<< problem 104 - Pandigital Fibonacci ends |
![]() |
Special subset sums: meta-testing - problem 106 >> |
Problem 105: Special subset sums: testing
(see projecteuler.net/problem=105)
Let S(A) represent the sum of elements in set A of size n. We shall call it a special sum set if for any two non-empty disjoint subsets,
B and C, the following properties are true:
i. S(B) != S(C); that is, sums of subsets cannot be equal.
ii. If B contains more elements than C then S(B) > S(C).
For example, { 81, 88, 75, 42, 87, 84, 86, 65 } is not a special sum set because 65 + 87 + 88 = 75 + 81 + 84,
whereas { 157, 150, 164, 119, 79, 159, 161, 139, 158 } satisfies both rules for all possible subset pair combinations and S(A) = 1286.
Using sets.txt (right click and "Save Link/Target As..."), a 4K text file with one-hundred sets containing seven to twelve elements (the two examples given above are the first two sets in the file), identify all the special sum sets, A1, A2, ..., Ak, and find the value of S(A1) + S(A2) + ... + S(Ak).
NOTE: This problem is related to Problem 103 and Problem 106.
My Algorithm
check
is pretty much the same code as in problem 103. See there for an explanation.
I spent most of the time writing code to parse the input (and did it twice because Project Euler has a CSV format while Hackerrank a simpler space-separated).
Modifications by HackerRank
The set can have up to 100 elements. That's 2^100 subsets ... my code can't handle that.
I always return "NO" whenever a set more 30 or more elements to avoid timeouts. Fun fact: that's the correct result for all tests.
Interactive test
You can submit your own input to my program and it will be instantly processed at my server:
This live test is based on the Hackerrank problem.
This is equivalent toecho "2 8 81 88 75 42 87 84 86 65 9 157 150 164 119 79 159 161 139 158" | ./105
Output:
(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, as well as the input data, too. Or just jump to my GitHub repository.
The code contains #ifdef
s to switch between the original problem and the Hackerrank version.
Enable #ifdef ORIGINAL
to produce the result for the original problem (default setting for most problems).
//#define ORIGINAL
#include <iostream>
#include <vector>
typedef std::vector<unsigned int> Sequence;
// return true if sequence is special
bool check(const Sequence& sequence)
{
// sum of all elements
unsigned int fullSum = 0;
for (auto x : sequence)
fullSum += x;
// mark each generated sum as true, no collisions allowed
std::vector<bool> sums(fullSum + 1, false);
// track the lowest and highest sum for each subset size
std::vector<unsigned int> maxSum(sequence.size() + 1, 0);
std::vector<unsigned int> minSum(sequence.size() + 1, fullSum + 1);
minSum[0] = maxSum[0] = 0; // empty set
unsigned int fullMask = (1 << sequence.size()) - 1;
// 2^elements iterations (actually, I ignore the empty set)
for (unsigned int mask = 1; mask <= fullMask; mask++)
{
unsigned int sum = 0;
unsigned int size = 0;
for (unsigned int element = 0; element < sequence.size(); element++)
{
// use that element ?
unsigned int bit = 1 << element;
if ((mask & bit) == 0)
continue;
sum += sequence[element];
// count subset size
size++;
}
// two subsets share the same sum ?
if (sums[sum])
return false;
sums[sum] = true;
// adjust lowest and highest sum of current subset
if (minSum[size] > sum)
minSum[size] = sum;
if (maxSum[size] < sum)
maxSum[size] = sum;
}
// make sure that no set will fewer elements has a higher sum
for (size_t i = 1; i < sequence.size(); i++)
if (maxSum[i] > minSum[i + 1])
return false;
// yes, have another solution
return true;
}
#ifdef ORIGINAL
// convert a line of Project Euler's format into a sequence
Sequence readLine()
{
Sequence result;
while (true)
{
result.push_back(0);
char oneByte = 0;
while (true)
{
oneByte = std::cin.get();
// end of file ?
