Given a rectangular grid and a point, I need an algorithm for visiting all points in a zigzag manner.
So, I'm looking for a function f that generates the below plot if run like this:
loop:
new_x, new_y = f(x, y, minx, miny, maxx, maxy)
if new_x == x and new_y == y:
end loop
Can someone help me with such an algorithm?
Be warned, I count from 1:
If you are on an odd-numbered row step to the right.
If you are on an even-numbered row step to the left.
If you are at the end of a row step up.
This is a bit fiddly to code but I can't see any particular problems.
Assuming that 0<=X<=L, and 0<=Y, given an index N, you can find the coordinates as follows:
Y = floor(N/L)
X = (Y is even)? N mod L : L - (N mod L) - 1
--Edit--
I Notice that this doesn't comply with your loop structure constraint, but it may be helpful anyway.
Maybe something like this?
#include <iostream>
#include <utility>
std::pair<size_t, size_t>
foo (size_t N, size_t k) {
const auto r = k / N;
const auto c = (r & 1) == 0 ? k % N : N - k % N - 1;
return {r, c};
}
int
main () {
const size_t N = 10;
for (size_t i = 0; i < N * N; ++i) {
auto p = foo (N, i);
std::cout << "(" << p.first << ", " << p.second << ")\n";
}
std::cout << std::endl;
}
Related
Suppose I have n circles of radius r. I want to place them randomly inside a rectangle of size AxA.
It is guaranteed that they fit. One can suppose that the sum of the area of all circles is about 60% of the area of the rectangle.
I can try it by doing a backtracking, trying to place, going back, etc., but there should be a better way to do it.
One possibility is to generate random points inside the rectangle without further constraints, and then move the points/centres iteratively (by little steps) such that avoiding overlapping. If two points are too near one from each other, each point can bring pressure to the other, to make it going away a little bit. The higher the pressure, the higher the move.
This process was implemented in C++. In the following simple code, to facilitate implementation, points and vectors are represented par std::complex type.
Note that I used srandand rand for test purpose. You may used better random algorithms, depending on your constraints.
According to the tests that I have performed, convergence seems guaranteed for a density of 60%. I also made some tests with a density of 70%: sometimes convergence, sometimes not.
Complexity is O(n^2 n_iter), where nis the number of circles and n_iterthe number of iterations.
n_iteris generally between 100 and 300, for a density of 60%. It could be decreased with relaxing the convergence criteria.
It could be seems high complexity, compared to other proposals in comments. In practice, for n = 15, the work is performed in less than 30ms on my PC. Huge time or fast enough, depending on the context. I have included a figure to illustrate the algorithm.
#include <cstdlib>
#include <iostream>
#include <fstream>
#include <vector>
#include <ctime>
#include <complex>
#include <cmath>
#include <tuple>
#include <ios>
#include <iomanip>
using dcomplex = std::complex<double>;
void print (const std::vector<dcomplex>& centers) {
std::cout << std::setprecision (9);
std::cout << "\ncenters:\n";
for (auto& z: centers) {
std::cout << real(z) << ", " << imag(z) << "\n";
}
}
std::tuple<bool, int, double> process (double A, double R, std::vector<dcomplex>& centers, int n_iter_max = 100) {
bool check = true;
int n = centers.size();
std::vector<dcomplex> moves (n, 0.0);
double acceleration = 1.0001; // to accelerate the convergence, if density not too large
// could be made dependent of the iteration index
double dmin;
auto limit = [&] (dcomplex& z) {
double zx = real(z);
double zi = imag(z);
if (zx < R) zx = R;
if (zx > A-R) zx = A-R;
if (zi < R) zi = R;
if (zi > A-R) zi = A-R;
return dcomplex(zx, zi);
};
int iter;
for (iter = 0; iter < n_iter_max; ++iter) {
