PreComputation Technique on Arrays

Precomputation refers to the process of pre-calculating and storing the results of certain computations or data structures(array in this case) in advance, in order to speed up the execution time of a program. This can be useful in situations where the same calculations are needed multiple times, as it avoids the need to recalculate them every time.

This technique relies on fast retreival of already stored data rather than recalculation.

Some Common Pre-Computation Techniques used on Array:

Though pre-computation can be done in various ways over a wide range of questions, in this section we will look at some of the popular techniques of pre-computation:

Prefix/Suffix on 1-D Array:

A prefix array PRE[] is defined on an array ARR[], such that for each index ‘i‘ PRE[i] stores the information about the values from ARR[0] to ARR[i]. This information can be in the form of prefix sum, prefix max, prefix gcd, etc.

Let’s construct prefix array for ARR[] = {4, 2, 1, -1, 3}, to do this we can simply iterate ARR[] from 0 to size-1, and store PRE[i]=ARR[i]+PRE[i-1], after doing this we have PRE[] = {4, 6, 7, 6, 9}

Why do we need this Prefix Array?

• For simplicity consider the example for Prefix Sum Array. Suppose we have an array ARR[] of size N, and Q queries and for each query, we have to output the sum of values between the index L to R.
• The Brute Force way of doing this will result in a time complexity of O(N*Q) where we iterate from L to R in each query to know the sum. In order to improve this time complexity we can use Pre-Computation i.e. calculate the Prefix sum array before processing the queries, and then for each query simply return PRE[R]-PRE[L-1] as the result in O(1), the overall complexity then would be O(max(N, Q))
• The Suffix array is similar to the Prefix array the only difference is that the Suffix array stores the information about the values from index ‘i’ to the end of the array rather that start of the array i..e. SUFFIX[i] stores information about values from ARR[i] to ARR[size-1].

Below is the code to construct a Prefix Sum and Suffix Sum array:

C++

#include <iostream>
using namespace std;

// This function prints the array
void print(int arr[], int N)
{
    for (int i = 0; i < N; i++) {
        cout << arr[i] << " ";
    }
    cout << endl;
}

// This function calculates the PrefixSum
// of the array
void PrefixSum(int ARR[], int N)
{

    // PRE[] array stores the result
    int PRE[N];
    for (int i = 0; i < N; i++) {

        // At index 0, no previous values
        // are present
        if (i == 0)
            PRE[i] = ARR[i];
        else
            PRE[i] = PRE[i - 1] + ARR[i];
    }
    cout << "Prefix Sum: ";
    print(PRE, N);
}

// This function finds the Suffix
// sum of the array
void SuffixSum(int ARR[], int N)
{

    // This array stores the result
    int SUF[N];
    for (int i = N - 1; i >= 0; i--) {

        // At index N-1, there is no suffix
        // value available
        if (i == N - 1)
            SUF[i] = ARR[i];
        else
            SUF[i] = SUF[i + 1] + ARR[i];
    }
    cout << "Suffix Sum: ";
    print(SUF, N);
}

// Driver code
int main()
{
    int N = 5;
    int ARR[N] = { 4, 2, 1, -1, 3 };
    cout << "Given Array: ";
    print(ARR, N);

    // Funtion call to calculate prefix sum
    PrefixSum(ARR, N);

    // Function call to calculate suffix sum
    SuffixSum(ARR, N);
    return 0;
}

Java

public class PrefixSuffixSum {

    // This function prints the array
    static void print(int[] arr, int N) {
        for (int i = 0; i < N; i++) {
            System.out.print(arr[i] + " ");
        }
        System.out.println();
    }

    // This function calculates the PrefixSum of the array
    static void prefixSum(int[] arr, int N) {
        // PRE[] array stores the result
        int[] PRE = new int[N];
        for (int i = 0; i < N; i++) {
            // At index 0, no previous values are present
            if (i == 0)
                PRE[i] = arr[i];
            else
                PRE[i] = PRE[i - 1] + arr[i];
        }
        System.out.print("Prefix Sum: ");
        print(PRE, N);
    }

    // This function finds the Suffix sum of the array
    static void suffixSum(int[] arr, int N) {
        // This array stores the result
        int[] SUF = new int[N];
        for (int i = N - 1; i >= 0; i--) {
            // At index N-1, there is no suffix value available
            if (i == N - 1)
                SUF[i] = arr[i];
            else
                SUF[i] = SUF[i + 1] + arr[i];
        }
        System.out.print("Suffix Sum: ");
        print(SUF, N);
    }

