I am trying to implement a hashtable using linear probing.
Before inserting a (key, value) pair into the hashtable, I want to check if it's half full. If it is, I need to double the size of the underlying array.
Obviously, there are two ways to do that:
One is to create another array with the doubled size, rehash all entries in the old one and add them to the new array. Then, rebind the old array to the new one. This way is easy to implement but uses a lot of space.
The other one is to double the array and do the rehashing in-place. It seems that this way may lead to longer running time because rehashing may cause collisions with both newly hashed slots and old slots.
Which way should I use?
Your second solution only saves space during the resize process if there is in fact room to expand the existing hash table in-place - I think the chances of that being the case for a large hash table are quite slim, so I would just go for your first solution.
Related
Looking around at some of the hash table implementations, separate chaining seems to be handled via a linked list or a tree. Is there a reason why a dynamic array is not used? I would imagine that having a dynamic array would have better cache performance as well. However, since I've not seen any such implementation, I'm probably missing something.
What am I missing?
One advantage of a linked list over a dynamic array is that rehashing can be accomplished more quickly. Rather than having to make a bunch of new dynamic arrays and then copy all the elements from the old dynamic arrays into the new, the elements from the linked lists can be redistributed into the new buckets without performing any allocations.
Additionally, if the load factor is small, the space overhead of using linked lists may be better than the space overhead for dynamic arrays. When using dynamic arrays, you usually need to store a pointer, a length, and a capacity. This means that if you have an empty dynamic array, you end up needing space for two integers and a pointer, plus any space preallocated to hold the elements. In an empty bucket, this space overhead is large compared to storing just a null pointer for a linked list. On the other hand, if the buckets have large numbers of elements in them, then dynamic arrays will be a bit more space-efficient and have higher performance due to locality of reference.
Hope this helps!
One advantage i can think of are the deletes ..while addition is done at the head of the hash..if i want to delete a value in the hash it will be difficult for array as it may be present in the middle of the array.
I have algorithms that works with dynamically growing lists (contiguous memory like a C++ vector, Java ArrayList or C# List). Until recently, these algorithms would insert new values into the middle of the lists. Of course, this was usually a very slow operation. Every time an item was added, all the items after it needed to be shifted to a higher index. Do this a few times for each algorithm and things get really slow.
My realization was that I could add the new items to the end of the list and then rotate them into position later. That's one option!
Another option, when I know how many items I'm adding ahead of time, is to add that many items to the back, shift the existing items and then perform the algorithm in-place in the hole I've made for myself. The negative is that I have to add some default value to the end of the list and then just overwrite them.
I did a quick analysis of these options and concluded that the second option is more efficient. My reasoning was that the rotation with the first option would result in in-place swaps (requiring a temporary). My only concern with the second option is that I am creating a bunch of default values that just get thrown away. Most of the time, these default values will be null or a mem-filled value type.
However, I'd like someone else familiar with algorithms to tell me which approach would be faster. Or, perhaps there's an even more efficient solution I haven't considered.
Arrays aren't efficient for lots of insertions or deletions into anywhere other than the end of the array. Consider whether using a different data structure (such as one suggested in one of the other answers) may be more efficient. Without knowing the problem you're trying to solve, it's near-impossible to suggest a data structure (there's no one solution for all problems). That being said...
The second option is definitely the better option of the two. A somewhat better option (avoiding the default-value issue): simply copy 789 to the end and overwrite the middle 789 with 456. So the only intermediate step would be 0123789789.
Your default-value concern is, however, (generally) not a big issue:
In Java, for one, you cannot (to my knowledge) even assign memory for an array that's not 0- or null-filled. C++ STL containers also enforce this I believe (but not C++ itself).
The size of a pointer compared to any moderate-sized class is minimal (thus assigning it to a default value also takes minimal time) (in Java and C# everything is pointers, in C++ you can use pointers (something like boost::shared_ptr or a pointer-vector is preferred above straight pointers) (N/A to primitives, which are small to start, so generally not really a big issue either).
I'd also suggest forcing a reallocation to a specified size before you start inserting to the end of the array (Java's ArrayList::ensureCapacity or C++'s vector::reserve). In case you didn't know - varying-length-array implementations tend to have an internal array that's bigger than what size() returns or what's accessible (in order to prevent constant reallocation of memory as you insert or delete values).
Also note that there are more efficient methods to copy parts of an array than doing it manually with for loops (e.g. Java's System.arraycopy).
