Dare2Solve
Given two integers a
and an array of integers b
, we need to compute (a^b) % 1337
, where b
is treated as a large exponent represented as an array of digits. The problem aims to calculate a large power efficiently under modulo constraints.
The challenge lies in managing the potentially huge size of the exponent b
and ensuring that we perform the exponentiation in a time-efficient manner.
Exponentiation with large numbers can lead to computational inefficiencies and potential overflows if done directly. Breaking down the problem, we recognize that powers of a
can be computed incrementally by breaking the array b
into smaller pieces and using properties of modular arithmetic. Specifically, using the property:
(a^b) % m = [(a^(b1)) % m * (a^(b2)) % m * ... ] % m
We can progressively compute powers of a
using smaller exponents derived from b
.
b
from the least significant digit to the most significant digit, computing powers of a
incrementally. For each digit in b
, we calculate a^digit % 1337
, update a
as a^10 % 1337
for the next iteration, and multiply it by the previously calculated result.pow(a, b)
to calculate the powers under modulo conditions. Multiply the result for each digit iteratively while maintaining the modulus.O(n * log(a))
, where n
is the size of the array b
and log(a)
is the time taken to calculate a^b % 1337
for each b[i]
. This is because for each digit in b
, we compute a^digit % 1337
which takes logarithmic time with respect to a
.
O(1)
since the space used does not scale with input size beyond a few variables for computation.
class Solution {
public:
const int MOD = 1337;
int pow(int a, int b) {
int result = 1;
a %= MOD; // Taking mod to prevent overflow
for (int i = 0; i < b; i++) {
result = (result * a) % MOD;
}
return result;
}
int superPow(int a, vector<int>& b) {
int result = 1;
for (int i = b.size() - 1; i >= 0; i--) {
result = (result * pow(a, b[i])) % MOD;
a = pow(a, 10); // Power up for the next iteration
}
return result;
}
};
class Solution:
MOD = 1337
def pow(self, a: int, b: int) -> int:
result = 1
a %= self.MOD # Taking mod to prevent overflow
for _ in range(b):
result = (result * a) % self.MOD
return result
def superPow(self, a: int, b: list[int]) -> int:
result = 1
for i in range(len(b) - 1, -1, -1):
result = (result * self.pow(a, b[i])) % self.MOD
a = self.pow(a, 10) # Power up for the next iteration
return result
class Solution {
private static final int MOD = 1337;
private int pow(int a, int b) {
int result = 1;
a %= MOD; // Taking mod to prevent overflow
for (int i = 0; i < b; i++) {
result = (result * a) % MOD;
}
return result;
}
public int superPow(int a, int[] b) {
int result = 1;
for (int i = b.length - 1; i >= 0; i--) {
result = (result * pow(a, b[i])) % MOD;
a = pow(a, 10); // Power up for the next iteration
}
return result;
}
}
/**
* @param {number} a
* @param {number[]} b
* @return {number}
*/
var superPow = function(a, b) {
const MOD = 1337;
const pow = (num, n) => {
let result = 1;
for (let index = 0; index < n; index++) {
result = result * num % MOD;
}
return result;
};
return b.reduceRight((result, n) => {
a %= MOD;
const powNum = result * pow(a, n) % MOD;
a = pow(a, 10);
return powNum;
}, 1);
};