Given a set of n data points generated from the same distribution, I want to "randomly partition" the set into k groups, where each contains n / k points randomly chosen from the original data set.
Alternatively, I can first divide the input data set into k contiguous chunks, where the first chunk contains 1, ..., n/k, and the second chunk contains n/k+1, ..., 2n/k, and so on. Then I "shuffle" the data points within each partition.
Are these two approaches always equal, given the data set are generated from the same distribution? If not, what assumptions do we need when these two approaches produces the same results?
Obviously they are not equivalent; the second restricts the values that can go in each partition, while the first does not.
If by "results" you mean what is done with these partitions, that would be wholly dependent on just what that is, which you provide no hint to.
Related
I would like to select the top n values from a dataset, but ignore elements based on what I have already selected - i.e., given a set of points (x,y), I would like to select the top 100 values of x (which are all distinct) but not select any points such that y equals the y of any already-selected point. I would like to make sure that the highest values of x are prioritized.
Is there any existing algorithm for this, or at least similar ones? I have a huge amount of data and would like to do this as efficiently as possible. Memory is not as much of a concern.
You can do this in O(n log k) time where n is the number of values in the dataset and k are the number of top values you'd like to get.
Store the values you wish to exclude in a hash table.
Make an empty min-heap.
Iterate over all of the values and for each value:
If it is in the hash table skip it.
If the heap contains fewer than k values, add it to the heap.
If the heap contains >=k values, if the value you're looking at is greater than the smallest member of the minheap, pop that value and add the new one.
I will share my thoughts and since the author still has not specified the scope of data to be processed, I will assume that it is too large to be handled by a single machine and I will also assume that the author is familiar with Hadoop.
So I would suggest using the MapReduce as follows:
Mappers simply emit pairs (x,y)
Combiners select k pairs with largest values of x (k=100 in this case) in the meantime maintaining the unique y's in the hashset to avoid duplicates, then emit k pairs found.
There should be only one reducer in this job since it has to get all pairs from combiners to finalize the job by selecting k pairs for the last time. Reducer's implementation is identical to combiner.
The number of combiners should be selected considering memory resources needed to select top k pairs out of incoming dataset since whichever method is used (sorting, heap or anything else) it is going to be done in-memory, as well as keeping that hashset with unique y's
I have a big sequence file storing the tfidf values for documents. Each line represents line and the columns are the value of tfidfs for each term (the row is a sparse vector). I'd like to pick the top-k words for each document using Hadoop. The naive solution is to loop through all the columns for each row in the mapper and pick the top-k but as the file becomes bigger and bigger I don't think this is a good solution. Is there a better way to do that in Hadoop?
1. In every map calculate TopK (this is local top K for each map)
2. Spawn a signle reduce , now top K from all mappers will flow to this reducer and hence global Top K will be evaluated.
Think of the problem as
1. You have been given the results of X number of horse races.
2. You need to find Top N fastest horse.
I have an array with, for example, 1000000000000 of elements (integers). What is the best approach to pick, for example, only 3 random and unique elements from this array? Elements must be unique in whole array, not in list of N (3 in my example) elements.
I read about Reservoir sampling, but it provides only method to pick random numbers, which can be non-unique.
If the odds of hitting a non-unique value are low, your best bet will be to select 3 random numbers from the array, then check each against the entire array to ensure it is unique - if not, choose another random sample to replace it and repeat the test.
If the odds of hitting a non-unique value are high, this increases the number of times you'll need to scan the array looking for uniqueness and makes the simple solution non-optimal. In that case you'll want to split the task of ensuring unique numbers from the task of making a random selection.
Sorting the array is the easiest way to find duplicates. Most sorting algorithms are O(n log n), but since your keys are integers Radix sort can potentially be faster.