You might want to consider changing your representation of the list from using a dynamic array to using some other structure. Here are two options that allow you to implement these operations efficiently:
An order statistic tree is a modified type of binary tree that supports insertions and selections anywhere in O(log n) time, as well as lookups in O(log n) time. This will increase your memory usage quite a bit because of the overhead for the pointers and extra bookkeeping, but should dramatically speed up insertions. However, it will slow down lookups a bit.
If you always know the insertion point in advance, you could consider switching to a linked list instead of an array, and just keep a pointer to the linked list cell where insertions will occur. However, this slows down random access to O(n), which could possibly be an issue in your setup.
Alternatively, if you always know where insertions will happen, you could consider representing your array as two stacks - one stack holding the contents of the array to the left of the insert point and one holding the (reverse) of the elements to the right of the insertion point. This makes insertions fast, and if you have the right type of stack implementation could keep random access fast.
Hope this helps!
HashMaps and Linked Lists were designed for the problem you are having. Given a indexed data structure with numbered items, the difficulty of inserting items in the middle requires a renumbering of every item in the list.
You need a data structure which is optimized to make inserts a constant O(1) complexity. HashMaps were designed to make insert and delete operations lightning quick regardless of dataset size.
I can't pretend to do the HashMap subject justice by describing it. Here is a good intro: http://en.wikipedia.org/wiki/Hash_table
Can somebody explain the main differences between (advantages / disadvantages) the two implementations?
For a library, what implementation is recommended?
Wikipedia's article on hash tables gives a distinctly better explanation and overview of different hash table schemes that people have used than I'm able to off the top of my head. In fact you're probably better off reading that article than asking the question here. :)
That said...
A chained hash table indexes into an array of pointers to the heads of linked lists. Each linked list cell has the key for which it was allocated and the value which was inserted for that key. When you want to look up a particular element from its key, the key's hash is used to work out which linked list to follow, and then that particular list is traversed to find the element that you're after. If more than one key in the hash table has the same hash, then you'll have linked lists with more than one element.
The downside of chained hashing is having to follow pointers in order to search linked lists. The upside is that chained hash tables only get linearly slower as the load factor (the ratio of elements in the hash table to the length of the bucket array) increases, even if it rises above 1.
An open-addressing hash table indexes into an array of pointers to pairs of (key, value). You use the key's hash value to work out which slot in the array to look at first. If more than one key in the hash table has the same hash, then you use some scheme to decide on another slot to look in instead. For example, linear probing is where you look at the next slot after the one chosen, and then the next slot after that, and so on until you either find a slot that matches the key you're looking for, or you hit an empty slot (in which case the key must not be there).
Open-addressing is usually faster than chained hashing when the load factor is low because you don't have to follow pointers between list nodes. It gets very, very slow if the load factor approaches 1, because you end up usually having to search through many of the slots in the bucket array before you find either the key that you were looking for or an empty slot. Also, you can never have more elements in the hash table than there are entries in the bucket array.
To deal with the fact that all hash tables at least get slower (and in some cases actually break completely) when their load factor approaches 1, practical hash table implementations make the bucket array larger (by allocating a new bucket array, and copying elements from the old one into the new one, then freeing the old one) when the load factor gets above a certain value (typically about 0.7).
There are lots of variations on all of the above. Again, please see the wikipedia article, it really is quite good.
For a library that is meant to be used by other people, I would strongly recommend experimenting. Since they're generally quite performance-crucial, you're usually best off using somebody else's implementation of a hash table which has already been carefully tuned. There are lots of open-source BSD, LGPL and GPL licensed hash table implementations.
If you're working with GTK, for example, then you'll find that there's a good hash table in GLib.
My understanding (in simple terms) is that both the methods has pros and cons, though most of the libraries use Chaining strategy.
Chaining Method:
Here the hash tables array maps to a linked list of items. This is efficient if the number of collision is fairly small. The worst case scenario is O(n) where n is the number of elements in the table.
Open Addressing with Linear Probe:
Here when the collision occurs, move on to the next index until we find an open spot. So, if the number of collision is low, this is very fast and space efficient. The limitation here is the total number of entries in the table is limited by the size of the array. This is not the case with chaining.
There is another approach which is Chaining with binary search trees. In this approach, when the collision occurs, they are stored in binary search tree instead of linked list. Hence, the worst case scenario here would be O(log n). In practice, this approach is best suited when there is a extremely nonuniform distribution.
