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We aim to meet on a **weekly basis** to discuss and work together on challenging and interesting programming problems, and get some valuable experience for technical interviews along the way! + +## Textbook + +Our main resource will be the [USACO Guide](https://usaco.guide/), which has a bunch of problems and walks through important topics in competitive programming. Another useful resource is [_Competitive Programming 3_](http://www.sso.sy/sites/default/files/competitive%20programming%203_1.pdf). + +## Useful Resources (thanks to Nathan!) + +- [Competitive Programming Handbook](https://cses.fi/book/book.pdf) which lists a lot of common topics, and also suitable for someone who has just started with C++. +- [A website](https://cp-algorithms.com/) with a bunch of algorithms for competitions. +- [Competitive programming course at NUS](https://www.comp.nus.edu.sg/~stevenha/cs3233.html). From this link you can get access to a bunch of Kattis problem sets for various topics. +- [Codeforces’ Gym](http://codeforces.com/gyms?filterContestType=Official+ACM-ICPC+Contest&order=ID_DESC), which is a place that has many ICPC problem sets from regions around the world. If we need team practice for ICPC, I reckon this is a very good place. +- Team notebooks: [KTH](https://github.com/kth-competitive-programming/kactl), [Stanford](https://github.com/jaehyunp/stanfordacm) +- If we want to construct our own team notebook later on, we can use [this website](https://judge.yosupo.jp) to check the correctness of our code. +- [Geometry for Competitive Programming](https://vlecomte.github.io/cp-geo.pdf) book by V. Lecomte. + +## Week 10: Bronze (Simulation) + +We'll be working through as many of the problems on [this page](https://usaco.guide/bronze/simulation), to get practice with simulation problems and parsing input/writing output manually (rather than just using the LeetCode interface). List of problems (same as on the website, note that submitting/checking solutions to these problems requires registering on the [USACO website](http://www.usaco.org/)): + +- [Shell Game](http://www.usaco.org/index.php?page=viewproblem2&cpid=891), Easy +- [Mixing Milk](http://www.usaco.org/index.php?page=viewproblem2&cpid=855), Easy +- [Speeding](http://www.usaco.org/index.php?page=viewproblem2&cpid=568), Easy +- [The Lost Cow](http://www.usaco.org/index.php?page=viewproblem2&cpid=735), Easy +- [The Bovine Shuffle](http://www.usaco.org/index.php?page=viewproblem2&cpid=760), Easy +- [Circular Barn](http://www.usaco.org/index.php?page=viewproblem2&cpid=616), Normal +- [Block Game](http://www.usaco.org/index.php?page=viewproblem2&cpid=664), Normal +- [Team Tic Tac Toe](http://www.usaco.org/index.php?page=viewproblem2&cpid=831), Normal +- [Why Did the Cow Cross the Road](http://www.usaco.org/index.php?page=viewproblem2&cpid=713), Normal +- [Mowing the Field](http://www.usaco.org/index.php?page=viewproblem2&cpid=593), Normal +- [Milk Measurement](http://www.usaco.org/index.php?page=viewproblem2&cpid=761), Hard +- [Angry Cows](http://www.usaco.org/index.php?page=viewproblem2&cpid=592), Hard + +# Past Problems + +## Week 4: Dynamic Programming + +We will work through the following problems, each based around dynamic programming. You are highly encouraged to try out more problems to get familiar with these graphs and traversals! + +**Easy: [Climbing Stairs](https://leetcode.com/problems/climbing-stairs/)** + +> You are climbing a stair case. It takes `n` steps to reach to the top. +> +> Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top? + +**"EaSy": [Maximum Subarray](https://leetcode.com/problems/maximum-subarray/)** + +> Given an integer array nums, find the contiguous subarray (containing at least one number) which has the largest sum and return its sum. + +**Medium: [Coin Change](https://leetcode.com/problems/coin-change/)** + +> You are given coins of different denominations and a total amount of money `amount`. Write a function to compute the fewest number of coins that you need to make up that amount. If that amount of money cannot be made up by any combination of the coins, return `-1`. + +**Hard (extension): [Edit Distance](https://leetcode.com/problems/edit-distance/)** + +> Given two words `word1` and `word2`, find the minimum number of operations required to convert `word1` to `word2`. +> +> You have the following 3 operations permitted on