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I understand the "Depth-First" maze geerating algorithm but I need a little help implementing it with Javascript.
Maze Generation at Rosetta Code contains many implementations to generate and show a maze, using the simple Depth-first search algorithm:
Code in JavaScript:
function maze(x,y) {
var n=x*y-1;
if (n<0) {alert("illegal maze dimensions");return;}
var horiz=[]; for (var j= 0; j<x+1; j++) horiz[j]= [];
var verti=[]; for (var j= 0; j<y+1; j++) verti[j]= [];
var here= [Math.floor(Math.random()*x), Math.floor(Math.random()*y)];
var path= [here];
var unvisited= [];
for (var j= 0; j<x+2; j++) {
unvisited[j]= [];
for (var k= 0; k<y+1; k++)
unvisited[j].push(j>0 && j<x+1 && k>0 && (j != here[0]+1 || k != here[1]+1));
}
while (0<n) {
var potential= [[here[0]+1, here[1]], [here[0],here[1]+1],
[here[0]-1, here[1]], [here[0],here[1]-1]];
var neighbors= [];
for (var j= 0; j < 4; j++)
if (unvisited[potential[j][0]+1][potential[j][1]+1])
neighbors.push(potential[j]);
if (neighbors.length) {
n= n-1;
next= neighbors[Math.floor(Math.random()*neighbors.length)];
unvisited[next[0]+1][next[1]+1]= false;
if (next[0] == here[0])
horiz[next[0]][(next[1]+here[1]-1)/2]= true;
else
verti[(next[0]+here[0]-1)/2][next[1]]= true;
path.push(here= next);
} else
here= path.pop();
}
return ({x: x, y: y, horiz: horiz, verti: verti});
}
function display(m) {
var text= [];
for (var j= 0; j<m.x*2+1; j++) {
var line= [];
if (0 == j%2)
for (var k=0; k<m.y*4+1; k++)
if (0 == k%4)
line[k]= '+';
else
if (j>0 && m.verti[j/2-1][Math.floor(k/4)])
line[k]= ' ';
else
line[k]= '-';
else
for (var k=0; k<m.y*4+1; k++)
if (0 == k%4)
if (k>0 && m.horiz[(j-1)/2][k/4-1])
line[k]= ' ';
else
line[k]= '|';
else
line[k]= ' ';
if (0 == j) line[1]= line[2]= line[3]= ' ';
if (m.x*2-1 == j) line[4*m.y]= ' ';
text.push(line.join('')+'\r\n');
}
return text.join('');
}
Code in Java:
public int[][] generateMaze() {
int[][] maze = new int[height][width];
// Initialize
for (int i = 0; i < height; i++)
for (int j = 0; j < width; j++)
maze[i][j] = 1;
Random rand = new Random();
// r for row、c for column
// Generate random r
int r = rand.nextInt(height);
while (r % 2 == 0) {
r = rand.nextInt(height);
}
// Generate random c
int c = rand.nextInt(width);
while (c % 2 == 0) {
c = rand.nextInt(width);
}
// Starting cell
maze[r][c] = 0;
// Allocate the maze with recursive method
recursion(r, c);
return maze;
}
public void recursion(int r, int c) {
// 4 random directions
int[] randDirs = generateRandomDirections();
// Examine each direction
for (int i = 0; i < randDirs.length; i++) {
switch(randDirs[i]){
case 1: // Up
// Whether 2 cells up is out or not
if (r - 2 <= 0)
continue;
if (maze[r - 2][c] != 0) {
maze[r-2][c] = 0;
maze[r-1][c] = 0;
recursion(r - 2, c);
}
break;
case 2: // Right
// Whether 2 cells to the right is out or not
if (c + 2 >= width - 1)
continue;
if (maze[r][c + 2] != 0) {
maze[r][c + 2] = 0;
maze[r][c + 1] = 0;
recursion(r, c + 2);
}
break;
case 3: // Down
// Whether 2 cells down is out or not
if (r + 2 >= height - 1)
continue;
if (maze[r + 2][c] != 0) {
maze[r+2][c] = 0;
maze[r+1][c] = 0;
recursion(r + 2, c);
}
break;
case 4: // Left
// Whether 2 cells to the left is out or not
if (c - 2 <= 0)
continue;
if (maze[r][c - 2] != 0) {
maze[r][c - 2] = 0;
maze[r][c - 1] = 0;
recursion(r, c - 2);
}
break;
}
}
}
/**
* Generate an array with random directions 1-4
* #return Array containing 4 directions in random order
*/
public Integer[] generateRandomDirections() {
ArrayList<Integer> randoms = new ArrayList<Integer>();
for (int i = 0; i < 4; i++)
randoms.add(i + 1);
Collections.shuffle(randoms);
return randoms.toArray(new Integer[4]);
}
Source, demo and some more explanations
Related
I'm trying to write a simple simplex solver for linear optimization problems, but I'm having trouble getting it working. Every time I run it I get a vector subscript out of range (which is quite easy to find), but I think that its probably a core issue somewhere else in my impl.
