13th International Conference on Membrane Computing - MTA Sztaki

Fast distributed DFS solutions for edge-disjoint paths in digraphs
70
60
Speed-up
gains
50
40
30
20
10
A* over A
B over A
C over A
C* over A
D over A
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Test case
Fig. 3. Speed-up gains of Algorithms A ∗ , B, C, C ∗ and D over A for thirty test cases.
is 38.0% faster than Algorithm A; (4) Algorithm D is 42.1% faster than Algorithm
A.
Analysing results (1–3), we found that the “dead” cells detected by Algorithms
C and C ∗ do cover all “dead” cells detected by Algorithm B, and even
a few more. However, not all detected “dead” cells can be effectively discarded
in “real-time”, unless we allow these two algorithms to run longer (which we do
not want). Thus, we can find (1) scenarios, such as shown in Figure 2, where
Algorithms C and C ∗ (and, of course, Algorithm D) outperform Algorithm B,
and (2) scenarios, such as shown in Figure 4, where Algorithm B runs faster
than Algorithms C and C ∗ (but not than D).
2 10
0 1
9
8
7
6
3 4 5
Fig. 4. An example, in which Algorithm B performs better than Algorithm C.
In Figure 4, Algorithm C does detect all “dead” cells detected by Algorithm
B, but, because of pruning propagating delays, does not effectively discard
them in “real-time”; briefly, it does not show the same runtime performance.
For Algorithm C, when round #1 search path τ = σ 0 .σ 1 .σ 3 .σ 4 .σ 5 .σ 6 .σ 7 .σ 8 .σ 9 ,
backtracks to the source cell, σ 0 , cells σ i , i ∈ {1}∪[3, 9] can be discarded, because
their reach-numbers are greater than σ 0 .depth = 0. Cell σ 0 triggers a discarding
notification, which follows the same path as the backtracked search τ. In round
#2, the new search path τ ′ , τ ′ = σ 0 .σ 2 visits σ 6 before this cell receives its
due discarding notification and then continues to σ 7 and further. Later, cell σ 6
receives its due discarding notification (started in round #1), discards itself and
sends an overdue backtrack token to σ 2 , which starts looking for other directions
to continue path τ ′ . However, several steps have been lost exploring “dead” nodes
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