a word: +> +> 1. Insert a character +> 2. Delete a character +> 3. Replace a character + +## Week 3: Graphs, Breadth-First Search and Depth-First Search + +We will work through the following problems, each based around graphs. You are highly encouraged to try out more problems to get familiar with these graphs and traversals! + +**Easy: [Flood Fill](https://leetcode.com/problems/flood-fill/)** + +> An image is represented by a 2-D array of integers, each integer representing the pixel value of the image (from 0 to 65535). +> +> Given a coordinate `(sr, sc)` representing the starting pixel (row and column) of the flood fill, and a pixel value `newColor`, "flood fill" the image. +> +> To perform a "flood fill", consider the starting pixel, plus any pixels connected 4-directionally to the starting pixel of the same color as the starting pixel, plus any pixels connected 4-directionally to those pixels (also with the same color as the starting pixel), and so on. Replace the color of all of the aforementioned pixels with the `newColor`. +> +> At the end, return the modified image. + +**Medium: [Number of Islands](https://leetcode.com/problems/number-of-islands/)** + +> Given a 2d grid map of `'1'`s (land) and `'0'`s (water), count the number of islands. An island is surrounded by water and is formed by connecting adjacent lands horizontally or vertically. You may assume all four edges of the grid are all surrounded by water. + +**Medium: [Course Schedule](https://leetcode.com/problems/course-schedule/)** + +> There are a total of `numCourses` courses you have to take, labeled from `0` to `numCourses-1`. +> +> Some courses may have prerequisites, for example to take course `0` you have to first take course `1`, which is expressed as a pair: `[0,1]` +> +> Given the total number of courses and a list of prerequisite pairs, is it possible for you to finish all courses? + +**Hard: [Bus Routes](https://leetcode.com/problems/bus-routes/)** + +> We have a list of bus routes. Each `routes[i]` is a bus route that the `i`-th bus repeats forever. For example if `routes[0] = [1, 5, 7]`, this means that the first bus (`0`-th indexed) travels in the sequence `1->5->7->1->5->7->1->...` forever. +> +> We start at bus stop `S` (initially not on a bus), and we want to go to bus stop `T`. Travelling by buses only, what is the least number of buses we must take to reach our destination? Return `-1` if it is not possible. + +## Week 2: Data Structures + +We will work through the following problems, each based around some data structure. You are highly encouraged to try out more problems to get familiar with these data structures (and others)! + +**Easy (Trees): [Invert a Binary Tree](https://leetcode.com/problems/invert-binary-tree/)** + +> Invert a binary tree. + +**Medium (Stacks): [Basic Calculator II](https://leetcode.com/problems/basic-calculator-ii/)** + +> Implement a basic calculator to evaluate a simple expression string. The expression string contains only non-negative integers, `+`, `-`, `*`, `/` operators and empty spaces. The integer division should truncate toward zero. + +**Medium (2D Arrays): [Valid Sudoku](https://leetcode.com/problems/valid-sudoku/)** + +> Determine if a 9x9 Sudoku board is valid. Only the filled cells need to be validated according to the following rules: +> +> 1. Each row must contain the digits 1-9 without repetition. +> 2. Each column must contain the digits 1-9 without repetition. +> 3. Each of the 9 3x3 sub-boxes of the grid must contain the digits 1-9 without repetition. + +**Hard (2D Arrays, for discussion): [Sudoku Solver](https://leetcode.com/problems/sudoku-solver)** + +> Write a program to solve a Sudoku puzzle by filling the empty cells. + +## Week 1: Beginnings + +We will work through the following problems: + +**Easy: [Two Sum](https://leetcode.com/problems/two-sum/)** + +> Given an array of integers, return indices of the two numbers such that they add up to a specific target. You may assume that each input would have exactly one solution, and you may not use the same element twice. + +**Medium: [Letter Combinations of a Phone Number](https://leetcode.com/problems/letter-combinations-of-a-phone-number/)** + +> Given a string containing digits from 2-9 inclusive, return all possible letter combinations that the number could represent. + +**Hard: [Median of Two Sorted Arrays](https://leetcode.com/problems/median-of-two-sorted-arrays/)** + +> There