Here is my simplex solver impl:
bool pivot(vector<vector<double>>& tableau, int row, int col) {
int n = tableau.size();
int m = tableau[0].size();
double pivot_element = tableau[row][col];
if (pivot_element == 0) return false;
for (int j = 0; j < m; j++) {
tableau[row][j] /= pivot_element;
}
for (int i = 0; i < n; i++) {
if (i != row) {
double ratio = tableau[i][col];
for (int j = 0; j < m; j++) {
tableau[i][j] -= ratio * tableau[row][j];
}
}
}
return true;
}
int simplex(vector<vector<double>>& tableau, vector<double>& basic, vector<double>& non_basic) {
int n = tableau.size() - 1;
int m = tableau[0].size() - 1;
while (true) {
int col = -1;
for (int j = 0; j < m; j++) {
if (tableau[n][j] > 0) {
col = j;
break;
}
}
if (col == -1) break;
int row = -1;
double min_ratio = numeric_limits<double>::infinity();
for (int i = 0; i < n; i++) {
if (tableau[i][col] > 0) {
double ratio = tableau[i][m] / tableau[i][col];
if (ratio < min_ratio) {
row = i;
min_ratio = ratio;
}
}
}
if (row == -1) return -1;
if (!pivot(tableau, row, col)) return -1;
double temp = basic[row];
basic[row] = non_basic[col];
non_basic[col] = temp;
}
return 1;
}
Given an array and some value X, find the number of pairs such that i < j , a[i] = a[j] and (i * j) % X == 0
Array size <= 10^5
I am thinking of this problem for a while but only could come up with the brute force solution(by checking all pairs) which will obviously time-out [O(N^2) time complexity]
Any better approach?
First of all, store separate search structures for each distinct A[i] as we iterate.
i * j = k * X
i = k * X / j
Let X / j be some fraction. Since i is an integer, k would be of the form m * least_common_multiple(X, j) / X, where m is natural.
Example 1: j = 20, X = 60:
lcm(60, 20) = 60
matching `i`s would be of the form:
(m * 60 / 60) * 60 / 20
=> m * q, where q = 3
Example 2: j = 6, X = 2:
lcm(2, 6) = 6
matching `i`s would be of the form:
(m * 6 / 2) * 2 / 6
=> m * q, where q = 1
Next, I would consider how to efficiently query the number of multiples of a number in a sorted list of arbitrary naturals. One way is to hash the frequency of divisors of each i we add to the search structure of A[i]. But first consider i as j and add to the result the count of divisors q that already exist in the hash map.
JavaScript code with brute force testing at the end:
function gcd(a, b){
return b ? gcd(b, a % b) : a;
}
function getQ(X, j){
return X / gcd(X, j);
}
function addDivisors(n, map){
let m = 1;
while (m*m <= n){
if (n % m == 0){
map[m] = -~map[m];
const l = n / m;
if (l != m)
map[l] = -~map[l];
}
m += 1;
}
}
function f(A, X){
const Ais = {};
let result = 0;
for (let j=1; j<A.length; j++){
if (A[j] == A[0])
result += 1;
// Search
if (Ais.hasOwnProperty(A[j])){
const q = getQ(X, j);
result += Ais[A[j]][q] || 0;
// Initialise this value's
// search structure
} else {
Ais[A[j]] = {};
}
// Add divisors for j
addDivisors(j, Ais[A[j]]);
}
return result;
}
function bruteForce(A, X){
let result = 0;
for (let j=1; j<A.length; j++){
for (let i=0; i<j; i++){
if (A[i] == A[j] && (i*j % X) == 0)
result += 1;
}
}
return result;
}
var numTests = 1000;
var n = 100;
var m = 50;
var x = 100;
for (let i=0; i<numTests; i++){
const A = [];
for (let j=0; j<n; j++)
A.push(Math.ceil(Math.random() * m));
const X = Math.ceil(Math.random() * x);
const _brute = bruteForce(A, X);
const _f = f(A, X);
if (_brute != _f){
console.log("Mismatch!");
console.log(X, JSON.stringify(A));
console.log(_brute, _f);
break;
}
}
console.log("Done testing.")
