Update 0_Silver_Complexity.md

content/4_Silver/0_Silver_Complexity.md

@@ -1,10 +1,10 @@
 
 
-\chapter{Time/Space Complexity and Algorithm Analysis}
+# Time/Space Complexity and Algorithm Analysis 
 In programming contests, there is a strict limit on program runtime. This means that in order to pass, your program needs to finish running within a certain timeframe. For USACO, this limit is 4 seconds for Java submissions. A conservative estimate for the number of operations the grading server can handle per second is {10}^8 (but is really closer to 5 \cdot 10^8 given good constant factors).
 
 
-\section{Big O Notation and Complexity Calculations}
+# Big O Notation and Complexity Calculations
 We want a method of how many operations it takes to run each algorithm, in terms of the input size n. Fortunately, this can be done relatively easily using Big O notation, which expresses worst-case complexity as a function of n, as n gets arbitrarily large. Complexity is an upper bound for the number of steps an algorithm requires, as a function of the input size. In Big O notation, we denote the complexity of a function as O(f(n)), where f(n) is a function without constant factors or lower-order terms. We'll see some examples of how this works, as follows.
 
 The following code is O(1), because it executes a constant number of operations.
@@ -91,43 +91,36 @@ for(int i = 1; i <= n; i++){
 }
 ```
 
-\section{Common Complexities and Constraints}
+# Common Complexities and Constraints
 Complexity factors that come from some common algorithms and data structures are as follows:
 
-    - Mathematical formulas that just calculate an answer: O(1)
-    - Unordered set/map: O(1) per operation
-    - Binary search: O(log n)
-    - Ordered set/map or priority queue: O(log n) per operation
-    - Prime factorization of an integer, or checking primality or compositeness of an integer: O(\sqrt{n})
-    - Reading in n items of input: O(n)
-    - Iterating through an array or a list of n elements: O(n)
-    - Sorting: usually O(n log n) for default sorting algorithms (mergesort, for example Collections.sort or Arrays.sort on objects)
-    - Java Quicksort Arrays.sort function on primitives on pathological worst-case data sets, don't use this in CodeForces rounds
-    - Iterating through all subsets of size k of the input elements: O(n^k). For example, iterating through all triplets is O(n^3).
-    - Iterating through all subsets: O(2^n)
-    - Iterating through all permutations: O(n!)
+- Mathematical formulas that just calculate an answer: O(1)
+- Unordered set/map: O(1) per operation
+- Binary search: O(log n)
+- Ordered set/map or priority queue: O(log n) per operation
+- Prime factorization of an integer, or checking primality or compositeness of an integer: O(\sqrt{n})
+- Reading in n items of input: O(n)
+- Iterating through an array or a list of n elements: O(n)
+- Sorting: usually O(n log n) for default sorting algorithms (mergesort, for example Collections.sort or Arrays.sort on objects)
+- Java Quicksort Arrays.sort function on primitives on pathological worst-case data sets, don't use this in CodeForces rounds
+- Iterating through all subsets of size k of the input elements: O(n^k). For example, iterating through all triplets is O(n^3).
+- Iterating through all subsets: O(2^n)
+- Iterating through all permutations: O(n!)
 
 
 Here are conservative upper bounds on the value of n for each time complexity. You can probably get away with more than this, but this should allow you to quickly check whether an algorithm is viable.
 
-\begin{center}
-    \begin{tabular}{c c}
-    \toprule
-        n & Possible complexities \\
-    \midrule
-        n \leq 10 & O(n!), O(n^7), O(n^6) \\
-        n \leq 20 & O(2^n \cdot n), O(n^5) \\
-        n \leq 80 & O(n^4) \\
-        n \leq 400 & O(n^3) \\
-        n \leq 7500 & O(n^2) \\
-        n \leq 7 \cdot 10^4 & O(n \sqrt n) \\
-        n \leq 5 \cdot 10^5 & O(n \log n) \\
-        n \leq 5 \cdot 10^6 & O(n) \\
-        n \leq 10^{12} & O(\sqrt{n} \log n), O(\sqrt{n}) \\
-        n \leq 10^{18} & O(\log^2 n), O(\log n), O(1) \\
-    \bottomrule
-    \end{tabular}
-\end{center}
+- n | Possible complexities 
+- n <= 10 | O(n!), O(n^7), O(n^6) 
+- n <= 20 | O(2^n \cdot n), O(n^5) 
+- n <= 80 | O(n^4) 
+- n <= 400 | O(n^3) 
+- n <= 7500 | O(n^2) 
+- n <= 7 * 10^4 | O(n \sqrt n) 
+- n <= 5 * 10^5 | O(n \log n) 
+- n <= 5 * 10^6 | O(n) 
+- n <= 10^18 | O(\log^2 n), O(\log n), O(1) 
+