for (int i = 0; i < n; ++i) moves[i] = 0.0;
dmin = A;
for (int i = 0; i < n; ++i) {
for (int j = i+1; j < n; ++j) {
auto vect = centers[i] - centers[j];
double dist = std::abs(vect);
if (dist < dmin) dmin = dist;
double x = std::max (0.0, 2*R*acceleration - dist) / 2.0;
double coef = x / (dist + R/10000);
moves[i] += coef * vect;
moves[j] -= coef * vect;
}
}
std::cout << "iteration " << iter << " dmin = " << dmin << "\n";
if (dmin/R >= 2.0 - 1.0e-6) break;
for (int i = 0; i < n; ++i) {
centers[i] += moves[i];
centers[i] = limit (centers[i]);
}
}
dmin = A;
for (int i = 0; i < n; ++i) {
for (int j = i+1; j < n; ++j) {
auto vect = centers[i] - centers[j];
double dist = std::abs(vect);
if (dist < dmin) dmin = dist;
}
}
std::cout << "Final: dmin/R = " << dmin/R << "\n";
check = dmin/R >= 2.0 - 1.0e-6;
return {check, iter, dmin};
}
int main() {
int n = 15; // number of circles
double R = 1.0; // ray of each circle
double density = 0.6; // area of all circles over total area A*A
double A; // side of the square
int n_iter = 1000;
A = sqrt (n*M_PI*R*R/density);
std::cout << "number of circles = " << n << "\n";
std::cout << "density = " << density << "\n";
std::cout << "A = " << A << std::endl;
std::vector<dcomplex> centers (n);
std::srand(std::time(0));
for (int i = 0; i < n; ++i) {
double x = R + (A - 2*R) * (double) std::rand()/RAND_MAX;
double y = R + (A - 2*R) * (double) std::rand()/RAND_MAX;
centers[i] = {x, y};
}
auto [check, n_iter_eff, dmin] = process (A, R, centers, n_iter);
std::cout << "check = " << check << "\n";
std::cout << "Relative min distance = " << std::setprecision (9) << dmin/R << "\n";
std::cout << "nb iterations = " << n_iter_eff << "\n";
print (centers);
return 0;
}
A day at the beach. A group of n people are lying on the beach. The beach is represented by the real line R and the location of the i-th person is some integer xi ∈ Z. Your task is to prevent people from getting sunburned by covering them with umbrellas. Each umbrella corresponds to a closed interval I = [a, a + L] of length L ∈ N, and the i-th person is covered by that umbrella if xi ∈ I. Design a greedy algorithm for covering all people with the minimum number of umbrellas. The input consists of the integers x1,...,xn, and L. The output of your algorithm should be the positions of umbrellas.
Forexample,iftheinputisx1 =1,x2 =3,x3 =5,andL=2,thenanoptimumsolutionisthe set of two umbrellas placed at positions 2 and 5, covering intervals [1, 3] and [4, 6]. The running time should be polynomial in n
The main idea is that when we see someone that is not covered yet, we will place the umbrella at position ( uncoveredPersonPosition + L/2 ), and the umbrella reach will be covering the range (uncoveredPersonPosition, uncoveredPersonPosition + L) .
Here is the code for this idea.
#include <vector>
#include <iostream>
#include <algorithm>
int solve(std::vector<double>& people, int umbrellaCoverage) {
std::sort(people.begin(), people.end() );
int answer = 0;
double umbrellaReach;
for(const auto& p : people) {
if(answer == 0) {
answer += 1;
umbrellaReach = p + umbrellaCoverage;
}
else {
if(p < umbrellaReach) continue;
else {
answer++;
umbrellaReach = p + umbrellaCoverage;
}
}
}
return answer;
}
int main() {
std::vector<double> v(10);
for(int i = 0; i < 10; ++i) v[i] = (double) i;
std::cout << solve(v, 3) << std::endl;
return 0;
}
Main DNA sequence(a string) is given (let say string1) and another string to search for(let say string2). You have to find the minimum length window in string1 where string2 is subsequence.
string1 = "abcdefababaef"
string2 = "abf"
Approaches that i thought of, but does not seem to be working:
1. Use longest common subsequence(LCS) approach and check if the (length of LCS = length of string2). But this will give me whether string2 is present in string1 as subsequence, but not smallest window.