if (!std::cin)
return result;
// not a digit ?
if (oneByte < '0' || oneByte > '9')
break;
// append digit
result.back() *= 10;
result.back() += oneByte - '0';
}
// end of line
if (oneByte != ',')
break;
}
return result;
}
#else
// convert a line of Project Euler's format into a sequence
Sequence readLine()
{
// read number elements
unsigned int size;
std::cin >> size;
// read elements
Sequence result(size);
for (auto& x : result)
std::cin >> x;
return result;
}
#endif
int main()
{
unsigned int tests = 100;
#ifdef ORIGINAL
unsigned int sum = 0;
#else
std::cin >> tests;
#endif
while (tests--)
{
auto sequence = readLine();
#ifdef ORIGINAL
// special ?
if (check(sequence))
// yes !
for (auto x : sequence)
sum += x;
#else
// special ?
if (sequence.size() < 30) // 2^30 has already one billion subsets ...
std::cout << (check(sequence) ? "YES" : "NO") << std::endl;
else
std::cout << "NO" << std::endl; // just make a guess
#endif
}
#ifdef ORIGINAL
std::cout << sum << std::endl;
#endif
return 0;
}
This solution contains 28 empty lines, 22 comments and 13 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
May 16, 2017 submitted solution
May 16, 2017 added comments
Hackerrank
see https://www.hackerrank.com/contests/projecteuler/challenges/euler105
My code solves 17 out of 17 test cases (score: 100%)
Difficulty
Project Euler ranks this problem at 45% (out of 100%).
Hackerrank describes this problem as easy.
Note:
Hackerrank has strict execution time limits (typically 2 seconds for C++ code) and often a much wider input range than the original problem.
In my opinion, Hackerrank's modified problems are usually a lot harder to solve. As a rule thumb: brute-force is rarely an option.
Links
projecteuler.net/thread=105 - the best forum on the subject (note: you have to submit the correct solution first)
Code in various languages:
C# www.mathblog.dk/project-euler-105-sum-special-sum-sets-file/ (written by Kristian Edlund)
C# github.com/HaochenLiu/My-Project-Euler/blob/master/105.cs (written by Haochen Liu)
Python github.com/hughdbrown/Project-Euler/blob/master/euler-105.py (written by Hugh Brown)
Python github.com/nayuki/Project-Euler-solutions/blob/master/python/p105.py (written by Nayuki)
C++ github.com/Meng-Gen/ProjectEuler/blob/master/105.cc (written by Meng-Gen Tsai)
C++ github.com/roosephu/project-euler/blob/master/105.cpp (written by Yuping Luo)
Java github.com/nayuki/Project-Euler-solutions/blob/master/java/p105.java (written by Nayuki)
Java github.com/thrap/project-euler/blob/master/src/Java/Problem105.java (written by Magnus Solheim Thrap)
Go github.com/frrad/project-euler/blob/master/golang/Problem105.go (written by Frederick Robinson)
Mathematica github.com/steve98654/ProjectEuler/blob/master/105.nb
Scala github.com/samskivert/euler-scala/blob/master/Euler105.scala (written by Michael Bayne)
Perl github.com/gustafe/projecteuler/blob/master/105-special-subsets-testing.pl (written by Gustaf Erikson)
Perl github.com/shlomif/project-euler/blob/master/project-euler/105/euler-105.pl (written by Shlomi Fish)
Rust github.com/gifnksm/ProjectEulerRust/blob/master/src/bin/p105.rs
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.
