Cascade Merge

CPSC 461: Copyright © 2002 Katrin Becker 1998-2002 Last Modified July 31, 2000 04:21 PM

Cascade Merge

This is another way to distribute initial runs for a merge:

- it's another form of K-way merge (using K+1 devices)

There are several steps to each phase:

First we do a K-way merge,
then a K-1 way merge,
then a K-2 way merge
.......
till we have one 2-way merge (which we do)

Device 1 Device 2 Device 3 Device k-1 Device k # of runs

1 0 0 ..... 0 0 1

1 1 1 ..... 1 1 K

n₁ n₂ n₃ ..... n_k-1 n_k

..... n₁ + n₂ n₁

Device 1 Device 2 Device 3 Device 4 Device 5 Device 6

initial distribution: 55¹ 50¹ 41¹ 29¹ 15¹ 0

5-way 40¹ 35¹ 26¹ 14¹ 0 15⁵

4-way 26¹ 21¹ 12¹ 0 14⁴ --

3-way 14¹ 9¹ 0 12³ -- --

2-way 5¹ 0 9² -- -- --

now.. 0 5¹ 9² 12³ 14⁴ 15⁵

5-way 5¹⁵ 0 4² 7³ 9⁴ 10⁵

4-way -- 4¹⁴ 0 3³ 5⁴ 6⁵

3-way -- -- 3¹² 0 2⁴ 3⁵

2-way -- -- -- 2⁹ 0 1⁵

now.. 5¹⁵ 4¹⁴ 3¹² 2⁹ 1⁵ 0

5-way 4¹⁵ 3¹⁴ 2¹² 1⁹ 0 1⁵⁵

4-way 3¹⁵ 2¹⁴ 1¹² 0 1³⁹ --

3-way 2¹⁵ 1¹⁴ 0 1⁴¹ -- --

2-way 1¹⁵ 0 1²⁹ -- -- --

finally.. 0 1¹⁵ 1²⁹ 1⁴¹ 1³⁹ 1⁵⁵

1¹⁷⁹

Perfect Distribution:

Level Device 1 Device 2 Device 3 Device 4 Device 5

0 1 0 0 0 0

1 1 1 1 1 1

2 5 4 3 2 1

3 15 14 12 9 5

4 55 50 41 29 15

5 190 175 146 105 55

n a_n b_n c_n d_n e_n

n + 1 a_{n +} b_{n +}c_{n +}d_{n +}e_n a_{n +} b_{n +}c_{n +}d_n a_{n +} b_{n +}c_n a_{n +} b_n a_n



6th Order Tables for:
Cascade Merge
0 1 2 3 4 5 6 total 1 1 1 1 1 1 1 6 2 6 5 4 3 2 1 21 3 21 20 18 15 11 6 91 4 91 85 74 59 41 21 371 5 371 350 309 250 176 91 1547 6 1547 1456 1280 1030 721 371 6405

In general, for > 5 tapes the Cascade Merge makes fewer passes than the Fibinacci (Polyphase) Merge

Device 1	Device 2	Device 3		Device k-1	Device k	# of runs
1	0	0	.....	0	0	1
1	1	1	.....	1	1	K
n₁	n₂	n₃	.....	n_k-1	n_k
			.....	n₁ + n₂	n₁

Level	Device 1	Device 2	Device 3	Device 4	Device 5
0	1	0	0	0	0
1	1	1	1	1	1
2	5	4	3	2	1
3	15	14	12	9	5
4	55	50	41	29	15
5	190	175	146	105	55
n	a_n	b_n	c_n	d_n	e_n
n + 1	a_{n +} b_{n +}c_{n +}d_{n +}e_n	a_{n +} b_{n +}c_{n +}d_n	a_{n +} b_{n +}c_n	a_{n +} b_n	a_n

0	1	2	3	4	5	6	total
1	1	1	1	1	1	1	6
2	6	5	4	3	2	1	21
3	21	20	18	15	11	6	91
4	91	85	74	59	41	21	371
5	371	350	309	250	176	91	1547
6	1547	1456	1280	1030	721	371	6405