Another possibility is to use a hash table to find duplicates, but that will require significant space. You can use a smaller hash table or Bloom filter to identify potential duplicates, then use another method to go through that smaller list.
counts = [0] * (MAXINT-MININT+1)
for value in Elements:
counts[value] += 1
uniques = [c for c in counts where c==1]
result = random.pick_3_from(uniques)
I assume that you have a reasonable idea what fraction of the array values are likely to be unique. So you would know, for instance, that if you picked 1000 random array values, the odds are good that one is unique.
Step 1. Pick 3 random hash algorithms. They can all be the same algorithm, except that you add different integers to each as a first step.
Step 2. Scan the array. Hash each integer all three ways, and for each hash algorithm, keep track of the X lowest hash codes you get (you can use a priority queue for this), and keep a hash table of how many times each of those integers occurs.
Step 3. For each hash algorithm, look for a unique element in that bucket. If it is already picked in another bucket, find another. (Should be a rare boundary case.)
That is your set of three random unique elements. Every unique triple should have even odds of being picked.
(Note: For many purposes it would be fine to just use one hash algorithm and find 3 things from its list...)
This algorithm will succeed with high likelihood in one pass through the array. What is better yet is that the intermediate data structure that it uses is fairly small and is amenable to merging. Therefore this can be parallelized across machines for a very large data set.
Lets say there are N numbers grouped into K disjoint sets. The problem is to create a key for each of these disjoint sets such that given any number, a simple operation on these keys and the number should be able to give the set containing the number.
A simple approach and it’s limitations:
For ex. Let the N numbers be 34,35,36….321 Let set 1 be made of 63,66,77,89,122,222 And set 2 be made of 53,69,137,230,280,299,300,306 And so on..
sol:
first an array of prime numbers containing (321-34=287) items is created. To create a key for the first set, the prime numbers corresponding to positions (63-34),(66-34),…(222-34) in the array are multiplied. Now this key is divisible only by prime numbers corresponding to the numbers in set 1 and not otherwise. So, given 77, [if(key1%(primeArray[77-34]==0)], 77 belongs to set1
But since I’m dealing with large number of data values, the keys cannot be represented by even 64 bit integers(and I don’t want to split the keys). Is there a better way of doing this?
Seems like a classic case to use union find data structure.
Union-find data structure
Give each number a rank. Trivially, the number itself in this case. The highest rank in a set represents the set.
I have n sorted lists (5 < n < 300). These lists are quite long (300000+ tuples). Selecting the top k of the individual lists is of course trivial - they are right at the head of the lists.
Example for k = 2:
top2 (L1: [ 'a': 10, 'b': 4, 'c':3 ]) = ['a':10 'b':4]
top2 (L2: [ 'c': 5, 'b': 2, 'a':0 ]) = ['c':5 'b':2]
Where it gets more interesting is when I want the combined top k across all the sorted lists.
top2(L1+L2) = ['a':10, 'c':8]
Just combining of the top k of the individual list would not necessarily gives the correct results:
top2(top2(L1)+top2(L2)) = ['a':10, 'b':6]
The goal is to reduce the required space and keep the sorted lists small.
top2(topX(L1)+topX(L2)) = ['a':10, 'c':8]
The question is whether there is an algorithm to calculate the combined top k having the correct order while cutting off the long tail of the lists at a certain position. And if there is: How does one find the limit X where is is safe to cut?
Note: Correct counts are not important. Only the order is.
top2(magic([L1,L2])) = ['a', 'c']
This algorithm uses O(U) memory where U is the number of unique keys. I doubt a lower memory bounds can be achieved because it is impossible to tell which keys can be discarded until all the keys have been summed.
Make a master list of (key:total_count) tuples. Simply run through each list one item at a time, keeping a tally of how many times each key has been seen.
Use any top-k selection algorithm on the master list that does not use additional memory. One simple solution is to sort the list in place.
If I understand your question correctly, the correct output is the top 10 items, irrespective of the list from which each came. If that's correct, then start with the first 10 items in each list will allow you to generate the correct output (if you only want unique items in the output, but the inputs might contain duplicates, then you need 10 unique items in each list).