2. KMP algo, but not sure how to modify it.
3. Prepare a map of {characters: pos of characters} of string1 which are in string2. Like:
{ a : 0,6,8,10
b : 1,7,9
f : 5,12 }
And then some approach to find min window and still maintaining the order of "abf"
I am not sure whether I am thinking in right directions or am I totally off.
Is there a known algorithm for this, or does anyone know any approach? Kindly suggest.
Thanks in advance.
You can do LCS and find all the max subsequences in the String1 of String2 using recursion on the DP table of the LCS result. Then calculate the window length of each of LCS and you can get minimum of it. You can also stop a branch if it already exceeds size of current smallest window found.
check Reading out all LCS :-
http://en.wikipedia.org/wiki/Longest_common_subsequence_problem
Dynamic Programming!
Here is a C implementation
#include <iostream>
#include <vector>
using namespace std;
int main() {
string a, b;
cin >> a >> b;
int m = a.size(), n = b.size();
int inf = 100000000;
vector < vector < int > > dp (n + 1, vector < int > (m + 1, inf)); // length of min string a[j...k] such that b[i...] is a subsequence of a[j...k]
dp[n] = vector < int > (m + 1, 0); // b[n...] = "", so dp[n][i] = 0 for each i
for (int i = n - 1; i >= 0; --i) {
for (int j = m - 1; j >= 0; --j) {
if(b[i] == a[j]) dp[i][j] = 1 + dp[i+1][j+1];
else dp[i][j] = 1 + dp[i][j+1];
}
}
int l, r, min_len = inf;
for (int i = 0; i < m; ++i) {
if(dp[0][i] < min_len) {
min_len = dp[0][i];
l = i, r = i + min_len;
}
}
if(min_len == inf) {
cout << "no solution!\n";
} else {
for (int i = l; i < r; ++i) {
cout << a[i];
}
cout << '\n';
}
return 0;
}
I found a similar interview question on CareerCup , only difference being that its an array of integers instead of characters. I borrowed an idea and made a few changes, let me know if you have any questions after reading this C++ code.
What I am trying to do here is : The for loop in the main function is used to loop over all elements of the given array and find positions where I encounter the first element of the subarray, once found, I call the find_subsequence function where I recursively match the elements of the given array to the subarray at the same time preserving the order of elements. Finally, find_subsequence returns the position and I calculate the size of the subsequence.
Please excuse my English, wish I could explain it better.
#include "stdafx.h"
#include "iostream"
#include "vector"
#include "set"
using namespace std;
class Solution {
public:
int find_subsequence(vector<int> s, vector<int> c, int arrayStart, int subArrayStart) {
if (arrayStart == s.size() || subArrayStart ==c.size()) return -1;
if (subArrayStart==c.size()-1) return arrayStart;
if (s[arrayStart + 1] == c[subArrayStart + 1])
return find_subsequence(s, c, arrayStart + 1, subArrayStart + 1);
else
return find_subsequence(s, c, arrayStart + 1, subArrayStart);
}
};
int main()
{
vector<int> v = { 1,5,3,5,6,7,8,5,6,8,7,8,0,7 };
vector<int> c = { 5,6,8,7 };
Solution s;
int size = INT_MAX;
int j = -1;
for (int i = 0; i <v.size(); i++) {
if(v[i]==c[0]){
int x = s.find_subsequence(v, c, i-1, -1);
if (x > -1) {
if (x - i + 1 < size) {
size = x - i + 1;
j = i;
}
if (size == c.size())
break;
}
}
}
cout << size <<" "<<j;
return 0;
}
In the grid constructed by grid points (M*x, M*y) and given the point A(x1,y1) and point B(x2,y2) where all the variables are integers. I need to check how many grid points lie on the line segment from point A to point B. I know that it can be done by using the extended euclidean algorithm somehow, but I have no clue on how to do it. I would appreciate your help.
Rephrased, you want to determine how many numbers t satisfy
(1) M divides (1-t) x1 + t x2
(2) M divides (1-t) y1 + t y2
(3) 0 <= t <= 1.
Let's focus on (1). We introduce an integer variable q to represent the divisibility constraint and solve for t:
exists integer q, M q = (1-t) x1 + t x2
exists integer q, M q - x1 = (x2 - x1) t.