Just in case If someone needed the java version of this answer - https://stackoverflow.com/a/69690416/19325755 explanation has been provided in that answer.
I spent lot of time in understanding the javascript code so I thought the people who are comfortable with java can refer this for better understanding.
import java.util.HashMap;
public class ThisProblem {
public static void main(String[] args) {
int t = 1000;
int n = 100;
int m = 50;
int x = 100;
for(int i = 0; i<t; i++) {
int[] A = new int[n];
for(int j = 0; j<n; j++) {
A[j] = ((int)Math.random()*m)+1;
}
int X = ((int)Math.random()*x)+1;
int optR = createMaps(A, X);
int brute = bruteForce(A, X);
if(optR != brute) {
System.out.println("Wrong Answer");
break;
}
}
System.out.println("Test Completed");
}
public static int bruteForce(int[] A, int X) {
int result = 0;
int n = A.length;
for(int i = 1; i<n; i++) {
for(int j = 0; j<i; j++) {
if(A[i] == A[j] && (i*j)%X == 0)
result++;
}
}
return result;
}
public static int gcd(int a, int b) {
return b==0 ? a : gcd(b, a%b);
}
public static int getQ(int X, int j) {
return X/gcd(X, j);
}
public static void addDivisors(int n, HashMap<Integer, Integer> map) {
int m = 1;
while(m*m <= n) {
if(n%m == 0) {
map.put(m, map.getOrDefault(m, 0)+1);
int l = n/m;
if(l != m) {
map.put(l, map.getOrDefault(l, 0)+1);
}
}
m++;
}
}
public static int createMaps(int[] A, int X) {
int result = 0;
HashMap<Integer, HashMap<Integer, Integer>> contentsOfA = new HashMap<>();
int n = A.length;
for(int i = 1; i<n; i++) {
if(A[i] == A[0])
result++;
if(contentsOfA.containsKey(A[i])) {
int q = getQ(X, i);
result += contentsOfA.get(A[i]).getOrDefault(q, 0);
} else {
contentsOfA.put(A[i], new HashMap<>());
}
addDivisors(i, contentsOfA.get(A[i]));
}
return result;
}
}
How can I find the number of blobs in a 2d matrix? SIZE MxN
A blob is a block of continuous X pixels. where the matrix contains X and O
XOOOXO
OXOXOX
XXOOXO
I would like to use 8-neighbourship (see here). So I would expect 2 blobs to be found in above example.
The idea is simple: Mark each continuous blob and count how many blobs were marked.
Here is some pseudo-code (you did not specify a programming language) to get you started:
numBlobs = 0;
foreach(item in matrix)
{
res = Visit(item);
if(res > 0)
{
numBlobs = numBlobs + 1;
}
}
return numBlobs;
The Visit function/method looks like this:
Visit(item)
{
marked = 0;
if(item.IsX() && !IsItemMarked(neighbour))
{
MarkItemAsVisited(item);
marked = 1;
foreach(neighbour in GetNeighbours(item))
{
marked = marked + Visit(neighbour);
}
}
return marked;
}
All you have to do is to implement the other fucntions/methods but they are pretty straightforward.