    // Driver code
    public static void main(String[] args) {
        int N = 5;
        int[] ARR = { 4, 2, 1, -1, 3 };
        System.out.print("Given Array: ");
        print(ARR, N);

        // Function call to calculate prefix sum
        prefixSum(ARR, N);

        // Function call to calculate suffix sum
        suffixSum(ARR, N);
    }
}
//This code is contriuted by chinmaya121221

Python3

# This function prints the array
def print_array(arr, N):
    for i in range(N):
        print(arr[i], end = " ")
    print()

# This function calculates the PrefixSum
# of the array
def PrefixSum(ARR, N):

    # PRE[] array stores the result
    PRE = [0] * N
    for i in range(N):

        # At index 0, no previous values
        # are present
        if (i == 0):
            PRE[i] = ARR[i]
        else:
            PRE[i] = PRE[i - 1] + ARR[i]
    print("Prefix Sum: ", end = "")
    print_array(PRE, N)

# This function finds the Suffix
# sum of the array
def SuffixSum(ARR, N):

    # This array stores the result
    SUF = [0] * N
    for i in range(N - 1, -1, -1):

        # At index N-1, there is no suffix
        # value available
        if (i == N - 1):
            SUF[i] = ARR[i]
        else:
            SUF[i] = SUF[i + 1] + ARR[i]
    print("Suffix Sum: ", end = "")
    print_array(SUF, N)

# Driver code
if __name__ == "__main__":
    N = 5
    ARR = [4, 2, 1, -1, 3]
    print("Given Array: ", end = "")
    print_array(ARR, N)

    # Funtion call to calculate prefix sum
    PrefixSum(ARR, N)

    # Function call to calculate suffix sum
    SuffixSum(ARR, N)

C#

using System;

class Program
{
    // This function prints the array
    static void Print(int[] arr, int N)
    {
        for (int i = 0; i < N; i++)
        {
            Console.Write(arr[i] + " ");
        }
        Console.WriteLine();
    }

    // This function calculates the PrefixSum of the array
    static void PrefixSum(int[] ARR, int N)
    {
        // PRE array stores the result
        int[] PRE = new int[N];
        for (int i = 0; i < N; i++)
        {
            // At index 0, no previous values are present
            if (i == 0)
                PRE[i] = ARR[i];
            else
                PRE[i] = PRE[i - 1] + ARR[i];
        }
        Console.Write("Prefix Sum: ");
        Print(PRE, N);
    }

    // This function finds the Suffix sum of the array
    static void SuffixSum(int[] ARR, int N)
    {
        // SUF array stores the result
        int[] SUF = new int[N];
        for (int i = N - 1; i >= 0; i--)
        {
            // At index N-1, there is no suffix value available
            if (i == N - 1)
                SUF[i] = ARR[i];
            else
                SUF[i] = SUF[i + 1] + ARR[i];
        }
        Console.Write("Suffix Sum: ");
        Print(SUF, N);
    }

    static void Main()
    {
        int N = 5;
        int[] ARR = { 4, 2, 1, -1, 3 };
        Console.Write("Given Array: ");
        Print(ARR, N);

        // Function call to calculate prefix sum
        PrefixSum(ARR, N);

        // Function call to calculate suffix sum
        SuffixSum(ARR, N);
    }
}

Javascript

// This function prints the array
function print(arr, N) {
    for (let i = 0; i < N; i++) {
        console.log(arr[i] + " ");
    }
    console.log();
}


// This function calculates the PrefixSum
// of the array
function PrefixSum(ARR, N) {

    // PRE[] array stores the result
    let PRE = [];
    for (let i = 0; i < N; i++) {

        // At index 0, no previous values
        // are present
        if (i === 0)
            PRE[i] = ARR[i];
        else
            PRE[i] = PRE[i - 1] + ARR[i];
    }
    console.log("Prefix Sum: ");
    print(PRE, N);
}

// This function finds the Suffix
// sum of the array
function SuffixSum(ARR, N) {

    // This array stores the result
    let SUF = [];
    for (let i = N - 1; i >= 0; i--) {

        // At index N-1, there is no suffix
        // value available
        if (i === N - 1)
            SUF[i] = ARR[i];
        else
            SUF[i] = SUF[i + 1] + ARR[i];
    }
    console.log("Suffix Sum: ");
    print(SUF, N);
}


// Driver code
let N = 5;
let ARR = [4, 2, 1, -1, 3];
console.log("Given Array: ");
print(ARR, N);

PrefixSum(ARR, N);

SuffixSum(ARR, N);

Output

Given Array: 4 2 1 -1 3 
Prefix Sum: 4 6 7 6 9 
Suffix Sum: 9 5 3 2 3 

Prefix on 2-D Array:

The above idea of Prefix on 1-D array can be expanded to 2-D array, where we have a 2-D array ARR[][] of size NxM and PRE[i][j] will store the information about the values from row 0 to ‘i’ and and column 0 to ‘j’ . Let’s say we want to retrieve the sum of values of submatrice [i1, j1] to [i2, j2], we can simply use our 2-D prefix sum array to achieve this by simple Inclusion Exclusion principle PRE[i2][j2] – PRE[i2][j1-1] – PRE[i1-1][j2] + PRE[i1-1][j1-1].

