This is the pseudo code for calculating integer factorisation took from CLRS. But what is the point in calculating **GCD** involved in **Line 8** and the need for **doubling k** when **i == k** in **Line 13**.? Help please.

Answer:

That pseudocode is not Pollard-rho factorization despite the label. It is one trial of the related Brent's factorization method. In Pollard-rho factorization, in the ith step you compute x_i and x_(2i), and check the GCD of x_(2i)-x_i with n. In Brent's factorization method, you compute GCD(x_(2^a)-x_(2^a+b),n) for b=1,2, ..., 2^a. (I used the indices starting with 1 to agree with the pseudocode, but elsewhere the sequence is initialized with x_0.) In the code, k=2^a and i=2^a+b. When you detect that i has reached the next power of 2, you increase k to 2^(a+1).

GCDs can be computed very rapidly by Euclid's algorithm without knowing the factorizations of the numbers. Any time you find a nontrivial GCD with n, this helps you to factor n. In both Pollard-rho factorization and Brent's algorithm, one idea is that if you iterate a polynomial such as x^2-c, the differences between the values of the iterates mod n tend to be good candidates for numbers that share nontrivial factors with n. This is because (by the Chinese Remainder Theorem) iterating the polynomial mod n is the same as simultaneously iterating the polynomial mod each prime power in the prime factorization of n. If x_i=x_j mod p1^e1 but not mod p2^e2, then GCD(xi-xj,n) will have p1^e1 as a factor but not p2^e2, so it will be a nontrivial factor.

This is one trial because x_1 is initialized once. If you get unlucky, the value you choose for x_1 starts a preperiodic sequence that repeats at the same time mod each prime power in the prime factorization of n, even though n is not prime. For example, suppose n=1711=29*59, and x_1 = 4, x_2=15, x_3=224, x_4=556, x_5=1155, x_6=1155, ... This sequence does not help you to find a nontrivial factorization, since all of the GCDs of differences between distinct elements and 1711 are 1. If you start with x_1=5, then x_2=24, x_3=575, x_4=401, x_5=1677, x_6=1155, x_7=1155, ... In either factorization method, you would find that GCD(x_4-x_2,1711)=GCD(377,1711)=29, a nontrivial factor of 1711. Not only are some sequences not helpful, others might work, but it might be faster to give up and start with another initial value. So, normally you don't keep increasing i forever, normally there is a termination threshold where you might try a different initial value.

algorithm,graph

I am looking for an algorithm that finds minimal subset of vertices such that by removing this subset (and edges connecting these vertices) from graph all other vertices become unconnected (i.e. the graph won't have any edges). Is there such algorithm? If not: Could you recommend some kind of heuristics...

string,algorithm,palindrome

Given length L find the shortest string >= L formed only of as & bs such that adding some character (Either a or b) doesn't produce a new palindrome substring (never seen before palindrome) For example for L = 1 there is the string aabbaba, adding "a" to it to...

algorithm,optimization,dynamic-programming,frequency

I have the following problem and I only have a slight idea about it: Consider a tape storage problem. Given n files of length l1,...,ln and frequencies with which they are accessed f1,...,fn, where sum of all frequencies is 1 and 0<fi<1. "Optimal" means to minimize the average retrieval time...

algorithm,math,statistics,variance,standard-deviation

I'm successfully using Welford's method to compute running variance and standard deviation as described many times on Stack Overflow and John D Cook's excellent blog post. However in the stream of samples, sometimes I encounter a "rollback", or "remove sample" order, meaning that a previous sample is no longer valid...

math,javafx,geometry,coordinates

I have a line which has points (x1,y1) and (x2,y2). I wanted to attach a Button to it, it should align with the line by rotating based on the line segment points. I need some help in calculating the rotation angle for the Button.

python,regex,algorithm,python-2.7,datetime

If I knew the format in which a string represents date-time information, then I can easily use datetime.datetime.strptime(s, fmt). However, without knowing the format of the string beforehand, would it be possible to determine whether a given string contains something that could be parsed as a datetime object with the...

c,string,algorithm,data-structures

int myStrCmp (const char *s1, const char *s2) { const unsigned char *p1 = (const unsigned char *)s1; const unsigned char *p2 = (const unsigned char *)s2; while (*p1 != '\0') { if (*p2 == '\0') return 1; if (*p2 > *p1) return -1; if (*p1 > *p2) return 1;...

python,algorithm,permutation

I'm trying to write a function that would input a list (of pre-determined length) and output the permutation of that list according to my choice of re-ordering. For example, suppose I have the list [1,'a',12,'b','poppy'] and I have the permutation (3,4,5,2,1) Then the function output would be a list with...

c++,algorithm,matrix

I am trying to print the right hemisphere of a matrix. If we draw the main and the secondary diagonals in a matrix we see that we got 4 equal parts, the right part is called (in my algorithms textbook) the right hemisphere of a square matrix. For example, in...

python,algorithm

I am trying to calculate least completion time of a project schedule, which has n number of tasks and each task can have a dependency task. example Task ID Duration Days Dependent ID 1 10 0 2 3 0 3 6 1 4 5 2 5 10 1 Task with...

