add slide 26

lectures/parallelism/slides.qmd

@@ -217,8 +217,11 @@ Wikipedia
 
 
 ## Types of parallelism
-* Data-level parallelism - what we've been discussing
-* Task-level parallelism - Example: Atmosphere ocean coupling
+
+* Data-level parallelism
+  * what we've been discussing
+* Task-level parallelism
+  * Example: Atmosphere ocean coupling
   
 
 ## Precondition for parallel execution
@@ -331,6 +334,62 @@ d = c - b;  // S2
 ```
 :::
 
+## Bernstein's parallelism conditions {.alignleft}
+
+:::{.fragment .semi-fade-out}
+```c++
+c = a + b;  // S1
+d = a - b;  // S2
+```
+:::
+
+
+:::{.fragment}
+Replace `a` with `c` in statement 2
+```c++
+c = a + b;  // S1
+d = c - b;  // S2
+```
+:::
+
+:::{.smaller}
+
+:::: {.columns}
+
+::: {.column width="50%"}
+
+:::{.fragment}
+
+Read and write sets for S1 and S2:
+$$
+R_1 = \{a,b\} ; W_1 = \{c\} \\
+R_2 = \{c,b\} ; W_2 = \{d\} \\
+$$
+
+:::
+:::
+
+::: {.column width="50%"}
+
+:::{.fragment}
+Bernstein's conditions:
+
+$$
+R_1 \cap W_2 = \emptyset \\
+W_1 \cap R_2 = \{c\} \\
+W_1 \cap W_2 = \emptyset
+$$
+:::
+
+:::
+:::
+::::
+
+:::{.fragment}
+S1 and S2 can NOT be executed in parallel!
+:::
+
+
 ## Best practices
 
 * Parallelisation should not change the results! Exceptions to be discussed next week!