Since excellent explanation is given, I'd just add visualizations taken from CLRS for further illustration:
Open Addressing:
Chaining:
Open addressing vs. separate chaining
Linear probing, double and random hashing are appropriate if the keys are kept as entries in the hashtable itself...
doing that is called "open addressing"
it is also called "closed hashing"
Another idea: Entries in the hashtable are just pointers to the head of a linked list (“chain”); elements of the linked list contain the keys...
this is called "separate chaining"
it is also called "open hashing"
Collision resolution becomes easy with separate chaining: just insert a key in its linked list if it is not already there
(It is possible to use fancier data structures than linked lists for this; but linked lists work very well in the average case, as we will see)
Let’s look at analyzing time costs of these strategies
Source: http://cseweb.ucsd.edu/~kube/cls/100/Lectures/lec16/lec16-25.html
If the number of items that will be inserted in a hash table isn’t known when the table is created, chained hash table is preferable to open addressing.
Increasing the load factor(number of items/table size) causes major performance penalties in open addressed hash tables, but performance degrades only linearly in chained hash tables.
If you are dealing with low memory and want to reduce memory usage, go for open addressing. If you are not worried about memory and want speed, go for chained hash tables.
When in doubt, use chained hash tables. Adding more data than you anticipated won’t cause performance to slow to a crawl.
Is it possible using a certain hash function and method (the division method, or double hashing) to make a chained hash table that can be resized without having to reinsert (rehash) each element already in the table?
You would still need to reinsert, but some way to make that cheaper would be to store the hash value before the modulus was applied. That way, you can save a large part of the calculation cost of rehashing.
With this approach, it would be possible to shrink the table in size as well.
I can only assume the reason you want to avoid rehashing everything is that the resulting high latency operation is not an issue to throughput but is instead a problem for responsiveness (either human or in SLA sense)
In theory you could use a modified closed addressing hash table like so:
remember all previous sizes where elements were added
On resize keep the old buckets around linked to internally via a map of sizeWhenUsed -> buckets (obviously if the buckets are empty no need to bother)
Invariant a mapping of Key k exists in only one of the 'internal hash tables' at any time.
on addition of a value you must first look it up in all the other maps to determine if the entry already exists and is mapped. If it is remove it from the old one and add it to the new one.
if an internal map becomes empty/below a certain size it should be deleted and remaining elements moved into the current hash table.
so long as the number of internal hashes is kept constant this will not impact the big O behaviour of the data structure in time, though it will in memory.
This will however affect the actual performance as X additional checks must be made where X is the number of old hashes maintained.
If the wasted space of the list of buckets (the buckets themselves will be null if empty so are zero cost unless populated) becomes significant (use a fudge factor for this) then at some point on a rehash you may have to take the hit of moving things into the current table unless you are willing to expend essentially unlimited memory.
Downgrades in size of the hash will only function in the desired manner (releasing memory) if you are willing to rehash. This is unavoidable.
It is possible you could make use of some complex additional data within an open addressing scheme to 'flag' which of the internal hashes the cell was in use by but removals would be extremely complex to get right and would be very expensive unless you just left them as wasted space. I would never attempt this.
I would not suggest using the former method either unless the underlying data spent very little time in the hash, thus the related churn would tend to steadily 'erase' the older sized hashes. It is likely that a hash tuned for just this sort of behaviour and preset with an appropriate size would perform much better though.
Since the above scheme is simply trading wasted memory and throughput for reduction in the expensive operations with speculative (at best) chance of reducing this waste I would suggest simply pre-sizing your hash to be larger than required and thus never resized would be a more sensible option.
Probably not - the hash would have to not use any variety of modulus, which would mean that it would have a required table size depending on the data anyway.
All hash tables must deal with collisions, either through chaining or probing or whatever, so, I suspect that if upon table resize you simply resized the table (IE, you don't re-insert everything), you would have a functional, though highly non optimal, hash table.
I assume you're asking this question because you want to avoid the high cost of resizing a hash table. You want a hash table which has guaranteed constant time (assuming no collision problems, of course). This can be done.
The trick is to iteratively initialize the next-size hash table while the current one is filling up. By the time you need it, it's ready.
