Posts tagged as “priority queue”

LeetCode 1801. Number of Orders in the Backlog

By zxi on March 21, 2021

You are given a 2D integer array orders, where each orders[i] = [price_i, amount_i, orderType_i] denotes that amount_iorders have been placed of type orderType_i at the price price_i. The orderType_i is:

0 if it is a batch of buy orders, or
1 if it is a batch of sell orders.

Note that orders[i] represents a batch of amount_i independent orders with the same price and order type. All orders represented by orders[i] will be placed before all orders represented by orders[i+1] for all valid i.

There is a backlog that consists of orders that have not been executed. The backlog is initially empty. When an order is placed, the following happens:

If the order is a buy order, you look at the sell order with the smallest price in the backlog. If that sell order’s price is smaller than or equal to the current buy order’s price, they will match and be executed, and that sell order will be removed from the backlog. Else, the buy order is added to the backlog.
Vice versa, if the order is a sell order, you look at the buy order with the largest price in the backlog. If that buy order’s price is larger than or equal to the current sell order’s price, they will match and be executed, and that buy order will be removed from the backlog. Else, the sell order is added to the backlog.

Return the total amount of orders in the backlog after placing all the orders from the input. Since this number can be large, return it modulo 10⁹ + 7.

Example 1:

Input: orders = [[10,5,0],[15,2,1],[25,1,1],[30,4,0]]
Output: 6
Explanation: Here is what happens with the orders:
- 5 orders of type buy with price 10 are placed. There are no sell orders, so the 5 orders are added to the backlog.
- 2 orders of type sell with price 15 are placed. There are no buy orders with prices larger than or equal to 15, so the 2 orders are added to the backlog.
- 1 order of type sell with price 25 is placed. There are no buy orders with prices larger than or equal to 25 in the backlog, so this order is added to the backlog.
- 4 orders of type buy with price 30 are placed. The first 2 orders are matched with the 2 sell orders of the least price, which is 15 and these 2 sell orders are removed from the backlog. The 3^rd order is matched with the sell order of the least price, which is 25 and this sell order is removed from the backlog. Then, there are no more sell orders in the backlog, so the 4^th order is added to the backlog.
Finally, the backlog has 5 buy orders with price 10, and 1 buy order with price 30. So the total number of orders in the backlog is 6.

Example 2:

Input: orders = [[7,1000000000,1],[15,3,0],[5,999999995,0],[5,1,1]]
Output: 999999984
Explanation: Here is what happens with the orders:
- 10⁹ orders of type sell with price 7 are placed. There are no buy orders, so the 10⁹ orders are added to the backlog.
- 3 orders of type buy with price 15 are placed. They are matched with the 3 sell orders with the least price which is 7, and those 3 sell orders are removed from the backlog.
- 999999995 orders of type buy with price 5 are placed. The least price of a sell order is 7, so the 999999995 orders are added to the backlog.
- 1 order of type sell with price 5 is placed. It is matched with the buy order of the highest price, which is 5, and that buy order is removed from the backlog.
Finally, the backlog has (1000000000-3) sell orders with price 7, and (999999995-1) buy orders with price 5. So the total number of orders = 1999999991, which is equal to 999999984 % (10⁹ + 7).

Constraints:

1 <= orders.length <= 10⁵
orders[i].length == 3
1 <= price_i, amount_i <= 10⁹
orderType_i is either 0 or 1.

Solution: Treemap / PriorityQueue / Heap

buy backlog: max heap
sell backlog: min heap
Trade happens between the tops of two queues.

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

C++

// Author: Huahua

class Solution {

public:

int getNumberOfBacklogOrders(vector<vector<int>>& orders) {

constexpr int kMod = 1e9 + 7;

map<int64_t, int64_t> buys;

map<int64_t, int64_t> sells;

for (const auto& order : orders) {

auto& m = order[2] == 0 ? buys : sells;

m[order[0]] += order[1];

while (buys.size() && sells.size()) {

auto b = rbegin(buys);

auto s = begin(sells);

if (b->first < s->first) break;

const int k = min(b->second, s->second);

if (!(b->second -= k)) buys.erase((++b).base());

if (!(s->second -= k)) sells.erase(s);

}

int64_t ans = 0;

for (const auto& [p, c] : buys) ans += c;

for (const auto& [p, c] : sells) ans += c;

return ans % kMod;

}

};

花花酱 LeetCode 1792. Maximum Average Pass Ratio

By zxi on March 13, 2021

There is a school that has classes of students and each class will be having a final exam. You are given a 2D integer array classes, where classes[i] = [pass_i, total_i]. You know beforehand that in the i^th class, there are total_i total students, but only pass_i number of students will pass the exam.

You are also given an integer extraStudents. There are another extraStudents brilliant students that are guaranteed to pass the exam of any class they are assigned to. You want to assign each of the extraStudents students to a class in a way that maximizes the average pass ratio across all the classes.

The pass ratio of a class is equal to the number of students of the class that will pass the exam divided by the total number of students of the class. The average pass ratio is the sum of pass ratios of all the classes divided by the number of the classes.

