I would like to create a data structure or collection which will have O(1) complexity in adding, removing and calculating no. of elements. How am I supposed to start?
I have thought of a solution: I will use a Hashtable and for each key / value pair inserted, I will have only one hash code, that is: my hash code algorithm will generate a unique hash value every time, so the index at which the value is stored will be unique (i.e. no collisions).
Will that give me O(1) complexity?
Yes that will work, but as you mentioned your hashing function needs to be 100% unique. Any duplicates will result in you having to use some sort of conflict resolution. I would recommend linear chaining.
edit: Hashmap.size() allows for O(1) access
edit 2: Respopnse to the confusion Larry has caused =P
Yes, Hashing is O(k) where k is the keylength. Everyone can agree on that. However, if you do not have a perfect hash, you simply cannot get O(1) time. Your claim was that you do not need uniqueness to acheive O(1) deletion of a specific element. I guarantee you that is wrong.
Consider a worst case scenario: every element hashes to the same thing. You end up with a single linked list which as everyone knows does not have O(1) deletion. I would hope, as you mentioned, nobody is dumb enough to make a hash like this.
Point is, uniqueness of the hash is a prerequisite for O(1) runtime.
Even then, though, it is technically not O(1) Big O efficiency. Only using amortized analysis you will acheive constant time efficiency in the worst case. As noted on wikipedia's article on amortized analysis
The basic idea is that a worst case operation can alter the state in such a way that the worst case cannot occur again for a long time, thus "amortizing" its cost.
That is referring to the idea that resizing your hashtable (altering the state of your data structure) at certain load factors can ensure a smaller chance of collisions etc.
I hope this clears everything up.
Adding, Removing and Size (provided it is tracked separately, using a simple counter) can be provided by a linked list. Unless you need to remove a specific item. You should be more specific about your requirements.
Doing a totally non-clashing hash function is quite tricky even when you know exactly the space of things being hashed, and it's impossible in general. It also depends deeply on the size of the array that you're hashing into. That is, you need to know exactly what you're doing to make that work.
But if you instead relax that a bit so that identical hash codes don't imply equality1, then you can use the existing Java HashMap framework for all the other parts. All you need to do is to plug in your own hashCode() implementation in your key class, which is something that Java has always supported. And make sure that you've got equality defined right too. At that point, you've got the various operations being not much more expensive than O(1), especially if you've got a good initial estimation for the capacity and load factor.
1 Equality must imply equal hash codes, of course.
Even if your hashcodes are unique this doesn't guarentee a collision free collection. This is because your hash map is not of an unlimited size. The hashcode has to be reduced to the number of buckets in your hash map and after this reduction you can still get collisions.
e.g. Say I have three objects A (hash: 2), B (hash: 18), C (hash: 66) All unique.
Say you put them in a HashMap of with a capacity of 16 (the default). If they were mapped to a bucket with % 16 (actually is more complex that this) after reducing the hash codes we now have A (hash: 2 % 16 = 2), B (hash: 18 % 16 = 2), C (hash: 66 % 16 = 2)
HashMap is likely to be faster than Hashtable, unless you need thread safety. (In which case I suggest you use CopncurrentHashMap)
IMHO, Hashtable has been a legacy collection for 12 years, and I would suggest you only use it if you have to.
What functionality do you need that a linked list won't give you?
Surprisingly, your idea will work, if you know all the keys you want to put in the collection in advance. The idea is to generate a special hash function which maps each key to a unique value in the range (1, n). Then our "hash table" is just a simple array (+ an integer to cache the number of elements)
Implementing this is not trivial, but it's not rocket science either. I'll leave it to Steve Hanov to explain the ins-and-outs, as he gives a much better explanation than I ever could.
It's simple. Just use a hash map. You don't need to do anything special. Hashmap itself is O(1) for insertion, deletion, calculating number of elements.
Even if the keys are not unique, the algorithm will still be O(1) as long as the Hashmap is automatically expanded in size if the collection gets too large (most implementations will do this for you automatically).
So, just use the Hash map according to the given documentation, and all will be well. Don't think up anything more complicated, it will just be a waste of time.
Avoiding collisions is really impossible with a hash .. if it was possible, then it would basically just be an array or a mapping to an array, not a hash. But it isn't necessary to avoid collisions, it will still be O(1) with collisions.
Related
I am really confused about this. Having read the textbook and done exercises I still don't get how it works, and unfortunately I can't go in person to see the professor and it's somewhat difficult to get in touch (summer online course, different time zones). I feel like it would 'click' if I just understood how to do this problem. The textbook details hash functions and runtime individually but I feel like this question is outside the scope of what we've learned. If someone could point me at anything that might help, that would be great.
1) Consider the process of inserting m keys into a hash table T[0..m − 1], where m is a prime, and we use open addressing. The hash function we use is h(k, i) = (k + i) mod m. Give an example of m keys k1, k2 ... km, such that the following sequence of operations takes Ω(n^2) time:
insert(k1), insert(k2), ..., insert(km)
I understand that insert operations are supposed to take O(1) time or, in some cases, O(n). How exactly am I supposed to come up with keys that will turn that into Ω(n^2) time? I'm hoping to understand this and I feel like I'm missing some huge hint, because the textbook chapter seems simple, makes sense to me, and doesn't help with this at all. In the question it's stated that m is a prime, is this important? I'm just so lost, and Google for once fails me.
The keyword here is hash collision:
In order for a hash function to work well, you need the values for a certain input to be well-distributed over all m possible values the entries are stored in. If the hash table has about as many entries as elements were inserted, you can expect every element to be stored at (or near) its hash value (meaning only small amounts of probing are necessary), making access, insertion and deletion a constant-time operation.
If you however find different input values for which the hash function maps to the same value every time (collisions), during insertion the probing step will have to skip over all previously added elements, taking Ω(n) time per element on average. Thus we get a runtime of Ω(n²)
The algorithm I know about for calculating the hash code of containers works by combining the hash of all elements in it recursively. How the hashes are combined is irrelevant for my question. But because the algorithm recurses, the calculation can become very expensive. O(n), where n is the total number of elements reachable.
My question is if there are any more efficient methods to do it? For example, if you have an array with 100k elements, you could calculate the hash by combining the hash of only 100 of the elements contained. That would make the calculation 1000 times faster, while still being a good hash function, wouldn't it?
The 100 elements you pick could be the 100 first or every 1000th (in the above example) or picked using some other deterministic formula.
So to answer my question, can you either tell me why my idea can't work or tell me where my idea has already been investigated. Like has any programming language implemented "sub O(n) sequence hashing" like I'm proposing?
In general, designing an appropriate hash function requires trading off computation time against quality, and this will be particularly true for very large objects.
Hashing only a fixed-size subset of a large object is a valid strategy (Lua uses this strategy for hashing large strings, for example), but it can obviously lead to problems if the hashed objects have few differences and it happens that the differences are not in the hashed subset. That opens the possibility of denial-of-service attacks (or inputs which accidentally trigger the same problem), so it is not generally a good idea if you are hashing uncontrolled inputs. (And if you're using the hash as part of a cryptographic exercise, then omitting part of the object makes falsification trivial, so in that context it's a really bad idea.)
Assuming you're using the hash as part of a database indexing strategy (that is, a hash table), remember that in the end you will need to compare the value being looked up with each potential match in the table; those comparisons are necessarily O(n) (unless you believe that almost all lookups will fail). Each false positive requires an additional comparison, so the quality-versus-computation-time tradeoff may turn out to be a false economy.
But, in the end, there is no definitive answer; you will have to decide based on the precise use case you have, including a consideration of what you are using the hash for, what the distribution of the data is (or is likely to be) and so on.
I need a data structure that stores tuples and would allow me to do a query like: given tuple (x,y,z) of integers, find the next one (an upped bound for it). By that I mean considering the natural ordering (a,b,c)<=(d,e,f) <=> a<=d and b<=e and c<=f. I have tried MSD radix sort, which splits items into buckets and sorts them (and does this recursively for all positions in the tuples). Does anybody have any other suggestion? Ideally I would like the abouve query to happen within O(log n) where n is the number of tuples.
