Path in a Rectangle with Circles

There is a m*n rectangular matrix whose top-left(start) location is (1, 1) and bottom-right(end) location is (m*n). There are k circles each with radius r. Find if there is any path from start to end without touching any circle.
The input contains values of m, n, k, r and two array of integers X and Y, each of length k. (X[i], Y[i]) is the center of ith circle.
Source : Directi Interview

Examples: 

Input : m = 5, n = 5, k = 2, r = 1, 
         X = {1, 3}, Y = {3, 3}
Output : Possible

Here is a path from start to end point.

Input : m = 5, n = 5, k = 2, r = 1, 
         X = {1, 1}, Y = {2, 3}.
Output : Not Possible

Approach : Check if the centre of a cell (i, j) of the rectangle comes within any of the circles then do not traverse through that cell and mark that as ‘blocked’. Mark rest of the cells initially as ‘unvisited’. Then use BFS to find out shortest path of each cell from starting position. If the end cell is visited then we will return “Possible” otherwise “Not Possible”.

Algorithm : 



  1. Take an array of size m*n. Initialize all the cells to 0.
  2. For each cell of the rectangle check whether it comes within any circle or not (by calculating the distance of that cell from each circle). If it comes within any circle then change the value of that cell to -1(‘blocked’).
  3. Now, apply BFS from the starting cell and if a cell can be reached then change the value of that cell to 1.
  4. If the value of the ending cell is 1, then return ‘Possible’, otherwise return ‘Not Possible’.

 Below is implementation of the above idea:

C++

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// C++ program to find out path in
// a rectangle containing circles.
#include <iostream>
#include <math.h>
#include <vector>
 
using namespace std;
 
// Function to find out if there is
// any possible path or not.
bool isPossible(int m, int n, int k, int r, vector<int> X,
                vector<int> Y)
{
    // Take an array of m*n size and
    // initialize each element to 0.
    int rect[m][n] = { 0 };
 
    // Now using Pythagorean theorem find if a
    // cell touches or within any circle or not.
    for (int i = 0; i < m; i++) {
        for (int j = 0; j < n; j++) {
            for (int p = 0; p < k; p++) {
                if (sqrt((pow((X[p] - 1 - i), 2)
                          + pow((Y[p] - 1 - j), 2)))
                    <= r) {
                    rect[i][j] = -1;
                }
            }
        }
    }
 
    // If the starting cell comes within
    // any circle return false.
    if (rect[0][0] == -1)
        return false;
 
    // Now use BFS to find if there
    // is any possible path or not.
 
    // Initialize the queue which holds
    // the discovered cells whose neighbors
    // are not discovered yet.
    vector<vector<int> > qu;
 
    rect[0][0] = 1;
    qu.push_back({ 0, 0 });
 
    // Discover cells until queue is not empty
    while (!qu.empty()) {
        vector<int> arr = qu.front();
        qu.erase(qu.begin());
        int elex = arr[0];
        int eley = arr[1];
 
        // Discover the eight adjacent nodes.
        // check top-left cell
        if ((elex > 0) && (eley > 0)
            && (rect[elex - 1][eley - 1] == 0)) {
            rect[elex - 1][eley - 1] = 1;
            vector<int> v = { elex - 1, eley - 1 };
            qu.push_back(v);
        }
 
        // check top cell
        if ((elex > 0) && (rect[elex - 1][eley] == 0)) {
            rect[elex - 1][eley] = 1;
            vector<int> v = { elex - 1, eley };
            qu.push_back(v);
        }
 
        // check top-right cell
        if ((elex > 0) && (eley < n - 1)
            && (rect[elex - 1][eley + 1] == 0)) {
            rect[elex - 1][eley + 1] = 1;
            vector<int> v = { elex - 1, eley + 1 };
            qu.push_back(v);
        }
 
        // check left cell
        if ((eley > 0) && (rect[elex][eley - 1] == 0)) {
            rect[elex][eley - 1] = 1;
            vector<int> v = { elex, eley - 1 };
            qu.push_back(v);
        }
 
        // check right cell
        if ((eley < n - 1) && (rect[elex][eley + 1] == 0)) {
            rect[elex][eley + 1] = 1;
            vector<int> v = { elex, eley + 1 };
            qu.push_back(v);
        }
 
        // check bottom-left cell
        if ((elex < m - 1) && (eley > 0)
            && (rect[elex + 1][eley - 1] == 0)) {
            rect[elex + 1][eley - 1] = 1;
            vector<int> v = { elex + 1, eley - 1 };
            qu.push_back(v);
        }
 
        // check bottom cell
        if ((elex < m - 1) && (rect[elex + 1][eley] == 0)) {
            rect[elex + 1][eley] = 1;
            vector<int> v = { elex + 1, eley };
            qu.push_back(v);
        }
 
        // check bottom-right cell
        if ((elex < m - 1) && (eley < n - 1)
            && (rect[elex + 1][eley + 1] == 0)) {
            rect[elex + 1][eley + 1] = 1;
            vector<int> v = { elex + 1, eley + 1 };
            qu.push_back(v);
        }
    }
 
    // Now if the end cell (i.e. bottom right cell)
    // is 1(reachable) then we will send true.
    return (rect[m - 1][n - 1] == 1);
}
 
