The main idea in this is to divide your entire data set into segments as to speed up the actual searching which is usually time consuming. In our example above, each of the filing cabinet would statistically contain about records. Searching regardless the order through records is much faster than having to deal with 30, You may have noticed that some actually already do this.
But instead of devising a hashing methodology to generate a hash key, they will in most cases simply use the first letter of the last name. So if you have 26 filing cabinets each containing a letter from A to Z, you in theory have just segmented your data and enhanced the filing and retrieval process. Essentially, a hash function is just a function that takes things from one space say strings of arbitrary length and maps them to a space useful for indexing unsigned integers, say.
If you only have a small space of things to hash, you might get away with just interpreting those things as integers, and you're done e. Usually, though, you've got a much larger space. If the space of things you allow as keys is bigger than the space of things you are using to index your uint32's or whatever then you can't possibly have a unique value for each one.
When two or more things hash to the same result, you'll have to handle the redundancy in an appropriate way this is usually referred to as a collision, and how you handle it or don't will depend a bit on what you are using the hash for.
This implies you want it to be unlikely to have the same result, and you probably also would really like the hash function to be fast. In practice you usually should be able to find a function that is known to work well for your application and use that. Now to make this work as a hashtable: Imagine you didn't care about memory usage. Then you can create an array as long as your indexing set all uint32's, for example. As you add something to the table, you hash it's key and look at the array at that index.
If there is nothing there, you put your value there. If there is already something there, you add this new entry to a list of things at that address, along with enough information your original key, or something clever to find which entry actually belongs to which key.
So as you go a long, every entry in your hashtable the array is either empty, or contains one entry, or a list of entries. Retrieving is a simple as indexing into the array, and either returning the value, or walking the list of values and returning the right one. Of course in practice you typically can't do this, it wastes too much memory. So you do everything based on a sparse array where the only entries are the ones you actually use, everything else is implicitly null.
Lots of answers, but none of them are very visual , and hash tables can easily "click" when visualised. Hash tables are often implemented as arrays of linked lists. If the table size grows, hash tables implemented as above tend to resize themselves i. When we do this, we're using the hash table as an associative container aka map , and the values it stores can be deemed to consist of a key the name and one or more other fields still termed - confusingly - the value in my example, just the age.
A hash table implementation used as a map is known as a hash map. This contrasts with the example earlier in this answer where we stored discrete values like "sue", which you could think of as being its own key: that kind of usage is known as a hash set. A general purpose, worst-case collision-minimising hash function's job is to spray the keys around the hash table buckets effectively at random, while always generating the same hash value for the same key.
Even one bit changing anywhere in the key would ideally - randomly - flip about half the bits in the resultant hash value. This is normally orchestrated with maths too complicated for me to grok. I'll mention one easy-to-understand way - not the most scalable or cache friendly but inherently elegant like encryption with a one-time pad!
With this approach, it's easy to see that a bit in the binary digit sense changing anywhere in the double results in a different random number being looked up in one of the tables, and a totally uncorrelated final value.
Many libraries' hashing functions pass integers through unchanged known as a trivial or identity hash function ; it's the other extreme from the strong hashing described above. It's also great to save the time it takes to generate a strong hash, and if keys are looked up in order they'll be found in buckets nearby in memory, improving cache hits.
When the keys don't increment nicely, the hope is they'll be random enough they won't need a strong hash function to totally randomise their placement into buckets. You guys are very close to explaining this fully, but missing a couple things. The hashtable is just an array.
The array itself will contain something in each slot. At a minimum you will store the hashvalue or the value itself in this slot. You can also store a pointer or pointers to other data you want to retrieve out of this slot. It's important to note that the hashvalue itself generally does not indicate the slot into which to place the value. For example, a hashvalue might be a negative integer value. Obviously a negative number cannot point to an array location.
Additionally, hash values will tend to many times be larger numbers than the slots available. Thus another calculation needs to be performed by the hashtable itself to figure out which slot the value should go into.
This is done with a modulus math operation like:. This value is the slot the value will go into. I suppose there may be other more advanced methods for determining slot index, but this is the common one I've seen With the modulus method, if you have a table of say size , any hashvalue that is between 1 and will go into the corresponding slot. Any Negative values, and any values greater than will be potentially colliding slot values. The chances of that happening depend both on your hashing method, as well as how many total items you add to the hash table.
For some real fun, build your own hashfunction and try to get it to NEVER collide on the specific data you are hashing, run faster than GetHashCode, and have a fairly even distribution. I've done this using long instead of int size hashcode values and it's worked quite well on up to 32 million entires hashvalues in the hashtable with 0 collisions. Unfortunately I can't share the code as it belongs to my employer When you can achieve this, the hashtable is VERY fast.