Below is the code construct prefix sum for 2-D array:

C++

// C++ code for the above approach:
#include <bits/stdc++.h>
using namespace std;

void print(vector<vector<int> >& arr, int N, int M)
{
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < M; j++) {
            cout << arr[i][j] << " ";
        }
        cout << endl;
    }
}

void PrefixSum(vector<vector<int> >& ARR, int N, int M)
{
    vector<vector<int> > PRE(N, vector<int>(M));
    for (int i = 0; i < N; i++) {
        for (int j = 0; j < M; j++) {
            int x = 0;
            int y = 0;
            int z = 0;
            if (i - 1 >= 0)
                x = PRE[i - 1][j];
            if (j - 1 >= 0)
                y = PRE[i][j - 1];
            if (i - 1 >= 0 && j - 1 >= 0)
                z = PRE[i - 1][j - 1];
            PRE[i][j] = ARR[i][j] + x + y - z;
        }
    }
    cout << "Give Array:" << endl;
    print(ARR, N, M);
    cout << "Prefix Sum Array:" << endl;
    print(PRE, N, M);
}

// Drivers code
int main()
{
    int N = 4;
    int M = 4;

    vector<vector<int> > ARR = { { 1, 1, 1, 1 },
                                 { 1, 1, 1, 1 },
                                 { 1, 1, 1, 1 },
                                 { 1, 1, 1, 1 } };

    // Function Call
    PrefixSum(ARR, N, M);
    return 0;
}

Java

import java.util.Arrays;

public class PrefixSum {

    // Function to print a 2D array
    static void print(int[][] arr, int N, int M)
    {
        for (int i = 0; i < N; i++) {
            for (int j = 0; j < M; j++) {
                System.out.print(arr[i][j] + " ");
            }
            System.out.println();
        }
    }

    // Function to compute the prefix sum of a 2D array
    static void prefixSum(int[][] arr, int N, int M)
    {
        int[][] pre = new int[N][M];

        for (int i = 0; i < N; i++) {
            for (int j = 0; j < M; j++) {
                int x = 0, y = 0, z = 0;
                if (i - 1 >= 0)
                    x = pre[i - 1][j];
                if (j - 1 >= 0)
                    y = pre[i][j - 1];
                if (i - 1 >= 0 && j - 1 >= 0)
                    z = pre[i - 1][j - 1];
                pre[i][j] = arr[i][j] + x + y - z;
            }
        }

        System.out.println("Given Array:");
        print(arr, N, M);
        System.out.println("Prefix Sum Array:");
        print(pre, N, M);
    }

    // Driver code
    public static void main(String[] args)
    {
        int N = 4;
        int M = 4;

        int[][] arr = { { 1, 1, 1, 1 },
                        { 1, 1, 1, 1 },
                        { 1, 1, 1, 1 },
                        { 1, 1, 1, 1 } };

        // Function call
        prefixSum(arr, N, M);
    }
}

Python3

def print_matrix(matrix):
    # Function to print a 2D array
    for row in matrix:
        print(" ".join(map(str, row)))

def prefix_sum(matrix, n, m):
    # Function to compute the prefix sum of a 2D array
    prefix = [[0] * m for _ in range(n)]

    for i in range(n):
        for j in range(m):
            x = 0 if i - 1 < 0 else prefix[i - 1][j]
            y = 0 if j - 1 < 0 else prefix[i][j - 1]
            z = 0 if i - 1 < 0 or j - 1 < 0 else prefix[i - 1][j - 1]

            prefix[i][j] = matrix[i][j] + x + y - z

    print("Given Array:")
    print_matrix(matrix)
    print("Prefix Sum Array:")
    print_matrix(prefix)

if __name__ == "__main__":
    N = 4
    M = 4

    matrix = [
        [1, 1, 1, 1],
        [1, 1, 1, 1],
        [1, 1, 1, 1],
        [1, 1, 1, 1]
    ]

