2个月前

Is it possible to not brute force day 7?

I thought about it for 15 mins, but couldn't think of any mathematical tricks. I thought of lots of minor tricks, like comparing the number to the result and not adding any more multiplications if it's over, things that would cut 10%-20% here and there, but nothing which fundamentally changes big O running time.

For reference, here's my solution for part 2 in smalltalk. I just generated every possible permutation and tested it. Part 1 is similar, mainly I just used bit magic to avoid generating permutations.

(even if you haven't used it, smalltalk is fairly readable, everything is left to right, except in parens)

 smalltalk

    
day7p2: in
    | input | 
    
    input := in lines collect: [ :l | (l splitOn: '\:|\s' asRegex) reject: #isEmpty thenCollect: #asInteger ].
    
    ^ (input select: [ :line |
        (#(1 2 3) permutationsWithRepetitionsOfSize: line size - 2) 
            anySatisfy: [ :num | (self d7addmulcat: line ops: num) = (line at: 1)]
    ]) sum: #first.

 smalltalk

    
d7addmulcat: nums ops: ops
    | final |
    
    final := nums at: 2.
    ops withIndexDo: [ :op :i |
        op = 1 ifTrue: [ final := final * (nums at: i + 2) ].
        op = 2 ifTrue: [ final := final + (nums at: i + 2) ].
        op = 3 ifTrue: [ final := (final asString, (nums at: i+2) asString) asInteger ]
    ].

    ^ final

11 评论

I am not sure how much the formal complexity changes, but the pruning that occurs when working from right to left and bailing early when inverse multiplication or concatenation fails seriously cuts down on the number of combinations that you need to explore. With this pruning, the average fraction of the times (relative to the average 3^(n-1) / 2 times necessary for a naive brute force) that my code actually ended up checking whether the full inverted equation was equal to the target, was only 2.0% for equations that did not have solutions. Interestingly, the figure rose for equations that did have solutions to 6.6%.
- How about the overall number of checks you did? String cat is heavy, but addition and multiplication are absurdly fast, probably faster than the branches needed for early escape.
  
  Inverse concat isn't too heavy if you implement with logs and such. Certainly still heavier than integer add/mul (or sub/div in my case), but arithmetic is usually faster than memory allocation. However, predicting the performance hit due to branching a priori is tricky on modern hardware, which implements sophisticated branch prediction and speculative execution. Furthermore, branching happens anyway between terms to select the right operation, though a misprediction there is likely less significant unless you are doing string manipulation.
  "Overall number of checks" is a bit ambiguous; if taken to mean the number of times I check against the target for early escape, plus the final check on success, the figure is 15% relative to the average 3^(n-1) / 2 checks required by brute force (n = number of terms in the equation, giving n-1 operators). That's still almost a 7-fold decrease. If we instead look at the number of operator evaluations relative to the (n-1)/2 * 3^(n-1) evaluations expected from an average brute force search (3^(n-1) / 2 combinations with (n-1) operations conducted per combination), the figure is only 7.0%. In both cases, there is a significant amount of work not being done.

Probably the easiest optimization (which admittedly I didn't think of myself) is to work backwards: you can eliminate multiplication and concatenation early if you start with the answer and check terms from the right.
- I don't entirely see how, you still need every possible combination of the left side to see what they would become. Plus addition and multiplication are order independent anyway
  
  The point is to prune away search space as early as possible. If the rightmost operand is 5, say, and the answer ends in a 7, then the rightmost operator cannot be anything other than plus. This is a deduction you can't make going left to right. Remember that in this problem the usual order of operations does not apply.
- Call it optimization and people will assume it is magic. Instead I call this a simple algebra challenge.(With part two having that quirky concatenation operation)
  It is like solving for x. Let's take an example:
  3267: 81 40 27
  If you change it to equations:
  81 (+ or *) 40 (+ or *) 27 = 3267
  Which effectively is:
  81 (+ or *) 40 = x (+ or *) 27 = 3267
  So what is the x that either add or multiply would need to create 3267? Or solve for x for only two equations.
  x + 27 = 3267 -> x = 3267 - 27 = 3240
  x * 27 = 3267 -> x = 3267 / 27 = 121
  (Special note since multiplying and adding will never make a float then a division that will have a remainder(or float) is impossible for x, we can easily remove multiplying as a possible choice)
  Now with our example we go plug in x:
  81 (+ or *) 40 = (3240 or 121) (+ or *) 27 = 3267
  So now we see if either 40 or 81 can divide or subtract from x to get the other number. Since it is easier to just program to use the next number in the list, we will use 40.
  81 + 40 = 3240 -> x = 3240 - 40 = 3200 != 81
  81 * 40 = 3240 -> x = 3240 / 40 = 81
  81 + 40 = 121 -> x = 121 - 40 = 81
  81 * 40 = 121 -> x = 121 / 40 = 3.025 != 81
  This particular example from the website has two solutions.
  For the concatenation operation, you simply check if the number ends with the number in the list. If not then a concatenation cannot occur, and if true remove the number from the end and continue in the list.
  Call it pruning tree paths but it is simply algebraic.

11 评论