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# Queries to count groups of N students possible having sum of ratings within given range

Given integers N and K representing the number of batches and number of students in each batch respectively, and a 2D array ratings[][] of size N * K where each row has ratings for every K students and Q queries of type {a, b}. The task for each query is to count the number of groups of N students possible by selecting a student from each batch, such that sum of ratings in each group lies in the range [a, b] inclusively.

Examples:

Input: N = 2, K = 3, ratings[][]= { {1, 2, 3}, {4, 5, 6} }, Q = 2, Queries[][]={ {6, 6}, {1, 6} }
Output: 2 3
Explanation:
All possible groups of size N(=2) are:
1 + 4 = 5
1 + 5 = 6
1 + 6 = 7
2 + 4 = 6
2 + 5 = 7
2 + 6 = 8
3 + 4 = 7
3 + 5 = 8
3 + 6 = 9
Query 1: The groups whose sum in range of (6, 6) inclusive are (1 + 5), (2 + 4) is 2.
Query 2: The groups whose sum in range of (1, 6) inclusive are (1 + 4), (1 + 5), (2 + 4) is 3.

Input: N = 3, K = 3, ratings[][]={ {1, 2, 3}, {4, 5, 6}, {7, 8, 9} }, Q = 2, Queries[][]={ {10, 13}, {1, 7} }
Output: 4 0
Explanation:
Out of All possible groups of size N(=3):
Query 1: The groups whose sum in range (10, 13) inclusive is (1 + 4 + 7), (1 + 5 + 7), (2 + 4 + 7), (1 + 4 + 8) is 4.
Query 2: The groups whose sum in range of (1, 7) inclusive is  0.

Naive Approach: The simplest approach is to use recursion to generate all possible groups of size N. At each step of recursion calculate the sum that lies within the range of a query and find the number of groups that lie in the given range.

Time Complexity: O(NK)
Auxiliary Space: O(1)

Efficient Approach: The above approach can be optimized by using Dynamic Programming. The idea is that if the number of times the sum S appears in the ith row is known, then the Prefix Sum Technique can be used to answer all the queries in constant time. So in this way, the Overlapping Subproblems are calculated only once reducing the exponential time into polynomial time. Below are the steps:

• Initialize auxiliary array dp[][] where dp[i][sum] is the number of times a sum is present in the ith row.
• For each batch, i iterate through all possible sum S, and for each j students, if sum S is greater than the current rating ratings[i][j] then update the current dp state dp[i][S] as:

dp[i][S] = dp[i][S] + dp[i – 1][sum – rating[i][j]]

• After the above steps, dp[N – 1][sum], which is the number of times the sum appears in the (N – 1)th row.
• To answer each queries efficiently find the prefix sum of the last row i.e., dp[N – 1][S] for all values of sum S.
• Now, the number of ways to form groups in the given range [a, b] is dp[N – 1][b] – dp[N – 1][a – 1].

Below is the implementation of the above approach:

## C++

 `// C++ program for the above approach` `#include ``using` `namespace` `std;` `// Given n batches and k students``#define n 2``#define k 3` `// Function to count number of``// ways to get given sum groups``void` `numWays(``int` `ratings[n][k], ``int` `queries[])``{` `    ``// Initialise dp array``    ``int` `dp[n][10000 + 2];` `    ``// Mark all 1st row values as 1``    ``// since the mat[i] is all``    ``// possible sums in first row``    ``for` `(``int` `i = 0; i < k; i++)``        ``dp[ratings[i]] += 1;` `    ``// Fix the ith row``    ``for` `(``int` `i = 1; i < n; i++) {` `        ``// Fix the sum``        ``for` `(``int` `sum = 0; sum <= 10000; sum++)``        ``{``            ``// Iterate through all``            ``// values of ith row``            ``for` `(``int` `j = 0; j < k; j++)``            ``{``                ``// If sum can be obtained``                ``if` `(sum >= ratings[i][j])``                    ``dp[i][sum]``                        ``+= dp[i - 1][sum - ratings[i][j]];``            ``}``        ``}``    ``}` `    ``// Find the prefix sum of last row``    ``for` `(``int` `sum = 1; sum <= 10000; sum++)``    ``{``        ``dp[n - 1][sum] += dp[n - 1][sum - 1];``    ``}` `    ``// Traverse each query` `    ``for` `(``int` `q = 0; q < 2; q++)``    ``{``        ``int` `a = queries[q];``        ``int` `b = queries[q];` `        ``// No of ways to form groups``        ``cout << dp[n - 1][b] - dp[n - 1][a - 1] << ``" "``;``    ``}``}` `// Driver Code``int` `main()``{` `    ``// Given ratings``    ``int` `ratings[n][k] = { { 1, 2, 3 }, { 4, 5, 6 } };` `    ``// Given Queries``    ``int` `queries[] = { { 6, 6 }, { 1, 6 } };` `    ``// Function Call``    ``numWays(ratings, queries);` `    ``return` `0;``}`

