proof format updates using gfm

content/CSE347/CSE347_L10.md

@@ -82,7 +82,8 @@ Let's try $R=S$.
 
 Claim: The comparative ratio is $2$.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 Case 1: The optimal offline solution takes the elevator, then $T+E\leq S$.
 
@@ -96,13 +97,14 @@ We wait for $R$ times and then take the stairs. In worst case, we wait for $R$ t
 
 Competitive ratio = $\frac{2R}{R}=2$.
 
-QED
+</details>
 
 Let's try $R=S-E$ instead.
 
 Claim: The comparative ratio is $max\{1,2-\frac{E}{S}\}$.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 Case 1: The optimal offline solution takes the elevator, then $T+E\leq S$.
 
@@ -116,7 +118,7 @@ We wait for $R=S-E$ times and then take the stairs.
 
 Competitive ratio = $\frac{S-E+S}{S}=2-\frac{E}{S}$.
 
-QED
+</details>
 
 What if we wait less time? Let's try $R=S-E-\epsilon$ for some $\epsilon>0$
 
@@ -162,7 +164,8 @@ Cache: $D A C$, the evict $D$ for $B$. 1 miss.
 
 Claim: LRU is $k+1$-competitive.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 Split the sequence into subsequences such that each subsequence contains $k+1$ distinct blocks.
 
@@ -174,7 +177,7 @@ The optimal offline solution: In each subsequence, must have at least $1$ miss.
 
 So the competitive ratio is at most $k+1$.
 
-QED
+</details>
 
 Using similar analysis, we can show that LRU is $k$ competitive.
 
@@ -184,8 +187,6 @@ Split the sequence into subsequences such that each subsequence LRU has $k$ miss
 
 Argue that OPT has at least $1$ miss in each subsequence.
 
-QED
-
 #### Many sensible algorithms are $k$-competitive
 
 **Lower Bound**: No deterministic online algorithm is better than $k$-competitive.
@@ -196,7 +197,8 @@ QED
 
 Say $c=2$. LRU cache has twice as much as cache. LRU is $2$-competitive.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 LRU has cache of size $2k$.
 
@@ -210,7 +212,7 @@ So competitive ratio is at most $\frac{ck}{(c-1)k}=\frac{c}{c-1}$.
 
 _Actual competitive ratio is $\sim \frac{c}{c-1+\frac{1}{k}}$._
 
-QED
+</details>
 
 ### Conclusion
 
@@ -273,7 +275,8 @@ Claim: RAND is $k$-competitive.
 2. There exists $k$ pages each of which is in the cache with probability $1-\frac{1}{k}$
 3. All other pages are in the cache with probability $0$.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 By induction.
 
@@ -297,11 +300,12 @@ Let $P$ be a page in the cache with probability $1-\frac{1}{k}$.
 
 With probability $\frac{1}{k}$, $P$ is not in the cache and RAND evicts $P'$ in the cache and brings $P$ to the cache.
 
-QED
+</details>
 
 MRU is $k$-competitive.
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 Case 1: Access MRU page.
 
@@ -317,4 +321,4 @@ Let's define the random variable $X$ as the number of misses of RAND MRU.
 
 $E[X]\leq 1+\frac{1}{k}$.
 
-QED
+</details>