14 Dec, 2011, Idealiad wrote in the 1st comment:
So I've been reading a book lately and I'm about halfway through it. The chapter I just started poses a small but interesting problem.

Given a vector of floating-point numbers, find the largest sum among the contiguous subvectors.

So for example, in the vector x

[45 -34 8 7 48 -30]

The contiguous subvector with the largest sum is x[2…4] totaling 63.

Now this problem would not be so interesting but for the story behind it. The author tells a story that the original algorithm the programmer came up with was going to take 15 days to process a vector of 100,000 numbers (note that this book is relatively old).

However, redesigning the algorithm brought the time down to 5 milliseconds.

So, your challenge, if you choose to accept it, is to try your hand at solving this problem.

At the end of this post you'll find a vector of 5000 random numbers to use. We can make it bigger if necessary.

Time your tests and post the results in the thread. Any language is acceptable. I encourage multiple attempts to make it faster. If you wish, refrain from reading the thread until you post your solution, but if you want to see other people's attempts first feel free.

Good luck!
Quote
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14 Dec, 2011, Rarva.Riendf wrote in the 2nd comment:
`45	-34	8	7	48	-30	45	-34	8	7	48  <- max 55		45	-34	8	7     <- here I find the 63 			45	-34	8  <- max  45				45	-34 <-max 74					45 <- max 7445	11	19	26	74	44`

For me 74 is the largest sum, or did I miss something in the problem definition?
14 Dec, 2011, Runter wrote in the 3rd comment:
considering this set: [10, -10, 9, 2] the largest summed continuous range is [3..4]. or 9 + 2 = 11. No other continuous range has a higher sum. With a small set of data this is easy to eyeball. With a set like the one he gave us, this would be quite a lot of computation.
14 Dec, 2011, Rarva.Riendf wrote in the 4th comment:
Quote
So for example, in the vector x

[45 -34 8 7 48 -30]

He states 63 as the max (8 7 48), I find 74 (45 -34 8 7 48), so I do not get it.

My algorithm is sucky as it needs to find the max number as many times as there are elements in the vector, and off course as many sums (but this one is pretty free if you do it binary, it is a simple &)
14 Dec, 2011, Tyche wrote in the 5th comment:
[45 -34 8 7 48 -30] => 74 [0..4]

[10, -10, 9, 2] => 11 [0..3]

No?
14 Dec, 2011, Runter wrote in the 6th comment:
Yes, my contrived example is flawed. :p
14 Dec, 2011, Tyche wrote in the 7th comment:
Actually those are two good test data sets, since the results are known, and the second being a corner case.
15 Dec, 2011, Idealiad wrote in the 8th comment:
Sorry, RR is correct. That's what I get for posting late. In the first example the largest one is [0…4]. If a mod wants to annotate that first post feel free.
15 Dec, 2011, Ssolvarain wrote in the 9th comment:
By a show of hands, how many of you are actually robots posing as humans?
15 Dec, 2011, Idealiad wrote in the 10th comment:
So I took my first shot at this; I haven't read more of the chapter yet.

A possible solution in Clojure, where the vector in the OP is v:

`(apply   max  (flatten     (for       [i (range (count v))]      [(apply + (take i v)) (apply + (drop i v))]))))`

gives my first time, of 17214.598783 msecs, and an answer of 15966.
15 Dec, 2011, David Haley wrote in the 11th comment:
This is a super-classic entry-algorithms course problem… typically solved most efficiently using dynamic programming, no?

You should rephrase the challenge as finding the most asymptotically efficient algorithm, to eliminate things like language speed.

You should also impose constraints on memory; if you allow the allocation of a very large block of memory the problem becomes substantially easier.
15 Dec, 2011, Idealiad wrote in the 12th comment:
Explain that efficiency thing. I thought about language speed before the OP but I didn't really know how to deal with it, so I just punted.

'Easy' I'm not so worried about. How would you solve it DH?

edit: to add to the speed issue, I'm more interested in how people improve their speed relative to their previous attempts, than comparing speed among different folks' attempts, if that makes sense.
15 Dec, 2011, Tyche wrote in the 13th comment:
Ssolvarain said:
By a show of hands, how many of you are actually robots posing as humans?

I used to be human, but then I took an arrow to a knee.
15 Dec, 2011, David Haley wrote in the 14th comment:
See this as a starting point for what I meant about asymptotic efficiency.
http://en.wikipedia.org/wiki/Analysis_of...

Spoiler
The standard solution to this is iterating over the array, keeping track of the largest sum so far, and stopping when you hit a negative number.
This problem is so common it even has a Wikipedia page :-)
http://en.wikipedia.org/wiki/Maximum_sub...

