Update Merge Sort.md

Grade 10/Computer Science/ICS4U1/Sorting Methods/Merge Sort.md

@@ -23,3 +23,44 @@ def mergesort(left, right, array[]):
 - Latop computer can run close to $`10^8`$ per second.
 - Supercomputer can execute $`10^{12}`$ compares per second.
 
+### Insertion Sort $`(N^2)`$
+|computer|thousand $`(10^3)`$|million $`(10^6)`$|billion $`(10^9)`$|
+|:-------|:-------|:-------|:-------|
+|home|instant|2.8 hours|317 years|
+|super|instant|1 second|1 week|
+
+### Merge Sort $`(N \log N)`$
+
+|computer|thousand $`(10^3)`$|million $`(10^6)`$|billion $`(10^9)`$|
+|:-------|:-------|:-------|:-------|
+|home|instant|1 second|18 min|
+|super|instant|instant|instant|
+
+
+## Proof Sketch
+
+**Proposition** Merge sort uses $`\le N \log N`$ compares to sort an array of length $`N`$.
+
+**Proof Sketch** The number of compares $`C(N)`$ to mergesort an array of length $`N`$ satisfies the recurrence:
+
+```math
+C(N) \le C(\lceil N/2 \rceil) + C(\lfloor N/2 \rfloor) + N \text{ for } N \gt 1, \text{ with } C(1) = 0.
+```
+
+## Divide and conquer recurrence induction proof
+
+**Proposition**. If $`D(N)`$ satisfies $`D(N)=2D(N/2) + N`$ for $`N \gt 1`$, with $`D(1) = 0,`$ then $`D(N) = N \log N`$.
+
+**Proof** [assuming $`N`$ is a power of $`2`$]
+- Base case: $`N = 1`$
+- Inductive hypothesis: $`D(N) = N \log N`$.
+- Goal: show that $`D(2N) = (2N) = \log (2N)`$
+
+$`D(2N) = 2D(N) + 2N`$ <br>
+$`\quad = 2 N \log N + 2N`$ <br>
+$`\quad = 2 N (log (2N) - 1) + 2N`$ <br>
+$`\quad = 2 N \log (2N)`$
+
+
+## Resources
+- Princeton University: https://algs4.cs.princeton.edu/lectures/22Mergesort.pdf