diff --git a/content/3_Bronze/Time_Comp.mdx b/content/3_Bronze/Time_Comp.mdx index 5ed650d..35ee952 100644 --- a/content/3_Bronze/Time_Comp.mdx +++ b/content/3_Bronze/Time_Comp.mdx @@ -15,11 +15,11 @@ description: Measuring how long your algorithm takes to run in terms of the inpu # Time Complexity -In programming contests, your program needs to finish running within a certain timeframe in order to receive credit. 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 $10^8$ but could be closer to $5 \cdot 10^8$ given good constant factors). +In programming contests, your program needs to finish running within a certain timeframe in order to receive credit. For USACO, this limit is $2$ seconds for C++ submissions, and $4$ seconds for Java/Python submissions. A conservative estimate for the number of operations the grading server can handle per second is $10^8$, but it could be closer to $5 \cdot 10^8$ given good constant factorsIf you don't know what constant factors are, don't worry -- we'll explain them below.. ## 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](https://en.wikipedia.org/wiki/Big_O_notation), which expresses worst-case time 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 constant factors and lower-order terms are generally omitted from $f(n)$. We'll see some examples of how this works, as follows. +We want a method to calculate 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](https://en.wikipedia.org/wiki/Big_O_notation), which expresses worst-case time 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 constant factors and lower-order terms are generally omitted from $f(n)$. 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. @@ -40,7 +40,7 @@ for(int i = 1; i <= n; i++){ } ``` -
+
```cpp int i = 0; @@ -58,7 +58,7 @@ for(int i = 1; i <= 5*n + 17; i++){ } ``` -
+
```cpp for(int i = 1; i <= n + 457737; i++){ @@ -76,7 +76,7 @@ for(int i = 1; i <= n; i++){ } ``` -In this example, the outer loop runs $O(n)$ iterations, and the inner loop runs anywhere between $1$ and $n$ iterations (which is a maximum of $n$). Since Big O notation calculates worst-case time complexity, we must (?) take the factor of $n$ from the inner loop. Thus, this code is $O(n^2)$. +In this example, the outer loop runs $O(n)$ iterations, and the inner loop runs anywhere between $1$ and $n$ iterations (which is a maximum of $n$). Since Big O notation calculates worst-case time complexity, we treat the inner loop as a factor of $n$.We can also do some math to calculate exactly how many times the code runs: 1+2+...+n = n*(n-1)/2 = (n^2 - n)/2 = O(n^2) Thus, this code is $O(n^2)$. ```cpp for(int i = 1; i <= n; i++){ @@ -99,7 +99,7 @@ for(int i = 1; i <= n + 58834; i++){ } ``` -The following code is $O(n^2 + nm)$, because it consists of two blocks of complexity $O(n^2)$ and $O(nm)$, and neither of them is a lower order function with respect to the other. +The following code is $O(n^2 + m)$, because it consists of two blocks of complexity $O(n^2)$ and $O(m)$, and neither of them is a lower order function with respect to the other. ```cpp for(int i = 1; i <= n; i++){ @@ -107,13 +107,19 @@ for(int i = 1; i <= n; i++){ // constant time code here } } -for(int i = 1; i <= n; i++){ - for(int j = 1; j <= m; j++){ - // more constant time code here - } +for(int j = 1; j <= m; j++){ + // more constant time code here } ``` +## Constant Factor + +The "Constant Factor" of an algorithm refers to the coefficient of the complexity of an algorithm. If an algorithm runs in $O(kn)$ time, where $k$ is a constant and $n$ is the input size, then the "constant factor" would be $k$. + +Normally when using the big-O notation, we ignore the constant factor: $O(3n) = O(n)$. This is fine most of the time, but sometimes we have an algorithm that just barely gets TLE, perhaps by just a few hundred milliseconds. When this happens, it is worth optimizing the constant factor of our algorithm. For example, if our code currently runs in $O(n^2)$ time, perhaps we can modify our code to make it run in $O(n^2/32)$ by using a bitset. (Of course, with big-O notation, $O(n^2) = O(n^2/32)$.) + +For now, don't worry about how to optimize constant factors -- just be aware of them. + ## Common Complexities and Constraints Complexity factors that come from some common algorithms and data structures are as follows: @@ -125,7 +131,7 @@ Complexity factors that come from some common algorithms and data structures are - Prime factorization of an integer, or checking primality or compositeness of an integer naively: $O(\sqrt{n})$ - 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) +- Sorting: usually $O(n \log n)$ for default sorting algorithms (mergesort, `Collections.sort`, `Arrays.sort`) - Java Quicksort `Arrays.sort` function on primitives: $O(n^2)$ - See "Introduction to Data Structures" for details. - Iterating through all subsets of size $k$ of the input elements: $O(n^k)$. For example, iterating through all triplets is $O(n^3)$.