how to build a M-tree

1. M-tree D1 Guo Bo

2. Insert: Node N is leaf o10 o7 o2 Insert entry(𝑂𝑛) into a leaf 𝑁, 𝑜𝑝1 𝑁 is a leaf IF 𝑁 is not full Store entry(𝑂𝑛)in𝑁 o5 o11 o3 𝑂𝑛 o12 o8 o1 o6 o4 o9

3. Insert: Node N is leaf(con.) o10 o7 o2 o5 o11 o3 o12 o8 o1 o6 o4 𝑂𝑛 o9 Insert entry(𝑂𝑛) into a leaf 𝑁, 𝑜𝑝1 𝑁 is a leaf IF 𝑁 is full Split (𝑁,𝑒𝑛𝑡𝑟𝑦(𝑂𝑛))

4. Insert: Node N is not leaf1 o10 o7 o2 Insert(𝑁:node, entry(𝑂𝑛):M-Tree entry) Let 𝒩 be the set of entries in node 𝑁 𝑁=𝑜1,𝑜10 Let 𝒩𝑖𝑛= entries such that 𝑑𝑂𝑟,𝑂𝑛≤𝑟𝑂𝑟 𝑑𝑂1,𝑂𝑛>𝑟(𝑂𝑟),𝑑𝑂10,𝑂𝑛>𝑟(𝑂𝑟) 𝒩𝑖𝑛={0} If 𝒩𝑖𝑛=0 Let entry𝑂𝑟∗∈𝒩: 𝑑𝑂𝑟∗,𝑂𝑛−𝑟(𝑂𝑟∗)is minimum Let𝑟𝑂𝑟∗=𝑑𝑂𝑟∗,𝑂𝑛 Insert∗𝑝𝑡𝑟(𝑇𝑂𝑟∗,,𝑒𝑛𝑡𝑟𝑦(𝑂𝑛)) ∗𝑝𝑡𝑟(𝑇𝑂𝑟∗, o5 o11 o3 o12 o8 o1 o6 o4 o9 𝑂𝑛 ∗𝑝𝑡𝑟(𝑇𝑂𝑟∗,

5. o10 o10 o7 o7 o2 o2 Insert: Node N is not leaf o5 o5 o11 o11 o3 o3 o12 o12 o8 o8 o1 o1 o6 o6 o4 o4 o9 o9 𝑂𝑛 𝑂𝑛

6. Split: N is not root (con.) o10 o7 o2 Split (Node 𝑁 ;entry(𝑂𝑛)) Let 𝒩=entries of node 𝑁∪ Entry(𝑂𝑛) 𝒩={𝑂1,𝑂6,𝑂12,𝑂𝑛} 𝑁 is not the root Let 𝑂𝑝 be the parent of𝑁,stored in 𝑁𝑝 𝑂𝑝 is 𝑂1and 𝑁𝑝 (parent pivot of𝑁 ) Allocate a new node 𝑁′ Randomly choose two pivot objects 𝑂𝑝1,𝑂𝑝2 𝑓𝑟𝑜𝑚 𝒩. Partition(𝒩,𝑂𝑝1,𝑂𝑝2)->𝒩1,𝒩2 𝒩1{𝑂1,𝑂12}& 𝒩2{𝑂6,𝑂𝑛} Store 𝒩1’s entries in 𝑁;𝒩2’s entries in 𝑁′ 𝑁 is not the current root Replace entry(𝑂𝑝)with entry(𝑂𝑝1)in 𝑁𝑝 𝑁𝑝 is not full, store entry(𝑂𝑝2)in 𝑁𝑝 𝑁 o5 o11 o3 o12 𝑁 o1 o6 o4 o9 𝑂𝑛 𝑁′ 𝑁′

7. o10 o10 o7 o7 o2 o2 Split: N is not root o5 o5 o11 o11 o3 o3 o12 o12 𝑁 o1 o1 o6 o6 o4 o4 o9 o9 𝑂𝑛 𝑂𝑛 𝑁′ o12 𝑁′ 𝑁 o6 o1 𝑂𝑛