are two sorted arrays `nums1` and `nums2` of size `m` and `n` respectively. Find the median of the two sorted arrays. The overall run time complexity should be `O(log (m+n))`. You may assume `nums1` and `nums2` cannot be both empty. diff --git a/mock-team1/a.py b/2020/mock-team1/a.py similarity index 100% rename from mock-team1/a.py rename to 2020/mock-team1/a.py diff --git a/mock-team1/b.py b/2020/mock-team1/b.py similarity index 100% rename from mock-team1/b.py rename to 2020/mock-team1/b.py diff --git a/mock-team1/c.py b/2020/mock-team1/c.py similarity index 100% rename from mock-team1/c.py rename to 2020/mock-team1/c.py diff --git a/mock-team1/e.py b/2020/mock-team1/e.py similarity index 100% rename from mock-team1/e.py rename to 2020/mock-team1/e.py diff --git a/mock-team1/f-notworking1.py b/2020/mock-team1/f-notworking1.py similarity index 100% rename from mock-team1/f-notworking1.py rename to 2020/mock-team1/f-notworking1.py diff --git a/mock-team1/f-notworking2.py b/2020/mock-team1/f-notworking2.py similarity index 100% rename from mock-team1/f-notworking2.py rename to 2020/mock-team1/f-notworking2.py diff --git a/mock-team1/f-notworking3.java 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b/2020/week4/maximum_subarray_ba.cpp similarity index 100% rename from week4/maximum_subarray_ba.cpp rename to 2020/week4/maximum_subarray_ba.cpp diff --git a/week4/week4-slides.pdf b/2020/week4/week4-slides.pdf similarity index 100% rename from week4/week4-slides.pdf rename to 2020/week4/week4-slides.pdf diff --git a/README.md b/README.md index 9199fe9..37006e9 100644 --- a/README.md +++ b/README.md @@ -1,10 +1,29 @@ # UQCS Competitive Programming Group -The aim of this group is to get practice with programming questions such as those found on LeetCode, and to apply to this skills to programming competitions such as Google CodeJam, Advent of Code, Google Kickstart and (potentially) team-based competitions such as the International Collegiate Programming Contest (or ICPC). We aim to meet on a **weekly basis** to discuss and work together on challenging and interesting programming problems, and get some valuable experience for technical interviews along the way! +The aim of this group is to get practice with programming questions such as those found on LeetCode, and to apply to this skills to programming competitions such as Google Code Jam, Advent of Code, Google Kickstart and (potentially) team-based competitions such as the International Collegiate Programming Contest (or ICPC). We aim to meet on a **weekly basis** to discuss and work together on challenging and interesting programming problems, and get some valuable experience for technical interviews along the way! -## Textbook +## Resources -Our main resource will be the [USACO Guide](https://usaco.guide/), which has a bunch of problems and walks through important topics in competitive programming. Another useful resource is [_Competitive Programming 3_](http://www.sso.sy/sites/default/files/competitive%20programming%203_1.pdf). +The primary resource will be the [CSES problem set](https://cses.fi/problemset/), which is a massive set of problems sorted by topic. We will slowly go through these topics over the weeks: we'll upload/go through some notes and you can work through the problems/ask if you get stuck during the session. + +There is also the [USACO Guide](https://usaco.guide/) which is a great place to look if you are confused about a topic or problem (it has some solutions too). + +## Schedule + +> Note: We started in Week 4, semester 1. + +| Week | Topic | Notes| +|----|------|----| +| Sem 1 Week 4 | Introduction to Competitive Programming | [Notes](notes/1-04.md) | +| Sem 1 Week 5 | Introductory Problems | [Notes](notes/1-05.md) | +| Sem 1 Week 6 | Code Jam Qualification Round | [Notes](notes/1-06.md) | +| Sem 1 Week 7 | Sorting and Searching | [Notes](notes/1-07.md) | +| Sem 1 Week 8 | Greedy Algorithms | [Notes](notes/1-08.md) | +| Sem 1 Week 9 | Dynamic Programming I | [Notes](notes/1-09.md) | +| Sem 1 Week 10 | Dynamic Programming II | [Notes](notes/1-10.md) | +| Sem 1 Week 11 | Graph Algorithms | [Notes](notes/1-11.md) | +| Sem 1 Week 12 | Mathematics | [Notes](notes/1-12.md) | +| Sem 1 Week 13 | No CPG. Go study for exams! | [Notes](notes/1-13.md) | ## Useful Resources (thanks to Nathan!) @@ -14,123 +33,4 @@ Our main resource will be the [USACO Guide](https://usaco.guide/), which has a b - [Codeforces’ Gym](http://codeforces.com/gyms?filterContestType=Official+ACM-ICPC+Contest&order=ID_DESC), which is a place that has many ICPC problem sets from regions around the world. If we need team practice for ICPC, I reckon this is a very good place. - Team notebooks: [KTH](https://github.com/kth-competitive-programming/kactl), [Stanford](https://github.com/jaehyunp/stanfordacm) - If we want to construct our own team notebook later on, we can use [this website](https://judge.yosupo.jp) to check the correctness of our code. -- [Geometry for Competitive Programming](https://vlecomte.github.io/cp-geo.pdf) book by V. Lecomte. - -## Week 10: Bronze (Simulation) - -We'll be working through as many of the problems on [this page](https://usaco.guide/bronze/simulation), to get practice with simulation problems and parsing input/writing output manually (rather than just using the LeetCode interface). List of problems (same as on the website, note that submitting/checking solutions to these problems requires registering on the [USACO website](http://www.usaco.org/)): - -- [Shell Game](http://www.usaco.org/index.php?page=viewproblem2&cpid=891), Easy -- [Mixing Milk](http://www.usaco.org/index.php?page=viewproblem2&cpid=855), Easy -- [Speeding](http://www.usaco.org/index.php?page=viewproblem2&cpid=568), Easy -- [The Lost Cow](http://www.usaco.org/index.php?page=viewproblem2&cpid=735), Easy -- [The Bovine Shuffle](http://www.usaco.org/index.php?page=viewproblem2&cpid=760), Easy -- [Circular Barn](http://www.usaco.org/index.php?page=viewproblem2&cpid=616), Normal -- [Block Game](http://www.usaco.org/index.php?page=viewproblem2&cpid=664), Normal -- [Team Tic Tac Toe](http://www.usaco.org/index.php?page=viewproblem2&cpid=831), Normal -- [Why Did the Cow Cross the Road](http://www.usaco.org/index.php?page=viewproblem2&cpid=713), Normal -- [Mowing the Field](http://www.usaco.org/index.php?page=viewproblem2&cpid=593), Normal -- [Milk Measurement](http://www.usaco.org/index.php?page=viewproblem2&cpid=761), Hard -- [Angry Cows](http://www.usaco.org/index.php?page=viewproblem2&cpid=592), Hard - -# Past Problems - -## Week 4: Dynamic Programming - -We will work through the following problems, each based around dynamic programming. You are highly encouraged to try out more problems to get familiar with these graphs and traversals! - -**Easy: [Climbing Stairs](https://leetcode.com/problems/climbing-stairs/)** - -> You are climbing a stair case. It takes `n` steps to reach to the top. -> -> Each time you can either climb 1 or 2 steps. In how many distinct ways can you climb to the top? - -**"EaSy": [Maximum Subarray](https://leetcode.com/problems/maximum-subarray/)** - -> Given an integer array nums, find the contiguous subarray (containing at least one number) which has the largest sum and return its sum. - -**Medium: [Coin Change](https://leetcode.com/problems/coin-change/)** - -> You are given coins of different denominations and a total amount of money `amount`. Write a function to compute the fewest number of coins that you need to make up that amount. If that amount of money cannot be made up by any combination of the coins, return `-1`. - -**Hard (extension): [Edit Distance](https://leetcode.com/problems/edit-distance/)** - -> Given two words `word1` and `word2`, find the minimum number of operations required to convert `word1` to `word2`. -> -> You have the following 3 operations permitted on a word: -> -> 1. Insert a character -> 2. Delete a character -> 3. Replace a character - -## Week 3: Graphs, Breadth-First Search and Depth-First Search - -We will work through the following problems, each based around graphs. You are highly encouraged to try out more problems to get familiar with these graphs and traversals! - -**Easy: [Flood Fill](https://leetcode.com/problems/flood-fill/)** - -> An image is represented by a 2-D array of integers, each integer representing the pixel value of the image (from 0 to 65535). -> -> Given a coordinate `(sr, sc)` representing the starting pixel (row and column) of the flood fill, and a pixel value `newColor`, "flood fill" the image. -> -> To perform a "flood fill", consider the starting pixel, plus any pixels connected 4-directionally to the starting pixel of the same color as the starting pixel, plus any pixels connected 4-directionally to those pixels (also with the same color as the starting pixel), and so on. Replace the color of all of the