
Problem Statement
The task is to develop a function pow(x, n)
, which calculates the value of x
raised to the power of n
. This operation is represented mathematically as x^n
. Here, x
is a double precision floating-point number and n
is an integer. The goal is to compute x^n
under various boundary conditions like positive, negative, or zero values of n
, ensuring correct and efficient handling.
Examples
Example 1
Input:
x = 2.00000, n = 10
Output:
1024.00000
Example 2
Input:
x = 2.10000, n = 3
Output:
9.26100
Example 3
Input:
x = 2.00000, n = -2
Output:
0.25000
Explanation:
2-2 = 1/22 = 1/4 = 0.25
Constraints
-100.0 < x < 100.0
-231 <= n <= 231-1
n
is an integer.- Either
x
is not zero orn > 0
. -104 <= xn <= 104
Approach and Intuition
The computation of x
raised to the power n
encapsulates more than just repeatedly multiplying x
by itself n
times. Several insights and optimizations can be leveraged to efficiently calculate large powers, especially considering floating-point arithmetic and the limits imposed by the constraints.
Using fast exponentiation or exponentiation by squaring:
- Recursive approach:
- When
n
is even,x^n
can be reduced to(x^(n/2))^2
. This reduces the number of multiplications significantly. - When
n
is odd,x^n
can be written asx * x^(n-1)
, thus still exploiting the even case optimization by reducingn-1
to an even number.
- When
- Iterative approach (often more efficient for large
n
):- Calculate
x^n
by iteratively squaring a base while halving the exponent, accumulating the result when the exponent is odd.
- Calculate
Handling different cases of n
:
- For positive
n
, the repeated squaring method can be directly applied. - For negative
n
, computex^-n
(using the methods above) and then take the reciprocal to get the result. This accurately handles the transformation from a negative to a positive exponent. - If
n = 0
, irrespective of the value ofx
(except whenx
is also 0 which is an indeterminate form not allowed by the constraints), the result is1
. This is a basic property of exponents.
Efficiency and limitations:
- The recursive and iterative methods reduce the time complexity from O(n) in a naive implementation to O(log n) by reducing the number of multiplications required.
- Edge cases due to floating-point arithmetic need careful handling, especially when
n
is large or whenx
is very small or very large. - Constraints help outline the boundaries of
x
andn
to prevent overflow and underflow, ensuring the results remain within practical limits for applications.
Examples provided illustrate typical scenarios:
- Simple positive powers (Example 1 & 2).
- Dealing with negative powers using reciprocals (Example 3).
Solutions
- C++
- Java
- C
- JavaScript
- Python
class Solution {
public:
double exp_power(double base, long long exp) {
if (exp == 0) {
return 1.0;
}
if (exp < 0) {
exp = -exp;
base = 1.0 / base;
}
double product = 1.0;
while (exp > 0) {
if (exp % 2 == 1) {
product *= base;
exp--;
}
base *= base;
exp /= 2;
}
return product;
}
double myPow(double x, int n) { return exp_power(x, (long long)n); }
};
The provided C++ solution efficiently computes x
raised to the power n
using the function myPow
. This function internally calls exp_power
, a helper function that employs the method of exponentiation by squaring, which is both time-efficient and space-efficient.
The function
exp_power
accepts two parameters:base
(as double) andexp
(as long long), initiating a process to handle powers both positive and negative:- If
exp
is zero, it immediately returns1.0
indicating that any number raised to the power of zero is one. - For negative exponents, it inverts the base (
base = 1.0 / base
) and turnsexp
to positive to simplify calculation. - It initializes
product
to1.0
which will ultimately store the result of the exponentiation. - Through a while loop, if
exp
is odd (checked usingexp % 2 == 1
), it multipliesproduct
bybase
and decreasesexp
by 1 to makeexp
even. Then, it squares the base (for the next iteration with half of the currentexp
) and halvesexp
.
- If
The main function
myPow
simply forwards its inputs toexp_power
, effectively starting the power calculation process.
This method, by handling squared multiplications recursively and reducing the problem size with each iteration, allows for the computation of large exponents quickly and minimizes the number of multiplicative operations, thus optimizing performance.
class Solution {
private double fastPower(double base, long exponent) {
if (exponent == 0) {
return 1.0;
}
if (exponent < 0) {
exponent = -exponent;
base = 1.0 / base;
}
double ans = 1.0;
while (exponent != 0) {
if (exponent % 2 == 1) {
ans *= base;
exponent--;
}
base *= base;
exponent /= 2;
}
return ans;
}
public double myPow(double x, int n) {
return fastPower(x, (long) n);
}
}
The given Java solution efficiently computes the power of a number x
raised to an integer n
(x^n) using a method known as "exponentiation by squaring." This method optimizes the process by reducing the number of multiplications needed, which is crucial for large values of n
. Below, find the breakdown of how the code implements this algorithm:
A helper function
fastPower(double base, long exponent)
is defined to handle the computation. The function takes into account the possibility of negative exponents by converting them to positive and taking the reciprocal of the base.This function first checks if the exponent is zero, in which case it immediately returns 1.0, because any number raised to the power of zero is 1.
