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;g5?ceNX7m(t=fExpd4ft0DvvM=Da08h8rPLMEWg!5B z5Oa_JDD(@{MIoTQ-0~O6*q!*9FKhjUz-wy4ZEojiYvjytWN!~lvM&b~CZK7&On<++ z;IaX(HE;ncxdMx@k;T;q8wkG{l$Ytk-@wPK50@`5ivL{(P(x0RThz|P<-*xgqWrct ziV$&gWo59DoV2pIjTn=Mox6+>AK#^dzq4Mn0iZ}hf$b0RN-qKpz4={p^PP ze|KX*J|CGQ&`JMclMgsi{%gp%3KAkffUvQ%xQG##zWY0WzB$s}uuV_s2_fRbhz=DLv4A^J^SpK>t!yV8K1w_KZi~{=kN5am=2JCK_qhHck zK)^N?FwwdABY|Ggm^s;5K)_J+I}Kn1%yW$f1mxS-@WK4uU*=N~j?FxQ_nV6)|VSzsUNx?EOf zu8Zl~Z!%a}z!&>8ztjGyEGzrPB;G;qP)eIR0!02e5j- z&d2!&HdsKM%*=oCU2akTE`yWxVz1&F4G2Vk&^Xwx;Te#@`KL_*_5Q>G_|LLjKtQ-I zgNyYaXzbwYb^!yi=@)-EI~qZ3%p8B*7^MPvXm$~`fgwQI&d!8nRk5*?cVHH6cW{O?zni!ERrcXYlSBUr#7PA+y7N=h*W@&5-xAg`kU diff --git a/main.typ b/main.typ index c5b271f..e34b84f 100644 --- a/main.typ +++ b/main.typ @@ -38,8 +38,8 @@ logarithmic time. It is an extension of the B-tree and is extensively used in databases and filesystems for indexing. B+ tree is *Balanced*; Order (n): Defined such that each node (except root) can have at most $n$ children (pointers) and at least $⌈n/2⌉$ children; *Internal nodes hold* between -⌈n/2⌉−1⌈n/2⌉−1 and n−1n−1 keys; Leaf nodes also hold between $⌈n/2⌉−1$ and -$n−1$ keys but also store all data values corresponding to the keys; *Leaf +$⌈n/2⌉−1$ and $n−1$ keys (values); Leaf nodes hold between $⌈frac(n −1,2)⌉$ and +$n−1$ keys, but also store all data values corresponding to the keys; *Leaf Nodes Linked*: Leaf nodes are linked together, making range queries and sequential access very efficient. @@ -49,8 +49,8 @@ sequential access very efficient. key to the parent, and adjust pointers; + Leaf split: $1$ to $ceil(frac(n,2)) $ and $ceil(frac(n,2)) + 1 $ to $n$ as two leafs. Promote the lowest from the 2nd one. - + Node split: $1$ to $ceil(frac(n+1, 2)) - 1 $ and $ceil(frac(n,2)) + 1 $. - $ceil(frac(n+1, 2)) - 1 $ gets moved up. + + Node split: $1$ to $ceil(frac(n+1, 2)) - 1 $ and $ceil(frac(n,2)) + 1$ to $n$. + $ceil(frac(n+1, 2))$ gets moved up. - If a split propagates to the root and causes the root to overflow, split the root and create a new root. Note: root can contain less than $ceil(frac(n,2)) - 1$ keys. @@ -196,19 +196,19 @@ $b_b$ blocks are allocated for the input buffer and each output buffer. E_2) join_(theta_2 and theta_3) |||| E_3 ≡E_1 join_(theta_1 or theta_3) (E_2 join_theta_2 E_3)$ + Selection distribution: $σ_(θ_1)(E_1 ⋈_θ E_2) ≡(σ_(θ_1) (E_1)) ⋈_θ E_2$; - $σ_(θ_1∧θ_2)(E_1 ⋈_θ E_2) ≡(σ_(θ_1)(E_1)) ⋈_θ (σ_(θ_2)(E_2))$ + $σ_(θ_1∧θ_2)(E_1 ⋈_θ E_2) ≡ (σ_(θ_1)(E_1)) ⋈_θ (σ_(θ_2)(E_2))$ + Projection distribution: - $Π_(L_1∪L_2) (E_1 ⋈_θ E_2) ≡(Π_(L_1(E_1)) ⋈_θ (Π_(L_2)(E_2))$ |||| $Π(L_1∪L_2) (E_1 ⋈_θ E_2) ≡Π_(L_1∪L_2) ((Π_(L_1∪L_3) (E_1)) ⋈_θ (Π_(L_2∪L_4) (E_2)))$ -+ Union and intersection commmutativity: E1 ∪E2 ≡E2 ∪E1 |||| - E1 ∩E2 ≡E2 ∩E1 -+ Set union and intersection are associative: (E1 ∪E2) ∪E3 ≡E1 ∪(E2 ∪E3) |||| (E1 - ∩E2) ∩E3 ≡E1 ∩(E2 ∩E3); ++ Union and intersection commmutativity: $E_1 ∪E_2 ≡E_2 ∪E_1 |||| - E_1 ∩E_2 ≡E_2 ∩E_1$ ++ Set union and intersection are associative: $(E_1 ∪E_2) ∪E_3 ≡E_1 ∪(E_2 ∪E_3) |||| (E_1 + ∩E_2) ∩E_3 ≡E_1 ∩(E_2 ∩E_3)$; + The selection operation distributes over the union, intersection, and - set-difference operations: σθ(E1 ∪E2) ≡σθ(E1) ∪σθ(E2) |||| σθ(E1 ∩E2) ≡σθ(E1) - ∩σθ(E2) |||| σθ(E1 −E2) ≡σθ(E1) −σθ(E2) |||| σθ(E1 ∩E2) ≡σθ(E1) ∩E2 |||| σθ(E1 −E2) ≡σθ(E1) - −E2 |||| 12. -+ The projection operation distributes over the union operation - ΠL(E1 - ∪E2) ≡(ΠL(E1)) ∪(ΠL(E2)) + set-difference operations: $σ_θ(E_1 ∪E_2) ≡σ_θ(E_1) ∪σ_θ(E_2) |||| σ_θ(E_1 ∩E_2) ≡σ_θ(E_1) + ∩σ_θ(E_2) |||| σ_θ(E_1 −E_2) ≡σ_θ(E_1) −σ_θ(E_2) |||| σ_θ(E_1 ∩E_2) ≡σ_θ(E_1) ∩E_2 |||| σ_θ(E_1 −E_2) ≡σ_θ(E_1) + −E_2$; ++ The projection operation distributes over the union operation - $Π_L(E_1 + ∪E_2) ≡(Π_L_(E_1)) ∪(Π_L(E_2))$. // == Operations // @@ -306,8 +306,6 @@ conflict serializable. - *Read committed* allows only committed data to be read, but does not require re- peatable reads. - *Read uncommitted* allows uncommitted data to be read. Lowest isolation level allowed by SQL. - - == Protocols We say that a schedule S is *legal* under a given locking protocol if S is a possible @@ -318,6 +316,12 @@ is acyclic. === Lock-based +*Shared Lock* -- If a transaction $T_i$ has obtained a shared-mode lock (denoted by $S$) on +item Q, then Ti can read, but cannot write, $Q$. \ +*Exclusive Lock* -- If a transaction $T_i$ has obtained an exclusive-mode lock +(denoted by $X$) on item Q, then Ti can both read and write $Q$. + + ==== 2-phased lock protocol *The Two-Phase Locking (2PL)* Protocol is a concurrency control method used in @@ -357,7 +361,7 @@ must validate that no other transactions have modified the data it accessed. *Commit Phase:* If the validation is successful, the transaction commits and applies its changes. If not, it rolls back and may be restarted. -=== Version isolation +// === Version isolation = Logs @@ -403,8 +407,10 @@ It performs rollback by scanning the log backward from the end: - $$ -- an update log record, indicating that transaction $T_i$ has performed a write on data item $X_j$. $X_j$ had value $V_1$ before - the write and has value $V_2$ after the write. -- $$ -- $T_i$ has started. -- $$ -- $T_i$ has committed. -- $$ -- $T_i$ has aborted. + the write and has value $V_2$ after the write; +- $$ -- $T_i$ has started; +- $$ -- $T_i$ has committed; +- $$ -- $T_i$ has aborted; +- $<#[checkpoint] {T_0, T_1, dots, T_n}>$ -- a checkpoint with a list of active + transactions at the moment of checkpoint.

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