Question

I have the following problem: I am given a tree with N apples, for each apple I am given it’s weight and height. I can pick apples up to a given height H, each time I pick an apple the height of every apple is increased with U. I have to find out the maximum weight of apples I can pick.

1 ≤ N ≤ 100000
0 < {H, U, apples’ weight and height, maximum weight} < 2³¹

Example:

N=4  H=100  U=10  

height weight  
  82     30
  91     10
  93      5
  94     15

The answer is 45: first pick the apple with the weight of 15 then the one with the weight of 30.

Could someone help me approach this problem?

Answer 1

Work backwards. Start by deciding the last apple you will pick, then the second to last, etc.

import heapq

def solve(apples, H, U):
  """Solve apple-picking problem.

  apples must be a sequence of (H, W) pairs.
  """
  if U == 0:
    return sum(x[1] for x in apples if x[0] <= H)

  apples = sorted(apples, reversed=True)
  # also creates a copy, to not modify caller's data

  picked_weight = 0
  available_weights = []  # stored negative for heapq

  offset = U - H % U
  if offset == U: offset = 0
  top = offset - U

  while (apples or available_weights) and top <= H:
    if available_weights:
      picked_weight += -heapq.heappop(available_weights)
      top += U
    else:
      top += U * max(1, (apples[-1][0] - top) // U)

    while apples and apples[0][0] <= top:
      heapq.heappush(available_weights, -apples.pop()[1])

  return picked_weight

Simple test:

def test(expected, apples, H, U):
  actual = solve(apples, H, U)
  if expected != actual:
    print "expected=%r actual=%r | H=%r U=%r apples=%r" % (
              expected,   actual,     H,   U,   apples)

test(45, [(82, 30), (91, 10), (93,  5), (94, 15)], 100, 10)
test(22, [( 1,  1), ( 2,  1), (81, 10), (82, 10)], 100, 10)
test(20, [( 1, 10), ( 2, 10), (11,  1)], 20, 10)
test(20, [(81, 10), (82, 10), (90,  5)], 100, 10)
test(1, [(2**31 - 1, 1)], 2**31 - 1, 1)

---

Someone requested C++, so here it is. It’s nearly identical code and logic to the above Python, except for one change: C++ stdlib’s heap functions work with the max value instead of the min, so no need for the negation. (I kept this self-contained, but utilities such as a heap adapter and container inserter will make the code easier to use.)

#include <cstdio>
#include <algorithm>
#include <iostream>
#include <numeric>
#include <vector>


struct Apple {
  int h, w;

  friend bool operator<(Apple const& a, Apple const& b) {
    return a.h < b.h or (a.h == b.h and a.w < b.w);
  }
  friend bool operator>(Apple const& a, Apple const& b) {
    return b < a;
  }
  friend std::ostream& operator<<(std::ostream& s, Apple const& v) {
    s << '(' << v.h << ", " << v.w << ')';
    return s;
  }
};

template<class T, class C, T C::*M>
struct sum {
  T operator()(T const& cur, C const& next) { return cur + next.*M; }
};

int solve(std::vector<Apple> apples, int H, int U) {
  if (U == 0) {
    return std::accumulate(apples.begin(), apples.end(), 0, sum<int, Apple, &Apple::w>());
  }

  std::sort(apples.begin(), apples.end(), std::greater<Apple>());

  int picked_weight = 0;
  std::vector<int> available_weights;  // NOT stored negative, unlike Python code

  int offset = U - H % U;
  if (offset == U) offset = 0;
  int top = offset - U;

  while ((apples.size() or available_weights.size()) and top <= H) {
    if (available_weights.size()) {
      picked_weight += available_weights.front();
      std::pop_heap(available_weights.begin(), available_weights.end());
      available_weights.pop_back();
      top += U;
    }
    else {
      top += U * std::max(1, (apples.back().h - top) / U);
    }

    while (apples.size() and apples.back().h <= top) {
      available_weights.push_back(apples.back().w);
      std::push_heap(available_weights.begin(), available_weights.end());
      apples.pop_back();
    }
  }

  return picked_weight;
}

C++ tests:

template<int N>
void test(int expected, Apple (&apples)[N], int H, int U) {
  std::vector<Apple> v (apples, apples + N);
  int actual = solve(v, H, U);
  if (expected != actual) {
    std::printf("expected=%d actual=%d | H=%d U=%d apples=[",
                    expected,   actual,     H,   U);
    std::vector<Apple>::const_iterator i = v.begin();
    if (i != v.end()) {
      std::cout << *i;
      for (++i; i != v.end(); ++i) {
        std::cout << ", " << *i;
      }
    }
    std::cout << "]" << std::endl;
  }
}

int main() {
  {
    Apple data[] = {{82, 30}, {91, 10}, {93,  5}, {94, 15}};
    test(45, data, 100, 10);
  }
  {
    Apple data[] = {{ 1,  1}, { 2,  1}, {81, 10}, {82, 10}};
    test(22, data, 100, 10);
  }
  {
    Apple data[] = {{ 1, 10}, { 2, 10}, {11,  1}};
    test(20, data, 20, 10);
  }
  {
    Apple data[] = {{81, 10}, {82, 10}, {90,  5}};
    test(20, data, 100, 10);
  }
  {
    int n = 2147483647; // 2**31 - 1
    Apple data[] = {{n, 1}};
    test(1, data, n, 1);
  }
}