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                                                | Sieve of Erath
                        
                                    
                        | // Java program to print all primes smaller than or equal to
// n using Sieve of Eratosthenes
 
class SieveOfEratosthenes
{
    void sieveOfEratosthenes(int n)
    {
        // Create a boolean array "prime[0..n]" and initialize
        // all entries it as true. A value in prime[i] will
        // finally be false if i is Not a prime, else true.
        boolean prime[] = new boolean[n+1];
        for(int i=0;i<n;i++)
            prime[i] = true;
         
        for(int p = 2; p*p <=n; p++)
        {
            // If prime[p] is not changed, then it is a prime
            if(prime[p] == true)
            {
                // Update all multiples of p
                for(int i = p*2; i <= n; i += p)
                    prime[i] = false;
            }
        }
         
        // Print all prime numbers
        for(int i = 2; i <= n; i++)
        {
            if(prime[i] == true)
                System.out.print(i + " ");
        }
    }
 |  3 Types of Decrease and Conquer
                        
                                                                                    
                                                                                            | Decrease by constant | Decrease by constant factor | variable-Size decrease |  
                                                                                            | insertion sort | binary search and bisection method | Euclid's algoirthm |  
                                                                                            | topolgical sorting | exponentiation by squaring | selection by partition |  
                                                                                            | algorithms for generating permutations, subsets | multiplication a la russe | nim-like games |  |  |  |  | Brute-Force Problems
                        
                                                                                    
                                                                                            | Problem | Method | Complexity |  
                                                                                            | TSP | Exhaustive | N! |  
                                                                                            | KnapSack | Exhaustive | n*W |  Generate Permutations
                        
                                    
                        | Example n=3:start: 1
 12, 21
 123, 132, 312
 321, 231, 213
 finish
 |  Euclidean Algorithm
                        
                                    
                        | Example:GCD(270,192)
 270/192 = 1 R 78
 GCD(192, 78)
 192/78 = 2 R 36
 GCD(78,36)
 78/36 = 2 R 6
 GCD(36/6)
 36/6 = 6 R0
 since R = 0, 6 is GCD
 |  Brute Force Pros vs Cons
                        
                                                                                    
                                                                                            | Pros | Cons |  
                                                                                            | Wide applicability | Rarely Yields Efficient |  
                                                                                            | Simple | Unacceptably Slow |  
                                                                                            | Reasonable Algorithms | Not as constructive as others |  | 
            
                            
            
            
        
        
        
        
        
            
    
        
          
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