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:

#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; } 

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 

# 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) 

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);  } } 

// 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); 

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++ 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; } 

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);  } } 

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 

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);  } } 

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); 

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: