Make a program run linear in C - c

Question

So based in the following problem from cumulative sum query I created the solution. But is any other way to solve the problem in C with linear complexity O(N)?
Problem description:
William Macfarlane wants to look at an array.
You are given a list of N numbers and Q queries. Each query is
specified by two numbers i and j; the answer to each query is the sum
of every number between the range [i, j] (inclusive).
Note: the query ranges are specified using 0-based indexing.
Input
The first line contains N, the number of integers in our list (N <=
100,000). The next line holds N numbers that are guaranteed to fit
inside an integer. Following the list is a number Q (Q <= 10,000). The
next Q lines each contain two numbers i and j which specify a query
you must answer (0 <= i, j <= N-1). Output
Output
For each query, output the answer to that query on its own line in the
order the queries were made.
Here is the solution:
#include <stdio.h>
#include <stdlib.h>
#include <limits.h>
struct node {
int first;
int last;
};
int sum_array(int *array, int first, int last) {
int sum = 0;
for (int i = first; i <= last; i++) {
sum += array[i];
}
return sum;
}
int main() {
FILE* input = fopen("share.in","r");
int N = 0;
fscanf(input,"%d",&N);
int *array = (int*)malloc(N * sizeof(int));
for (int i = 0; i < N; i++) {
fscanf(input,"%d",&array[i]);
}
int Q = 0;
fscanf(input,"%d",&Q);
struct node query[Q];
for (int i=0; i < Q; i++) {
fscanf(input,"%d",&query[i].first);
fscanf(input,"%d",&query[i].last);
}
fclose(input);
int sum = 0;
for ( int i = 0; i < Q ; i++) {
int first = query[i].first;
int last = query[i].last;
sum = sum_array(array,first,last);
printf("Number of queries : %d , sum is %d\n",i ,sum);
}
free(array);
return 0;
}
Update:
The answer given is good. But for some reason I couldn't make it work.
So here is the code rewritten and if someone can explain me what I do wrong I will be happy! Keep in mind we want the range to be [first,last]
#include <stdio.h>
#include <stdlib.h>
#include <limits.h>
struct node {
int first;
int last;
};
int sum_array(int *array, int first, int last) {
int sum = 0;
for (int i = first; i <= last; i++) {
sum += array[i];
}
return sum;
}
int main() {
FILE* input = fopen("share.in","r");
int N = 0;
fscanf(input,"%d",&N);
int *array = (int*)malloc(N * sizeof(int));
int *integralArray = (int*)malloc(N * sizeof(int));
for (int i = 0; i < N; i++) {
fscanf(input,"%d",&array[i]);
integralArray[i] = array[i] + ((i > 0) ? array[i-1] : 0);
}
int Q = 0;
fscanf(input,"%d",&Q);
struct node query[Q];
for (int i=0; i < Q; i++) {
fscanf(input,"%d",&query[i].first);
fscanf(input,"%d",&query[i].last);
}
fclose(input);
int sum = 0;
for (int i = 0; i < Q ; i++) {
int first = query[i].first;
int last = query[i].last;
sum = integralArray[last] - integralArray[first - 1];
printf("Number of queries : %d , sum is %d\n",i ,sum);
}
free(array);
return 0;
}

Answer 1

You'd form the integral array. Modify to something like:
int *array = (int*)malloc(N * sizeof(int));
int *integralArray = (int*)malloc(N * sizeof(int));
for (int i = 0; i < N; i++) {
fscanf(input,"%d",&array[i]);
integralArray[i] = array[i] + ((i > 0) ? integralArray[i-1] : 0);
}
So the element at integralArray[i] is the sum of all elements in array from 0 to i.
Then, to get the sum from a to b, where a > b, integralArray[b] is the sum from 0 to b and integralArray[a] is the sum from 0 to a so you can just compute integralArray[b] - integralArray[a] to get the total from a to b. Intuitively, integralArray[b] includes the numbers you want but it also includes the numbers up to and including a. You don't want those so you take them off again.
Vary appropriately for inclusion or exclusion of the number at a and the number at b. That as given will include the number at b but not that at a. You could adjust your integralArray to be one earlier (so integralArray[b] is the sum from 0 to b-1) or adjust your indices.