public static void main(String[] args) {
int[][] matrix = new int[6][5];
System.out.println(matrix.length);
for (int i=0; i < matrix.length; i++) {
for (int j=0; j < matrix[i].length; j++) {
matrix[i][j] = 0;
}
}
matrix[0][3] = 1;
matrix[1][1] = 1;
matrix[1][3] = 1;
matrix[2][1] = 1;
matrix[2][2] = 1;
matrix[2][3] = 1;
matrix[4][0] = 1;
matrix[4][4] = 1;
matrix[5][2] = 1;
matrix[5][3] = 1;
matrix[5][4] = 1;
System.out.println(findBlobCount(matrix, matrix.length, matrix[0].length));
}
static int findBlobCount (int matrix[][], int rowCount, int colCount)
{
int visited[][] = new int[rowCount][colCount]; // all initialized to false
int count=0;
for (int i=0; i<rowCount; i++)
{
for (int j=0; j<colCount; j++)
{
if (matrix[i][j] == 1 && visited[i][j] == 0) // unvisited black cell
{
markVisited (i,j, matrix, visited, rowCount, colCount);
count++;
}
}
}
return count;
}
static int markVisited (int i, int j, int [][]matrix, int [][]visited, int rowCount, int colCount)
{
if (i < 0 || j < 0)
return 0;
if (i >= rowCount || j >= colCount)
return 0;
if (visited[i][j] == 1) // already visited
return 1;
if (matrix[i][j] == 0) // not a black cell
return 0;
visited[i][j] = 1;
// recursively mark all the 4 adjacent cells - right, left, up and down
return markVisited (i+1, j, matrix, visited, rowCount, colCount)
+ markVisited (i-1, j, matrix, visited, rowCount, colCount)
+ markVisited (i, j+1, matrix, visited, rowCount, colCount)
+ markVisited (i, j-1, matrix, visited, rowCount, colCount);
}
my question is based from topcoder's recent SRM (628 500 point question), i'm trying to solve the string parsing question by brute force since the number of solutions cannot exceed 5^5 and I'm very close. My question is how would I go about generating not only n! permutations but n^r permutations in c#. Using an adaptation of Knuth's lexicographical permutation algorithm my solution works but not for cases when the answer (the winning permutation) has repeated characters.
input example: "X{}]X{X{}]X]"
public string ifPossible(string expression)
{
char[] c = new char[6] { '(', ')', '{', '}', '[', ']' };
StringBuilder sb = new StringBuilder(expression);
int j = 0;
special(sb);
while (my_next_permutation(c) && sb.ToString().Contains('X'))
{
for(int i = 0; i < sb.Length;i++)
{
if (sb[i] == 'X')
sb[i] = c[j++];
}
special(sb);
if (sb.Length > 0)
{
sb.Clear();
sb.Append(expression);
special(sb);
j = 0;
}
else
{
break;
}
}
if (sb.Length > 0)
return "impossible";
return "possible";
}
void special(StringBuilder sb)
{
while (sb.ToString().Contains("()") || sb.ToString().Contains("[]") || sb.ToString().Contains("{}"))
{
while (sb.ToString().Contains("()"))
sb.Replace("()", "");
while (sb.ToString().Contains("[]"))
sb.Replace("[]", "");
while (sb.ToString().Contains("{}"))
sb.Replace("{}", "");
}
}
public Boolean my_next_permutation(char[] a)
{
int N = a.Length, i = N - 2;
for (; i >= 0; i--)
if (a[i] < a[i + 1])
break;
if (i < 0) return false;
for (int j = N - 1; j >= i; j--)
{
if (a[j] > a[i])
{
var temp = a[i];
a[i] = a[j];
a[j] = temp;
break;
}
}
for (int j = i + 1; j < (N + i + 1) / 2; j++)
{
var temp = a[j];
a[j] = a[N + i - j];
a[N + i - j] = temp;
}
return true;
}
I could have used the array idea as gmch and jrh suggested but I
ended up using this:
IEnumerable<string> GetPermutations(char[] list, int? resultSize = null, bool withRepetition = false)
{
List<string> l = new List<string>();
char[] result = new char[resultSize.HasValue ? resultSize.Value : list.Length];
var indices = new int[result.Length];
for (int i = 0; i < indices.Length; i++)
indices[i] = withRepetition ? -1 : indices.Length - i - 2;
int curIndex = 0;
while (curIndex != -1)
{
indices[curIndex]++;
if (indices[curIndex] == list.Length)
{
indices[curIndex] = withRepetition ? -1 : curIndex - 1;
curIndex--;
}
else
{
result[curIndex] = list[indices[curIndex]];
if (curIndex < indices.Length - 1)
curIndex++;
else
yield return new string(result);
}
}
}
I got the idea from this page http://noldorin.com/programming/CombinatoricsUtilities.cs.txt
All credit given to Alex Regueiro.
Can someone give an example for finding greatest common divisor algorithm for more than two numbers?
I believe programming language doesn't matter.