Quick pseudo-code to add an element:
if resizing then
smallTable = bigTable
bigTable = new T[smallTable.length * 2] //if allocation zeroes memory, we lose O(1)
set state to zeroing
elseif zeroing then
zero a small amount of the bigTable memory
if done zeroing then set state to transfering
elseif transfering then
transfer a few values in the small table to the big table
if done transfering then set state to resizing
end if
add new item to small array
add new item to large array
What is the best way to remove an entry from a hashtable that uses linear probing? One way to do this would be to use a flag to indicate deleted elements? Are there any ways better than this?
An easy technique is to:
Find and remove the desired element
Go to the next bucket
If the bucket is empty, quit
If the bucket is full, delete the element in that bucket and re-add it to the hash table using the normal means. The item must be removed before re-adding, because it is likely that the item could be added back into its original spot.
Repeat step 2.
This technique keeps your table tidy at the expense of slightly slower deletions.
It depends on how you handle overflow and whether (1) the item being removed is in an overflow slot or not, and (2) if there are overflow items beyond the item being removed, whether they have the hash key of the item being removed or possibly some other hash key. [Overlooking that double condition is a common source of bugs in deletion implementations.]
If collisions overflow into a linked list, it is pretty easy. You're either popping up the list (which may have gone empty) or deleting a member from the middle or end of the linked list. Those are fun and not particularly difficult. There can be other optimizations to avoid excessive memory allocations and freeings to make this even more efficient.
For linear probing, Knuth suggests that a simple approach is to have a way to mark a slot as empty, deleted, or occupied. Mark a removed occupant slot as deleted so that overflow by linear probing will skip past it, but if an insertion is needed, you can fill the first deleted slot that you passed over [The Art of Computer Programming, vol.3: Sorting and Searching, section 6.4 Hashing, p. 533 (ed.2)]. This assumes that deletions are rather rare.
Knuth gives a nice refinment as Algorithm R6.4 [pp. 533-534] that instead marks the cell as empty rather than deleted, and then finds ways to move table entries back closer to their initial-probe location by moving the hole that was just made until it ends up next to another hole.
Knuth cautions that this will move existing still-occupied slot entries and is not a good idea if pointers to the slots are being held onto outside of the hash table. [If you have garbage-collected- or other managed-references in the slots, it is all right to move the slot, since it is the reference that is being used outside of the table and it doesn't matter where the slot that references the same object is in the table.]
The Python hash table implementation (arguable very fast) uses dummy elements to mark deletions. As you grow or shrink or table (assuming you're not doing a fixed-size table), you can drop the dummies at the same time.
If you have access to a copy, have a look at the article in Beautiful Code about the implementation.
The best general solutions I can think of include:
If you're can use a non-const iterator (ala C++ STL or Java), you should be able to remove them as you encounter them. Presumably, though, you wouldn't be asking this question unless you're using a const iterator or an enumerator which would be invalidated if the underlying collection is modified.
As you said, you could mark a deleted flag within the contained object. This doesn't release any memory or reduce collisions on the key, though, so it's not the best solution. Also requires the addition of a property on the class that probably doesn't really belong there. If this bothers you as much as it would me, or if you simply can't add a flag to the stored object (perhaps you don't control the class), you could store these flags in a separate hash table. This requires the most long-term memory use.
Push the keys of the to-be-removed items into a vector or array list while traversing the hash table. After releasing the enumerator, loop through this secondary list and remove the keys from the hash table. If you have a lot of items to remove and/or the keys are large (which they shouldn't be), this may not be the best solution.
If you're going to end up removing more items from the hash table than you're leaving in there, it may be better to create a new hash table, and as you traverse your original one, add to the new hash table only the items you're going to keep. Then replace your reference(s) to the old hash table with the new one. This saves a secondary list iteration, but it's probably only efficient if the new hash table will have significantly fewer items than the original one, and it definitely only works if you can change all the references to the original hash table, of course.
If your hash table gives you access to its collection of keys, you may be able to iterate through those and remove items from the hash table in one pass.
If your hash table or some helper in your library provides you with predicate-based collection modifiers, you may have a Remove() function to which you can pass a lambda expression or function pointer to identify the items to remove.
A common technique when time is a factor is to have a second table of deleted items, and clean up the main table when you have time. Commonly used in search engines.
How about enhancing the hash table to contain pointers like a linked list?
When you insert, if the bucket is full, create a pointer from this bucket to the bucket where the new field in stored.
While deleting something from the hashtable, the solution will be equivalent to how you write a function to delete a node from linkedlist.