Multiple solutions of Fibonacci (Python or Java)

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Violence law(Top-down)

It can be solved directly according to the known conditions (f (0) = 0, f (1) = 1 F(N) = F(N - 1) + F(N - 2), for N > 1)

Python Code

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class Solution:

def fib(self, N: int) -> int:

if N == 1 or N == 2: return N

return self.fib(N - 1) + self.fib(N - 2)

Java Code

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class Solution {

public int fib(int N) {

if (N == 1 || N == 2) return 1;

return fib(N - 1) + fib(N - 2);

}

}

class Solution {

public int fib(int N) {

return N < 2 ? N : fib(N - 1) + fib(N - 2);

}

}

Violence law add cache(Pruning)

We know that if we don’t do any processing, we will repeat too many calculations, which is very bad The processing idea will avoid repeated calculation

Python Code

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class Solution2:

@functools.lru_cache()

def fib(self, N: int) -> int:

if N <= 1: return N

else: return self.fib(N - 1) + self.fib(N - 2)

Java Code

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class Solution {

private Integer[] cache = new Integer[31];

public int fib(int N) {

if (N <= 1) return N;

cache[0] = 0;

cache[1] = 1;

return memoize(N);

}

public int memoize(int N) {

if (cache[N] != null) return cache[N];

cache[N] = memoize(N-1) + memoize(N-2);

return memoize(N);

}

}

Divide and conquer solution

Recursion, iteration, divide and conquer, backtracking, they do not have a clear distinction Recursion：The core idea is to govern separately and unify the officials

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class Solution:

def fib(self, N: int) -> int:

memo = {}

if N < 2: return N

if N-1 not in memo: memo[N-1] = self.fib(N-1)

if N-2 not in memo: memo[N-2] = self.fib(N-2)

return memo[N-1] + memo[N-2]

Dynamic recursion(Bottom up)

Basic solutions

More initial value, continuous dynamic recursive

Python Code

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class Solution:

def fib(self, N: int) -> int:

if N < 2: return N

dp = [0 for _ in range(N + 1)]

dp[0], dp[1] = 0, 1

for i in range(2, N + 1):

dp[i] = dp[i - 1] + dp[i - 2]

return dp[- 1]

class Solution:

def fib(self, N: int) -> int:

if N == 0: return 0

memo = [0,1]

for _ in range(2,N+1):

memo = [memo[-1], memo[-1] + memo[-2]]

return memo[-1]

Java Code

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class Solution {

public int fib(int N) {

if (N <= 1) return N;

if (N == 2) return 1;

int curr = 0, prev1 = 1, prev2 = 1;

for (int i = 3; i <= N; i++) {

curr = prev1 + prev2;

prev2 = prev1;

prev1 = curr;

}

return curr;

}

}

Use better base types (tuples) to improve performance

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class Solution:

def fib(self, N: int) -> int:

if N == 0: return 0

memo = (0,1)

for _ in range(2,N+1):

memo = (memo[-1], memo[-1] + memo[-2])

return memo[-1]

Better solutions

Python Code

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class Solution:

def fib(self, N: int) -> int:

curr, prev1, prev2 = 0, 1, 1

for i in range(3, N + 1):

curr = prev1 + prev2

prev2 = prev1

prev1 = curr

return curr

class Solution5:

def fib(self, N: int) -> int:

prev, now = 0, 1

for i in range(N):

prev, now = now, now + prev

return prev

Java Code

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class Solution {

public int fib(int N) {

if (N == 0) return 0;

if (N == 2 || N == 1) return 1;

int prev = 1, curr = 1;

for (int i = 3; i <= N; i++) {

int sum = prev + curr;

prev = curr;

curr = sum;

}

return curr;

}

}

Mathematical conclusion method

Python Code

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class Solution:

def fib(self, N: int) -> int:

sqrt5 = 5 ** 0.5

fun = pow((1 + sqrt5) / 2, n + 1) - pow((1 - sqrt5) / 2, n + 1)

return int(fun / sqrt5)

Java Code

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class Solution {

public int fib(int N) {

double sqrt5 = (1 + Math.sqrt(5)) / 2;

return (int)Math.round(Math.pow(sqrt5, N)/ Math.sqrt(5));

}

}