Here's an example: picture the entire table as a series of buckets. Suppose you have an implementation with alpha-numeric hash-codes and have one bucket for each letter of the alphabet. This implementation puts each item whose hash code begins with a particular letter in the corresponding bucket. Let's say you have objects, but only 15 of them have hash codes that begin with the letter 'B.
As far as calculating the hash code, there is nothing magical about it. The goal is just to have different objects return different codes and for equal objects to return equal codes. You could write a class that always returns the same integer as a hash-code for all instances, but you would essentially destroy the usefulness of a hash-table, as it would just become one giant bucket.
A hash table wraps up an array, lets call it internalArray. Items are inserted into the array in this way:. Sometimes two keys will hash to the same index in the array, and you want to keep both values. I like to store both values in the same index, which is simple to code by making internalArray an array of linked lists:. Delete operations are just as simple to write. As you can tell, inserts, lookups, and removal from our array of linked lists is nearly O 1.
The maximum size of this array is typically smaller than the number of items in the set of possible values for the type of data being stored in the hashtable. The hash algorithm is used to generate an index into that array based on the values of the item that will be stored in the array.
Because the set of values that can be indexes in the array is typically smaller than the number of all possible values that the type can have, it is possible that your hash algorithm is going to generate the same value for two separate keys. A good hash algorithm will prevent this as much as possible which is why it is relegated to the type usually because it has specific information which a general hash algorithm can't possibly know , but it's impossible to prevent.
Because of this, you can have multiple keys that will generate the same hash code. When that happens, the items in the vector are iterated through, and a direct comparison is done between the key in the vector and the key that is being looked up. If it is found, great and the value associated with the key is returned, otherwise, nothing is returned.
For each thing, you make up an index for it, called a hash. The important thing about the hash is that it 'scatter' a lot; you don't want two similar things to have similar hashes. You put your things into the array at position indicated by the hash. More than one thing can wind up at a given hash, so you store the things in arrays or something else appropriate, which we generally call a bucket. When you're looking things up in the hash, you go through the same steps, figuring out the hash value, then seeing what's in the bucket at that location and checking whether it's what you're looking for.
When your hashing is working well and your array is big enough, there will only be a few things at most at any particular index in the array, so you won't have to look at very much. For bonus points, make it so that when your hash table is accessed, it moves the thing found if any to the beginning of the bucket, so next time it's the first thing checked.
All of the answers so far are good, and get at different aspects of how a hashtable works. Here is a simple example that might be helpful. Lets say we want to store some items with lower case alphabetic strings as a keys. As simon explained, the hash function is used to map from a large space to a small space. A simple, naive implementation of a hash function for our example could take the first letter of the string, and map it to an integer, so "alligator" has a hash code of 0, "bee" has a hash code of 1, "zebra" would be 25, etc.
Next we have an array of 26 buckets could be ArrayLists in Java , and we put the item in the bucket that matches the hash code of our key. If we have more than one item that has a key that begins with the same letter, they will have the same hash code, so would all go in the bucket for that hash code so a linear search would have to be made in the bucket to find a particular item. In our example, if we just had a few dozen items with keys spanning the alphabet, it would work very well.
However, if we had a million items or all the keys all started with 'a' or 'b', then our hash table would not be ideal. I assume you understand the concept of an array A. That's something that supports the operation of indexing, where you can get to the Ith element, A[I], in one step, no matter how large A is. So, for example, if you want to store information about a group of people who all happen to have different ages, a simple way would be to have an array that is large enough, and use each person's age as an index into the array.
Thay way, you could have one-step access to any person's information. But of course there could be more than one person with the same age, so what you put in the array at each entry is a list of all the people who have that age. So you can get to an individual person's information in one step plus a little bit of search in that list called a "bucket". It only slows down if there are so many people that the buckets get big. Then you need a larger array, and some other way to get more identifying information about the person, like the first few letters of their surname, instead of using age.
That's the basic idea. Instead of using age, any function of the person that produces a good spread of values can be used. That's the hash function. Like you could take every third bit of the ASCII representation of the person's name, scrambled in some order. All that matters is that you don't want too many people to hash to the same bucket, because the speed depends on the buckets remaining small. A hash table totally works on the fact that practical computation follows random access machine model i.
So, if I have a universe of keys set of all possible keys that I can use in a application, e. One way to address this situation is to use modular hashing on the binary representation of the key this is what Java does. Modular hashing works for long keys such as strings, too: we simply treat them as huge integers.
For example, the code below computes a modular hash function for a String s, where R is a small prime integer Java uses If the key type has multiple integer fields, we can typically mix them together in the way just described for String values. For example, suppose that search keys are of type USPhoneNumber.
Java helps us address the basic problem that every type of data needs a hash function by requiring that every data type must implement a method called hashCode which returns a bit integer. The implementation of hashCode for an object must be consistent with equals.
