diff --git a/cs3223/cheatsheet.pdf b/cs3223/cheatsheet.pdf
index c1e0afe..fe45048 100644
Binary files a/cs3223/cheatsheet.pdf and b/cs3223/cheatsheet.pdf differ
diff --git a/cs3223/cheatsheet.typ b/cs3223/cheatsheet.typ
index 9e5d74a..71c1259 100644
--- a/cs3223/cheatsheet.typ
+++ b/cs3223/cheatsheet.typ
@@ -296,3 +296,72 @@ Cost of index scan = $nin + #nle + nlo$
   - $N_"bucket"$: max no of index's primary/overflow pages accessed
   - $nlo= nso + min{||sigma_p_c (R)||, |R|}$ if I is not covering index for $sigma_p (R)$
 ]}
+
+= Join Algorithms
+Considerations when choosing join Algorithm
+- Types of join predicates (Equality / inequality)
+- Sizes of join operands
+- Allocated memory pages
+- Available Access Methods
+- *Notation*: $R join_(A) S$
+  - $R$ is outer relation, $S$ is inner relation
+- Nested Loop Join(NLJ) and Partition based join
+== Tuple-based NLJ
+Iterate through each page R, for each tuple in page, iterate through each page S, for each tuple in S, check if matches
+- *Cost*: $|R| + ||R|| times |S|$ - Read $S$ for each tuple in $R$
+- *Optimised*: Page based, Iterate through page R, iterate page S, then iterate tuples and check matching
+  - *Cost*: $|R| + |R| times |S|$ - Read $S$ for each page in $R$
+
+== Main Memory NLJ
+Assuming $|S| < |R|$, for optimal IO, compute $R join S$ with smaller operand as inner relation
+- Min pages needed: $B = |S| + 2$
+- 1 for $B_"outer"$, 1 for $B_"join"$, rest for $B_"inner"$ to read $S$
+- *Cost*: $|R| + |S|$
+
+== Block NLJ
+- $R join S = union.big^k_(i=1) (R_i join S)$, $k = ceil((|R|)/B_"outer")$
+- To min IO Cost, we min $|R|$ or max $B_"outer"$
+- Choose smaller table as outer, ($R "if" |R| < |S|$)
+- IO Cost: $|R| + ceil((|R|)/(B-2)) times |S|$
+
+== Index NLJ
+- Inner column: Table with index
+- Cost: $|R| + ||R|| times (N_"internal" + N_"leaf" + N_"lookup")$
+  - Scan $R$ + search index for each tuple in $S$
+
+== Sort-Merge Join
+- $R join S = union.big_(i in J) (R_i times S_i), "where" J = {i | R_i != emptyset, S_i != emptyset}$
+  - $X_i subset.eq X$ is partition of $X$ where all records have join attribute value $i$
+- *Cost*: Sort $R$ + Sort $S$ + Merging cost
+  - Sorting cost: $0$ if sorted, or internal sorting
+  - Min merging cost: max $|B_"inner"|$
+  - $S$ to be inner relation if $|"Max"P_S|<= |"Max"P_R|$
+    - $"Max"P_x$: largest matching $X$-partition
+  - If $|"Max"P_S|<= B-2$, Cost: $|R| + |S|$
+  - else $|S| + ceil((|S|)/(B-2)) |R|$
+== Optimized SMJ
+- $S$ to be inner relation if $|"Max"P_S|<= |"Max"P_R|$
+- Find $i$ & $j$, $B > N(R, i) + N(S, j)$
+  - $N(X, 0) = ceil((|X|) / B)$ and $N(X, k) = ceil((N(X, k-1))/(B-1))$
+- *Cost*: $2|R|(i+1) + 2|S|(j+1) + |R| + |S|$
+  - Partial sort R + Partial sort S + merge & join
+
+== Grace Hash Join
+- Partition $R$: $R_1, ..., R_k$, Partition $S$: $S_1, ..., S_k$
+- Read $R_i$ to build hash table (Build relation)
+- Read $S_i$ to probe hash table (Probe relation)
+- $R_i$ overflows if hash table is larger than memory page allocated
+  - Recursively partition $R_i$ and $S_i$
+- *To avoid overflow*
+- Pick smaller operand $R$ as build relation $(|R| <= |S|)$
+- Partitioning: Max build partitions to min size
+  - 1 page to read build $R$
+  - 1 page to output $B-1$ partitions
+- Probing: Max memory allocated for hash table
+  - 1 page to read probe $S_i$
+  - 1 page to output $S_i join R_i$
+  - $B-2$ pages for $R_i$'s hash table
+- $B > sqrt(f times |R|)$: size to avoid overflow
+- *Cost*: $2(|R| + |S|) + (|R| + |S|) = 3(|R| + |S|)$
+  - 2 for partitioning, 1 for probing
+