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 |
26 | 27 | 28 | 29 | 30 | 31 | 32 | 33 | 34 | 35 | 36 | 37 | 38 | 39 | 40 | 41 | 42 | 43 | 44 | 45 | 46 | 47 | 48 | 49 | 50 |
51 | 52 | 53 | 54 | 55 | 56 | 57 | 58 | 59 | 60 | 61 | 62 | 63 | 64 | 65 | 66 | 67 | 68 | 69 | 70 | 71 | 72 | 73 | 74 | 75 |
76 | 77 | 78 | 79 | 80 | 81 | 82 | 83 | 84 | 85 | 86 | 87 | 88 | 89 | 90 | 91 | 92 | 93 | 94 | 95 | 96 | 97 | 98 | 99 | 100 |
101 | 102 | 103 | 104 | 105 | 106 | 107 | 108 | 109 | 110 | 111 | 112 | 113 | 114 | 115 | 116 | 117 | 118 | 119 | 120 | 121 | 122 | 123 | 124 | 125 |
126 | 127 | 128 | 129 | 130 | 131 | 132 | 133 | 134 | 135 | 136 | 137 | 138 | 139 | 140 | 141 | 142 | 143 | 144 | 145 | 146 | 147 | 148 | 149 | 150 |
151 | 152 | 153 | 154 | 155 | 156 | 157 | 158 | 159 | 160 | 161 | 162 | 163 | 164 | 165 | 166 | 167 | 168 | 169 | 170 | 171 | 172 | 173 | 174 | 175 |
176 | 177 | 178 | 179 | 180 | 181 | 182 | 183 | 184 | 185 | 186 | 187 | 188 | 189 | 190 | 191 | 192 | 193 | 194 | 195 | 196 | 197 | 198 | 199 | 200 |
201 | 202 | 203 | 204 | 205 | 206 | 207 | 208 | 209 | 210 | 211 | 212 | 213 | 214 | 215 | 216 | 217 | 218 | 219 | 220 | 221 | 222 | 223 | 224 | 225 |
226 | 227 | 228 | 229 | 230 | 231 | 232 | 233 | 234 | 235 | 236 | 237 | 238 | 239 | 240 | 241 | 242 | 243 | 244 | 245 | 246 | 247 | 248 | 249 | 250 |
251 | 252 | 253 | 254 | 255 | 256 | 257 | 258 | 259 | 260 | 261 | 262 | 263 | 264 | 265 | 266 | 267 | 268 | 269 | 270 | 271 | 272 | 273 | 274 | 275 |
276 | 277 | 278 | 279 | 280 | 281 | 282 | 283 | 284 | 285 | 286 | 287 | 288 | 289 | 290 | 291 | 292 | 293 | 294 | 295 | 296 | 297 | 298 | 299 | 300 |
301 | 302 | 303 | 304 | 305 | 306 | 307 | 308 | 309 | 310 | 311 | 312 | 313 | 314 | 315 | 316 | 317 | 318 | 319 | 320 | 321 | 322 | 323 | 324 | 325 |
326 | 327 | 328 | 329 | 330 | 331 | 332 | 333 | 334 | 335 | 336 | 337 | 338 | 339 | 340 | 341 | 342 | 343 | 344 | 345 | 346 | 347 | 348 | 349 | 350 |
351 | 352 | 353 | 354 | 355 | 356 | 357 | 358 | 359 | 360 | 361 | 362 | 363 | 364 | 365 | 366 | 367 | 368 | 369 | 370 | 371 | 372 | 373 | 374 | 375 |
376 | 377 | 378 | 379 | 380 | 381 | 382 | 383 | 384 | 385 | 386 | 387 | 388 | 389 | 390 | 391 | 392 | 393 | 394 | 395 | 396 | 397 | 398 | 399 | 400 |
401 | 402 | 403 | 404 | 405 | 406 | 407 | 408 | 409 | 410 | 411 | 412 | 413 | 414 | 415 | 416 | 417 | 418 | 419 | 420 | 421 | 422 | 423 | 424 | 425 |
426 | 427 | 428 | 429 | 430 | 431 | 432 | 433 | 434 | 435 | 436 | 437 | 438 | 439 | 440 | 441 | 442 | 443 | 444 | 445 | 446 | 447 | 448 | 449 | 450 |
