### Table 1. Current bounds for algorithms scheduling jobs one by one with no constraints.

"... In PAGE 10: ... The analysis of all these algorithms is relatively complicated. The current state of our knowledge is summarized in Table1 . For comparison we include also the competitive ratio of List Scheduling.... In PAGE 14: ... It essentially tries to preserve the invariant above, with some special considerations for large jobs. Thus, in this model both deterministic and randomized cases are completely solved, giving the same bounds as the randomized lower bounds in Table1 . More- over, we know that randomization does not help.... In PAGE 17: ... Joel Wein observed that for preemptive open shop scheduling there exists a 2-competitive algorithm for arbitrary m. Gerhard Woeginger and the author ob- served that the randomized lower bound from the basic model which approaches e=(e ? 1) 1:5819 (see Table1 ) can be modi ed to work for open shop, too. 4.... ..."

### Table 6 Running times of the branch-and-bound algorithm for different problem sizes and multiple precedence constraints

1999

### Table 7 Comparison of the branch-and-bound algorithm with the CM-5 phase III parallelization in respect to CPU time in minutes (with three precedence constraints)

1999

"... In PAGE 12: ... 3. The CPU times of the branch-and-bound and parallel neural network (Phase III parallelization) approaches are compared in Table7 . The run times for the neural network approach have been calculated by multiplying the number of simulations required to obtain an optimal solution with 99% confidence (data extracted from the third column of Table 3) with the average time required for 1 simulation using Phase III parallelization (data extracted from the third column of Table 4).... ..."

### Table 1: Table showing time in seconds and number of backtracks taken by various algorithms for 100 variable problems with K=4, T=4. C:number of constraints and i is the i-bound used. The quantity in the bracket alongside each constraint indicates the number of instances on which the results are based on.

"... In PAGE 6: ... While it is not reflected in the time measure, the number of backtracks required by our al- gorithms is significantly lower than MAC(table 1). Table1 shows the results for insoluble instances. From this table, we can see that for insoluble instances in the 100-variable-set, IJGP(2)-SC performs better than MBTE(2)-SC which in turn is better than MBTE(2)- MC both in terms of cpu time and the number of back- tracks.... ..."

### Table 2: A Tighter Upper Bound on Number of Consumption Channels Required from Topology and Routing Constraints. Virtual Channel Characteristics and Usage of Two Routing Algorithms are Shown. Number of virtual Number of messages that Number of messages that can

"... In PAGE 18: ...Table2 summarizes the virtual channel usage characteristics of two routing algorithms: a) dimension-order [12] as discussed above and b) Duato apos;s fully adaptive routing [15] on three di erent topologies. The last column in this table shows the maximum number of messages that can simultaneously enter a node from all dimensions for di erent topologies.... In PAGE 28: ... Hence, with increase in dimensionality of hypercubes more consumption channels are not useful. c) It is also reassuring to observe that the values for ca2o t;r derived here are much smaller than the corresponding values of ct;r obtained in Table2 in Sec.... ..."

### Table 3: Upper bounds on (fk) ? (bk) obtained from the TSE algorithm.

"... In PAGE 15: ... This analysis, however, does not indicate whether or not our production line will operate correctly given the temporal constraint which must be observed. Table3 summarizes the results of the TSE algorithm which is used to determine an upper bound on the maximum separation in time between b and f, i.e.... ..."

### Table 12: Constraint Programming results.

1998

"... In PAGE 27: ... It can be seen that usually only a few seconds are needed by the Tabu code to find a solution a solution as good as that found by the LP heuristic. For purposes of comparison, in Table12 we reproduce the best results obtained by the Constraint Programming algorithm of Heipcke and Colombani ([16]) as reported in [6]. The columns LB and UB represent respectively the lower and upper bounds produced by the algorithm while column Cr.... ..."

Cited by 10

### Table 2 shows statistics on the grouping algorithm, performed on the same machine. Notice the drastic reduction in the number of cells after the grouping algorithm (# cells, left), and the effect of the lower bound of the constraint (# cells, right). The video shows (in sequence 2) the same walk- through after applying our method to the model, clearly demonstrating the usefulness of the new decomposition.

"... In PAGE 18: ... Table2 : Statistics for the grouping algorithm, with a merging threshold of 50 polygons and no splitting threshold. Column validation shows the time needed to compute the valid separators and to construct the final cell-and-portal graph (although the time for constructing this graph is negligible).... ..."

### Table 1: Well-known algorithms for tree layout along with their runtime and the fulfilled layout constraints. n designates the number of nodes; h designates the drawing height of the tree, which is measured in unit bounding boxes for example (cf. Section 3).

"... In PAGE 3: ... The compilation considers all of the mentioned papers and shall serve as a unification of the language usage; more- over, it classifies the constraints with respect to a layout problem class. Table1 relates the constraints to algorithms while Figure 2 illustrates selected constraints. Boolean layout constraints for binary trees with uniform-sized nodes: (1) The vertical coordinate of a node corresponds to its level in the tree [see 2].... ..."