## Java

 `// Java program for the above approach` `import` `java.util.*;``public` `class` `Main {` `    ``// Function to count number of``    ``// ways to get given sum groups``    ``public` `static` `void``    ``numWays(``int``[][] ratings,``            ``int` `queries[][],``            ``int` `n, ``int` `k)``    ``{` `        ``// Initialise dp array``        ``int` `dp[][] = ``new` `int``[n][``10000` `+ ``2``];` `        ``// Mark all 1st row values as 1``        ``// since the mat[i] is all``        ``// possible sums in first row``        ``for` `(``int` `i = ``0``; i < k; i++)``            ``dp[``0``][ratings[``0``][i]] += ``1``;` `        ``// Fix the ith row``        ``for` `(``int` `i = ``1``; i < n; i++)``        ``{` `            ``// Fix the sum``            ``for` `(``int` `sum = ``0``; sum <= ``10000``; sum++)``            ``{` `                ``// Iterate through all``                ``// values of ith row``                ``for` `(``int` `j = ``0``; j < k; j++)``                ``{` `                    ``// If sum can be obtained``                    ``if` `(sum >= ratings[i][j])``                        ``dp[i][sum]``                            ``+= dp[i - ``1``]``                               ``[sum - ratings[i][j]];``                ``}``            ``}``        ``}` `        ``// Find the prefix sum of last row``        ``for` `(``int` `sum = ``1``; sum <= ``10000``; sum++) {``            ``dp[n - ``1``][sum] += dp[n - ``1``][sum - ``1``];``        ``}` `        ``// Traverse each query``        ``for` `(``int` `q = ``0``; q < queries.length; q++) {` `            ``int` `a = queries[q][``0``];``            ``int` `b = queries[q][``1``];` `            ``// No of ways to form groups``            ``System.out.print(dp[n - ``1``][b] - dp[n - ``1``][a - ``1``]``                             ``+ ``" "``);``        ``}``    ``}` `    ``// Driver Code``    ``public` `static` `void` `main(String args[])``    ``{``        ``// Given N batches and K students``        ``int` `N = ``2``, K = ``3``;` `        ``// Given ratings``        ``int` `ratings[][] = { { ``1``, ``2``, ``3` `}, { ``4``, ``5``, ``6` `} };` `        ``// Given Queries``        ``int` `queries[][] = { { ``6``, ``6` `}, { ``1``, ``6` `} };` `        ``// Function Call``        ``numWays(ratings, queries, N, K);``    ``}``}`

## Python3

 `# Python3 program for the``# above approach` `# Function to count number of``# ways to get given sum groups``def` `numWays(ratings, queries,``            ``n, k):``  ` `    ``# Initialise dp array``    ``dp ``=` `[[``0` `for` `i ``in` `range``(``10002``)]``             ``for` `j ``in` `range``(n)];` `    ``# Mark all 1st row values as 1``    ``# since the mat[i] is all``    ``# possible sums in first row``    ``for` `i ``in` `range``(k):``        ``dp[``0``][ratings[``0``][i]] ``+``=` `1``;` `    ``# Fix the ith row``    ``for` `i ``in` `range``(``1``, n):` `        ``# Fix the sum``        ``for` `sum` `in` `range``(``10001``):` `            ``# Iterate through all``            ``# values of ith row``            ``for` `j ``in` `range``(k):` `                ``# If sum can be obtained``                ``if` `(``sum` `>``=` `ratings[i][j]):``                    ``dp[i][``sum``] ``+``=` `dp[i ``-` `1``][``sum` `-``                                            ``ratings[i][j]];` `    ``# Find the prefix sum of``    ``# last row``    ``for` `sum` `in` `range``(``1``, ``10001``):``        ``dp[n ``-` `1``][``sum``] ``+``=` `dp[n ``-` `1``][``sum` `-` `1``];` `    ``# Traverse each query``    ``for` `q ``in` `range``(``len``(queries)):``        ``a ``=` `queries[q][``0``];``        ``b ``=` `queries[q][``1``];` `        ``# No of ways to form groups``        ``print``(dp[n ``-` `1``][b] ``-``              ``dp[n ``-` `1``][a ``-` `1``],``              ``end ``=` `" "``);` `# Driver Code``if` `__name__ ``=``=` `'__main__'``:``  ` `    ``# Given N batches and``    ``# K students``    ``N ``=` `2``;``    ``K ``=` `3``;` `    ``# Given ratings``    ``ratings ``=` `[[``1``, ``2``, ``3``],``               ``[``4``, ``5``, ``6``]];` `    ``queries ``=` `[[``6``, ``6``],``               ``[``1``, ``6``]];` `    ``# Function Call``    ``numWays(ratings, queries, N, K);` `# This code is contributed by 29AjayKumar`

