In this tutorial we will be discussing dynamic programming on trees, a very popular algorithmic technique that solves many problems involving trees. We'll be learning this technique by example. We'll take a problem solving approach in this tutorial, not just describing what the final solution looks like, but walking through how one might go about solving such problems.

A first example

Task

Consider a tree in which every node has a positive value written on it. The task is to select several nodes from the tree (you may also select none), so that the sum of values of the nodes you select is maximized, and no two adjacent nodes are selected. The desired solution has complexity O(n).

General approach

Dynamic programming is a fancy name for storing intermediate results and re-using the stored result instead of re-computing them each time.

Let's see an example of a dynamic programming problem. Once we solve the problem using dynamic programming, the formal technical definitions will be easier to follow.

Tiling problem

Problem: You are given a grid of size n \times 2 and n tiles of size 2 \times 1. In how many different ways can you tile the grid such that the entire grid is covered and no tiles overlap. (The tiles look identical to each other. Two ways of tiling are different based on whether the tiles are placed horizontally or vertically).

Example: There are 3 possible ways of tiling a 3 \times 2 grid.

Tiling 1: 
Three vertical tiles
 __ __ __

Recursion is a fundamental technique used in a variety of disciplines ranging from linguistics to logic. The most common application of recursion is in mathematics and computer science, where a function being defined is applied within its own definition.

In computer science, a common method of simplification is to divide a problem into subproblems of the same type. As a computer programming technique, this is called divide and conquer and is key to the design of many important algorithms. Divide and conquer serves as a top-down approach to problem solving, where problems are solved by solving smaller and smaller instances. A contrary approach is Dynamic Programming. This approach serves as a bottom-up approach, where problems are solved by solving larger and larger instances, until the desired size is reached.

It's recommended that you learn mathematical induction before learning recursion. You can find a tutorial here:

Mathematical Induction

If you are familiar with mathematical induction, rec...

Introduction

Computers can only do a limited number of operations in a given time, usually about a billion operations per second. Here, by an individual operation, we mean addition, multiplication, assigning a value to a variable, etc. Hence, if we want our algorithm to finish executing in a second or so, then we need to make sure that the total number of operations we are asking it to do is less than 1 billion.

Estimating the exact run-time of an algorithm without implementing and executing it is very tough since it depends on so many factors - including the programming language being used, the compiler, the hardware of the computer itself, and more. Instead, what we'd like to do most of the time, is to estimate the run-time approximately.

Rate of growth

In algorithms, we focus on estimating the execution ...

Sieve of Eratosthenes is a way to compute prime factors of a number by pre-computing the smallest prime-factors of all numbers till a limit.

A great video tutorial by Khan Academy -

Also I recommend you to read this blog on CodeForces...