algorithm,big-o

I was doing some reading on logarithms and the rate of growth of the running time of algorithms. I have, however, a problem understanding the Big-Ω (Big-Omega) notation. I know that we use it for 'asymptotic lower bounds', and that we can express the idea that an algorithm takes at...

ruby,algorithm,search,recursion

I'm working through a toy problem in Ruby: how to produce all possible 10-digit phone numbers where each successive number is adjacent to the last on the keypad. I've represented the adjacent relationships between numbers, and have a recursive function, but my method isn't iterating through the whole solution space....

algorithm

I attended a programming competition(it has ended now). I don't know why my solution is giving WA, I read the editorial, saw other people solution but unable to find a flaw in my solution. Obviously I am missing somewhere. Please help! Question Link My Solution My approach If the pattern...

algorithm,dynamic-programming,knapsack-problem

I am familiar with the 0-1 knapsack problem and when you are given a certain number of copies from each item but I can figure out how to solve it when you are given infinite copies of each item using dynamic programming. I am trying to solve it by hand...

algorithm

You are managing a software project that involves building a computer-assisted instrument for medical surgery. The exact placement of the surgical knife is dependent on a number of different parameters, usually at least 25, sometimes more. Your programmer has developed two algorithms for positioning the cutting tool, and is seeking...

c++,algorithm,parallel-processing,c++14

Proposal N3554 (A Parallel Algorithms Library) for C++14, proposes (among other things), what seem to be parallel versions of the current std::partial_sum, e.g.: template< class ExecutionPolicy, class InputIterator, class OutputIterator, class BinaryOperation> OutputIterator inclusive_scan( ExecutionPolicy &&exec, InputIterator first, InputIterator last, OutputIterator result, BinaryOperation binary_op); With the explanation Effects: For each...

algorithm,recursion,permutation

Assuming I have a list of elements [1,2,3,4,] and a number of bins (let's assume 2 bins), I want to come up with a list of all combinations of splitting up items 1-4 into the 2 bins. Solution should look something like this [{{1}, {2,3,4}}, {{2}, {1,3,4}}, {{3}, {1,2,4}}, {{4},...

math,cordic

How can we use cordic to tanh(x+1)/tanh(x) I can't get a idea about how to apply cordic to above function. In other word, which point on the above function, can we apply cordic?...

algorithm,graph,cycle,dfs

I came around one of those questions in my exams: Topologocial sorting using Kahn's Algorithm requires the graph to be DAG (Directed Acyclic Graph). How can we determine if a graph contains no cycles without using DFS/BFS first? I am trying to answer that for too long now and I...

python,math,rounding

This question already has an answer here: How do you round UP a number in Python? 9 answers Python round up integer to next hundred 6 answers so i'm making a pizza program in Python 3.3 that takes input from the user and prints the amount of pizza's needed....

java,math,set,linkedhashset

I am trying to do a homework in math which is find a subset of collection {1,2,..,n} where n is a number given in the code, I cannot get it done with the sublist so I need to get your help with a math programming. For example for n =...

c++,algorithm,math,recursion

I'm trying to find all possible solutions to the 3X3 magic square. There should be exactly 8 solutions. My code gets them all but there are a lot of repeats. I'm having a hard time tracking the recursive steps to see why I'm getting all the repeats. // This program...

c++,arrays,algorithm,recursion

I'm trying to write a recursive algorithm that returns true if at least one array[i] == i. And false if there is no array[i] = i. Also, the parameters needed are (int * arr, int start, int end). So i'll be traversing the array with a pointer. For example: int...

c++,algorithm,graph

Someone can help me to think in the better way to adapt Dijkstra's Algorithm in these conditions? All I thought didn't was good. Example of input: GP4578 MADRID 01:00 PORTO 02:00 IK6587 PORTO 03:00 VALENCIA 05:00 05:30 TENERIFE 08:00 AB5874 VALENCIA 05:40 BERLIM 10:00 "VALENCIA 05:00 05:30" This is a...

c,algorithm,security,math,encryption

I'm trying to reverse the following code in order to provide a function which takes the buffer and decrypts it. void crypt_buffer(unsigned char *buffer, size_t size, char *key) { size_t i; int j; j = 0; for(i = 0; i < size; i++) { if(j >= KEY_SIZE) j = 0;...

algorithm,coordinates,coordinate-systems,coordinate

I'm writing a program for visualizing crystals. As a part of the program, I have to generate all different basic points in a lattice structure. For those that aren't familiar with crystallography, you can find the most general cases of these structures here: https://en.wikipedia.org/wiki/Hermann%E2%80%93Mauguin_notation#Lattice_types The problem was that I wanted...

c#,regex,algorithm

I'm making a script to try and hack into an account whose login password is at least 8 characters long and includes at least 1 number, 1 special character and 1 capital letter. I will use brute force. Is there a compact, elegant and efficient way to iterate through every...