In the most extreme case, all the top items come from one list, and all items from the other lists are ignored. In this case, having 10 items in the one list will be sufficient to produce the correct result.
Associate an index with each of your n lists. Set it to point to the first element in each case.
Create a list-of-lists, and sort it by the indexed elements.
The indexed item on the top list in your list-of-lists is your first element.
Increment the index for the topmost list and remove that list from the list-of-lists and re-insert it based on the new value of its indexed element.
The indexed item on the top list in your list-of-lists is your next element
Goto 4 and repeat until done.
You didn't specify how many lists you have. If n is small, then step 4 can be done very simply (just re-sort the lists). As n grows you may want to think about more efficient ways to resort and almost-sorted list-of-lists.
I did not understand if an 'a' appears in two lists, their counts must be combined. Here is a new memory-efficient algorithm:
(New) Algorithm:
(Re-)sort each list by ID (not by count). To release memory, the list can be written back to disk. Only enough memory for the longest list is required.
Get the next lowest unprocessed ID and find the total count across all lists.
Insert the ID into a priority queue of k nodes. Use the total count as the node's priority (not the ID). This priority queue drops the lowest node if more than k nodes are inserted.
Go to step 2 until all ID's have been exhausted.
Analysis: This algorithm can be implemented using only O(k) additional memory to store the min-heap. It makes several trade-offs to accomplish this:
The lists are sorted by ID in place; the original orderings by counts are lost. Otherwise O(U) additional memory is required to make a master list with ID: total_count tuples where U is number of unique ID's.
The next lowest ID is found in O(n) time by checking the first tuple of each list. This is repeated U times where U is the number of unique ID's. This might be improved by using a min-heap to track the next lowest ID. This would require O(n) additional memory (and may not be faster in all cases).
Note: This algorithm assumes ID's can be quickly compared. String comparisons are not trivial. I suggest hashing string ID's to integers. They do not have to be unique hashes, but collisions must be checked so all ID's are properly sorted/compared. Of course, this would add to the memory/time complexity.
The perfect solution requires all tuples to be inspected at least once.
However, it is possible to get close to the perfect solution without inspecting every tuple. Discarding the "long tail" introduces a margin of error. You can use some type of heuristic to calculate when the margin of error is acceptable.
For example, if there are n=100 sorted lists and you have inspected down each list until the count is 2, the most the total count for a key could increase by is 200.
I suggest taking an iterative approach:
Tally each list until a certain lower count threshold L is reached.
Lower L to include more tuples.
Add the new tuples to the counts tallied so far.
Go to step 2 until lowering L does not change the top k counts by more than a certain percentage.
This algorithm assumes the counts for the top k keys will approach a certain value the further long tail is traversed. You can use other heuristics instead of the certain percentage like number of new keys in the top k, how much the top k keys were shuffled, etc...
There is a sane way to implement this through mapreduce:
http://www.yourdailygeekery.com/2011/05/16/top-k-with-mapreduce.html
In general, I think you are in trouble. Imagine the following lists:
['a':100, 'b':99, ...]
['c':90, 'd':89, ..., 'b':2]
and you have k=1 (i.e. you want only the top one). 'b' is the right answer, but you need to look all the way down to the end of the second list to realize that 'b' beats 'a'.
Edit:
If you have the right distribution (long, low count tails), you might be able to do better. Let's keep with k=1 for now to make our lives easier.
The basic algorithm is to keep a hash map of the keys you've seen so far and their associated totals. Walk down the lists processing elements and updating your map.
The key observation is that a key can gain in count by at most the sum of the counts at the current processing point of each list (call that sum S). So on each step, you can prune from your hash map any keys whose total is more than S below your current maximum count element. (I'm not sure what data structure you would need to prune as you need to look up keys given a range of counts - maybe a priority queue?)
When your hash map has only one element in it, and its count is at least S, then you can stop processing the lists and return that element as the answer. If your count distribution plays nice, this early exit may actually trigger so you don't have to process all of the lists.