    # Function call
    prefix_sum(matrix, N, M)

# This code is contributed by shivamgupta0987654321

C#

using System;

class Program {
    // Function to print a 2D array
    static void Print(int[, ] arr, int N, int M)
    {
        for (int i = 0; i < N; i++) {
            for (int j = 0; j < M; j++) {
                Console.Write(arr[i, j] + " ");
            }
            Console.WriteLine();
        }
    }

    // Function to compute the prefix sum of a 2D array
    static void PrefixSum(int[, ] ARR, int N, int M)
    {
        int[, ] PRE = new int[N, M];

        for (int i = 0; i < N; i++) {
            for (int j = 0; j < M; j++) {
                int x = 0;
                int y = 0;
                int z = 0;

                // Calculate the sum of elements above the
                // current element
                if (i - 1 >= 0)
                    x = PRE[i - 1, j];

                // Calculate the sum of elements to the left
                // of the current element
                if (j - 1 >= 0)
                    y = PRE[i, j - 1];

                // Calculate the sum of elements in the
                // diagonal (top-left)
                if (i - 1 >= 0 && j - 1 >= 0)
                    z = PRE[i - 1, j - 1];

                // Compute the prefix sum for the current
                // element
                PRE[i, j] = ARR[i, j] + x + y - z;
            }
        }

        // Print the given array
        Console.WriteLine("Given Array:");
        Print(ARR, N, M);

        // Print the computed prefix sum array
        Console.WriteLine("Prefix Sum Array:");
        Print(PRE, N, M);
    }

    // Driver code
    static void Main()
    {
        int N = 4;
        int M = 4;

        int[, ] ARR = { { 1, 1, 1, 1 },
                        { 1, 1, 1, 1 },
                        { 1, 1, 1, 1 },
                        { 1, 1, 1, 1 } };

        // Function Call
        PrefixSum(ARR, N, M);
    }
}

Javascript

function print(arr) {
    for (let i = 0; i < arr.length; i++) {
        console.log(arr[i].join(' '));
    }
    console.log();
}

function calculatePrefixSum(ARR) {
    const N = ARR.length;
    const M = ARR[0].length;

    // Initialize the prefix sum array
    const PRE = new Array(N);
    for (let i = 0; i < N; i++) {
        PRE[i] = new Array(M).fill(0);
    }

    for (let i = 0; i < N; i++) {
        for (let j = 0; j < M; j++) {
            let x = 0;
            let y = 0;
            let z = 0;

            if (i - 1 >= 0)
                x = PRE[i - 1][j];
            if (j - 1 >= 0)
                y = PRE[i][j - 1];
            if (i - 1 >= 0 && j - 1 >= 0)
                z = PRE[i - 1][j - 1];

            PRE[i][j] = ARR[i][j] + x + y - z;
        }
    }

    console.log("Given Array:");
    print(ARR);
    console.log("Prefix Sum Array:");
    print(PRE);
}

// Driver's code
const N = 4;
const M = 4;

const ARR = [
    [1, 1, 1, 1],
    [1, 1, 1, 1],
    [1, 1, 1, 1],
    [1, 1, 1, 1]
];

// Function Call
calculatePrefixSum(ARR);

Output

Give Array:
1 1 1 1 
1 1 1 1 
1 1 1 1 
1 1 1 1 
Prefix Sum Array:
1 2 3 4 
2 4 6 8 
3 6 9 12 
4 8 12 16 

Hashing the Arrays data:

There are times when Hashing can be used for pre-computation and drastically reduce the time complexity of program. Suppose for various query we would require the first index of an element ‘e‘ in the array. Instead of calculating this again and again, we can simply iterate on array once and hash the element to the first index it occurs to and finally retrieve this value in O(1) for each query.

Moreover Hashing can be combined with other precomputation techniques like prefix_sum in order to enhance the usability.

Practice Problems:

Problem	Difficulty
Queries for the product of first N factorials	Easy
Range sum queries without updates	Easy
Range Queries for Frequencies of array elements	Easy
Count Primes in Ranges	Easy
Check in binary array the number represented by a subarray is odd or even	Easy
GCDs of given index ranges in an array	Easy
Mean of range in array	Medium
Difference Array | Range update query in O(1)	Medium
Range sum query using Sparse Table	Medium
Number whose sum of XOR with given array range is maximum	Medium
Number of primes in a subarray (with updates)	Medium
Sum of Interval and Update with Number of Divisors	Medium
Print modified array after multiple array range increment operations	Medium
Sparse Table	Hard
Range Minimum Query (Square Root Decomposition and Sparse Table)	Hard
Range Query on array whose each element is XOR of index value and previous element	Hard
Queries for GCD of all numbers of an array except elements in a given range	Hard
Queries of nCr%p in O(1) time complexity	Hard