8. o10 o2 Split detail A(Update) Split (Node 𝑁 ;entry(𝑂𝑛)) Let 𝒩=entries of node 𝑁∪ Entry(𝑂𝑛) 𝒩={𝑂4,𝑂3,𝑂9,𝑂𝑛} 𝑁 is not the root Let 𝑂𝑝 be the pivot object of𝑁,stored in 𝑁𝑝 𝑂𝑝 is 𝑂4 Allocate a new node 𝑁′ Randomly choose two pivot objects 𝑂𝑝1,𝑂𝑝2 𝑓𝑟𝑜𝑚 𝒩. 𝑂𝑝1=𝑂4,𝑂𝑝2=𝑂9 Partition(𝒩,𝑂𝑝1,𝑂𝑝2)->𝒩1,𝒩2 𝒩1{𝑂𝑛,𝑂4}& 𝒩2{𝑂9,𝑂3} Store 𝒩1’s entries in 𝑁;𝒩2’s entries in 𝑁′ 𝑁 is not the current root Replace entry(𝑂𝑝)with entry(𝑂𝑝1)in 𝑁𝑝 𝑁𝑝 is not full, store entry(𝑂𝑝2)in 𝑁𝑝 o7 o11 o12 o8 o3 𝒩2 o1 𝑂𝑛 o6 o4 o9 𝒩1 𝑁 𝑁′

9. o7 o2 Split detail B(Update) Split (Node 𝑁 ;entry(𝑂𝑛)) Let 𝒩=entries of node 𝑁∪ Entry(𝑂𝑛) 𝒩={𝑂3,𝑂4,𝑂9,𝑂𝑛} 𝑁 is not the root Let 𝑂𝑝 be the parent of𝑁,stored in 𝑁𝑝 𝑂𝑝 is 𝑂4(parent pivot of𝑁 ) Allocate a new node 𝑁′ Randomly choose two pivot objects 𝑂𝑝1,𝑂𝑝2 𝑓𝑟𝑜𝑚 𝒩. 𝑂𝑝1=𝑂𝑛,𝑂𝑝2=𝑂9 Partition(𝒩,𝑂𝑝1,𝑂𝑝2)->𝒩1,𝒩2 𝒩1{𝑂𝑛,𝑂4}& 𝒩2{𝑂9,𝑂3} Store 𝒩1’s entries in 𝑁;𝒩2’s entries in 𝑁′ 𝑁 is not the current root Replace entry(𝑂𝑝)with entry(𝑂𝑝1)in 𝑁𝑝 Node 𝑁𝑝 is full. Split (𝑁𝑝, entry(𝑂𝑝2)) o10 o5 o6 𝒩2 o8 o3 𝑂𝑛 o4 𝒩1 o9 𝑁

10. o7 o2 Split detail B(Update)(Con.) Split (Node 𝑁 ;entry(𝑂𝑛)) Let 𝒩=entries of node 𝑁∪ Entry(𝑂𝑛) 𝒩={𝑂2,𝑂7,𝑂𝑛,𝑂9} 𝑁 is the root Allocate a new node 𝑁′ Randomly choose two pivot objects 𝑂𝑝1,𝑂𝑝2 𝑓𝑟𝑜𝑚 𝒩. 𝑂𝑝1=𝑂𝑛,𝑂𝑝2=𝑂2 Partition(𝒩,𝑂𝑝1,𝑂𝑝2)->𝒩1,𝒩2 𝒩1{𝑂𝑛,𝑂9}& 𝒩2{𝑂2,𝑂7} Store 𝒩1’s entries in 𝑁;𝒩2’s entries in 𝑁′ 𝑁 is the current root Allocate a new root node, 𝑁𝑝 Store entry 𝑂𝑝1 and entry 𝑂𝑝2 in 𝑁𝑝 𝒩2 o10 o5 o6 o8 o3 𝒩1 𝑂𝑛 o4 o9 𝑁 𝑁′ 𝑁𝑝