Return the maximum possible average pass ratio after assigning the extraStudents students. Answers within 10^-5 of the actual answer will be accepted.

Example 1:

Input: classes = [[1,2],[3,5],[2,2]], extraStudents = 2
Output: 0.78333
Explanation: You can assign the two extra students to the first class. The average pass ratio will be equal to (3/4 + 3/5 + 2/2) / 3 = 0.78333.

Example 2:

Input: classes = [[2,4],[3,9],[4,5],[2,10]], extraStudents = 4
Output: 0.53485

Constraints:

1 <= classes.length <= 10⁵
classes[i].length == 2
1 <= pass_i <= total_i <= 10⁵
1 <= extraStudents <= 10⁵

Solution: Greedy + Heap

Sort by the ratio increase potential (p + 1) / (t + 1) – p / t.

Time complexity: O((m+n)logn)
Space complexity: O(n)

C++

// Author: Huahua

class Solution {

public:

double maxAverageRatio(vector<vector<int>>& classes, int extraStudents) {

const int n = classes.size();

auto ratio = [&](int i, int delta = 0) {

return static_cast<double>(classes[i][0] + delta) /

(classes[i][1] + delta);

};

priority_queue<pair<double, int>> q;

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

q.emplace(ratio(i, 1) - ratio(i), i);

while (extraStudents--) {

const auto [r, i] = q.top(); q.pop();

++classes[i][0];

++classes[i][1];

q.emplace(ratio(i, 1) - ratio(i), i);

}

double total_ratio = 0;

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

total_ratio += ratio(i);

return total_ratio / n;

}

};

Python3

# Author: Huahua

class Solution:

def maxAverageRatio(self, classes: List[List[int]], extraStudents: int) -> float:

def ratio(i, delta=0):

return (classes[i][0] + delta) / (classes[i][1] + delta)

q = []

for i, c in enumerate(classes):

heapq.heappush(q, (-(ratio(i, 1) - ratio(i)), i))

for _ in range(extraStudents):

_, i = heapq.heappop(q)

classes[i][0] += 1

classes[i][1] += 1

heapq.heappush(q, (-(ratio(i, 1) - ratio(i)), i))

return mean(ratio(i) for i, _ in enumerate(classes))

花花酱 LeetCode 1705. Maximum Number of Eaten Apples

By zxi on December 27, 2020

There is a special kind of apple tree that grows apples every day for n days. On the i^th day, the tree grows apples[i] apples that will rot after days[i] days, that is on day i + days[i] the apples will be rotten and cannot be eaten. On some days, the apple tree does not grow any apples, which are denoted by apples[i] == 0 and days[i] == 0.

You decided to eat at most one apple a day (to keep the doctors away). Note that you can keep eating after the first n days.

Given two integer arrays days and apples of length n, return the maximum number of apples you can eat.

Example 1:

Input: apples = [1,2,3,5,2], days = [3,2,1,4,2]
Output: 7
Explanation: You can eat 7 apples:
- On the first day, you eat an apple that grew on the first day.
- On the second day, you eat an apple that grew on the second day.
- On the third day, you eat an apple that grew on the second day. After this day, the apples that grew on the third day rot.
- On the fourth to the seventh days, you eat apples that grew on the fourth day.

Example 2:

Input: apples = [3,0,0,0,0,2], days = [3,0,0,0,0,2]
Output: 5
Explanation: You can eat 5 apples:
- On the first to the third day you eat apples that grew on the first day.
- Do nothing on the fouth and fifth days.
- On the sixth and seventh days you eat apples that grew on the sixth day.

Constraints:

apples.length == n
days.length == n
1 <= n <= 2 * 10⁴
0 <= apples[i], days[i] <= 2 * 10⁴
days[i] = 0 if and only if apples[i] = 0.

Solution: PriorityQueue

Sort by rotten day in ascending order, only push onto the queue when that day has come (be able to grow apples).

Time complexity: O((n+ d)logn)
Space complexity: O(n)

C++

// Author: Huahua

class Solution {

public:

int eatenApples(vector<int>& apples, vector<int>& days) {

const int n = apples.size();

using P = pair<int, int>;

priority_queue<P, vector<P>, greater<P>> q; // {rotten_day, index}

int ans = 0;

for (int d = 0; d < n || !q.empty(); ++d) {

if (d < n && apples[d]) q.emplace(d + days[d], d);

while (!q.empty()

&& (q.top().first <= d || apples[q.top().second] == 0)) q.pop();

if (q.empty()) continue;

--apples[q.top().second];

++ans;

}

return ans;

}

};

花花酱 LeetCode 1675. Minimize Deviation in Array

By zxi on November 29, 2020

You are given an array nums of n positive integers.

You can perform two types of operations on any element of the array any number of times:

If the element is even, divide it by 2.
- For example, if the array is [1,2,3,4], then you can do this operation on the last element, and the array will be [1,2,3,2].
If the element is odd, multiply it by 2.
- For example, if the array is [1,2,3,4], then you can do this operation on the first element, and the array will be [2,2,3,4].

