Având în vedere o matrice dreptunghiulară, putem trece de la celula curentă în 4 direcții cu probabilitate egală. Cele 4 direcții sunt dreapta stânga sus sau inferioară. Calculați probabilitatea ca după n să se deplaseze dintr -o poziție dată (i j) în matrice, nu vom traversa limitele matricei în niciun moment.
caracter în șir în java
Vă recomandăm cu tărie să vă minimizați browserul și să încercați acest lucru mai întâi.
Ideea este de a efectua ceva similar cu DFS. Traversăm recursiv în fiecare dintre cele 4 direcții permise și pentru fiecare celulă întâlnită calculăm probabilitatea necesară cu o mișcare mai mică. Deoarece fiecare direcție are probabilitatea egală, fiecare direcție va contribui la 1/4 din probabilitatea totală a acelei celule, adică 0,25. Revenim 0 dacă ieșim în afara matricei și returnăm 1 dacă n pași sunt finalizați fără a traversa limitele matricei.
Mai jos este implementarea ideii de mai sus:
C++/// C++ program to find the probability // that we do not cross boundary of a // matrix after N moves. #include
/// C program to find the probability // that we do not cross boundary of a // matrix after N moves. #include
// Java program to find the probability // that we do not cross boundary // of a matrix after N moves. import java.io.*; class GFG { // check if (x y) is valid // matrix coordinate static boolean isSafe(int x int y int m int n) { return (x >= 0 && x < m && y >= 0 && y < n); } // Function to calculate probability // that after N moves from a given // position (x y) in m x n matrix // boundaries of the matrix will // not be crossed. static double findProbability(int m int n int x int y int N) { // boundary crossed if (! isSafe(x y m n)) return 0.0; // N steps taken if (N == 0) return 1.0; // Initialize result double prob = 0.0; // move up prob += findProbability(m n x - 1 y N - 1) * 0.25; // move right prob += findProbability(m n x y + 1 N - 1) * 0.25; // move down prob += findProbability(m n x + 1 y N - 1) * 0.25; // move left prob += findProbability(m n x y - 1 N - 1) * 0.25; return prob; } // Driver code public static void main (String[] args) { // matrix size int m = 5 n = 5; // coordinates of starting point int i = 1 j = 1; // Number of steps int N = 2; System.out.println('Probability is ' + findProbability(m n i j N)); } } // This code is contributed by KRV.
# Python3 program to find the probability # that we do not cross boundary of a # matrix after N moves. # check if (x y) is valid matrix coordinate def isSafe(x y m n): return (x >= 0 and x < m and y >= 0 and y < n) # Function to calculate probability # that after N moves from a given # position (x y) in m x n matrix # boundaries of the matrix will # not be crossed. def findProbability(m n x y N): # boundary crossed if (not isSafe(x y m n)): return 0.0 # N steps taken if (N == 0): return 1.0 # Initialize result prob = 0.0 # move up prob += findProbability(m n x - 1 y N - 1) * 0.25 # move right prob += findProbability(m n x y + 1 N - 1) * 0.25 # move down prob += findProbability(m n x + 1 y N - 1) * 0.25 # move left prob += findProbability(m n x y - 1 N - 1) * 0.25 return prob # Driver code if __name__ == '__main__': # matrix size m = 5 n = 5 # coordinates of starting po i = 1 j = 1 # Number of steps N = 2 print('Probability is' findProbability(m n i j N)) # This code is contributed by PranchalK
// C# program to find the probability // that we do not cross boundary // of a matrix after N moves. using System; class GFG { // check if (x y) is valid // matrix coordinate static bool isSafe(int x int y int m int n) { return (x >= 0 && x < m && y >= 0 && y < n); } // Function to calculate probability // that after N moves from a given // position (x y) in m x n matrix // boundaries of the matrix will // not be crossed. static double findProbability(int m int n int x int y int N) { // boundary crossed if (! isSafe(x y m n)) return 0.0; // N steps taken if (N == 0) return 1.0; // Initialize result double prob = 0.0; // move up prob += findProbability(m n x - 1 y N - 1) * 0.25; // move right prob += findProbability(m n x y + 1 N - 1) * 0.25; // move down prob += findProbability(m n x + 1 y N - 1) * 0.25; // move left prob += findProbability(m n x y - 1 N - 1) * 0.25; return prob; } // Driver code public static void Main () { // matrix size int m = 5 n = 5; // coordinates of starting point int i = 1 j = 1; // Number of steps int N = 2; Console.Write('Probability is ' + findProbability(m n i j N)); } } // This code is contributed by nitin mittal.
// PHP program to find the probability // that we do not cross boundary of a // matrix after N moves. // check if (x y) is valid // matrix coordinate function isSafe($x $y $m $n) { return ($x >= 0 && $x < $m && $y >= 0 && $y < $n); } // Function to calculate probability // that after N moves from a given // position (x y) in m x n matrix // boundaries of the matrix will // not be crossed. function findProbability($m $n $x $y $N) { // boundary crossed if (!isSafe($x $y $m $n)) return 0.0; // N steps taken if ($N == 0) return 1.0; // Initialize result $prob = 0.0; // move up $prob += findProbability($m $n $x - 1 $y $N - 1) * 0.25; // move right $prob += findProbability($m $n $x $y + 1 $N - 1) * 0.25; // move down $prob += findProbability($m $n $x + 1 $y $N - 1) * 0.25; // move left $prob += findProbability($m $n $x $y - 1 $N - 1) * 0.25; return $prob; } // Driver code // matrix size $m = 5; $n = 5; // coordinates of starting point $i = 1; $j = 1; // Number of steps $N = 2; echo 'Probability is ' findProbability($m $n $i $j $N); // This code is contributed by nitin mittal. ?> <script> // JavaScript program to find the probability // that we do not cross boundary of a // matrix after N moves. // check if (x y) is valid matrix coordinate function isSafe(x y m n){ return (x >= 0 && x < m && y >= 0 && y < n); } // Function to calculate probability // that after N moves from a given // position (x y) in m x n matrix // boundaries of the matrix will not be crossed. function findProbability( m n x y N){ // boundary crossed if (!isSafe(x y m n)) return 0.0; // N steps taken if (N == 0) return 1.0; // Initialize result let prob = 0.0; // move up prob += findProbability(m n x - 1 y N - 1) * 0.25; // move right prob += findProbability(m n x y + 1 N - 1) * 0.25; // move down prob += findProbability(m n x + 1 y N - 1) * 0.25; // move left prob += findProbability(m n x y - 1 N - 1) * 0.25; return prob; } // Driver code // matrix size let m = 5 n = 5; // coordinates of starting point let i = 1 j = 1; // Number of steps let N = 2; document.write( 'Probability is ' +findProbability(m n i j N)); </script>
Ieșire
Probability is 0.875