A difference array transforms sequential update operations into constant-time modifications by recording only the boundary changes between adjacent elements. Given an original sequence A, its corresponding difference sequence D is defined such that D[0] = A[0] and D[i] = A[i] - A[i-1] for i > 0. Recovering the original sequence simply requires computing the prefix sum of D.

int* build_delta_sequence(const int* source, size_t n) {
    int* delta = malloc(n * sizeof(int));
    if (!delta) return NULL;
    
    delta[0] = source[0];
    for (size_t i = 1; i < n; i++) {
        delta[i] = source[i] - source[i - 1];
    }
    return delta;
}

int* reconstruct_original(const int* delta, size_t n) {
    int* restored = malloc(n * sizeof(int));
    if (!restored) return NULL;
    
    restored[0] = delta[0];
    for (size_t i = 1; i < n; i++) {
        restored[i] = restored[i - 1] + delta[i];
    }
    return restored;
}

The computational advantage emerges during interval modifications. To increase every element within indices [l, r] by a value v, the delta array requires exactly two point updates: delta[l] += v and delta[r + 1] -= v (if r + 1 is within bounds). Because subsequent prefix sum calculations inherently accumulate these boundary adjustments, the cumulative effect precisely targets the specified subesgment while leaving outside elements unchanged. This reduces each range modification from linear time to amortized constant time.

Implementing multiple simultaneous updates benefits from encapsulating the delta structure alongside utility routines. The following architecture manages dynamic range adjustments and final state reconstruction:

#include <stdio.h>
#include <stdlib.h>

typedef struct {
    int* values;
    size_t cap;
} DeltaBuffer;

DeltaBuffer* create_delta(size_t n) {
    DeltaBuffer* buf = malloc(sizeof(DeltaBuffer));
    buf->values = calloc(n, sizeof(int));
    buf->cap = n;
    return buf;
}

void apply_offset(DeltaBuffer* buf, size_t lhs, size_t rhs, int shift) {
    if (lhs < buf->cap) {
        buf->values[lhs] += shift;
    }
    if (rhs + 1 < buf->cap) {
        buf->values[rhs + 1] -= shift;
    }
}

int* finalize_state(DeltaBuffer* buf) {
    int* output = malloc(buf->cap * sizeof(int));
    int current_sum = 0;
    for (size_t i = 0; i < buf->cap; i++) {
        current_sum += buf->values[i];
        output[i] = current_sum;
    }
    return output;
}

void destroy_delta(DeltaBuffer* buf) {
    free(buf->values);
    free(buf);
}

int* execute_range_operations(size_t length, const int ops[][3], int op_count) {
    DeltaBuffer* buffer = create_delta(length);
    
    for (int k = 0; k < op_count; k++) {
        apply_offset(buffer, (size_t)ops[k][0], (size_t)ops[k][1], ops[k][2]);
    }
    
    int* final_array = finalize_state(buffer);
    destroy_delta(buffer);
    return final_array;
}

Spatial capacity verification across transit routes follows identical boundary-tracking principles. Consider a scenario where passenger boarding and disembarkation events are recorded as coordinate pairs. Each event modifies the onboard occupancy count. By treating station indices as array positions, the algorithm tracks net load variations without simulating individual passenger movements.

int verify_vehicle_limits(const int schedule[][3], int route_events, int max_seats) {
    DeltaBuffer* load_tracker = create_delta(1001);
    
    for (int idx = 0; idx < route_events; idx++) {
        int pax_count = schedule[idx][0];
        int board_at = schedule[idx][1];
        int alight_at = schedule[idx][2] - 1; // Passengers exit at alight_at
        
        apply_offset(load_tracker, (size_t)board_at, (size_t)alight_at, pax_count);
    }
    
    int current_load = 0;
    for (size_t i = 0; i < 1001; i++) {
        current_load += load_tracker.values[i];
        if (current_load > max_seats) {
            destroy_delta(load_tracker);
            return 0;
        }
    }
    
    destroy_delta(load_tracker);
    return 1;
}

Boundary checks prevent out-of-bounds memory access during index arithmetic. The prefix accumulation loop guarantees linear traversal regardless of update frequency. Memory allocation follows standard C conventions, requiring explicit deallocation upon completion.