Two options.
Use binary search on a sorted array. If you build the keys ( assuming 32bit int)' with (a<<64)|(b<<32)|c and hold them in a simple array, packed one beside the other, you can use binary search to locate the value you are searching for ( if using C, there is even a library function to do this), and the next one is simply one position along. Worst case Performance is O(logN), and if you can do http://en.wikipedia.org/wiki/Interpolation_search then you might even approach O(log log N)
Problem with binary keys is might be tricky to add new values, might need gyrations if you will exceed available memory. But it is fast, only a few random memory accesses on average.
Alternatively, you could build a hash table by generating a key with a|b|c in some form, and then have the hash data pointing to a structure that contains the next value, whatever that might be. Possibly a little harder to create in the first place as when generating the table you need to know the next value already.
Problems with hash approach are it will likely use more memory than binary search method, performance is great if you don't get hash collisions, but then starts to drop off, although there a variations around this algorithm to help in some cases. Hash approach is possibly much easier to insert new values.
I also see you had a similar question along these lines, so I guess the guts of what I am saying is combine A,b,c to produce a single long key, and use that with binary search, hash or even b-tree. If the length of the key is your problem (what language), could you treat it as a string?
If this answer is completely off base, let me know and I will see if I can delete this answer, so you questions remains unanswered rather than a useless answer.
In cases where I have a key for each element and I don't know the index of the element into an array, hashtables perform better than arrays (O(1) vs O(n)).
Why is that? I mean: I have a key, I hash it.. I have the hash.. shouldn't the algorithm compare this hash against every element's hash? I think there's some trick behind the memory disposition, isn't it?
In cases where I have a key for each element and I don't know the
index of the element into an array, hashtables perform better than
arrays (O(1) vs O(n)).
The hash table search performs O(1) in the average case. In the worst case, the hash table search performs O(n): when you have collisions and the hash function always returns the same slot. One may think "this is a remote situation," but a good analysis should consider it. In this case you should iterate through all the elements like in an array or linked lists (O(n)).
Why is that? I mean: I have a key, I hash it.. I have the hash..
shouldn't the algorithm compare this hash against every element's
hash? I think there's some trick behind the memory disposition, isn't
it?
You have a key, You hash it.. you have the hash: the index of the hash table where the element is present (if it has been located before). At this point you can access the hash table record in O(1). If the load factor is small, it's unlikely to see more than one element there. So, the first element you see should be the element you are looking for. Otherwise, if you have more than one element you must compare the elements you will find in the position with the element you are looking for. In this case you have O(1) + O(number_of_elements).
In the average case, the hash table search complexity is O(1) + O(load_factor) = O(1 + load_factor).
Remember, load_factor = n in the worst case. So, the search complexity is O(n) in the worst case.
I don't know what you mean with "trick behind the memory disposition". Under some points of view, the hash table (with its structure and collisions resolution by chaining) can be considered a "smart trick".
Of course, the hash table analysis results can be proven by math.
With arrays: if you know the value, you have to search on average half the values (unless sorted) to find its location.
With hashes: the location is generated based on the value. So, given that value again, you can calculate the same hash you calculated when inserting. Sometimes, more than 1 value results in the same hash, so in practice each "location" is itself an array (or linked list) of all the values that hash to that location. In this case, only this much smaller (unless it's a bad hash) array needs to be searched.
Hash tables are a bit more complex. They put elements in different buckets based on their hash % some value. In an ideal situation, each bucket holds very few items and there aren't many empty buckets.
Once you know the key, you compute the hash. Based on the hash, you know which bucket to look for. And as stated above, the number of items in each bucket should be relatively small.
Hash tables are doing a lot of magic internally to make sure buckets are as small as possible while not consuming too much memory for empty buckets. Also, much depends on the quality of the key -> hash function.
Wikipedia provides very comprehensive description of hash table.
A Hash Table will not have to compare every element in the Hash. It will calculate the hashcode according to the key. For example, if the key is 4, then hashcode may be - 4*x*y. Now the pointer knows exactly which element to pick.