// Driver Program
int main()
{
 
    // Test case 1
    int m1 = 5, n1 = 5, k1 = 2, r1 = 1;
    vector<int> X1 = { 1, 3 };
    vector<int> Y1 = { 3, 3 };
   
    // Function call
    if (isPossible(m1, n1, k1, r1, X1, Y1))
        cout << "Possible" << endl;
    else
        cout << "Not Possible" << endl;
 
    // Test case 2
    int m2 = 5, n2 = 5, k2 = 2, r2 = 1;
    vector<int> X2 = { 1, 1 };
    vector<int> Y2 = { 2, 3 };
   
    // Function call
    if (isPossible(m2, n2, k2, r2, X2, Y2))
        cout << "Possible" << endl;
    else
        cout << "Not Possible" << endl;
 
    return 0;
}

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Python3

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# Python3 program to find out path in
# a rectangle containing circles.
import math
import queue
 
# Function to find out if there is
# any possible path or not.
 
 
def isPossible(m, n, k, r, X, Y):
 
    # Take an array of m*n size and
    # initialize each element to 0.
    rect = [[0] * n for i in range(m)]
 
    # Now using Pythagorean theorem find if a
    # cell touches or within any circle or not.
    for i in range(m):
        for j in range(n):
            for p in range(k):
                if (math.sqrt((pow((X[p] - 1 - i), 2) +
                               pow((Y[p] - 1 - j), 2))) <= r):
                    rect[i][j] = -1
 
    # If the starting cell comes within
    # any circle return false.
    if (rect[0][0] == -1):
        return False
 
    # Now use BFS to find if there
    # is any possible path or not.
 
    # Initialize the queue which holds
    # the discovered cells whose neighbors
    # are not discovered yet.
    qu = queue.Queue()
 
    rect[0][0] = 1
    qu.put([0, 0])
 
    # Discover cells until queue is not empty
    while (not qu.empty()):
        arr = qu.get()
        elex = arr[0]
        eley = arr[1]
 
        # Discover the eight adjacent nodes.
        # check top-left cell
        if ((elex > 0) and (eley > 0) and
                (rect[elex - 1][eley - 1] == 0)):
            rect[elex - 1][eley - 1] = 1
            v = [elex - 1, eley - 1]
            qu.put(v)
 
        # check top cell
        if ((elex > 0) and
                (rect[elex - 1][eley] == 0)):
            rect[elex - 1][eley] = 1
            v = [elex - 1, eley]
            qu.put(v)
 
        # check top-right cell
        if ((elex > 0) and (eley < n - 1) and
                (rect[elex - 1][eley + 1] == 0)):
            rect[elex - 1][eley + 1] = 1
            v = [elex - 1, eley + 1]
            qu.put(v)
 
        # check left cell
        if ((eley > 0) and
                (rect[elex][eley - 1] == 0)):
            rect[elex][eley - 1] = 1
            v = [elex, eley - 1]
            qu.put(v)
 
        # check right cell
        if ((eley < n - 1) and
                (rect[elex][eley + 1] == 0)):
            rect[elex][eley + 1] = 1
            v = [elex, eley + 1]
            qu.put(v)
 
        # check bottom-left cell
        if ((elex < m - 1) and (eley > 0) and
                (rect[elex + 1][eley - 1] == 0)):
            rect[elex + 1][eley - 1] = 1
            v = [elex + 1, eley - 1]
            qu.put(v)
 
        # check bottom cell
        if ((elex < m - 1) and
                (rect[elex + 1][eley] == 0)):
            rect[elex + 1][eley] = 1
            v = [elex + 1, eley]
            qu.put(v)
 
        # check bottom-right cell
        if ((elex < m - 1) and (eley < n - 1) and
                (rect[elex + 1][eley + 1] == 0)):
            rect[elex + 1][eley + 1] = 1
            v = [elex + 1, eley + 1]
            qu.put(v)
 
    # Now if the end cell (i.e. bottom right cell)
    # is 1(reachable) then we will send true.
    return (rect[m - 1][n - 1] == 1)
 
 
# Driver Code
if __name__ == '__main__':
 
    # Test case 1
    m1 = 5
    n1 = 5
    k1 = 2
    r1 = 1
    X1 = [1, 3]
    Y1 = [3, 3]
     
    # Function call
    if (isPossible(m1, n1, k1, r1, X1, Y1)):
        print("Possible")
    else:
        print("Not Possible")
 
    # Test case 2
    m2 = 5
    n2 = 5
    k2 = 2
    r2 = 1
    X2 = [1, 1]
    Y2 = [2, 3]
     
    # Function call
    if (isPossible(m2, n2, k2, r2, X2, Y2)):
        print("Possible")
    else:
        print("Not Possible")
 
# This code is contributed by PranchalK

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Output

Possible
Not Possible

Time Complexity : It takes O(m*n*k) time to compute whether a cell is within or not in any circle. And it takes O(V+E) time in BFS. Here, number of edges in m*n grid is m*(n-1)+n*(m-1) and vertices m*n. So it takes O(m*n) time in DFS. Hence, the time complexity is O(m*n*k). The complexity can be improved if we iterate through each circles and mark -1 the cells which are coming within it.
 

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