### Table 7: Timing in hours taken by our implementations of different external memory BFS algo- rithms.

"... In PAGE 10: ...2 Line graph with random layout on disk 47 3.3 Table 6: Timing in hours for computing the deterministic preprocessing of MM BFS by the two implementations of MM BFS D Comparing MM BFS D with other external memory BFS algorithm implemen- tations Table7 shows the performances of our implementations of different external memory BFS algo- rithms. While MR BFS performs better than the other two on random graphs saving a few hours, MM BFS D outperforms MR BFS and MM BFS R on line graphs with random layout on disk saving a few months and a few days, respectively.... ..."

### Table I. Algorithms and data structures of the external memory libraries.

2005

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### Table 2: Comparison of the external-memory versions of structured duplicate detection (SDD) and parallel structured duplicate detection (PSDD) with 1, 2, 3, and 4 threads. Columns show solution length (Len), peak number of nodes stored in RAM for both SDD and PSDD (RAM), peak number of nodes stored on disk (Disk), number of node expansions (Exp), and running time in wall clock seconds (Secs).

"... In PAGE 6: ... Note that the search graphs for planning problems have many duplicate paths, and are especially challenging for duplicate detection. Table2 compares running times (in wall-clock seconds) for SDD and PSDD with up to 4 threads on problems that do not fit in the amount of RAM given to the algorithms (specified in the column labeled RAM ). Since the timing results for external-memory PSDD using 4 threads are less negatively affected by the overhead of the OS and other pro- grams running on the same machine (due to I/O parallelism), we include these results to give a better sense of scalabil- ity.... ..."

### Table 2: Comparison of the external-memory versions of structured duplicate detection (SDD) and parallel structured duplicate detection (PSDD) with 1, 2, 3, and 4 threads. Columns show solution length (Len), peak number of nodes stored in RAM for both SDD and PSDD (RAM), peak number of nodes stored on disk (Disk), number of node expansions (Exp), and running time in wall clock seconds (Secs).

"... In PAGE 6: ... Note that the search graphs for planning problems have many duplicate paths, and are especially challenging for duplicate detection. Table2 compares running times (in wall-clock seconds) for SDD and PSDD with up to 4 threads on problems that do not fit in the amount of RAM given to the algorithms (specified in the column labeled RAM ). Since the timing results for external-memory PSDD using 4 threads are less negatively affected by the overhead of the OS and other pro- grams running on the same machine (due to I/O parallelism), we include these results to give a better sense of scalabil- ity.... ..."

### Table 9: Timing in hours taken by our implementations of different external memory BFS algorithms.

### TABLE 1: INTERNAL AND EXTERNAL MEMORY REFERENCES #Internal mem. refs #External mem. refs Algorithm

### Table 9: Timing in hours taken by our implementations of different external memory BFS algorithms.

### Table 13: The best total running time (in hours) for BFS traversal on different graphs with the best external memory BFS implementations

"... In PAGE 3: ... Also, on low diameter graphs, the time taken by our improved MR BFS is around one-third of that in [3]. Towards the end, we summarize our results ( Table13 ) by giving the state of the art implementations of external memory BFS on different graph classes. 2 Improvements in the previous implementat- ions of MR BFS and MM BFS R The computation of each level of MR BFS involves sorting and scanning of neighbours of the nodes in the previous level.... In PAGE 8: ...2 Summary. Table13 gives the current state of the art implementations of external memory BFS on different graph classes. Our improved MR BFS implementation outper- forms the other external memory BFS implementations on low diameter graphs or when the nodes of a graph are arranged on the disk in the order required for BFS traversal.... ..."

### Table 13: The best total running time (in hours) for BFS traversal on different graphs with the best external memory BFS implementations

"... In PAGE 3: ... Also, on low diameter graphs, the time taken by our improved MR BFS is around one-third of that in [3]. Towards the end, we summarize our results ( Table13 ) by giving the state of the art implementations of external memory BFS on different graph classes. 2 Improvements in the previous implementat- ions of MR BFS and MM BFS R The computation of each level of MR BFS involves sorting and scanning of neighbours of the nodes in the previous level.... In PAGE 8: ...2 Summary. Table13 gives the current state of the art implementations of external memory BFS on different graph classes. Our improved MR BFS implementation outper- forms the other external memory BFS implementations on low diameter graphs or when the nodes of a graph are arranged on the disk in the order required for BFS traversal.... ..."

### Table 11: The best total running time (in hours) for BFS traversal on different graphs with the best external memory BFS implementations

"... In PAGE 4: ... Also, on low diameter graphs, the time taken by our improved MR BFS is around one-third of that in [3]. Towards the end, we summarize our results ( Table11 ) by giving the state of the art implementations of external memory BFS on different graph classes. Our implementations can be downloaded from http://www.... ..."