##stl cheat sheet https://github.com/gibsjose/cpp-cheat-sheet/blob/master/Data%20Structures%20and%20Algorithms.md https://www.hackerearth.com/practice/notes/standard-template-library/
- A Linked List is a linear collection of data elements, called nodes, each pointing to the next node by means of a pointer. It is a data structure consisting of a group of nodes which together represent a sequence.
- Singly-linked list: linked list in which each node points to the next node and the last node points to null
- Doubly-linked list: linked list in which each node has two pointers, p and n, such that p points to the previous node and n points to the next node; the last node's n pointer points to null
- Circular-linked list: linked list in which each node points to the next node and the last node points back to the first node
- Time Complexity:
- Access:
O(n) - Search:
O(n) - Insert:
O(1) - Remove:
O(1)
- Access:
- A Stack is a collection of elements, with two principle operations: push, which adds to the collection, and pop, which removes the most recently added element
- Last in, first out data structure (LIFO): the most recently added object is the first to be removed
- Time Complexity:
- Access:
O(n) - Search:
O(n) - Insert:
O(1) - Remove:
O(1)
- Access:
- A Queue is a collection of elements, supporting two principle operations: enqueue, which inserts an element into the queue, and dequeue, which removes an element from the queue
- First in, first out data structure (FIFO): the oldest added object is the first to be removed
- Time Complexity:
- Access:
O(n) - Search:
O(n) - Insert:
O(1) - Remove:
O(1)
- Access:
- A Tree is an undirected, connected, acyclic graph
- A Binary Tree is a tree data structure in which each node has at most two children, which are referred to as the left child and right child
- Full Tree: a tree in which every node has either 0 or 2 children
- Perfect Binary Tree: a binary tree in which all interior nodes have two children and all leave have the same depth
- Complete Tree: a binary tree in which every level except possibly the last is full and all nodes in the last level are as far left as possible
- A binary search tree, sometimes called BST, is a type of binary tree which maintains the property that the value in each node must be greater than or equal to any value stored in the left sub-tree, and less than or equal to any value stored in the right sub-tree
- Time Complexity:
- Access:
O(log(n)) - Search:
O(log(n)) - Insert:
O(log(n)) - Remove:
O(log(n))
- Access:
- A trie, sometimes called a radix or prefix tree, is a kind of search tree that is used to store a dynamic set or associative array where the keys are usually Strings. No node in the tree stores the key associated with that node; instead, its position in the tree defines the key with which it is associated. All the descendants of a node have a common prefix of the String associated with that node, and the root is associated with the empty String.
- A Fenwick tree, sometimes called a binary indexed tree, is a tree in concept, but in practice is implemented as an implicit data structure using an array. Given an index in the array representing a vertex, the index of a vertex's parent or child is calculated through bitwise operations on the binary representation of its index. Each element of the array contains the pre-calculated sum of a range of values, and by combining that sum with additional ranges encountered during an upward traversal to the root, the prefix sum is calculated
- Time Complexity:
- Range Sum:
O(log(n)) - Update:
O(log(n))
- Range Sum:
- A Segment tree, is a tree data structure for storing intervals, or segments. It allows querying which of the stored segments contain a given point
- Time Complexity:
- Range Query:
O(log(n)) - Update:
O(log(n))
- Range Query:
- A Heap is a specialized tree based structure data structure that satisfies the heap property: if A is a parent node of B, then the key (the value) of node A is ordered with respect to the key of node B with the same ordering applying across the entire heap. A heap can be classified further as either a "max heap" or a "min heap". In a max heap, the keys of parent nodes are always greater than or equal to those of the children and the highest key is in the root node. In a min heap, the keys of parent nodes are less than or equal to those of the children and the lowest key is in the root node
- Time Complexity:
- Access Max / Min:
O(1) - Insert:
O(log(n)) - Remove Max / Min:
O(log(n))
- Access Max / Min:
- Hashing is used to map data of an arbitrary size to data of a fixed size. The values returned by a hash function are called hash values, hash codes, or simply hashes. If two keys map to the same value, a collision occurs
- Hash Map: a hash map is a structure that can map keys to values. A hash map uses a hash function to compute an index into an array of buckets or slots, from which the desired value can be found.
