE

18. Isomorphism Checking in GROOVE
Conclusion
Experience: performance acceptable
Large fraction of total exploration time
Pays off for problems with many symmetries
Certificates are very good predictor for iso
Related work: McKay (Nauty)
Based on the same principles
Idea: in case of non-injective element certificates, break symmetry and observe effects
Not straightforward to reuse
Future work: avoid the iso check!
Take confluence into account
Optionally switch off isomorphism checking
GRABATS 2006, Natal, Brazil
Isomorphism Checking in GROOVE