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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.
1function isPrime(num) {
2 if (num <= 1) return false;
3 if (num <= 3) return true;
4 if (num % 2 === 0 || num % 3 === 0) return false;
5 let i = 5;
6 while (i * i <= num) {
7 if (num % i === 0 || num % (i + 2) === 0) return false;
8 i += 6;
9 }
10 return true;
11}
12
13function canBeStrictlyIncreasing(nums) {
14 for (let i = 1; i < nums.length; i++) {
15 if (nums[i] <= nums[i - 1]) {
16 let diff = nums[i] - nums[i - 1];
17 for (let p = diff + 1; p < nums[i]; p++) {
18 if (isPrime(p)) {
19 nums[i] -= p;
20 break;
21 }
22 }
23 if (nums[i] <= nums[i - 1]) {
24 return false;
25 }
26 }
27 }
28 return true;
29}
30
31let nums = [4, 9, 6, 10];
32console.log(canBeStrictlyIncreasing(nums));The JavaScript solution efficiently ensures each element in the array is larger than its predecessor by decreasing the current number with the largest possible 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.
This C solution uses a sieve to precompute all primes less than a given maximum value. This approach allows us to quickly identify potential subtraction values to transform the array into one that is strictly increasing.