Whereas if it has been an array, it will have to traverse through the whole array to search for this element.
Why is [it] that [hashtables perform lookups by key better than arrays (O(1) vs O(n))]? I mean: I have a key, I hash it.. I have the hash.. shouldn't the algorithm compare this hash against every element's hash? I think there's some trick behind the memory disposition, isn't it?
Once you have the hash, it lets you calculate an "ideal" or expected location in the array of buckets: commonly:
ideal bucket = hash % num_buckets
The problem is then that another value may have already hashed to that bucket, in which case the hash table implementation has two main choice:
1) try another bucket
2) let several distinct values "belong" to one bucket, perhaps by making the bucket hold a pointer into a linked list of values
For implementation 1, known as open addressing or closed hashing, you jump around other buckets: if you find your value, great; if you find a never-used bucket, then you can store your value in there if inserting, or you know you'll never find your value when searching. There's a potential for the searching to be even worse than O(n) if the way you traverse alternative buckets ends up searching the same bucket multiple times; for example, if you use quadratic probing you try the ideal bucket index +1, then +4, then +9, then +16 and so on - but you must avoid out-of-bounds bucket access using e.g. % num_buckets, so if there are say 12 buckets then ideal+4 and ideal+16 search the same bucket. It can be expensive to track which buckets have been searched, so it can be hard to know when to give up too: the implementation can be optimistic and assume it will always find either the value or an unused bucket (risking spinning forever), it can have a counter and after a threshold of tries either give up or start a linear bucket-by-bucket search.
For implementation 2, known as closed addressing or separate chaining, you have to search inside the container/data-structure of values that all hashed to the ideal bucket. How efficient this is depends on the type of container used. It's generally expected that the number of elements colliding at one bucket will be small, which is true of a good hash function with non-adversarial inputs, and typically true enough of even a mediocre hash function especially with a prime number of buckets. So, a linked list or contiguous array is often used, despite the O(n) search properties: linked lists are simple to implement and operate on, and arrays pack the data together for better memory cache locality and access speed. The worst possible case though is that every value in your table hashed to the same bucket, and the container at that bucket now holds all the values: your entire hash table is then only as efficient as the bucket's container. Some Java hash table implementations have started using binary trees if the number of elements hashing to the same buckets passes a threshold, to make sure complexity is never worse than O(log2n).
Python hashes are an example of 1 = open addressing = closed hashing. C++ std::unordered_set is an example of closed addressing = separate chaining.
The purpose of hashing is to produce an index into the underlying array, which enables you to jump straight to the element in question. This is usually accomplished by dividing the hash by the size of the array and taking the remainder index = hash%capacity.
The type/size of the hash is typically that of the smallest integer large enough to index all of RAM. On a 32 bit system this is a 32 bit integer. On a 64 bit system this is a 64 bit integer. In C++ this corresponds to unsigned int and unsigned long long respectively. To be pedantic C++ technically specifies minimum sizes for its primitives i.e. at least 32 bits and at least 64 bits, but that's beside the point. For the sake of making code portable C++ also provides a size_t primative which corresponds to the appropriate unsigned integer. You'll see that type a lot in for loops which index into arrays, in well written code. In the case of a language like Python the integer primitive grows to whatever size it needs to be. This is typically implemented in the standard libraries of other languages under the name "Big Integer". To deal with this the Python programming language simply truncates whatever value you return from the __hash__() method down to the appropriate size.
On this score I think it's worth giving a word to the wise. The result of arithmetic is the same regardless of whether you compute the remainder at the end or at each step along the way. Truncation is equivalent to computing the remainder modulo 2^n where n is the number of bits you leave intact. Now you might think that computing the remainder at each step would be foolish due to the fact that you're incurring an extra computation at every step along the way. However this is not the case for two reasons. First, computationally speaking, truncation is extraordinarily cheap, far cheaper than generalized division. Second, and this is the real reason as the first is insufficient, and the claim would generally hold even in its absence, taking the remainder at each step keeps the number (relatively) small. So instead of something like product = 31*product + hash(array[index]), you'll want something like product = hash(31*product + hash(array[index])). The primary purpose of the inner hash() call is to take something which might not be a number and turn it into one, where as the primary purpose of the outer hash() call is to take a potentially oversized number and truncate it. Lastly I'll note that in languages like C++ where integer primitives have a fixed size this truncation step is automatically performed after every operation.
