Posts tagged as “math”

花花酱 LeetCode 973. K Closest Points to Origin

By zxi on January 13, 2019

We have a list of points on the plane. Find the K closest points to the origin (0, 0).

(Here, the distance between two points on a plane is the Euclidean distance.)

You may return the answer in any order. The answer is guaranteed to be unique (except for the order that it is in.)

Example 1:

Input: points = [[1,3],[-2,2]], K = 1 
Output: [[-2,2]] 
Explanation:  The distance between (1, 3) and the origin is sqrt(10). The distance between (-2, 2) and the origin is sqrt(8). Since sqrt(8) < sqrt(10), (-2, 2) is closer to the origin. We only want the closest K = 1 points from the origin, so the answer is just [[-2,2]].

Example 2:

Input: points = [[3,3],[5,-1],[-2,4]], K = 2 
Output: [[3,3],[-2,4]] (The answer [[-2,4],[3,3]] would also be accepted.)

Note:

1 <= K <= points.length <= 10000
-10000 < points[i][0] < 10000
-10000 < points[i][1] < 10000

Solution: Sort

Time complexity: O(nlogn)
Space complexity: O(n)

C++

// Author: Huahua, Running time: 180 ms

class Solution {

public:

vector<vector<int>> kClosest(vector<vector<int>>& points, int K) {

vector<long> s(points.size());

for (int i = 0; i < points.size(); ++i)

s[i] = (static_cast<long>(points[i][0] * points[i][0] +

points[i][1] * points[i][1]) << 32) | i;

std::sort(begin(s), end(s));

vector<vector<int>> ans(K);

for (int i = 0; i < K; ++i)

ans[i] = points[static_cast<int>(s[i])];

return ans;

}

};

Python3

# Author: Huahua, running time: 528 ms

class Solution:

def kClosest(self, points, K):

return sorted(points, key = lambda p: p[0]**2 + p[1]**2)[0:K]

花花酱 LeetCode 972. Equal Rational Numbers

By zxi on January 5, 2019

Given two strings S and T, each of which represents a non-negative rational number, return True if and only if they represent the same number. The strings may use parentheses to denote the repeating part of the rational number.

In general a rational number can be represented using up to three parts: an integer part, a non-repeating part, and a repeating part. The number will be represented in one of the following three ways:

<IntegerPart> (e.g. 0, 12, 123)
<IntegerPart><.><NonRepeatingPart> (e.g. 0.5, 1., 2.12, 2.0001)
<IntegerPart><.><NonRepeatingPart><(><RepeatingPart><)> (e.g. 0.1(6), 0.9(9), 0.00(1212))

The repeating portion of a decimal expansion is conventionally denoted within a pair of round brackets. For example:

1 / 6 = 0.16666666… = 0.1(6) = 0.1666(6) = 0.166(66)

Both 0.1(6) or 0.1666(6) or 0.166(66) are correct representations of 1 / 6.

Example 1:

Input: S = "0.(52)", T = "0.5(25)"
Output: true
Explanation:
Because "0.(52)" represents 0.52525252..., and "0.5(25)" represents 0.52525252525..... , the strings represent the same number.

Example 2:

Input: S = "0.1666(6)", T = "0.166(66)"
Output: true

Example 3:

Input: S = "0.9(9)", T = "1."
Output: true
Explanation: 
"0.9(9)" represents 0.999999999... repeated forever, which equals 1.  [See this link for an explanation.]
"1." represents the number 1, which is formed correctly: (IntegerPart) = "1" and (NonRepeatingPart) = "".

Note:

Each part consists only of digits.
The <IntegerPart> will not begin with 2 or more zeros. (There is no other restriction on the digits of each part.)
1 <= <IntegerPart>.length <= 4
0 <= <NonRepeatingPart>.length <= 4
1 <= <RepeatingPart>.length <= 4

Solution1: Expend the string

Extend the string to 16+ more digits and covert it to double.