The deviation of the array is the maximum difference between any two elements in the array.

Return the minimum deviation the array can have after performing some number of operations.

Example 1:

Input: nums = [1,2,3,4]
Output: 1
Explanation: You can transform the array to [1,2,3,2], then to [2,2,3,2], then the deviation will be 3 - 2 = 1.

Example 2:

Input: nums = [4,1,5,20,3]
Output: 3
Explanation: You can transform the array after two operations to [4,2,5,5,3], then the deviation will be 5 - 2 = 3.

Example 3:

Input: nums = [2,10,8]
Output: 3

Constraints:

n == nums.length
2 <= n <= 10⁵
1 <= nums[i] <= 10⁹

Solution: Priority Queue

If we double an odd number it becomes an even number, then we can only divide it by two which gives us back the original number. So we can pre-double all the odd numbers and only do division in the following process.

We push all numbers including pre-doubled odd ones onto a priority queue, and track the difference between the largest and smallest number.

Each time, we pop the largest number out and divide it by two then put it back to the priority queue, until the largest number becomes odd. We can not discard it and divide any other smaller numbers by two will only increase the max difference, so we can stop here.

ex1: [3, 5, 8] => [6, 8, 10] (pre-double) => [5, 6, 8] => [4, 5, 6] => [3, 4, 5] max diff is 5 – 3 = 2
ex2: [4,1,5,20,3] => [2, 4, 6, 10, 20] (pre-double) => [2, 4, 6, 10] => [2, 4, 5, 6] => [2,3,4,5] max diff = 5-2 = 3

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

C++/Set

class Solution {

public:

int minimumDeviation(vector<int>& nums) {

set<int> s;

for (int x : nums)

s.insert(x & 1 ? x * 2 : x);

int ans = *rbegin(s) - *begin(s);

while (*rbegin(s) % 2 == 0) {

s.insert(*rbegin(s) / 2);

s.erase(*rbegin(s));

ans = min(ans, *rbegin(s) - *begin(s));

}

return ans;

}

};

C++/PQ

class Solution {

public:

int minimumDeviation(vector<int>& nums) {

priority_queue<int> q;

int lo = INT_MAX;

for (int x : nums) {

x = x & 1 ? x * 2 : x;

q.push(x);

lo = min(lo, x);

}

int ans = q.top() - lo;

while (q.top() % 2 == 0) {

int x = q.top(); q.pop();

q.push(x / 2);

lo = min(lo, x / 2);

ans = min(ans, q.top() - lo);

}

return ans;

}

};

花花酱 LeetCode 1439. Find the Kth Smallest Sum of a Matrix With Sorted Rows

By zxi on May 5, 2020

You are given an m * n matrix, mat, and an integer k, which has its rows sorted in non-decreasing order.

You are allowed to choose exactly 1 element from each row to form an array. Return the Kth smallest array sum among all possible arrays.

Example 1:

Input: mat = [[1,3,11],[2,4,6]], k = 5
Output: 7
Explanation: Choosing one element from each row, the first k smallest sum are:
[1,2], [1,4], [3,2], [3,4], [1,6]. Where the 5th sum is 7.

Example 2:

Input: mat = [[1,3,11],[2,4,6]], k = 9
Output: 17

Example 3:

Input: mat = [[1,10,10],[1,4,5],[2,3,6]], k = 7
Output: 9
Explanation: Choosing one element from each row, the first k smallest sum are:
[1,1,2], [1,1,3], [1,4,2], [1,4,3], [1,1,6], [1,5,2], [1,5,3]. Where the 7th sum is 9.

Example 4:

Input: mat = [[1,1,10],[2,2,9]], k = 7
Output: 12

Constraints:

m == mat.length
n == mat.length[i]
1 <= m, n <= 40
1 <= k <= min(200, n ^ m)
1 <= mat[i][j] <= 5000
mat[i] is a non decreasing array.

Solution 1: Priority Queue

Generate the arrays in order.

Each node is {sum, idx_0, idx_1, …, idx_m},

Start with {sum_0, 0, 0, …, 0}.

For expansion, pick one row and increase its index

Time complexity: O(k * m ^ 2* log k)
Space complexity: O(k)

C++

// Author: Huahua

class Solution {

public:

int kthSmallest(vector<vector<int>>& mat, int k) {

const int m = mat.size();

const int n = mat[0].size();

priority_queue<vector<int>> q;

vector<int> node(m + 1); // {-sum, idx_0, idx_1, ..., idx_m}

for (int i = 0; i < m; ++i) node[0] -= mat[i][0];

q.push(node);

set<vector<int>> seen{{node}};

while (!q.empty()) {

const vector<int> cur = q.top(); q.pop();

if (--k == 0) return -cur[0];

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

if (cur[i] == n - 1) continue;

vector<int> nxt(cur);

++nxt[i]; // increase i-th row's index.

// Update sum.

nxt[0] -= (mat[i - 1][nxt[i]] - mat[i - 1][nxt[i] - 1]);

if (!seen.insert(nxt).second) continue;

q.push(move(nxt));

}

return -1;

}

};