Now for the elephant in the room. You've probably realized that hash codes being generally speaking smaller than the objects they correspond to, not to mention that the indices derived from them are again generally speaking even smaller still, it's entirely possible for two objects to hash to the same index. This is called a hash collision. Data structures backed by a hash table like Python's set or dict or C++'s std::unordered_set or std::unordered_map primarily handle this in one of two ways. The first is called separate chaining, and the second is called open addressing. In separate chaining the array functioning as the hash table is itself an array of lists (or in some cases where the developer feels like getting fancy, some other data structure like a binary search tree), and every time an element hashes to a given index it gets added to the corresponding list. In open addressing if an element hashes to an index which is already occupied the data structure probes over to the next index (or in some cases where the developer feels like getting fancy, an index defined by some other function as is the case in quadratic probing) and so on until it finds an empty slot, of course wrapping around when it reaches the end of the array.
Next a word about load factor. There is of course an inherent space/time trade off when it comes to increasing or decreasing the load factor. The higher the load factor the less wasted space the table consumes; however this comes at the expense of increasing the likelihood of performance degrading collisions. Generally speaking hash tables implemented with separate chaining are less sensitive to load factor than those implemented with open addressing. This is due to the phenomenon known as clustering where by clusters in an open addressed hash table tend to become larger and larger in a positive feed back loop as a result of the fact that the larger they become the more likely they are to contain the preferred index of a newly added element. This is actually the reason why the afore mentioned quadratic probing scheme, which progressively increases the jump distance, is often preferred. In the extreme case of load factors greater than 1, open addressing can't work at all as the number of elements exceeds the available space. That being said load factors greater than 1 are exceedingly rare in general. At time of writing Python's set and dict classes employ a max load factor of 2/3 where as Java's java.util.HashSet and java.util.HashMap use 3/4 with C++'s std::unordered_set and std::unordered_map taking the cake with a max load factor of 1. Unsurprisingly Python's hash table backed data structures handle collisions with open addressing where as their Java and C++ counterparts do it with separate chaining.
Last a comment about table size. When the max load factor is exceeded, the size of the hash table must of course be grown. Due to the fact that this requires that every element there in be reindexed, it's highly inefficient to grow the table by a fixed amount. To do so would incur order size operations every time a new element is added. The standard fix for this problem is the same as that employed by most dynamic array implementations. At every point where we need to grow the table we simply increase its size by its current size. This unsurprisingly is known as table doubling.
I think you answered your own question there. "shouldn't the algorithm compare this hash against every element's hash". That's kind of what it does when it doesn't know the index location of what you're searching for. It compares each element to find the one you're looking for:
E.g. Let's say you're looking for an item called "Car" inside an array of strings. You need to go through every item and check item.Hash() == "Car".Hash() to find out that that is the item you're looking for. Obviously it doesn't use the hash when searching always, but the example stands. Then you have a hash table. What a hash table does is it creates a sparse array, or sometimes array of buckets as the guy above mentioned. Then it uses the "Car".Hash() to deduce where in the sparse array your "Car" item is actually. This means that it doesn't have to search through the entire array to find your item.
So if I have to choose between a hash table or a prefix tree what are the discriminating factors that would lead me to choose one over the other. From my own naive point of view it seems as though using a trie has some extra overhead since it isn't stored as an array but that in terms of run time (assuming the longest key is the longest english word) it can be essentially O(1) (in relation to the upper bound). Maybe the longest english word is 50 characters?
Hash tables are instant look up once you get the index. Hashing the key to get the index however seems like it could easily take near 50 steps.
Can someone provide me a more experienced perspective on this? Thanks!