0.9(9) => 0.99999999999999
0.(52) => 0.525252525252525
0.5(25) => 0.5252525252525

C++

// Author: Huahua, running time: 4 ms

class Solution {

public:

bool isRationalEqual(string S, string T) {

auto convert = [](string s) {

// 0.1234(567)

// 01234567890

// i = 1, p = 6

auto i = s.find('.');

auto p = s.find('(');

if (p != string::npos) {

string r = s.substr(p + 1, s.length() - p - 2);

s = s.substr(0, p);

while (s.length() < 16)

s += r;

}

return stod(s);

};

return abs(convert(S) - convert(T)) < 1e-10;

}

};

Python3

class Solution:

def isRationalEqual(self, S, T):

def convert(s):

i = s.find('.')

p = s.find('(')

if p >= 0:

r = s[p+1:-1]

s = s[0:p]

while len(s) < 16:

s += r

return float(s)

return abs(convert(S) - convert(T)) < 1e-10

Solution 2: Convert to a friction number

C++

// Author: Huahua, running time: 24 ms.

class Friction {

public:

Friction(long n = 0, long d = 1) {

long g = __gcd(n, d);

n_ = n / g;

d_ = d / g;

}

Friction operator+(const Friction& o) const {

return Friction(n_ * o.d_ + d_ * o.n_, d_ * o.d_);

}

bool operator==(const Friction& o) const {

return this->n_ * o.d_ == this->d_ * o.n_;

}

private:

long n_;

long d_;

};

class Solution {

public:

bool isRationalEqual(string S, string T) {

auto convert = [](string s) {

std::regex re("(\\d+)\\.?(\\d+)?(\\((\\d+)\\))?");

std::smatch matches;

std::regex_match(s, matches, re);

string int_part = matches[1].str();

string nr_part = matches[2].str();

string r_part = matches[4].str();

Friction f(stoi(int_part), 1);

const int base = pow(10, nr_part.length());

if (nr_part.length())

f = f + Friction(stoi(nr_part), base);

if (r_part.length())

f = f + Friction(stoi(r_part), (pow(10, r_part.length()) - 1) * base);

return f;

};

return convert(S) == convert(T);

}

};

花花酱 LeetCode 970. Powerful Integers

By zxi on January 5, 2019

Given two non-negative integers x and y, an integer is powerful if it is equal to x^i + y^j for some integers i >= 0 and j >= 0.

Return a list of all powerful integers that have value less than or equal to bound.

You may return the answer in any order. In your answer, each value should occur at most once.

Example 1:

Input: x = 2, y = 3, bound = 10
Output: [2,3,4,5,7,9,10]
Explanation: 
2 = 2^0 + 3^0
3 = 2^1 + 3^0
4 = 2^0 + 3^1
5 = 2^1 + 3^1
7 = 2^2 + 3^1
9 = 2^3 + 3^0
10 = 2^0 + 3^2

Example 2:

Input: x = 3, y = 5, bound = 15
Output: [2,4,6,8,10,14]

Note:

1 <= x <= 100
1 <= y <= 100
0 <= bound <= 10^6

Solution: Brute Force

Time complexity: O(log(bound) / log(x) * log(bound) / log(y))
Space complexity: O(log(bound) / log(x) * log(bound) / log(y))

C++

// Author: Huahua, running time: 4 ms

class Solution {

public:

vector<int> powerfulIntegers(int x, int y, int bound) {

unordered_set<int> ans;

for (int a = 1; a < bound; a *= x) {

for (int b = 1; a + b <= bound; b *= y) {

ans.insert(a + b);

if (y == 1) break;

}

if (x == 1) break;

}

return {begin(ans), end(ans)};

}

};

花花酱 LeetCode 313. Super Ugly Number

By zxi on January 2, 2019

Write a program to find the n^th super ugly number.

Super ugly numbers are positive numbers whose all prime factors are in the given prime list primes of size k.