## C#

 `// C# program for the above approach``using` `System;` `class` `GFG{` `// Function to count number of``// ways to get given sum groups``public` `static` `void` `numWays(``int``[,] ratings,``                           ``int` `[,]queries,``                           ``int` `n, ``int` `k)``{``    ` `    ``// Initialise dp array``    ``int` `[,]dp = ``new` `int``[n, 10000 + 2];` `    ``// Mark all 1st row values as 1``    ``// since the mat[0,i] is all``    ``// possible sums in first row``    ``for``(``int` `i = 0; i < k; i++)``        ``dp[0, ratings[0, i]] += 1;` `    ``// Fix the ith row``    ``for``(``int` `i = 1; i < n; i++)``    ``{``        ` `        ``// Fix the sum``        ``for``(``int` `sum = 0; sum <= 10000; sum++)``        ``{``            ` `            ``// Iterate through all``            ``// values of ith row``            ``for``(``int` `j = 0; j < k; j++)``            ``{` `                ``// If sum can be obtained``                ``if` `(sum >= ratings[i, j])``                    ``dp[i, sum] += dp[i - 1,``                                   ``sum - ratings[i, j]];``            ``}``        ``}``    ``}` `    ``// Find the prefix sum of last row``    ``for``(``int` `sum = 1; sum <= 10000; sum++)``    ``{``        ``dp[n - 1, sum] += dp[n - 1, sum - 1];``    ``}` `    ``// Traverse each query``    ``for``(``int` `q = 0; q < queries.GetLength(0); q++)``    ``{``        ``int` `a = queries[q, 0];``        ``int` `b = queries[q, 1];` `        ``// No of ways to form groups``        ``Console.Write(dp[n - 1, b] -``                      ``dp[n - 1, a - 1] + ``" "``);``    ``}``}` `// Driver Code``public` `static` `void` `Main(String []args)``{``    ` `    ``// Given N batches and K students``    ``int` `N = 2, K = 3;` `    ``// Given ratings``    ``int` `[,]ratings = { { 1, 2, 3 }, { 4, 5, 6 } };` `    ``// Given Queries``    ``int` `[,]queries = { { 6, 6 }, { 1, 6 } };` `    ``// Function Call``    ``numWays(ratings, queries, N, K);``}``}` `// This code is contributed by Amit Katiyar`

## Javascript

 ``

Output

`2 3 `

Time Complexity: O(N*maxSum*K), where maxSum is the maximum sum.
Auxiliary Space: O(N*maxSum), where maxSum is the maximum sum.

Efficient approach : Space optimization

In previous approach the dp[i][j] is depend upon the current and previous row of 2D matrix. So to optimize space we use two vectors curr and prev that keep track of current and previous row of DP.

Implementation Steps:

• Initialize two vectors curr and prev to keep track of only current and previous row of Dp with 0.
• Now iterative over subproblems and get the current computation.
• While iteration initialize curr vector.
• Now compute the current value by the help of prev vector and store that value in curr.
• After every iteration store values of curr vector in prev vector for further iteration.
• At last return the answer stored in curr.

Implementation:

## C++

 `// C++ program for the above approach` `#include ``using` `namespace` `std;` `// Given n batches and k students``#define n 2``#define k 3` `// Function to count number of``// ways to get given sum groups``void` `numWays(``int` `ratings[n][k], ``int` `queries[])``{` `    ``// Initialise dp array``    ``// int dp[n][10000 + 2];``    ``vector<``int``> prev(10000 + 2);``    ``vector<``int``> curr(10000 + 2);` `    ``// Mark all 1st row values as 1``    ``// since the prev[i] is all``    ``// possible sums in first row``    ``for` `(``int` `i = 0; i < k; i++)``        ``prev[ratings[i]] += 1;` `    ``// Fix the ith row``    ``for` `(``int` `i = 1; i < n; i++) {` `        ``// Fix the sum``        ``for` `(``int` `sum = 0; sum <= 10000; sum++) {``            ``// Iterate through all``            ``// values of ith row``            ``for` `(``int` `j = 0; j < k; j++) {``                ``// If sum can be obtained``                ``if` `(sum >= ratings[i][j])``                    ``curr[sum] += prev[sum - ratings[i][j]];``            ``}``        ``}` `        ``// assigning values of curr vector to prev vector``        ``// for further iteration``        ``prev = curr;``    ``}` `    ``// Find the prefix sum of last row``    ``for` `(``int` `sum = 1; sum <= 10000; sum++) {``        ``curr[sum] += curr[sum - 1];``    ``}` `    ``// Traverse each query` `    ``for` `(``int` `q = 0; q < 2; q++) {``        ``int` `a = queries[q];``        ``int` `b = queries[q];` `        ``// No of ways to form groups``        ``cout << curr[b] - curr[a - 1] << ``" "``;``    ``}``}` `// Driver Code``int` `main()``{` `    ``// Given ratings``    ``int` `ratings[n][k] = { { 1, 2, 3 }, { 4, 5, 6 } };` `    ``// Given Queries``    ``int` `queries[] = { { 6, 6 }, { 1, 6 } };` `    ``// Function Call``    ``numWays(ratings, queries);` `    ``return` `0;``}`

