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The idea is to ensure every element is greater than its previous one by subtracting a suitable prime. We iterate through each element and subtract the smallest possible prime until the condition is met.
For each element in the array, find if there's a prime that can be subtracted to make the current element greater than the previous one. Repeat this for all elements until the array becomes strictly increasing.
Time Complexity: O(n * log(n)), primarily dominated by the prime-checking function.
Space Complexity: O(1), since we are modifying the array in place and not using any additional data structures.
1import java.util.*;
2
3public class PrimeSubtraction {
4 private static boolean isPrime(int num) {
5 if (num <= 1) return false;
6 if (num <= 3) return true;
7 if (num % 2 == 0 || num % 3 == 0) return false;
8 for (int i = 5; i * i <= num; i += 6) {
9 if (num % i == 0 || num % (i + 2) == 0) return false;
10 }
11 return true;
12 }
13
14 public static boolean canBeStrictlyIncreasing(int[] nums) {
15 for (int i = 1; i < nums.length; i++) {
16 if (nums[i] <= nums[i - 1]) {
17 int diff = nums[i] - nums[i - 1];
18 for (int p = diff + 1; p < nums[i]; p++) {
19 if (isPrime(p)) {
20 nums[i] -= p;
21 break;
22 }
23 }
24 if (nums[i] <= nums[i - 1]) {
25 return false;
26 }
27 }
28 }
29 return true;
30 }
31
32 public static void main(String[] args) {
33 int[] nums = {4, 9, 6, 10};
34 System.out.println(canBeStrictlyIncreasing(nums));
35 }
36}In Java, this solution adheres to the same logical approach - checking the condition and modifying elements with the smallest prime necessary to meet the requirements for a strictly increasing order.
This approach involves using the Sieve of Eratosthenes to precompute primes up to the maximum value in the array. For each element, we maintain a list of unpicked primes smaller than the current number and dynamically reduce the current element with the largest viable prime, proceeding until a strictly increasing order is established.
We use this optimized list to attempt to produce a valid increasing order to minimize operation count and improve efficiency.
Time Complexity: O(n * sqrt(m)), but preprocessing the prime sieve to reduce runtime cost per loop.
Space Complexity: O(m), for storing the boolean list to evaluate primes.
Using Java, we leverage the Sieve of Eratosthenes to store prime numbers so that during iteration, adjusting current elements are quicker. This produces a more efficient Alpha over a smaller computation differential.