Example:

Input: n = 12, primes = [2,7,13,19] 
Output: 32  
Explanation: [1,2,4,7,8,13,14,16,19,26,28,32] is the sequence of the first 12               super ugly numbers given primes = [2,7,13,19] of size 4.

Note:

1 is a super ugly number for any given primes.
The given numbers in primes are in ascending order.
0 < k ≤ 100, 0 < n ≤ 10⁶, 0 < primes[i] < 1000.
The n^th super ugly number is guaranteed to fit in a 32-bit signed integer.

Solution 1: Set

Maintain an ordered set of super ugly numbers, each time extract the smallest one, and multiply it with all primes and insert the new number into set.

Time complexity: O(n*k*logn)
Space complexity: O(n)

C++

// Author: Huahua, running time: 484 ms

class Solution {

public:

int nthSuperUglyNumber(int n, vector<int>& primes) {

if (n == 1) return 1;

set<int> q{begin(primes), end(primes)};

for (int i = 0; i < n - 2; ++i) {

auto it = begin(q);

int t = *it;

q.erase(it);

for (int p : primes) {

if (INT_MAX / t < p) continue;

q.insert(p * t);

}

return *begin(q);

}

};

Solution 2: Priority Queue

Time complexity: O(nlogn)
Space complexity: O(n)

C++

// Author: Huahua, 40 ms

struct Node {

int num;

int index;

int prime;

bool operator>(const Node& o) const {

if (this->num == o.num) return this->index > o.index;

return this->num > o.num;

}

};

class Solution {

public:

int nthSuperUglyNumber(int n, vector<int>& primes) {

if (n == 1) return 1;

vector<int> nums(n, 1);

priority_queue<Node, vector<Node>, greater<Node>> q;

for (int p : primes)

q.push({p, 1, p});

for (int i = 1; i < n; ++i) {

nums[i] = q.top().num;

while (nums[i] == q.top().num) {

Node node = std::move(q.top()); q.pop();

node.num = nums[node.index++] * node.prime;

q.push(std::move(node));

}

return nums.back();

}

};

花花酱 LeetCode 954. Array of Doubled Pairs

By zxi on December 9, 2018

Problem

Given an array of integers A with even length, return true if and only if it is possible to reorder it such that A[2 * i + 1] = 2 * A[2 * i] for every 0 <= i < len(A) / 2.

Example 1:

Input: [3,1,3,6]
Output: false

Example 2:

Input: [2,1,2,6]
Output: false

Example 3:

Input: [4,-2,2,-4]
Output: true
Explanation: We can take two groups, [-2,-4] and [2,4] to form [-2,-4,2,4] or [2,4,-2,-4].

Example 4:

Input: [1,2,4,16,8,4]
Output: false

Note:

0 <= A.length <= 30000
A.length is even
-100000 <= A[i] <= 100000

Solution 1:

Time complexity: O(N + 100000 * 2)

Space complexity: O(100000 * 2)

C++

// Author: Huahua, 108 ms

class Solution {

public:

bool canReorderDoubled(vector<int>& A) {

const int kMaxN = 100000;

vector<int> p(kMaxN + 1);

vector<int> n(kMaxN + 1);

for (int a : A) {

if (a >= 0) ++p[a];

else ++n[-a];

}

for (int i = 0; i <= kMaxN / 2; ++i) {

if (p[i] && (p[i * 2] -= p[i]) < 0) return false;

if (n[i] && (n[i * 2] -= n[i]) < 0) return false;

}

return true;

}

};

Solution 2:

Time complexity: O(NlogN)

Space complexity: O(N)

C++

// Author: Huahua, 108 ms

class Solution {

public:

bool canReorderDoubled(vector<int>& A) {

unordered_map<int, int> c;

for (int a : A) ++c[a];

vector<int> keys;

for (const auto &kv : c) keys.push_back(kv.first);

sort(begin(keys), end(keys), [](int a, int b) { return abs(a) < abs(b); });

for (int key : keys)

if (c[key] && (c[key * 2] -= c[key]) < 0) return false;

return true;

}

};