451 | 452 | 453 | 454 | 455 | 456 | 457 | 458 | 459 | 460 | 461 | 462 | 463 | 464 | 465 | 466 | 467 | 468 | 469 | 470 | 471 | 472 | 473 | 474 | 475 |
476 | 477 | 478 | 479 | 480 | 481 | 482 | 483 | 484 | 485 | 486 | 487 | 488 | 489 | 490 | 491 | 492 | 493 | 494 | 495 | 496 | 497 | 498 | 499 | 500 |
501 | 502 | 503 | 504 | 505 | 506 | 507 | 508 | 509 | 510 | 511 | 512 | 513 | 514 | 515 | 516 | 517 | 518 | 519 | 520 | 521 | 522 | 523 | 524 | 525 |
526 | 527 | 528 | 529 | 530 | 531 | 532 | 533 | 534 | 535 | 536 | 537 | 538 | 539 | 540 | 541 | 542 | 543 | 544 | 545 | 546 | 547 | 548 | 549 | 550 |
551 | 552 | 553 | 554 | 555 | 556 | 557 | 558 | 559 | 560 | 561 | 562 | 563 | 564 | 565 | 566 | 567 | 568 | 569 | 570 | 571 | 572 | 573 | 574 | 575 |
576 | 577 | 578 | 579 | 580 | 581 | 582 | 583 | 584 | 585 | 586 | 587 | 588 | 589 | 590 | 591 | 592 | 593 | 594 | 595 | 596 | 597 | 598 | 599 | 600 |
601 | 602 | 603 | 604 | 605 | 606 | 607 | 608 | 609 | 610 | 611 | 612 | 613 | 614 | 615 | 616 | 617 | 618 | 619 | 620 | 621 | 622 | 623 | 624 | 625 |
626 | 627 | 628 | 629 | 630 | 631 | 632 | 633 | 634 | 635 | 636 | 637 | 638 | 639 | 640 | 641 | 642 | 643 | 644 | 645 | 646 | 647 | 648 | 649 | 650 |
651 | 652 | 653 | 654 | 655 | 656 | 657 | 658 | 659 | 660 | 661 | 662 | 663 | 664 | 665 | 666 | 667 | 668 | 669 | 670 | 671 | 672 | 673 | 674 | 675 |
676 | 677 | 678 | 679 | 680 | 681 | 682 | 683 | 684 | 685 | 686 | 687 | 688 | 689 | 690 | 691 | 692 | 693 | 694 | 695 | 696 | 697 | 698 | 699 | 700 |
701 | 702 | 703 | 704 | 705 | 706 | 707 | 708 | 709 | 710 | 711 | 712 | 713 | 714 | 715 | 716 | 717 | 718 | 719 | 720 | 721 | 722 | 723 | 724 | 725 |
726 | 727 | 728 | 729 | 730 | 731 | 732 | 733 | 734 | 735 | 736 | 737 | 738 | 739 | 740 | 741 | 742 | 743 | 744 | 745 | 746 | 747 | 748 | 749 | 750 |
751 | 752 | 753 | 754 | 755 | 756 | 757 | 758 | 759 | 760 | 761 | 762 | 763 | 764 | 765 | 766 | 767 | 768 | 769 | 770 | 771 | 772 | 773 | 774 | 775 |
776 | 777 | 778 | 779 | 780 | 781 | 782 | 783 | 784 | 785 | 786 | 787 | 788 | 789 | 790 | 791 | 792 | 793 | 794 | 795 | 796 | 797 | 798 | 799 | 800 |
801 | 802 | 803 | 804 | 805 | 806 | 807 | 808 | 809 | 810 | 811 | 812 | 813 | 814 | 815 | 816 | 817 | 818 | 819 | 820 | 821 | 822 | 823 | 824 | 825 |
826 | 827 | 828 |
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 104 - Pandigital Fibonacci ends |
![]() |
Special subset sums: meta-testing - problem 106 >> |