c++,algorithm,inheritance,time-complexity

Here I have 800 derived classes of Base and a list of 8000000 objects of these types, which can be of any order. The goal is to separate the list into the 800 types as efficiently as possible. Here I have written two functions to do that. The first is...

javascript,arrays,math

i was playing around in code academy and for some reason when i change the last slaying i the else statement to true instead of making the person invincible it just crashes my browser any ideas why thank you in advance var slaying = true; var youHit = Math.floor(Math.random()*2); var...

python,algorithm,time-complexity,longest-substring

Someone asked me a question Find the longest alphabetically increasing or equal string composed of those letters. Note that you are allowed to drop unused characters. So ghaaawxyzijbbbklccc returns aaabbbccc. Is an O(n) solution possible? and I implemented it code [in python] s = 'ghaaawxyzijbbbklccc' lst = [[] for i...

algorithm,discrete-mathematics

I'm doing an online course and i'm stuck on this question. I know there are similar questions but they don't help me. What is the order of growth of the worst case running time of the following code fragment as a function of N? int sum = 0; for (int...

math,vector,3d,cube

I'm trying to figure out the math to find a random point inside a cube. I have something small but it can't take into account the rotation of the cube. Here are some images of my results. Here you can see the cube is rotated to some degree but when...

java,math

I did a code like this: public static double myPow(double x, int n){ if(n==0) return 1; double t = myPow(x, n/2); if(n % 2 != 0){ if(n < 0){ return (1 / t * t * x); } else { return t * t * x; } } else {...

python,math,combinations,itertools

Other than doing this: from itertools import combinations def brute_force(x): for l in range (1,len(x)+1): for f in list(combinations(range(0,len(x)),l)): yield f x = range(1,18) len(list(brute_force(x))) [out]: 131071 How could I mathematically calculate the number of all possible combinations? Is there a way to do it computationally without enumerating the possible...

arrays,matlab,math,for-loop,while-loop

This is my one dimensional array A, containing 10 numbers: A = [-8.92100000000000 10.6100000000000 1.33300000000000 ... -2.57400000000000 -4.52700000000000 9.63300000000000 ... 4.26200000000000 16.9580000000000 8.16900000000000 4.75100000000000]; I want the loop to go through like this; (calculating mean interval wise) - Interval length of 2,4,8 (a(1)+a(2))/2 - value stored in one block of...

c++,algorithm

The project Euler problem 5 is stated as : "2520 is the smallest number that can be divided by each of the numbers from 1 to 10 without any remainder. What is the smallest positive number that is evenly divisible by all of the numbers from 1 to 20?" Here's...

algorithm,go

I am trying to solve the following problem from project euler (please take a look at description and the example in the link, but here is the short explanation). in the matrix, find the minimal path sum from the top left to the bottom right, by moving left, right, up,...

algorithm,data-structures

In an exam I see a question: Which one of the following is true? For a binary search, the best-case occurs when the target item is in the beginning of the search list. For a binary search, the best-case occurs when the target is at the end of the search...

javascript,algorithm

I accidentally got undefined == true and undefined == false that both of them returns false. But !undefined returns true. And this is the question: What's the algorithmic difference(s) between !someVariable and someVariable == false? If i want to explain it more, type undefined == false ? 't' : 'f'...

python,math

I'm writing a python script to find out some things about randomization. I have the following code: from random import randint one = 0 two = 0 olddiff = 0 diff = 0 sumdiff = 0 avgdiff = 0 headcounter = 0 counter = 0 while (headcounter < 500000): while...

c++,algorithm,sorting,c++11

I have implemented three different sorting algorithms and now I want to confirm that my approach of counting the total number of comparisons is correct. In my mind, the number of comparisons shouldn't be tied to the conditional branches because if the condition isn't met, the comparison was still made...

algorithm,numerical-methods,numerical,numerical-integration

Consider the following differential equation f(x) = g'(x) I have a build a code that spits out values of the function f(x) for the variable x, where x goes from 0 to very large. Now, I'm looking for a scheme that will analyse these values of f(x) in order to...

python,algorithm,sorting,math

I am looking for a basic algorithm that gives more weigh to the recent reviews. So, the output value of the algorithm is mutable. For example, two reviews with exactly the same score, will have a different ranking based on the timestamp of the creation. Review_1 Score 10 creation 10/5/2014...

php,algorithm,digits

I'm following the programming test, and there are questions like this From this consecutive digits: 123456789101112131415161718192021.... For example The 10th digit is 1 The 11th digit is 0 and so on What is the 1,000,000th digit? What is the 1,000,000,000th digit? What is the 1,000,000,000,000th digit? Your solution should run...

algorithm,bit-manipulation

Given a 64 bit number, I need to extract every other bit from it, and convert it into a number: decimal: 357 binary: 0000 0001 0110 0101 odd bits: 0 0 0 1 1 0 1 1 decimal: 27 Any idea of a good algorithmic way to do it? And...

javascript,algorithm,recursion

While working through some Coderbyte challenges, I was able to solve the following problem recursively, but was hoping to get some feedback on how I can improve it. Have the function AdditivePersistence(num) take the num parameter being passed which will always be a positive integer and return its additive persistence...