1. (a) Insert key = 32
n=3
100 30 3 5 11 30 31 32

2. (a) Insert key = 7
n=3
100 30 7 3 5 11 30 31 3 5 7

3. (c) Insert key = 160
n=3
100
160
120
150
180
180
150
156
179
180
200
160
179

4. (d) New root, insert 45 n=3 30 new root 10 20 30 40 1 2 3 10 12 20 2532 40 40 45

5. Deletion from B+tree (a) Simple case - no example
(b) Coalesce with neighbor (sibling)
(c) Re-distribute keys
(d) Cases (b) or (c) at non-leaf

6. (b) Coalesce with sibling
Delete 50
n=4 10 40
100 40 10 20 30 40 50

7. (c) Redistribute keys
Delete 50
n=4 10 40
100 35 10 20 30 35 40 50

8. n=4 (d) Non-leaf coalese Delete 37 new root 25 25 10 20 30 40 40 30 2526 1 3 10 14 20 22 30 37 40 45

9. Example: h(k) is 4 bits; 2 keys/bucket1
i =
0001 1 1
1001
1100
“slide” conventions:
slide shows h(k), while actual directory has key+pointer

10. Example: h(k) is 4 bits; 2 keys/bucket
New directory 2 00 01 10 11
i = 1
i =
0001 1 1
1001 1
1100
1010
1100
Insert 1010

11. Example continued 2
0000
0111
0001
i = 2 00 01 10 11 1
0001
0111 2
1001
1010
Insert:
0111
0000 2
1100

12. Example continued i = 3 2 0000 0001 i = 2 2 0111 1001 1010 2 1001 1010
110
111 3
i =
0000 2
i =
0001 2 00 01 10 11
0111 2
1001
1010 2
1001
1010
Insert:
1001 2
1100

13. Example b=4 bits, i =2, 2 keys/bucket
0101
can have overflow chains!
insert 0101
Future
growth
buckets
0000
0101
1010
1111
m = 01 (max used block)
If h(k)[i ]  m, then
look at bucket h(k)[i ]
else, look at bucket h(k)[i ] - 2i -1
Rule

14. Example b=4 bits, i =2, 2 keys/bucket
0101
insert 0101
1111
0101
Future
growth
buckets 11
0000
1010
0101 10
1010
1111
m = 01 (max used block)

15. Example Continued: How to grow beyond this? 3
i = 2
0000
100
0101
101
0101
1010
1111
0101
. . .
m = 11 (max used block)