Advantages of tries:
The basics:
Predictable O(k) lookup time where k is the size of the key
Lookup can take less than k time if it's not there
Supports ordered traversal
No need for a hash function
Deletion is straightforward
New operations:
You can quickly look up prefixes of keys, enumerate all entries with a given prefix, etc.
Advantages of linked structure:
If there are many common prefixes, the space they require is shared.
Immutable tries can share structure. Instead of updating a trie in place, you can build a new one that's different only along one branch, elsewhere pointing into the old trie. This can be useful for concurrency, multiple simultaneous versions of a table, etc.
An immutable trie is compressible. That is, it can share structure on the suffixes as well, by hash-consing.
Advantages of hashtables:
Everyone knows hashtables, right? Your system will already have a nice well-optimized implementation, faster than tries for most purposes.
Your keys need not have any special structure.
More space-efficient than the obvious linked trie structure (see comments below)
It all depends on what problem you're trying to solve. If all you need to do is insertions and lookups, go with a hash table. If you need to solve more complex problems such as prefix-related queries, then a trie might be the better solution.
Everyone knows hash table and its uses but it is not exactly constant look up time , it depends on how big the hash table is , the computational complexity of the hash function.
Creating huge hash tables for efficient lookup is not an elegant solution in most of the industrial scenarios where even small latency/scalability matters (e.g.: high frequency trading). You have to care about the data structures to be optimized for space it takes up in memory too to reduce cache miss.
A very good example where trie better suits the requirements is messaging middleware . You have a million subscribers and publishers of messages to various categories (in JMS terms - Topics or exchanges) , in such cases if you want to filter out messages based on topics (which are actually strings) , you definitely do not want create hash table for the million subscriptions with million topics . A better approach is store the topics in trie , so when filtering is done based on topic match , its complexity is independent of number of topics/subscriptions/publishers (only depends on the length of string). I like it because you can be creative with this data structure to optimize space requirements and hence have lower cache miss.
Use a tree:
If you need auto complete feature
Find all words beginning with 'a' or 'axe' so on.
A suffix tree is a special form of a tree. Suffix trees have a whole list of advantages that hash cannot cover.
Insertion and lookup on a trie is linear with the lengh of the input string O(s).
A hash will give you a O(1) for lookup ans insertion, but first you have to calculate the hash based on the input string which again is O(s).
Conclussion, the asymptotic time complexity is linear in both cases.
The trie has some more overhead from data perspective, but you can choose a compressed trie which will put you again, more or less on a tie with the hash table.
To break the tie ask yourself this question: Do i need to lookup for full words only? Or do I need to return all words matching a prefix? (As in a predictive text input system ). For the first case, go for a hash. It is simpler and cleaner code. Easier to test and maintain. For a more ellaborated use case where prefixes or sufixes matter, go for a trie.
And if you do it just for fun, implementing a trie would put a Sunday afternoon to a good use.
There's something I haven't seen anyone mention explicitly that I think is important to keep in mind. Both hash tables and tries of various kinds will typically have O(k) operations, where k is the length of the string in bits (or equivalently in chars).
This is assuming you have a good hash function. If you don't want "farm" and "farm animals" to hash to the same value, then the hash function will have to use all the bits of the key, and so hashing "farm animals" should take about twice as long as "farm" (unless you're in some sort of rolling hash scenario, but there are somewhat similar operation-saving scenarios with tries too). And with a vanilla trie, it's clear why inserting "farm animals" will take about twice as long as just "farm". In the long run it's true with compressed tries as well.
HashTable implementation is space efficient as compared to basic Trie implementation. But with strings, ordering is necessary in most of the practical applications. But HashTable totally disturbs the lexographical order. Now, if your application is doing operations based on lexographical order (like partial search, all strings with given prefix, all words in sorted order), you should use Tries. For only lookup, HashTable should be used (as arguably, it gives minimum lookup time).
P.S.: Other than these, Ternary Search Trees (TSTs) would be an excellent choice. Its lookup time is more than HashTable, but is time-efficient in all other operations. Also, its more space efficient than tries.
Some (usually embedded, real-time) applications require that the processing time be independent of the data. In that case, a hash table can guarantee a known execution time, while a trie varies based on the data.