Combining coins to reach a target sum (C, Python)

http://users.softlab.ece.ntua.gr/~ttsiod/euler31.html

by ttsiodras • 15 years ago

33 21 15 years ago

users.softlab.ece.ntua.gr

gwern• 15 years ago

Very nice example of memoization.

yoklov• 15 years ago

Specifically of the memoization technique known as dynamic programming.

owenmarshall3• 15 years ago

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.

cperciva4• 15 years ago

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

ttsiodras2• 15 years ago

Wow - speechless :-)

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

cperciva4• 15 years ago

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.

pmiller2• 15 years ago

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

smcdow5• 15 years ago

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.

nkurz• 15 years ago

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?

smcdow5• 15 years ago

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.

smcdow5• 15 years ago

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.

d0mine• 15 years ago

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

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

ttsiodras2• 15 years ago

Thank you, didn't know this!

Code updated.

sur• 15 years ago

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]

Combining coins to reach a target sum (C, Python)

Related Stories