- Collision Resolution
- Separate Chaining: in separate chaining, each bucket is independent, and contains a list of entries for each index. The time for hash map operations is the time to find the bucket (constant time), plus the time to iterate through the list
- Open Addressing: in open addressing, when a new entry is inserted, the buckets are examined, starting with the hashed-to-slot and proceeding in some sequence, until an unoccupied slot is found. The name open addressing refers to the fact that the location of an item is not always determined by its hash value
- A Graph is an ordered pair of G = (V, E) comprising a set V of vertices or nodes together with a set E of edges or arcs, which are 2-element subsets of V (i.e. an edge is associated with two vertices, and that association takes the form of the unordered pair comprising those two vertices)
- Undirected Graph: a graph in which the adjacency relation is symmetric. So if there exists an edge from node u to node v (u -> v), then it is also the case that there exists an edge from node v to node u (v -> u)
- Directed Graph: a graph in which the adjacency relation is not symmetric. So if there exists an edge from node u to node v (u -> v), this does not imply that there exists an edge from node v to node u (v -> u)
- Stable:
No - Time Complexity:
- Best Case:
O(nlog(n)) - Worst Case:
O(n^2) - Average Case:
O(nlog(n))
- Best Case:
- Mergesort is also a divide and conquer algorithm. It continuously divides an array into two halves, recurses on both the left subarray and right subarray and then merges the two sorted halves
- Stable:
Yes - Time Complexity:
- Best Case:
O(nlog(n)) - Worst Case:
O(nlog(n)) - Average Case:
O(nlog(n))
- Best Case:
- Bucket Sort is a sorting algorithm that works by distributing the elements of an array into a number of buckets. Each bucket is then sorted individually, either using a different sorting algorithm, or by recursively applying the bucket sorting algorithm
- Time Complexity:
- Best Case:
Ω(n + k) - Worst Case:
O(n^2) - Average Case:
Θ(n + k)
- Best Case:
- Radix Sort is a sorting algorithm that like bucket sort, distributes elements of an array into a number of buckets. However, radix sort differs from bucket sort by 're-bucketing' the array after the initial pass as opposed to sorting each bucket and merging
- Time Complexity:
- Best Case:
Ω(nk) - Worst Case:
O(nk) - Average Case:
Θ(nk)
- Best Case:
- Depth First Search is a graph traversal algorithm which explores as far as possible along each branch before backtracking
- Time Complexity:
O(|V| + |E|)
- Breadth First Search is a graph traversal algorithm which explores the neighbor nodes first, before moving to the next level neighbors
- Time Complexity:
O(|V| + |E|)
- Topological Sort is the linear ordering of a directed graph's nodes such that for every edge from node u to node v, u comes before v in the ordering
- Time Complexity:
O(|V| + |E|)
- Dijkstra's Algorithm is an algorithm for finding the shortest path between nodes in a graph
- Time Complexity:
O(|V|^2)
- Bellman-Ford Algorithm is an algorithm that computes the shortest paths from a single source node to all other nodes in a weighted graph
- Although it is slower than Dijkstra's, it is more versatile, as it is capable of handling graphs in which some of the edge weights are negative numbers
- Time Complexity:
- Best Case:
O(|E|) - Worst Case:
O(|V||E|)
- Best Case:
- Floyd-Warshall Algorithm is an algorithm for finding the shortest paths in a weighted graph with positive or negative edge weights, but no negative cycles
- A single execution of the algorithm will find the lengths (summed weights) of the shortest paths between all pairs of nodes
- Time Complexity:
- Best Case:
O(|V|^3) - Worst Case:
O(|V|^3) - Average Case:
O(|V|^3)
- Best Case:
- Prim's Algorithm is a greedy algorithm that finds a minimum spanning tree for a weighted undirected graph. In other words, Prim's find a subset of edges that forms a tree that includes every node in the graph
- Time Complexity:
O(|V|^2)
- Kruskal's Algorithm is also a greedy algorithm that finds a minimum spanning tree in a graph. However, in Kruskal's, the graph does not have to be connected
- Time Complexity:
O(|E|log|V|)
- Greedy Algorithms are algorithms that make locally optimal choices at each step in the hope of eventually reaching the globally optimal solution
- Problems must exhibit two properties in order to implement a Greedy solution:
- Optimal Substructure
- An optimal solution to the problem contains optimal solutions to the given problem's subproblems
- The Greedy Property
- An optimal solution is reached by "greedily" choosing the locally optimal choice without ever reconsidering previous choices
- Example - Coin Change
- Given a target amount V cents and a list of denominations of n coins, i.e. we have coinValue[i] (in cents) for coin types i from [0...n - 1], what is the minimum number of coins that we must use to represent amount V? Assume that we have an unlimited supply of coins of any type
- Coins - Penny (1 cent), Nickel (5 cents), Dime (10 cents), Quarter (25 cents)
- Assume V = 41. We can use the Greedy algorithm of continuously selecting the largest coin denomination less than or equal to V, subtract that coin's value from V, and repeat.
