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pages/CSE347/Exam_reviews/CSE347_E2.md

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+# Exam 2 Review
+
+## Reductions
+
+We say that a problem $A$ reduces to a problem $B$ if there is a **polynomial time** reduction function $f$ such that for all $x$, $x \in A \iff f(x) \in B$.
+
+To prove a reduction, we need to show that the reduction function $f$:
+
+1. runs in polynomial time
+2. $x \in A \iff f(x) \in B$.
+
+### Useful results from reductions
+
+1. $B$ is at least as hard as $A$ if $A \leq B$.
+2. If we can solve $B$ in polynomial time, then we can solve $A$ in polynomial time.
+3. If we want to solve problem $A$, and we already know an efficient algorithm for $B$, then we can use the reduction $A \leq B$ to solve $A$ efficiently.
+4. If we want to show that $B$ is NP-hard, we can do this by showing that $A \leq B$ for some known NP-hard problem $A$.
+
+$P$ is the class of problems that can be solved in polynomial time. $NP$ is the class of problems that can be verified in polynomial time.
+
+We know that $P \subseteq NP$.
+
+### NP-complete problems
+
+A problem is NP-complete if it is in $NP$ and it is also NP-hard.
+
+#### NP
+
+A problem is in $NP$ if 
+
+1. there is a polynomial size certificate for the problem, and
+2. there is a polynomial time verifier for the problem that takes the certificate and checks whether it is a valid solution.
+
+#### NP-hard
+
+A problem is NP-hard if every instance of $NP$ hard problem can be reduced to it in polynomial time.
+
+List of known NP-hard problems:
+
+1. 3-SAT (or SAT)
+2. Independent Set
+3. Vertex Cover
+4. 3-coloring
+5. Hamiltonian Cycle
+6. Hamiltonian Path
+
+## Approximation Algorithms
+
+## Randomized Algorithms
+
+## Online Algorithms
+
+
+

pages/CSE347/_meta.js

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 export default {
-    index: {
+    index: "Course Description",
-        display: 'hidden'
+    "---":{
+        type: 'separator'
     },
     Exam_reviews: "Exam reviews",
     CSE347_L1: "Lecture 1",

pages/CSE442T/_meta.js

@@ -1,6 +1,7 @@
 export default {
-    index: {
+    index: "Course Description",
-        display: 'hidden'
+    "---":{
+        type: 'separator'
     },
     Exam_reviews: "Exam reviews",
     CSE442T_L1: "Lecture 1",

pages/Math4111/Math4111_L1.md

@@ -2,7 +2,7 @@
 
 ## Introduction
 
-Reading is not recommended before class, it;s hard.
+Reading is not recommended before class, it's hard.
 
 ## Chapter 1: The real number and complex number systems
 

pages/Math4111/_meta.js

@@ -1,6 +1,7 @@
 export default {
-    index: {
+    index: "Course Description",
-        display: 'hidden'
+    "---":{
+        type: 'separator'
     },
     Exam_reviews: "Exam reviews",
     Math4111_L1: "Lecture 1",

pages/Math429/_meta.js

@@ -1,6 +1,7 @@
 export default {
-    index: {
+    index: "Course Description",
-        display: 'hidden'
+    "---":{
+        type: 'separator'
     },
     Math429_L1: "Lecture 1",
     Math429_L2: "Lecture 2",