"... In PAGE 2: ... This is the first algorithm that breaks the O(n2) update barrier for all graphs with o(n2) edges. The new algorithm and some of the previously existing algorithms for the dynamic reachability problem are compared in Table1 . Our algorithm has the fastest update time, among all algorithms that work on all graphs, but alas the slowest query time.... ..."

"... In PAGE 2: ... This is the first algorithm that breaks the O(n2) update barrier for all graphs with o(n2) edges. The new algorithm and some of the previously existing algorithms for the dynamic reachability problem are compared in Table1 . Our algorithm has the fastest update time, among all algorithms that work on all graphs, but alas the slowest query time.... ..."

"... In PAGE 111: ... Those results are in alignment with figure 3 except that ResvConf and PathErr OCC graph SCC graph Model Nodes Arcs Secs Nodes Arcs Secs ResvSetup 11 10 0 11 10 0 ResvSetup WithErrors 17 16 0 17 16 0 PathTear 67 105 0 67 105 0 ResvTear 182 400 0 182 400 1 Refresh 613 3584 6 10 379 0 RSVP 13570 96837 536 2601 43493 69 RSVP (with multiple sessions) 24576 199360 18105 1 0 988 Table 1: A comparison of several versions of RSVP model. Home Markings [6] Dead Markings [6] Dead Transitions Instances None Live Transitions Instances None Table2 : Home and liveness properties.... In PAGE 126: ...net distributed with PROD without and with unfolding, and three variants of a corresponding model with MARIA: without and with capacity constraints, and with redundant places indicated. The figures in Table2 are for models where the redundant place unused has been removed. The space consumption drops to less than a fifth when this place is omitted.... In PAGE 146: ... Since the running time of the reachability graph generation grows very signi cantly into an almost trashing situation even when increasing only by 1 the number of applications, then wehave tried the discrete event simulation available in GreatSPN, to compute performance indices for our LV model while varying the number N of applications. Table2 shows the list of timing parameters to be assigned to transitions to perform a simulation. For a timing assignment in whichvalues di er for up to one order of magnitude the simulator was able to provide estimates for throughput and mean value of tokens in places in a few minutes, for an accuracy level set to 10 percent and a con dence level of 0.... ..."

"... In PAGE 4: ... The branching block, B1, is executed n + 1 times. This infor- mation is captured by the directed graph of basic blocks in Table2 . The directed graph is parameterized on the dimensionality of the problem: ie.... In PAGE 6: ... 2.2 Real-time analysis of SVM decision function An analysis the directed graph (same approach as in Table2 ) of the SVM decision function shows that basic block B1, B2, and B3 have conditional branches. B1 is conditioned on the number of training examples, B2 is conditioned on a support vector being non-zero, and B3 is conditioned on the dimension of the training problem, n.... ..."

"... In PAGE 11: ...Table 1. Symbolic Interpretation of Reachability Logic To read the rules of Table1 some notation needs to be explained. For D aconstraintsystemandra set of variables (to be reset) r(D) denotes the set of variable assignments fr(v) j v 2 Dg.... In PAGE 12: ...directed graphs (with clock and data variables as nodes), these operations as well as testing for inclusion between constraint systems may be e ectively im- plemented in O(n2)andO(n3) using shortest path algorithms [11, 12, 6]. Now, by applying the proof rules of Table1 in a goal directed manner we obtain an algorithm (see also [13]) for deciding whether a given symbolic network con guration [l;D] satis es a property 93 . To ensure termination (and e ciency), we maintain a (past{) list L of the symbolic network con gurations encountered.... ..."

"... In PAGE 9: ... Algorithm A1 has clear performance advantages in dense graphs, but when the density is under 20% A2 is the faster. Table1 shows our results when we varied the number of nodes from 1000 to 8000 while maintaining a constant density of 2%. The times for both algorithms increase almost linearly as the number of nodes is increased.... In PAGE 9: ... The inverse is true for A2 whose time to nd the connected components increases almost linearly with the increased number of edges. Table 3 also shows what happens when the density is kept constant at 50%, but the number of nodes is increased (as we did in Table1 with 2% dense graphs). 5.... ..."