13th International Conference on Membrane Computing - MTA Sztaki

Fast distributed DFS solutions for edge-disjoint paths in digraphs
3.2. Frontier: cf. Lines 4.5-26
1. S 20 → min.min S 20 r i(X) (r i(X))↕ ∀ | f j h(X) ι i ¬ v
2. S 20 f j → min.min S 20 v s ′ j (v i)↕ ∀ f | ι i ¬ v
3. S 20 f j → min.min S 20 (b i)↕ j | v ι i
4. S 20 r k (X) r k(Y ′ ) → min.min S 20 r k (Y ) | b k
5. S 20 → min.min S 20 (x)↕ k | h(X) r k (XY ) b k ι i ¬ w k
6. S 20 b k s ′′
k → min.min S 20 f z ′′
k
7. S 20 b k → max.max S 20
8. S 20 → min.min S 20 n(X) | h(X) f ι i
9. S 20 n(XY ) → dyn.min S 20 n(X) | r j(X) n ′′
j v j f ι i
10. S 20 n(XY ) → dyn.min S 20 n(X) | r j(X) d ′ j v j f ι i
11. S 20 n(X) r i(XY ) → min.min S 20 r i(X) (r i(X))↕ ′ ∀ | ι i
12. S 20 f → min.min S 20 v k s ′′
k (f i h(Xc))↓ k | ι i n ′′
k h(X) ¬ v k d ′ k d ′′
k
13. S 20 f → min.min S 20 v k s ′′
k (f i h(Xc))↑ k | ι i d ′ k h(X) ¬ v k
14. S 20 f s ′ j → min.min S 20 (b i)↕ j | ι i
15. S 20 n(X) → min.min S 20
3.3. Path confirmation: cf. Lines 3.16-19
1. S 20 a s ′ j s ′′
k → min.min S 20 d ′ j d ′′
k (a)↕ j
2. S 20 a → max S 20
3. S 20 d ′′
k d ′ k → min.min S 20
3.4. End-of-round resets: cf. Lines 7.15, 3.20
1. S 20 → min S 21 (q)↕ ∀ | q
2. S 20 → min S 21 w | q v ¬ w
3. S 20 → max.min S 21 w k | q v k ¬ w k
4. S 20 z k ′′ → min S 21 | q
5. S 20 → min S 21 (p)↕ ∀ | p
6. S 20 v → min S 21 | p ¬ w
7. S 20 v k → max.min S 21 | p ¬ w k
3.5. Transit to the end of a search round
1. S 21 → min S 40
3.6. Update: cf. Lines 6.4-15
1. S 20 r j(X) → max.max S 20 | r j(X)
2. S 20 r j(X) ′ → max.max S 20 | r j(X)
′
3. S 20 r j(X) r j(Y ′ ) → min.min S 20 r j(Y )
4. S 20 → min.min S 20 n(X) | h(X) u k ι i
5. S 20 n(XY ) → dyn.min
6. S 20 n(XY ) → dyn.min
S 20 n(X) | r j(X) n ′′
j v j u k ι i
S 20 n(X) | r j(X) d ′ j v j u k ι i
7. S 20 n(XY ) r i(X) → min.min S 20 r i(XY ) (r i(XY ′ ) u i)↕ ∀ | u k n ′′
k v k ι i ¬ w
8. S 20 n(XY ) r i(X) → min.min S 20 r i(XY ) (r i(XY ′ ) u i)↕ ∀ | u k d ′ k v k ι i ¬ w
9. S 20 n(X) → min.min S 20
10. S 20 u k → min.min S 20 (x)↕ k | h(X) r k (XY ) ι i ¬ w k w
11. S 20 u k → max.max S 20
3.7. Discard: cf. Lines 5.3-15
1. S 20 z k ′′ → max.min S 20 (x)↕ k | x ι i ¬ w k
2. S 20 x → min.min S 20 w r i(∞) (w i r i(∞) ′ u i)↕ ∀ | ι i ¬ w
3. S 20 z k ′′ → max.max S 20 | w
4. S 20 s ′ j s ′′
k → min.min S 20 (b i)↕ j | r i(X) w ι i
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