66 lines
2.1 KiB
Java
66 lines
2.1 KiB
Java
package DynamicProgramming;
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// Given N dice each with M faces, numbered from 1 to M, find the number of ways to get sum X.
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// X is the summation of values on each face when all the dice are thrown.
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/* The Naive approach is to find all the possible combinations of values from n dice and
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keep on counting the results that sum to X. This can be done using recursion. */
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// The above recursion solution exhibits overlapping subproblems.
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/* Hence, storing the results of the solved sub-problems saves time.
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And it can be done using Dynamic Programming(DP).
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Following is implementation of Dynamic Programming approach. */
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// Code ---->
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// Java program to find number of ways to get sum 'x' with 'n'
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// dice where every dice has 'm' faces
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class DP {
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/* The main function that returns the number of ways to get sum 'x' with 'n' dice and 'm' with m faces. */
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public static long findWays(int m, int n, int x){
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/* Create a table to store the results of subproblems.
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One extra row and column are used for simplicity
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(Number of dice is directly used as row index and sum is directly used as column index).
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The entries in 0th row and 0th column are never used. */
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long[][] table = new long[n+1][x+1];
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/* Table entries for only one dice */
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for(int j = 1; j <= m && j <= x; j++)
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table[1][j] = 1;
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/* Fill rest of the entries in table using recursive relation
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i: number of dice, j: sum */
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for(int i = 2; i <= n;i ++){
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for(int j = 1; j <= x; j++){
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for(int k = 1; k < j && k <= m; k++)
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table[i][j] += table[i-1][j-k];
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}
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}
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return table[n][x];
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}
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public static void main (String[] args) {
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System.out.println(findWays(4, 2, 1));
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System.out.println(findWays(2, 2, 3));
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System.out.println(findWays(6, 3, 8));
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System.out.println(findWays(4, 2, 5));
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System.out.println(findWays(4, 3, 5));
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}
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}
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/*
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OUTPUT:
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0
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2
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21
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4
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6
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*/
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// Time Complexity: O(m * n * x) where m is number of faces, n is number of dice and x is given sum.
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