aforementioned pixels with the `newColor`. -> -> At the end, return the modified image. - -**Medium: [Number of Islands](https://leetcode.com/problems/number-of-islands/)** - -> Given a 2d grid map of `'1'`s (land) and `'0'`s (water), count the number of islands. An island is surrounded by water and is formed by connecting adjacent lands horizontally or vertically. You may assume all four edges of the grid are all surrounded by water. - -**Medium: [Course Schedule](https://leetcode.com/problems/course-schedule/)** - -> There are a total of `numCourses` courses you have to take, labeled from `0` to `numCourses-1`. -> -> Some courses may have prerequisites, for example to take course `0` you have to first take course `1`, which is expressed as a pair: `[0,1]` -> -> Given the total number of courses and a list of prerequisite pairs, is it possible for you to finish all courses? - -**Hard: [Bus Routes](https://leetcode.com/problems/bus-routes/)** - -> We have a list of bus routes. Each `routes[i]` is a bus route that the `i`-th bus repeats forever. For example if `routes[0] = [1, 5, 7]`, this means that the first bus (`0`-th indexed) travels in the sequence `1->5->7->1->5->7->1->...` forever. -> -> We start at bus stop `S` (initially not on a bus), and we want to go to bus stop `T`. Travelling by buses only, what is the least number of buses we must take to reach our destination? Return `-1` if it is not possible. - -## Week 2: Data Structures - -We will work through the following problems, each based around some data structure. You are highly encouraged to try out more problems to get familiar with these data structures (and others)! - -**Easy (Trees): [Invert a Binary Tree](https://leetcode.com/problems/invert-binary-tree/)** - -> Invert a binary tree. - -**Medium (Stacks): [Basic Calculator II](https://leetcode.com/problems/basic-calculator-ii/)** - -> Implement a basic calculator to evaluate a simple expression string. The expression string contains only non-negative integers, `+`, `-`, `*`, `/` operators and empty spaces. The integer division should truncate toward zero. - -**Medium (2D Arrays): [Valid Sudoku](https://leetcode.com/problems/valid-sudoku/)** - -> Determine if a 9x9 Sudoku board is valid. Only the filled cells need to be validated according to the following rules: -> -> 1. Each row must contain the digits 1-9 without repetition. -> 2. Each column must contain the digits 1-9 without repetition. -> 3. Each of the 9 3x3 sub-boxes of the grid must contain the digits 1-9 without repetition. - -**Hard (2D Arrays, for discussion): [Sudoku Solver](https://leetcode.com/problems/sudoku-solver)** - -> Write a program to solve a Sudoku puzzle by filling the empty cells. - -## Week 1: Beginnings - -We will work through the following problems: - -**Easy: [Two Sum](https://leetcode.com/problems/two-sum/)** - -> Given an array of integers, return indices of the two numbers such that they add up to a specific target. You may assume that each input would have exactly one solution, and you may not use the same element twice. - -**Medium: [Letter Combinations of a Phone Number](https://leetcode.com/problems/letter-combinations-of-a-phone-number/)** - -> Given a string containing digits from 2-9 inclusive, return all possible letter combinations that the number could represent. - -**Hard: [Median of Two Sorted Arrays](https://leetcode.com/problems/median-of-two-sorted-arrays/)** - -> There are two sorted arrays `nums1` and `nums2` of size `m` and `n` respectively. Find the median of the two sorted arrays. The overall run time complexity should be `O(log (m+n))`. You may assume `nums1` and `nums2` cannot be both empty. +- [Geometry for Competitive Programming](https://vlecomte.github.io/cp-geo.pdf) book by V. Lecomte. \ No newline at end of file diff --git a/notes/1-04.md b/notes/1-04.md new file mode 100644 index 0000000..8e19259 --- /dev/null +++ b/notes/1-04.md @@ -0,0 +1,3 @@ +# Introduction session + +We followed some slides from the [South Pacific Competitive Programming Training Camp](https://pcs.org.au/sppc), specifically the slide deck on **"Your first solve!"