If x1 is not equal to x2, then this gives a periodic set of possibilities of the form t in {a + b q | q integer}, where a and b are fractions. Otherwise, all t or no t are solutions.
The extended Euclidean algorithm is useful for intersecting the solution sets arising from (1) and (2). Suppose that we want to compute the intersection
{a + b q | q integer} intersect {c + d s | s integer}.
By expressing a generic common element in two different ways, we arrive at a linear Diophantine equation:
a + b q = c + d s,
where a, b, c, d are constant and q, s are integer. Let's clear denominators and gather terms into one equation
A q + B s = C,
where A, B, C are integers. This equation has solutions if and only if the greatest common divisor g of A and B also divides C. Use the extended Euclidean algorithm to compute integer coefficients u, v such that
A u + B v = g.
Then we have a solution
q = u (C/g) + k (B/g)
s = v (C/g) - k (A/g)
for each integer k.
Finally, we have to take constraint (3) into consideration. This should boil down to some arithmetic and one floor division, but I'd rather not work out the details (the special cases of what I've written so far already will take quite a lot of your time).
Let's dX = Abs(x2-x1) and dY = Abs(y2 - y1)
Then number of lattice points on the segment is
P = GCD(dX, dY) + 1
(including start and end points)
where GCD is the greatest common divisor (through usual (not extended) Euclidean algorithm)
(See last Properties here)
Following instructions of Mr. David Eisenstat I have managed to write a program in c++ that calculates the answer.
#include <iostream>
#include <math.h>
using namespace std;
int gcd (int A, int B, int &u, int &v)
{
int Ad = 1;
int Bd = 1;
if (A < 0) { Ad = -1; A = -A; }
if (B < 0) { Bd = -1; B = -B; }
int x = 1, y = 0;
u = 0, v = 1;
while (A != 0)
{
int q = B/A;
int r = B%A;
int m = u-x*q;
int n = v-y*q;
B = A;
A = r;
u = x;
v = y;
x = m;
y = n;
}
u *= Ad;
v *= Bd;
return B;
}
int main(int argc, const char * argv[])
{
int t;
scanf("%d", &t);
for (int i = 0; i<t; i++) {
int x1, x2, y1, y2, M;
scanf("%d %d %d %d %d", &M, &x1, &y1, &x2, &y2);
if ( x1 == x2 ) // vertical line
{
if (x1%M != 0) { // in between the horizontal lines
cout << 0 << endl;
} else
{
int r = abs((y2-y1)/M); // number of points
if (y2%M == 0 || y1%M == 0) // +1 if the line starts or ends on the point
r++;
cout << r << endl;
}
} else {
if (x2 < x1)
{
int c;
c = x1;
x1 = x2;
x2 = c;
}
int A, B, C;
C = x1*y2-y1*x2;
A = M*(y2-y1);
B = -M*(x2-x1);
int u, v;
int g = gcd(A, B, u, v);
//cout << "A: " << A << " B: " << B << " C: " << C << endl;
//cout << A << "*" << u <<"+"<< B <<"*"<<v<<"="<<g<<endl;
double a = -x1/(double)(x2-x1);
double b = M/(double)(x2-x1);
double Z = (-a-C*b/g*u)*g/(B*b);
double Y = (1-a-C*b/g*u)*g/(B*b);
cout << floor(Z) - ceil(Y) + 1 << endl;
}
}
return 0;
}
Thank you for your help! Please check if all special cases are taken into consideration.
I'm trying to use the repeated squaring algorithm (using recursion) to perform matrix exponentiation. I've included header files from the NEWMAT library instead of using arrays. The original matrix has elements in the range (-5,5), all numbers being of type float.
