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Combining coins to reach a target sum (C, Python) (ntua.gr)
33 points by ttsiodras on Dec 29, 2011 | hide | past | web | favorite | 21 comments



Alternatively, you can ask Wolfram Alpha to compute a power series: http://www.wolframalpha.com/input/?i=series+1%2F%28%281-x%29...


Wow - speechless :-)

The correlation with my program's results is clear (the coefficients are my last column) - any references as to why?


The power series for 1/(1-x^n) is 1 + x^n + x^2n + x^3n + ... in which the term x^i is the number of ways of forming i as a sum of zero or more n.

Multiplying two power series maintains that property -- the x^j term in the product of two power series comes from x^i in one power series and x^(j-i) in the other power series, just like a sum of j pence made out of two types of coins has to be i pence worth of one type of coin and j-i pence worth of the other type of coin.

Converting (1/(1-x)) * (1/(1-x^2)) * ... into 1/((1-x) * (1-x^2) * ...) is just routine algebraic manipulation, which I claim without proof is valid for power series.


This is why you should treat Project Euler more as a way to learn mathematics and less as a way to practice code. Most of their problems can, and IMO should, be solved by pen and paper alone.


To be fair, I wouldn't want to compute the 200th term of a power series by hand. I could do it, but it would probably take me several attempts to do that much arithmetic without making a mistake somewhere along the way.


(sorry to necrobump!)

That's a fair point. I should revisit my critique: IMO, PE should be used to practice conceptual mathematics; if you're trying to perform an exhaustive search, for example, you might want to see if there's a better way.


That's because the series is the generating function for the number of ways to make 2 pounds with those coins.


FWIW, printf() is not async-signal-safe. It's bad form to call such routines from within a signal handler.

Here's a list of async-signal-safe functions (scroll down): http://pubs.opengroup.org/onlinepubs/009695399/functions/xsh...

I know I'm missing the point of the post, but it is a particular nit that I pick. You'd be surprised at how often this happens in production code, and how often it causes mysterious, hard-to-reproduce bugs.


I don't disagree with your logic, but how would you recommend that he solve this in a safer manner? Fork() and printf()? Ignore signals before printf()? Use write()? Some other manner of triggering besides Ctrl-C?


I'd keep an int flag, initialized to zero. The sighandler sets the flag. The flag is checked (and reset) on entry to newCoin().

Not exactly pretty, but using signals to trigger to produce reports isn't exactly pretty either.

Unfortunately, signals are often abused in this manner. Not a big deal for such for this small piece of software, but in larger systems this kind of abuse can cause very hard to fix bugs.


I'd keep an int flag, initialized to zero.

If you do, your code is buggy: sig_atomic_t is the only type you can safely use inside and outside a signal handler.


Please explain why this would be true for single-threaded applications.


Because your signal handler may get interrupted by another signal.


So what? Even then, there will be no concurrency issues in a single threaded application.


A (possibly) cleaner way would be to use Boost.asio. I haven't yet evaluated the POSIX Signal handing part of Boost.asio, but I'm hopeful that it will be useful for my purposes.


This problem is closely related to the Frobenius coin problem: http://en.wikipedia.org/wiki/Coin_problem

It's an interesting problem. My gut reaction was to solve X1 + X2 + (...) + X6 = 200; X1, ..., X6 < 0, which would be C(205, 200), but that ignores the denominations of each coin, so it doesn't really generate what you want.


It's possible to do this using just a one-dimensional array. Since we only need the last column of the matrix we calculated to calculate the next column, the array can be updated in-place.

  coins = [1, 2, 5, 10, 20, 50, 100, 200]
  total = 200
  matrix = [0] * (total + 1)
  matrix[0] = 1
  
  for coin in coins:
      for j in range(coin, len(matrix)):
          matrix[j] += matrix[j - coin]


`[(y,x)]` is the same thing as `[y,x]`

`,` creates a tuple, not parentheses (empty tuple is an exception).


Thank you, didn't know this!

Code updated.


Very nice example of memoization.


Specifically of the memoization technique known as dynamic programming.




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