- V = 41 | 0 coins used
- V = 16 | 1 coin used (41 - 25 = 16)
- V = 6 | 2 coins used (16 - 10 = 6)
- V = 1 | 3 coins used (6 - 5 = 1)
- V = 0 | 4 coins used (1 - 1 = 0)
- Using this algorithm, we arrive at a total of 4 coins which is optimal
- Bitmasking is a technique used to perform operations at the bit level. Leveraging bitmasks often leads to faster runtime complexity and helps limit memory usage
- Test kth bit:
s & (1 << k); - Set kth bit:
s |= (1 << k); - Turn off kth bit:
s &= ~(1 << k); - Toggle kth bit:
s ^= (1 << k); - Multiple by 2n:
s << n; - Divide by 2n:
s >> n; - Intersection:
s & t; - Union:
s | t; - Set Subtraction:
s & ~t; - Extract lowest set bit:
s & (-s); - Extract lowest unset bit:
~s & (s + 1); - Swap Values:
x ^= y; y ^= x; x ^= y;
- Big O Notation is used to describe the upper bound of a particular algorithm. Big O is used to describe worst case scenarios
- Little O Notation is also used to describe an upper bound of a particular algorithm; however, Little O provides a bound that is not asymptotically tight
- Big Omega Notation is used to provide an asymptotic lower bound on a particular algorithm
- Little Omega Notation is used to provide a lower bound on a particular algorithm that is not asymptotically tight
- Theta Notation is used to provide a bound on a particular algorithm such that it can be "sandwiched" between two constants (one for an upper limit and one for a lower limit) for sufficiently large values
Problem solving is the ability to reach a solution for a defined problem by, potentially, using mathematical or systematic operations (algorithms).
Not only an important skill to have in competitive programming, but also in the real world where problems arise everyday at jobs; being able to attack a problem quickly and efficiently is a skill desired by all companies.
The following steps should serve as a general timeline for solving a problem:
- Define the problem
- Come up with all possible cases
- Implement the solution
In competitive programming, having a strong problem solving ability is just as important as having a strong programming background.
- Even if you’ve been programming your entire life, if you don’t know how to attack a problem, coding isn’t going to help you
Being able to identify edge cases and worst case scenarios for a problem are also important in competitive programming.
- If you just rely on the input/output given to you in the problem statement, your code might not handle ALL cases (which is necessary for an accepted solution)
If we are given a problem to work on, what are the steps we must take in order to get an accepted solution? Here are the basic steps to follow:
- Read the problem
- Manually solve the problem
- Take note of the constraints
- Implement a solution
- Optimize your solution
- Handle edge cases
Yes, this is an obvious step in solving a problem, but it is a step that can cause turmoil in the long run.
Even if the background story behind a problem is silly or boring, you must read the problem all the way through to make sure that you didn’t miss any important details that are critical to solving the problem.
If you spent 45 minutes solving a problem with some algorithm, but realize later that the problem asked you to do something completely different, you don’t get that time back.
You will be given input and output for a problem, so you should take a few minutes to grab pen and paper and figure out how they got their answer.
Writing out how they derived their output from the given input will give you a general algorithm for your solution; start by taking the steps you took and turning those into lines of code.
This will also stop you from jumping into the problem straight away and saying “how did they get that answer?” when debugging your code down the road.
Step 3: Take note of the constraints
These are the size of the input variables that are given to you by the problem (e.g., X <= 1,000,000).
Knowing the sizes of the variables that are given can eliminate possible algorithms that you chose to use.
In order to know what is a good algorithm for a given constraint, you will need to understand Big O Notation.
For most problems, you can assume that your program will be able to process ~100,000,000 operations per second.
Additionally, most problems will have a time limit between 0.5 and 3 seconds, so the number of "steps" your algorithm takes cannot exceed ~300M.
Use the algorithm/pseudocode you came up with when manually solving the test cases to implement a basic solution.
Remember: this algorithm you came up with does not need to be optimal; as long as you can come up with something that will get you the correct answer, you can worry about optimizing afterwards.
Making sure that your code can handle the constraints of the input is key to getting AC on a problem; if you don’t take this step into account, you won’t be able to get anything but Time Limit Exceeded on a problem.
Thinks of ways to eliminate unnecessary loops and operations within your code in order to trim off some time so that you can efficiently solve the problem.
Being able to come up with tricky test cases that the judge possibly came up with can save you trouble in the long run (as well as avoid a Wrong Answer). Here are some things to keep in mind:
- Did you handle the minimum and maximum cases?
- Did you handle potential negative numbers?
- Is the problem using longs instead of ints?
- Is there potential for your code to throw a runtime error? (e.g., array out of bounds, stack overflow, etc.)
If you followed all of the previous steps, you can go ahead and submit your solution to the judge and see if you get AC.
If you happen to get Wrong Answer, Time Limit Exceeded, or Runtime Error, make sure that you read the problem correctly, handled all cases, have a fast enough algorithm, and that your program isn't crashing when being tested.