# include "C:\User\newmat10\newmat.h"
# include "C:\User\newmat10\newmatio.h"
# include "C:\User\newmat10\newmatap.h"
# include <iostream>
# include <time.h>
# include <ctime>
# include <cstdlib>
# include <iomanip>
using namespace std;
Matrix repeated_squaring(Matrix A, int exponent, int n) //Recursive function
{
A(n,n);
IdentityMatrix I(n);
if (exponent == 0) //Matrix raised to zero returns an Identity Matrix
return I;
else
{
if ( exponent%2 == 1 ) // if exponent is odd
return (A * repeated_squaring (A*A, (exponent-1)/2, n));
else //if exponent is even
return (A * repeated_squaring( A*A, exponent/2, n));
}
}
Matrix direct_squaring(Matrix B, int k, int no) //Brute Force Multiplication
{
B(no,no);
Matrix C = B;
for (int i = 1; i <= k; i++)
C = B*C;
return C;
}
//----Creating a matrix with elements b/w (-5,5)----
float unifRandom()
{
int a = -5;
int b = 5;
float temp = (float)((b-a)*( rand()/RAND_MAX) + a);
return temp;
}
Matrix initialize_mat(Matrix H, int ord)
{
H(ord,ord);
for (int y = 1; y <= ord; y++)
for(int z = 1; z<= ord; z++)
H(y,z) = unifRandom();
return(H);
}
//---------------------------------------------------
void main()
{
int exponent, dimension;
cout<<"Insert exponent:"<<endl;
cin>>exponent;
cout<< "Insert dimension:"<<endl;
cin>>dimension;
cout<<"The number of rows/columns in the square matrix is: "<<dimension<<endl;
cout<<"The exponent is: "<<exponent<<endl;
Matrix A(dimension,dimension),B(dimension,dimension);
Matrix C(dimension,dimension),D(dimension,dimension);
B= initialize_mat(A,dimension);
cout<<"Initial Matrix: "<<endl;
cout<<setw(5)<<setprecision(2)<<B<<endl;
//-----------------------------------------------------------------------------
cout<<"Repeated Squaring Result: "<<endl;
clock_t time_before1 = clock();
C = repeated_squaring (B, exponent , dimension);
cout<< setw(5) <<setprecision(2) <<C;
clock_t time_after1 = clock();
float diff1 = ((float) time_after1 - (float) time_before1);
cout << "It took " << diff1/CLOCKS_PER_SEC << " seconds to complete" << endl<<endl;
//---------------------------------------------------------------------------------
cout<<"Direct Squaring Result:"<<endl;
clock_t time_before2 = clock();
D = direct_squaring (B, exponent , dimension);
cout<<setw(5)<<setprecision(2)<<D;
clock_t time_after2 = clock();
float diff2 = ((float) time_after2 - (float) time_before2);
cout << "It took " << diff2/CLOCKS_PER_SEC << " seconds to complete" << endl<<endl;
}
I face the following problems:
The random number generator returns only "-5" as each element in the output.
The Matrix multiplication yield different results with brute force multiplication and using the repeated squaring algorithm.
I'm timing the execution time of my code to compare the times taken by brute force multiplication and by repeated squaring.
Could someone please find out what's wrong with the recursion and with the matrix initialization?
NOTE: While compiling this program, make sure you've imported the NEWMAT library.
Thanks in advance!
rand() returns an int so rand()/RAND_MAX will truncate to an integer = 0. Try your
repeated square algorithm by hand with n = 1, 2 and 3 and you'll find a surplus A *
and a gross inefficiency.
Final Working code has the following improvements:
Matrix repeated_squaring(Matrix A, int exponent, int n) //Recursive function
{
A(n,n);
IdentityMatrix I(n);
if (exponent == 0) //Matrix raised to zero returns an Identity Matrix
return I;
if (exponent == 1)
return A;
{
if (exponent % 2 == 1) // if exponent is odd
return (A*repeated_squaring (A*A, (exponent-1)/2, n));
else //if exponent is even
return (repeated_squaring(A*A, exponent/2, n));
}
}
Matrix direct_squaring(Matrix B, int k, int no) //Brute Force Multiplication
{
B(no,no);
Matrix C(no,no);
C=B;
for (int i = 0; i < k-1; i++)
C = B*C;
return C;
}
//----Creating a matrix with elements b/w (-5,5)----
float unifRandom()
{
int a = -5;
int b = 5;
float temp = (float) ((b-a)*((float) rand()/RAND_MAX) + a);
return temp;
}