If the current
exponent
is odd (checked usingexponent % 2 == 1
), thebase
is multiplied to theans
, and theexponent
is decremented by one to make it even. This is to ensure that the squaring process can proceed.Then, the
base
is squared, and theexponent
is halved. This process repeats until theexponent
becomes zero, at which pointans
will contain the correct result.The main
myPow(double x, int n)
method, which is accessible publicly, internally callsfastPower
by castingn
to long to avoid potential overflow issues with integer arithmetic.
By squaring the base and halving the exponent iteratively, this solution greatly reduces the number of operations compared to the naive approach of multiplying x
to itself n
times, especially for large n
. This makes it a highly efficient solution for power calculations in Java.
double power(double base, long exp) {
if (exp == 0) return 1.0;
if (exp < 0) {
exp = -exp;
base = 1.0 / base;
}
double res = 1.0;
while (exp > 0) {
if (exp % 2 == 1) {
res *= base;
exp--;
}
base = base * base;
exp = exp / 2;
}
return res;
}
double myPow(double base, int exp) { return power(base, (long)exp); }
This summary explains how to compute the power of a number x
raised to the exponent n
using a C function. Here's how the solution works:
Define a helper function
power(double base, long exp)
to compute the result:- Check if the exponent
exp
is zero, and if so, return 1.0 immediately, since any number raised to the power of zero is 1. - Handle negative exponents by converting
exp
to a positive equivalent and inverting thebase
(i.e.,base
becomes1.0 / base
). - Initialize a result variable
res
to 1.0. - Use a loop to compute the power for positive exponents:
- If the current exponent
exp
is odd (i.e.,exp % 2 == 1
), multiplyres
bybase
and decrementexp
by 1. - Square the
base
and halve theexp
(integer division by 2) untilexp
becomes zero.
- If the current exponent
- Return the computed result
res
.
- Check if the exponent
Implement the main function
myPow(double base, int exp)
that converts the exponentexp
to along
data type and calls the helper functionpower(base, (long)exp)
to get the final result.
This method of repeatedly squaring the base and halving the exponent, known as "exponentiation by squaring," optimizes the computation, especially for large exponents, as it reduces the number of multiplicative operations needed.
function expBySquaring(base, exp) {
if (exp === 0) {
return 1;
}
if (exp < 0) {
exp = -exp;
base = 1.0 / base;
}
let accumulator = 1;
while (exp !== 0) {
if (exp % 2 === 1) {
accumulator *= base;
exp -= 1;
}
base *= base;
exp = Math.floor(exp / 2);
}
return accumulator;
}
var power = function (base, exp) {
return expBySquaring(base, exp);
};
The provided JavaScript solution implements the power(base, exp)
function, which calculates the power of a number by exponentiation using the method of exponentiation by squaring. This efficient algorithm handles different scenarios including negative exponents. Here's a breakdown of how the solution works:
Define
expBySquaring(base, exp)
:- Return
1
ifexp
is zero—any number raised to the power of zero is1
. - If
exp
is negative, modifyexp
to be positive, and adjustbase
to1.0 / base
to account for the inversion. - Initialize an accumulator variable to
1
to hold the result of successive operations. - Use a
while
loop to process the exponentiation:- If
exp
is odd (exp % 2 === 1
), multiply the accumulator bybase
, then decrementexp
by 1 to make it even. - Square the
base
and halveexp
using floor division (Math.floor(exp / 2)
) to reduce the exponent systematically until it reaches zero.
- If
- Return the value of the accumulator once the loop terminates, yielding the final result of the exponentiation.
- Return
Define
power(base, exp)
to wrap aroundexpBySquaring(base, exp)
for simplicity and easier usage.
Apply this solution to efficiently compute the power of a number for both positive and negative exponents, leveraging the performance benefits of exponentiation by squaring particularly for larger exponents.
class Solution:
def power(self, base: float, exp: int) -> float:
if exp == 0:
return 1
if exp < 0:
exp = -exp
base = 1.0 / base
result = 1
while exp > 0:
if exp % 2 == 1:
result *= base
exp -= 1
base *= base
exp //= 2
return result
def exponentiate(self, base: float, exp: int) -> float:
return self.power(base, exp)
Implement the power function (x^n) in Python by following this approach where you calculate the power of base
to the exponent exp
. The solution uses a method that optimizes the calculation using an iterative version of exponentiation by squaring, which reduces the time complexity significantly.
- Define the function
power(base, exp)
which handles both positive and negative exponents. - Return 1 immediately if
exp
is 0 since any number raised to the power of 0 is 1. - If
exp
is negative, convert it to a positive by makingexp = -exp
and invert thebase
to1.0 / base
. - Initialize
result
to 1. Use a loop to handle the multiplication of the base when the exponent is greater than 0. - If
exp % 2 == 1
, multiply the currentresult
by thebase
and decreaseexp
by 1. - Square the
base
and halve theexp
using integer divisionexp //= 2
. - Return the computed
result
onceexp
becomes zero.
Additionally, define exponentiate(base, exp)
which simply calls power(base, exp)
to perform the actual computation. This division makes the code modular and easy to manage.
By using exponentiation by squaring, you ensure efficient computation even for large exponents, leveraging the halving of the exponent and squaring of the base at each step of the iterative process. This approach significantly reduces the number of multiplications needed compared to the straightforward approach of multiplying the base exp
times.
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