The solution is basically one-dimensional dynamic programming.
15 Dec, 2011, Idealiad wrote in the 15th comment:
OK I understand you now, you meant the O(n) stuff.
15 Dec, 2011, Runter wrote in the 16th comment:
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Might save a kindrid spirit a little time.
15 Dec, 2011, Rarva.Riendf wrote in the 17th comment:
My algo in pseudo code btw: (I gave up coding it, as it requires know how in bit fiddling and I dont even remember the syntx to do that…)

struct max {
int value;
int position;
}
data[nbElements] = {x,y, z, ….};
index = nbElements
max tempMax;
max finalMax;

while (nbElements) {
nbElements–;
sum (data, (switch(data, sizeof(int)) <- (if data is 1 2 3 it does 1 2 3 + 0 1 2 next loop it will be 1 3 5 + 0 0 1) I know it is possible in binary with the appropriate << and &
tempMax = findmax(data) <- find the highest value and its position, and this can be heavily and easily parallized if needed
finalMax = maxof (temMax,finalMax) <- pick the highest one
}

I think that translate in O(n^2) that is not vey nice but easy (for the one that know how to code the bit bashing part) to code and read :P
15 Dec, 2011, Sharmair wrote in the 18th comment:
So I took my first shot at this; I haven't read more of the chapter yet.

A possible solution in Clojure, where the vector in the OP is v:

`(apply   max  (flatten     (for       [i (range (count v))]      [(apply + (take i v)) (apply + (drop i v))]))))`

gives my first time, of 17214.598783 msecs, and an answer of 15966.

I got a bit different answer: Start = 58, end = 569, sum = 21494 in 31uS
using MSVC++ 6.0 running on my old (2005) windows XP computer.
The way I did it was with a single scan through the data, keeping track of
the max sum as I went. The scan code was:
`int rstart = 0;    int rend = 0;    int maxsum = 0;    int csum = 0;    int start = 0;    for(int current = 0; current < size; ++current){        csum += data[current];        if(csum < 1){            start = current+1;            csum = 0;        }else if(csum > maxsum){            rstart = start;            rend = current;            maxsum = csum;        }    }`

It works on an array called data, with a number of elements of size.
15 Dec, 2011, Twisol wrote in the 19th comment:
Well, first thing I notice is that you can treat contiguous sets of positive and negative numbers as single units. If a positive you've included in your solution is bordered by another positive that isn't, you can only add to the total by including it as well. Likewise, the total can only decrease in the same situation with negatives.

Incidentally, the first and last items in the solution will always be positive (unless there aren't any positives in the set to begin with). If you have a negative on the edge of your set, simply shrinking to un-include it would increase your total.

So… okay. I would start by going over the whole thing and partitioning it up into like-signed sets. And I'd keep a list mapping the ranges of these sets with their sums. Then I would repeatedly iterate over these islands, attempting to join positives separated by a smaller negative so that the "islands" consistently increase over time. When I hit the point where no island can be joined with a neighbor without losing value, we've found our maxima. From there, it's a matter of comparing them and finding out who has the largest magnitude.

<time passes>

Okay! Here's my work. It seems to work properly, though I don't see an easy way of checking the answer for the giant vector of numbers given.

\$ time ruby vector.rb
[3728…3823, 14004]

real 0m0.285s
user 0m0.340s
sys 0m0.028s

`# Isolate regions of positive and negative numbersdef partition (v)  sets = []    left = 0  i = 1  v.each_cons(2) do |l, r|    # are the signs different?    if (l >= 0) != (r >= 0)      sets.push [(left…i), v[left…i].reduce {|a, n| a + n}]      left = i    end    i += 1  end    sets.push [(left…i), v[left…i].reduce {|a, n| a + n}]end# Remove negative regions from the front and back - they'll never be included.def trim! (sets)  sets.shift if sets.first[1] < 0  sets.pop if sets.last[1] < 0end# Merge adjacent regions of positive over a lesser region of negative.def merge! (sets)  original_length = sets.length    (sets.length-3).step(0, -2) do |i|    left, neg, right = sets[i], sets[i+1], sets[i+2]        if [left[1], right[1]].min + neg[1] > 0      sets[i] = [(left[0].first…right[0].last), left[1] + neg[1] + right[1]]      sets.delete_at(i+2)      sets.delete_at(i+1)    end  end    sets.length != original_lengthenddef calculate (v)  # Partition the vector into regions by sign.  sets = partition v    # If there's only one entry, return it, whether it's positive or negative.  return sets[0] if sets.length == 1  # Remove negatives from the front and back.  trim! sets  # If there's only one now, it's the only positive island, so return it.  return sets[0] if sets.length == 1    # Iteratively merge adjacent regions to create larger regions  nil while merge! sets    # We have our maxima. Return the largest of our remaining regions.  sets.reduce {|a, x| x[1] > a[1] ? x : a}end`

(EDIT: Well, based on other answers, it looks like this isn't complete yet. From some debug output, I think it needs to be able to merge groups that are separated by more than one negative region… time to work on that a bit.)
15 Dec, 2011, Omega wrote in the 20th comment:
Tyche said:
Ssolvarain said:
By a show of hands, how many of you are actually robots posing as humans?

I used to be human, but then I took an arrow to a knee.

Skyrim arrow to the knee meme FTW!
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