Givet en rektangulær matrix kan vi bevæge os fra den nuværende celle i 4 retninger med lige sandsynlighed. De 4 retninger er højre venstre øverste eller bund. Beregn sandsynligheden for, at vi efter N bevæger sig fra en given position (i J) i matrixen, vil vi ikke krydse grænserne for matrixen på noget tidspunkt.
sjakal vs ulv
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Ideen er at udføre noget, der ligner DFS. Vi gennemgår rekursivt i hver af de 4 tilladte retning, og for hver celle, der er stødt på, beregner vi den krævede sandsynlighed med et mindre træk. Da hver retning har lige sandsynlighed, vil hver retning bidrage til 1/4 af den samlede sandsynlighed for den celle, dvs. 0,25. Vi returnerer 0, hvis vi træder uden for matrixen og returnerer 1, hvis n trin er afsluttet uden at krydse matrixgrænser.
Nedenfor er implementeringen af ovenstående idé:
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>
Produktion
Probability is 0.875