<< problem 218 - Perfect right-angled triangles | Sphere Packing - problem 222 >> |
Problem 219: Skew-cost coding
(see projecteuler.net/problem=219)
Let A and B be bit strings (sequences of 0's and 1's).
If A is equal to the leftmost length(A) bits of B, then A is said to be a prefix of B.
For example, 00110 is a prefix of 001101001, but not of 00111 or 100110.
A prefix-free code of size n is a collection of n distinct bit strings such that no string is a prefix of any other.
For example, this is a prefix-free code of size 6:
0000, 0001, 001, 01, 10, 11
Now suppose that it costs one penny to transmit a '0' bit, but four pence to transmit a '1'.
Then the total cost of the prefix-free code shown above is 35 pence, which happens to be the cheapest possible for the skewed pricing scheme in question.
In short, we write Cost(6) = 35.
What is Cost(10^9) ?
My Algorithm
The basic idea behind the construction of prefix-free codes is outlined on the Wikipedia page about Huffman codes (see en.wikipedia.org/wiki/Huffman_coding):
- add all codes to a priority-queue sorted by their weight
- pick the code from the queue's front and create two need codes: append a 0 and a 1 and insert those two codes in the priority queue
The algorithm then has to find the remaining 10^9 - 2 codes and keep track of their cost.
Even though the correct result is found, this algorithm is pretty slow (
queue
needs 147 seconds).When I looked at the lengths of the codes I saw that they are pretty short. That means that their cost is pretty low, too.
queue
was repeatedly picking codes with the same cost from its storage. And adding children to the same two categories: plus 1 and plus 4 pence.That's why I wrote a different approach called
array
:Don't keep track of every single code - just count how many codes with a certain weight exists.
Initially there is one code with weight 1 and one code with weight 4.
Then the algorithm is as follows:
- pick all codes with the lowest weight from
costs[x]
- append a zero and a one in order to create their children:
costs[x + 1] += costs[x]
andcosts[x + 4] += costs[x]
Note
The STL's priority_queue
is a max-heap, that means that top()
always returns the largest element.
However, I need the smallest element for my program and therefore needed to use std::greater
for comparisons.
The peak memory usage of my first algorithm was about 1 GByte.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This is equivalent toecho 6 | ./219
Output:
Note: the original problem's input 1000000000
cannot be entered
because just copying results is a soft skill reserved for idiots.
(this interactive test is still under development, computations will be aborted after one second)
My code
… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too.
#include <iostream>
#include <queue>
#include <vector>
#include <functional>
// bit sequence
typedef unsigned char Cost;
// find result using a priority_queue
unsigned long long queue(unsigned int limit)
{
std::priority_queue<Cost, std::vector<Cost>, std::greater<Cost>> codes; // min-heap instead of default max-heap
// first two codes
codes.push(1);
codes.push(4);
unsigned long long totalCost = 5; // sum of the first two codes: 1+4
// until enough codes generated
unsigned int numCodes = 2;
while (numCodes < limit)
{
// pick the first
auto current = codes.top();
codes.pop();
// add two new children codes
codes.push(current + 1);
codes.push(current + 4);
// keep track of the cost
numCodes++;
totalCost += current + 1 + current + 4 - current; // same as current + 5
}
return totalCost;
}
// find result using a bit-length counters
unsigned long long array(unsigned int limit)
{
std::vector<unsigned long long> costs(70, 0);
// initial single-bit codes: "0" => weight 1, "1" => weight 4
costs[1] = 1;
costs[4] = 1;
unsigned long long totalCost = 1 + 4;
// start with the lowest weight (which is 1)
auto current = 1;
// number of codes that I need to generate
auto remaining = limit - 2;
while (remaining > 0)
{
// all codes "used" of the current weight ? => look at higher weights
while (costs[current] == 0) // no gaps: "if" instead of "while" works as well
current++;
// try to process all codes of a certain weight at once
auto block = costs[current];
// except when I don't need all of them
if (block > remaining)
block = remaining;
// adjust counters
remaining -= block;
costs[current] -= block;
costs[current + 1] += block;
costs[current + 4] += block;
// weight is block * (current + 1 + current + 4 - current)
totalCost += block * (unsigned long long)(current + 5);
}
return totalCost;
}
int main()
{
unsigned int limit = 1000000000;
std::cin >> limit;
// slow algorithm
//std::cout << queue(limit) << std::endl;
// fast algorithm
std::cout << array(limit) << std::endl;
return 0;
}
This solution contains 14 empty lines, 19 comments and 4 preprocessor commands.
Benchmark
The correct solution to the original Project Euler problem was found in less than 0.01 seconds on an Intel® Core™ i7-2600K CPU @ 3.40GHz.
(compiled for x86_64 / Linux, GCC flags: -O3 -march=native -fno-exceptions -fno-rtti -std=gnu++11 -DORIGINAL
)
See here for a comparison of all solutions.
Note: interactive tests run on a weaker (=slower) computer. Some interactive tests are compiled without -DORIGINAL
.
Changelog
July 30, 2017 submitted solution
July 30, 2017 added comments
Difficulty
Project Euler ranks this problem at 70% (out of 100%).
Links
projecteuler.net/thread=219 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
Python github.com/LaurentMazare/ProjectEuler/blob/master/e219.py (written by Laurent Mazare)
Python github.com/Meng-Gen/ProjectEuler/blob/master/219.py (written by Meng-Gen Tsai)
Python github.com/smacke/project-euler/blob/master/python/219.py (written by Stephen Macke)
C++ github.com/roosephu/project-euler/blob/master/219.cpp (written by Yuping Luo)
Those links are just an unordered selection of source code I found with a semi-automatic search script on Google/Bing/GitHub/whatever.
You will probably stumble upon better solutions when searching on your own.
Maybe not all linked resources produce the correct result and/or exceed time/memory limits.
Heatmap
Please click on a problem's number to open my solution to that problem:
green | solutions solve the original Project Euler problem and have a perfect score of 100% at Hackerrank, too | |
yellow | solutions score less than 100% at Hackerrank (but still solve the original problem easily) | |
gray | problems are already solved but I haven't published my solution yet | |
blue | solutions are relevant for Project Euler only: there wasn't a Hackerrank version of it (at the time I solved it) or it differed too much | |
orange | problems are solved but exceed the time limit of one minute or the memory limit of 256 MByte | |
red | problems are not solved yet but I wrote a simulation to approximate the result or verified at least the given example - usually I sketched a few ideas, too | |
black | problems are solved but access to the solution is blocked for a few days until the next problem is published | |
[new] | the flashing problem is the one I solved most recently |
I stopped working on Project Euler problems around the time they released 617.
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I scored 13526 points (out of 15700 possible points, top rank was 17 out of ≈60000 in August 2017) at Hackerrank's Project Euler+.
My username at Project Euler is stephanbrumme while it's stbrumme at Hackerrank.
Look at my progress and performance pages to get more details.
Copyright
I hope you enjoy my code and learn something - or give me feedback how I can improve my solutions.
All of my solutions can be used for any purpose and I am in no way liable for any damages caused.
You can even remove my name and claim it's yours. But then you shall burn in hell.
The problems and most of the problems' images were created by Project Euler.
Thanks for all their endless effort !!!
<< problem 218 - Perfect right-angled triangles | Sphere Packing - problem 222 >> |