## Javascript

 `// Given n batches and k students``const n = 2;``const k = 3;` `// Function to count number of ways to get given sum groups``function` `numWays(ratings, queries) {``  ``// Initialize dp array``  ``let prev = ``new` `Array(10000 + 2).fill(0);``  ``let curr = ``new` `Array(10000 + 2).fill(0);` `  ``// Mark all 1st row values as 1 since the prev[i] is all possible sums in first row``  ``for` `(let i = 0; i < k; i++) prev[ratings[i]] += 1;` `  ``// Fix the ith row``  ``for` `(let i = 1; i < n; i++) {``    ``// Fix the sum``    ``for` `(let sum = 0; sum <= 10000; sum++) {``      ``// Iterate through all values of ith row``      ``for` `(let j = 0; j < k; j++) {``        ``// If sum can be obtained``        ``if` `(sum >= ratings[i][j]) curr[sum] += prev[sum - ratings[i][j]];``      ``}``    ``}` `    ``// Assigning values of curr vector to prev vector for further iteration``    ``prev = [...curr];``  ``}` `  ``// Find the prefix sum of last row``  ``for` `(let sum = 1; sum <= 10000; sum++) {``    ``curr[sum] += curr[sum - 1];``  ``}` `  ``// Traverse each query``  ``for` `(let q = 0; q < queries.length; q++) {``    ``let a = queries[q];``    ``let b = queries[q];` `    ``// No of ways to form groups``    ``console.log(curr[b] - curr[a - 1]);``  ``}``}` `// Given ratings``const ratings = [``  ``[1, 2, 3],``  ``[4, 5, 6],``];` `// Given Queries``const queries = [``  ``[6,6],``  ``[1,6]``];` `// Function Call``numWays(ratings, queries);`

## Python3

 `# Python program for the above approach` `# Given n batches and k students``n ``=` `2``k ``=` `3` `# Function to count number of``# ways to get given sum groups`  `def` `numWays(ratings, queries):` `    ``# Initialise dp array``    ``prev ``=` `[``0``] ``*` `(``10000` `+` `2``)``    ``curr ``=` `[``0``] ``*` `(``10000` `+` `2``)` `    ``# Mark all 1st row values as 1``    ``# since the prev[i] is all``    ``# possible sums in first row``    ``for` `i ``in` `range``(k):``        ``prev[ratings[``0``][i]] ``+``=` `1` `    ``# Fix the ith row``    ``for` `i ``in` `range``(``1``, n):` `        ``# Fix the sum``        ``for` `s ``in` `range``(``0``, ``10001``):` `            ``# Iterate through all``            ``# values of ith row``            ``for` `j ``in` `range``(k):` `                ``# If sum can be obtained``                ``if` `s >``=` `ratings[i][j]:``                    ``curr[s] ``+``=` `prev[s ``-` `ratings[i][j]]` `        ``# assigning values of curr list to prev list``        ``# for further iteration``        ``prev ``=` `curr.copy()``        ``curr ``=` `[``0``] ``*` `(``10000` `+` `2``)` `    ``# Find the prefix sum of last row``    ``for` `s ``in` `range``(``1``, ``10001``):``        ``prev[s] ``+``=` `prev[s ``-` `1``]` `    ``# Traverse each query``    ``for` `q ``in` `range``(``2``):` `        ``a ``=` `queries[q][``0``]``        ``b ``=` `queries[q][``1``]` `        ``# No of ways to form groups``        ``print``(prev[b] ``-` `prev[a ``-` `1``], end``=``' '``)`  `# Driver Code``if` `__name__ ``=``=` `'__main__'``:` `    ``# Given ratings``    ``ratings ``=` `[[``1``, ``2``, ``3``], [``4``, ``5``, ``6``]]` `    ``# Given Queries``    ``queries ``=` `[[``6``, ``6``], [``1``, ``6``]]` `    ``# Function Call``    ``numWays(ratings, queries)`

Output

`2 3`

Time Complexity: O(N*maxSum*K), where maxSum is the maximum sum.
Auxiliary Space: O(maxSum)