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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.
1using System;
2
3public class PrimeSubtraction {
4 public static bool IsPrime(int n) {
5 if (n <= 1) return false;
6 if (n <= 3) return true;
7 if (n % 2 == 0 || n % 3 == 0) return false;
8 for (int i = 5; i * i <= n; i += 6) {
9 if (n % i == 0 || n % (i + 2) == 0) return false;
10 }
11 return true;
12 }
13
14 public static bool 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() {
33 int[] nums = {4, 9, 6, 10};
34 Console.WriteLine(CanBeStrictlyIncreasing(nums));
35 }
36}This C# solution evaluates two consecutive elements and assures the resultant sequence remains strictly increasing by reducing a valid prime.
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.
public class PrimeSubtraction {
public static void Sieve(int max, bool[] isPrime) {
Array.Fill(isPrime, true);
isPrime[0] = isPrime[1] = false;
for (int p = 2; p * p <= max; ++p) {
if (isPrime[p]) {
for (int i = p * p; i <= max; i += p) {
isPrime[i] = false;
}
}
}
}
public static bool CanBeStrictlyIncreasing(int[] nums) {
bool[] isPrime = new bool[1001];
Sieve(1000, isPrime);
for (int i = 1; i < nums.Length; ++i) {
if (nums[i] <= nums[i - 1]) {
int diff = nums[i] - nums[i - 1];
for (int p = diff + 1; p < nums[i]; ++p) {
if (isPrime[p]) {
nums[i] -= p;
break;
}
}
if (nums[i] <= nums[i - 1]) {
return false;
}
}
}
return true;
}
public static void Main() {
int[] nums = {4, 9, 6, 10};
Console.WriteLine(CanBeStrictlyIncreasing(nums));
}
}The C# solution uses the sieve to maintain up-to-date info on prime candidates handling elements via efficient means that accommodate timely transformations.