**. \ No newline at end of file diff --git a/notes/1-05.md b/notes/1-05.md new file mode 100644 index 0000000..039cc1c --- /dev/null +++ b/notes/1-05.md @@ -0,0 +1,34 @@ +# Agenda + +- ANZAC registration details (if they are finalised by the time CPG starts tonight…) +- Google Code Jam qualification round registration details (the competition is open registration and you can do it anytime this weekend, just need to get 30 points) +- some new problems to try out on [cses.fi](http://cses.fi/) and also last year’s [Google Code Jam Qualification Round](https://codingcompetitions.withgoogle.com/codejam/round/000000000019fd27) + +# ANZAC registration details + +- Online +- This Saturday 11AM–4PM +- Can meet up at uni if you wanna +- Registration here +- Maybe a single division, so it might be a bit difficult? + +# Google Code Jam + +- Online +- Friday 11PM to Sunday 5AM +- Won't take the whole time (unless you do all of the questions) +- Need 30 points to qualify (around 2 and a half problems) +- Registration here https://codingcompetitions.withgoogle.com/ + +# Some problems to try out today + +- cses.fi (more starter problems) + - Missing Number + - Repetitions + - Increasing Array + - Permutations + - Number Spiral + - etc. +- Google Code Jam problems from last year + - Qualification round https://codingcompetitions.withgoogle.com/codejam/round/000000000019fd27 +- ANZAC problems https://prog4fun.csse.canterbury.ac.nz/ \ No newline at end of file diff --git a/notes/1-06.md b/notes/1-06.md new file mode 100644 index 0000000..0422395 --- /dev/null +++ b/notes/1-06.md @@ -0,0 +1,6 @@ +# Code Jam overview (lockdown edition!) + +We were in lockdown, so we just went over the problems of the Qualification Round, available [here](https://codingcompetitions.withgoogle.com/codejam/round/000000000043580a). There are some great solution walkthroughs available from prominent competitive programmers: + +- [Errichto](https://www.youtube.com/watch?v=56yh1YF-TqA) +- [Neal Wu](https://www.youtube.com/watch?v=HOoJ0TxQDuA) \ No newline at end of file diff --git a/notes/1-07.md b/notes/1-07.md new file mode 100644 index 0000000..59b8538 --- /dev/null +++ b/notes/1-07.md @@ -0,0 +1,94 @@ +# Sorting and searching + +## Agenda + +- How was Code Jam? +- Next round 1B April 26 2am (...lol) +- Round 1C May 1 7pm + +## Agenda + +- A crash course in complexity +- Sorting +- Binary search +- Two pointers +- Do some problems! + +## A crash course in complexity + +A graph of various complexities. + + + +`O(...)` is a rough estimate of how an algorithm performs as input size increases. + +``` +O(1) < O(log n) < O(n) < O(n log n) < O(n^2) < O(2^n) < O(n!) +``` + +Examples of these complexities in practice: + +- `O(1)`: lookup table, check if number is odd. +- `O(log n)`: binary search (see soon). +- `O(n)`: naive search through an array. +- `O(n log n)`: fastest sort. +- `O(n^2)`: search through 2D array. +- `O(2^n)`: list all combinations of a list. +- `O(n!)`: list all permutations of a list. + +## Sorting + +- Not gonna go over sorting algorithms (see COMP3506 for this). +- Sometimes knowledge of different sorting algorithms may be useful, but for now, know that _sorting can sometimes be useful_. +- Why? +- **Binary search**. + +## Binary search + +Say you have a list + +``` +[1, 2, 3, 4, 10, 8, 7, 9, 3, 2, 5] +``` + +How do I find 7? No way to do so easily. I have to iterate over the whole list. This is `O(n)` for a list of length `n`, so not ideal. + +Instead, what if we sort through the list? + +``` +[1, 2, 2, 3, 4, 5, 7, 8, 9, 10] +``` + +...and then go to the middle, and split the list in two based on whether the number is bigger? + +> *Note:* We don't actually need to split the list, we can just adjust our bounds. See below implementation + +``` +[1, 2, 2, 3, 4] [5, 7, 8, 9, 10] +[5, 7] [8, 9, 10] +[5] [7] +[7] +``` +Here it took us only four steps to find 7, whereas above it took us 7! + +**Binary search is `O(log n)`.** This is faster than `O(n)`. + +> *Note:* how you split a list is implementation-specific, just make sure you're roughly splitting it in half. + +Can be implemented as follows: + +```python +def binary_search(arr, low, high, x): + if high >= low: + mid = (high + low) // 2 + if arr[mid] == x: + return mid + elif arr[mid] > x: + return binary_search(arr, low, mid - 1, x) + else: + return binary_search(arr, mid + 1, high, x) + else: + return -1 +``` + +## Two pointers \ No newline at end of file diff --git a/notes/complexity.png b/notes/complexity.png new file mode 100644 index 0000000000000000000000000000000000000000..da9e2d46bcedf0b8234c8ba6ce303d186875bacc GIT binary patch literal 523512 zcmeFYV|Zm-vnafh?xbUPY}+i3S3)^@LHrYqW*v 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