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Given is the B+-tree shown below (n = 3). Execute the following operations and write down the resulting B+-tree after each op

The rules are:
• Leaf Split: In case a leaf node needs to be split during insertion and n is even, the left node should get the extra key. E.g, if n = 2 and we insert a key 4 into a node [1,5], then the resulting nodes should be [1,4] and [5]. For odd values of n we can always evenly split the keys between the two nodes. In both cases the value inserted into the parent is the smallest value of the right node.

• Non-Leaf Split: In case a non-leaf node needs to be split and n is odd, we cannot split the node evenly (one of the new nodes will have one more key). In this case the “middle” value inserted into the parent should be taken from the right node. E.g., if n = 3 and we have to split a non-leaf node [1,3,4,5], the resulting nodes would be [1,3] and [5]. The value inserted into the parent would be 4.

• Node Underflow: In case of a node underflow you should first try to redistribute values from a sibling and only if this fails merge the node with one of its siblings. Both approaches should prefer the left sibling. E.g., if we can borrow values from both the left and right sibling, you should borrow from the left one.

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Homework Answers

Answer #1

Given B+ Tree is

109 DE25 |U011 ལགས་ཀྱི་ ༼UDD2 JZ2 » JUu5 JUue JUU7 • Up11 Up17 » JU1 JU23 • up2 Up31 །

Insert 9:

·002 - - DC15 ) 2011 | 0002 0003 + ac05 006 0007 + 0011 0047 + s01e 0023 + 0029 0031 ||

ܕ101 0113ܘ ܙ ܘܘܘܘ | ܗ 02a _ poz1 ܘ ܝ023_015ܩ ܝ 0011 _ oe17 occ2 _ dona , nona _ ness _ ooo7 |

0019 00050011 000200030005 0008 00070011 00170019002300290031

Found Leaf Node now Insert 9

- 00050011 0002 0003 +0005 2006 2007 00000011 0017 0019 00230029 0031

Above Node Contains too many keys so Perform Splitting

0005 0007 0011 0002 0003 0005 0008 +0007 0009 0011 0017 0019 0023 00290031

Delete 5:

0019 0005 0007 0011 0002 0003 0005 0008 +0007 0009 0011 0017 0019 0023 00290031

1800 8200 + szoo 0100 21001L00 2000 2000 2000 2000 2000 2000 0011 0005, 0007

ܓܙܘܢ ܕܗܘ tot ܢ 7007 ,0005- [dos1_028ܘ ܝ cons _ o023 ܝ oo11 _ o017 ܝ ܗ00 oops _ code = ooo7 nos2 _ cons

Found Leaf Node now Delete 5

1800 8200 + szoo 0100 21001L00 2000 2000 2000 2000 2000 0011 - 00050007

| 1SDU szun + ecug stup + ZLUD JIUn + spun ZUuz | ZUn – SUUg zucz བ DUDe/DDD7 ༽ UD1 1 07

Insert 10:

0019 - 0006 0007 0011 0002 0003+ 00087 0007 00090011 00170019 00230029 00310019 - 0006 0007 0011 0002 0003+ 00087 000700090011 00170019 00230029 0031

1800 6200 - Ezoo 0100 21001100 0000 2000 2000 2000 2000 0011 0007 0006 0019

Found Leaf Node now Insert 10

LEDO 8.00 8700 8100 21001100 0100 0000 2000 2000 2000 2000 0029 0006 00070011

Insert 8:

LEDO 8.00 8700 8100 21001100 0100 0000 2000 2000 2000 2000 0029 000600070011 0019

0019 000800070011 0029 00020003 +0008 +0007000900100011 00170019 0023 +00290031

0006 00070011 0029 0002000300080007 0009 00100011 00170019 002300290031

Found Leaf Node now Insert 8

0006 00070011 0002 00030008 +0007 2008 2009 0010 0011 00170019 00230029 0031

Above Node Contains too many keys so Perform Splitting

0019 0006 0007 0009 0011 0029 0002 0003000800070008 +0000 00100011 00170019 0023 +00290031

Above Node Contains too many keys so Perform Splitting

- UDDg ༽DD19 J017 0029 u002 JUs • J017 Up11 J017 + 001 J023 • J02 J031 །

Delete 23:

1800 8200 + E200 6100 2100 1100 +0100 8000 8000 2000 2000 2000 2000 6200 2000 9000 BLOD 6000

| Lego ਸੁਣ00 4 ਣop 500 * 20p up t op 5000 - 800D LOOD + 9੦੦੦ 4 coun zoop | 3 ਚ BਟOD ZDVDDBDXJQ Blop BDDD -

[ 1SDE szun Eczun sun + ZLOn EIUE + 0LUg spun + BUDD ZUJU + BZug + SDUD zupn ཐZ00 ། ་་་་ L ZUDE BUDD BLOp\ DJD -

Now Delete 23

DUDg ༽DD19 LUDE BUD0+ ༼0711 E :༈ TT [upp2 JUn + once + DE07 JUs upu c010 + 0011 J017 ; 001 + 002 DE31 །

Delete 19:

- 0029 ༽ 2013 LUDE BUD0+ ། Updf 0029 [upp2 JUn + once + DE07 JUs upu c010 + 0011 J017 ; 001 + 002 DE31 །

ཐLop\ 3ZUJ་ LUDE BUD0+ ༼0711 E :༈ JU29 ཐ JUn + once + DE07 c010 + 0011 J017 001 + 002 DE31 །

DUDg ༽DD19 052e0707 ༼0711 E :༈ 2g [upp2 JUn + once + DE07 JUs upu c010 + 0011 J017 001 002 DE31 །

Now delete 19

བ པ JU2 ༼0711 :༈ E LUDE BUD0+ ཐLop\ 3ZUJ་

Stealing from Right sibling

[ 1SUn + szun + ZLUE EIUE + 0LUE spun + BUDD ZUJU + BZug + Sun zupp བ བ བ བ བ བ བ བ པ » 2017 19070 scop་ UDD +

Delete 11:

1800 - 6200 +2100 1100 +01000000 3000 2000 2000 2000 2000 0031 0006, 0007 -B700 6000

- 0009 0029 0006, 0007 0011 0031 000200030008 +000700080009 00100011 001700290031

- 0009 0029 0006, 0007 0011 0031 000200030008 +000700080009 00100011 001700290031

Now Delete 11

8000+ + B200 9000 2000 0017 0031 0002 0003 + 0008 + 00070008 + 0000 0010 0 017 0 029 + 0031

Which is Required B+ Tree after performing all Operations

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