Start with the first pair and get their GCD, then take the GCD of that result and the next number. The obvious optimization is you can stop if the running GCD ever reaches 1. I'm watching this one to see if there are any other optimizations. :)
Oh, and this can be easily parallelized since the operations are commutative/associative.
The GCD of 3 numbers can be computed as gcd(a, b, c) = gcd(gcd(a, b), c). You can apply the Euclidean algorithm, the extended Euclidian or the binary GCD algorithm iteratively and get your answer. I'm not aware of any other (smarter?) ways to find a GCD, unfortunately.
A little late to the party I know, but a simple JavaScript implementation, utilising Sam Harwell's description of the algorithm:
function euclideanAlgorithm(a, b) {
if(b === 0) {
return a;
}
const remainder = a % b;
return euclideanAlgorithm(b, remainder)
}
function gcdMultipleNumbers(...args) { //ES6 used here, change as appropriate
const gcd = args.reduce((memo, next) => {
return euclideanAlgorithm(memo, next)}
);
return gcd;
}
gcdMultipleNumbers(48,16,24,96) //8
I just updated a Wiki page on this.
[https://en.wikipedia.org/wiki/Binary_GCD_algorithm#C.2B.2B_template_class]
This takes an arbitrary number of terms.
use GCD(5, 2, 30, 25, 90, 12);
template<typename AType> AType GCD(int nargs, ...)
{
va_list arglist;
va_start(arglist, nargs);
AType *terms = new AType[nargs];
// put values into an array
for (int i = 0; i < nargs; i++)
{
terms[i] = va_arg(arglist, AType);
if (terms[i] < 0)
{
va_end(arglist);
return (AType)0;
}
}
va_end(arglist);
int shift = 0;
int numEven = 0;
int numOdd = 0;
int smallindex = -1;
do
{
numEven = 0;
numOdd = 0;
smallindex = -1;
// count number of even and odd
for (int i = 0; i < nargs; i++)
{
if (terms[i] == 0)
continue;
if (terms[i] & 1)
numOdd++;
else
numEven++;
if ((smallindex < 0) || terms[i] < terms[smallindex])
{
smallindex = i;
}
}
// check for exit
if (numEven + numOdd == 1)
continue;
// If everything in S is even, divide everything in S by 2, and then multiply the final answer by 2 at the end.
if (numOdd == 0)
{
shift++;
for (int i = 0; i < nargs; i++)
{
if (terms[i] == 0)
continue;
terms[i] >>= 1;
}
}
// If some numbers in S are even and some are odd, divide all the even numbers by 2.
if (numEven > 0 && numOdd > 0)
{
for (int i = 0; i < nargs; i++)
{
if (terms[i] == 0)
continue;
if ((terms[i] & 1) == 0)
terms[i] >>= 1;
}
}
//If every number in S is odd, then choose an arbitrary element of S and call it k.
//Replace every other element, say n, with | n−k | / 2.
if (numEven == 0)
{
for (int i = 0; i < nargs; i++)
{
if (i == smallindex || terms[i] == 0)
continue;
terms[i] = abs(terms[i] - terms[smallindex]) >> 1;
}
}
} while (numEven + numOdd > 1);
// only one remaining element multiply the final answer by 2s at the end.
for (int i = 0; i < nargs; i++)
{
if (terms[i] == 0)
continue;
return terms[i] << shift;
}
return 0;
};
For golang, using remainder
func GetGCD(a, b int) int {
for b != 0 {
a, b = b, a%b
}
return a
}
func GetGCDFromList(numbers []int) int {
var gdc = numbers[0]
for i := 1; i < len(numbers); i++ {
number := numbers[i]
gdc = GetGCD(gdc, number)
}
return gdc
}
In Java (not optimal):
public static int GCD(int[] a){
int j = 0;
boolean b=true;
for (int i = 1; i < a.length; i++) {
if(a[i]!=a[i-1]){
b=false;
break;
}
}
if(b)return a[0];
j=LeastNonZero(a);
System.out.println(j);
for (int i = 0; i < a.length; i++) {
if(a[i]!=j)a[i]=a[i]-j;
}
System.out.println(Arrays.toString(a));
return GCD(a);
}
public static int LeastNonZero(int[] a){
int b = 0;
for (int i : a) {
if(i!=0){
if(b==0||i<b)b=i;
}
}
return b;
}