# Compute the height of a tower made of buckets

I need to speed up my algorithm. It is about finding the height of a tower. The tower is built from buckets. Each bucket has height and radius (1 <= height,radius <= 1000). Variable bucketCount describe how many buckets are placed on tower (1 <= bucketCount <= 10^6). We set buckets in sequence. Thickness of bucket is 0 (for simplicity)

Image of example tower:

I decided to use a stack. My algorithm for each bucket:

1. If stack is empty, then push bucket to stack.
2. If the bucket I hold is narrower then bucket on top then put it on the stack.
3. Else while bucket I hold is wider: pop and find maximum height (for new bucket ground), after that push bucket which I hold.

For each bucket I added additional an variable ground, which specifies the height at which the bucket is placed. Meantime I keep maximum height of tower in variable.

This task is automatically checked and gives output: time limit exceeded. I tried to optimize code but nothing gives positive result.

Example input: 2 20 20 30 30
Output: 50

#include <iostream>
#include <stack>

using namespace std;

struct Bucket{
public:
int height;
int ground;
};

int main()
{
int ile;
cin >> ile;
for(int c = 0; c < ile; c++){
stack<Bucket> tower;
int hightestPoint = 0;
int bucketCount;
cin >> bucketCount;
Bucket temp;
Bucket temporary; //Used to speed up a bit
int maksimum;
int sum;
int counter = 0; //Its size of the Stack. Used it because thought its faster then empty() function
for(int i = 0; i < bucketCount; i++){
maksimum = -1;
sum = 0;
if(counter == 0){
temp.ground = 0;
tower.push(temp);
counter++;
} else {
temp.ground = tower.top().ground;
tower.push(temp);
counter++;
} else { //If bucket is wider
while((counter > 0) && temp.radius >= (temporary = tower.top()).radius){ //Pop and search for new ground (maximum)
sum = temporary.height + temporary.ground;
if(maksimum < sum){
maksimum = sum;
}
tower.pop();
counter--;
}
temp.ground = maksimum; //Set ground for new bucket
tower.push(temp);
counter++;
}
}
sum = temp.height + temp.ground; //Meantime find highest point in stack
if(hightestPoint < sum){
hightestPoint = sum;
}
}
cout << hightestPoint << endl;
}

return 0;
}

• I used radius to determine if bucket will be placed inside another bucket or on top of another bucket. There is no relationship. Radius and height can be any number less then 1000. U can replace word "radius" with "width" – Th3NiKo Apr 29 '18 at 13:39
• Is radius an integer? – vnp Apr 29 '18 at 18:01
• @vnp Yes, radius and height are integers. – Th3NiKo Apr 29 '18 at 18:21

You don't need to remember every single bucket you've seen.

Once a bucket is added to the top of the pile, any wider bucket that has its rim lower than the rim of the new bucket becomes irrelevant, as do all the narrower buckets it sits on.

So a better data structure would just store the altitudes of all rims currently in play; this can be initialised to (0,0). When you add a bucket, find the widest rim not greater than the new bucket's width; we now set height[0] to that height and height[bucket_width] to height[0]+bucket_height. Remove all narrower buckets, and all other buckets up to a the new height - they can no longer affect the result.

I've added a working demonstration, with tests, below.

Other items for review:

• using namespace std;


Bringing all names in from a namespace is problematic; namespace std particularly so. See Why is “using namespace std” considered bad practice?.

• Spellings: highestPoint, maximum.

• Naming: temp and temporary - these are declared far from their use, so it's hard to appreciate the difference.
• You can omit return 0; in main().

## Working solution

#include <algorithm>
#include <iostream>
#include <istream>
#include <iterator>
#include <map>
#include <sstream>

// Assume that unsigned int is large enough to represent the tallest
// tower
unsigned tower_height(std::istream& input)
{
unsigned count;
input >> count;
if (!input) { std::cerr << "reading count" << std::endl; return 0; }

std::map<unsigned,unsigned> m{ {0,0} }; // radius -> altitude

while (count-->0) {
if (!input) { std::cerr << "reading size" << std::endl; return 0; }
// find the widest bucket this sits on top of
// floor height is now smaller's height
m[0] = smaller->second;
// floor extends out over smaller
if (smaller != m.begin()) {
m.erase(std::next(m.begin()), std::next(smaller));
}
// put this bucket in place (it will always be second)
// find the first wider bucket that's taller
auto height_test = [current](auto const& e){ return e.second > current->second; };
auto wider = std::find_if(current, m.end(), height_test);
// and remove values up to it
m.erase(std::next(current), wider);
}

return m.rbegin()->second;
}

static bool test(unsigned expected, const std::string& input)
{
std::istringstream is{input};
auto actual = tower_height(is);
if (actual != expected) {
std::cerr << "Got " << actual
<< " instead of " << expected
<< " for input " << input
<< std::endl;
return false;
}
return true;
}

int main()
{
int errors = 0;
errors += !test(0, "0");
errors += !test(10, "1  20 10");
errors += !test(20, "2  20 10  20 10");
errors += !test(15, "2  20 10  10 15");
errors += !test(25, "2  10 15  20 10");
errors += !test(25, "3  10 15  20 10   5 10");
errors += !test(20, "3  20 10   5  5  10 15");
return errors;
}


### A note on complexity

At first glance, we might think that the code here is O(n²), because we used std::find_if() each time we add a bucket. But find_if() returns at the first match, and we then remove all the elements we've seen, thus reducing work for subsequent iterations. Therefore on average, we scale as O(n), and that's demonstrable:

std::mt19937 gen{std::random_device()()};
std::uniform_int_distribution dist{1, 1000};
for (unsigned n: {10000, 100000, 1000000}) {
std::stringstream s;
s << n;
for (unsigned i = 0;  i < n;  ++i) {
s << ' ' << dist(gen) << ' ' << dist(gen);
}
auto start_time = std::chrono::high_resolution_clock::now();
auto height = tower_height(s);
auto end_time = std::chrono::high_resolution_clock::now();
auto millis = std::chrono::duration_cast<std::chrono::milliseconds>(end_time - start_time);
std::clog << "Measured " << n << "-bucket tower as "
<< height << " in " << millis.count() << "ms"
<< std::endl;
}

Measured 10000-bucket tower as 2897237 in 1ms
Measured 100000-bucket tower as 28587991 in 13ms
Measured 1000000-bucket tower as 285125976 in 136ms


The very worst case is where each bucket is narrower and shorter than the previous, meaning that the map continuously grows in size. This introduces an O(log n) term, since that's how map operations scale. So we have worst-case performance that scales as O(n log n):

for (unsigned n: {10000, 100000, 1000000}) {
std::stringstream s;
s << n;
for (unsigned i = n;  i > 0;  --i) {
s << ' ' << i << ' ' << i;
}
auto start_time = std::chrono::high_resolution_clock::now();
auto height = tower_height(s);
auto end_time = std::chrono::high_resolution_clock::now();
auto millis = std::chrono::duration_cast<std::chrono::milliseconds>(end_time - start_time);
std::clog << "Measured " << n << "-bucket tower as "
<< height << " in " << millis.count() << "ms"
<< std::endl;
}

Measured 10000-bucket tower as 10000 in 2ms
Measured 100000-bucket tower as 100000 in 36ms
Measured 1000000-bucket tower as 1000000 in 539ms