That is, if a. If the hashCode values are the same, the objects may or may not be equal, and we must use equals to decide which condition holds. Converting a hashCode to an array index. User-defined hashCode. Client code expects that hashCode disperses the keys uniformly among the possible bit result values. That is, for any object x , you can write x.
Java provides hashCode implementations that aspire to this functionality for many common types including String , Integer , Double , Date , and URL , but for your own type, you have to try to do it on your own. Program PhoneNumber. Program Transaction. We have three primary requirements in implementing a good hash function for a given data type: It should be deterministic —equal keys must produce the same hash value. It should be efficient to compute.
It should uniformly distribute the keys. To analyze our hashing algorithms and develop hypotheses about their performance, we make the following idealized assumption. Assumption J uniform hashing assumption. The hash function that we use uniformly distributes keys among the integer values between 0 and M Hashing with separate chaining. A hash function converts keys into array indices.
The second component of a hashing algorithm is collision resolution: a strategy for handling the case when two or more keys to be inserted hash to the same index. A straightforward approach to collision resolution is to build, for each of the M array indices, a linked list of the key-value pairs whose keys hash to that index. The basic idea is to choose M to be sufficiently large that the lists are sufficiently short to enable efficient search through a two-step process: hash to find the list that could contain the key, then sequentially search through that list for the key.
It maintains an array of SequentialSearchST objects and implements get and put by computing a hash function to choose which SequentialSearchST can contain the key and then using get and put from SequentialSearchST to complete either job. Proposition K. Assumes an idealistic hash function. This classical mathematical result is compelling, but it completely depends on Assumption J.
Still, in practice, the same behavior occurs. Property L. Hashing with linear probing. Using the same technique, deletion can also be implemented in constant average time. The number of records in each list must remain small, and the records must be evenly distributed over the lists.
To achieve this we just need to change the hash function , the function which selects the list where a key belongs. It takes a key a positive integer as input and produces a number in the interval In general, a hash function is a function from E to We want this function to be uniform: it should map the expected inputs as evenly as possible over its output range. The overall expected time of such algorithm -"for each element x in the list, if k - y is in the initially empty hash set H , then return yes , otherwise insert x in H "- is therefore also O n , corresponding to the O n insertions and lookups in the hash set, in the worst case.
A similar approach can be used to determine the intersection of two lists A , B. Create a hash set H and populate it using the elements of one of the lists, say A. For a trivial solution, we may simply test each quadruple of distinct elements of the set A. A more efficient solution can be achieved by sorting. Now, all pairs of elements whose images under f sum up to a same value d will constitute a contiguous sublist of L.
Therefore, they can be easily combined to produce the intended quadruples in constant time per quadruple. Now we look into a hashing-based approach. We start with an empty hash map H. If d is not a key stored in H , we insert the key, value pair d, S d in H , where S d is a collection which can be implemented as a linked list, a hash set or whatever other structure containing only the pair x , y initially. On the other hand, if d is already stored in H , then we simply add x , y to the non-empty collection S d which d already maps to in H.
The pseudocode of the proposed hash-based solution is given as Algorithm 1. Algorithm 1 Sum Functions A, f. Now, each of the q quadruples that satisfy the equality can be obtained, by combining pairs in a same list, in O 1 time.
If we tried to achieve the same performance by using an array and direct addressing, such array would have to be twice as large as the maximum element in the image of f.
The use of hashing avoids that issue seamlessly. In the standard game of bingo, each of the contestants receives a card with some integers.
Then, subsequent numbers are drawn at random from A , without replacement, until all numbers in the card of a contestant have been drawn. That contestant wins the prize. In our variation of the game, the contestants receive an empty card. As an example, the sequence. Even though the set B is huge, the distribution of the drawing probability along B is not uniform, and the game finishes in reasonable time almost surely. We want to write a program to play the permutation bingo.
Moreover, we want our program to win the game when it plays against other programs. In other words, we need an efficientalgorithm to check, after each number is added to S , whether S happens to be a winning sequence. One possible approach would be as follows. Keep S in memory all the time. We can do better by performing a single traversal of S , during which we add each subsequence of three consecutive distinct elements of S into a bin.
To make sure that two permutations go to the same bin if and only if they have the same elements in different orders , we may label each bin using the ascending sequence of its elements. Now, for each subsequence of three distinct elements in S , we compute the label of its bin, find the bin, and place the subsequence therein, making sure we do not add the same subsequence twice in a bin.
Whenever a bin has size six, we have a winner. As a matter of fact, the way we implement each bin is not important, since its size will never exceed six. Moreover, rather than building the bins from scratch by traversing the whole S each time a new element s n is added, we can keep the bins in memory and proceed to place only each freshly formed subsequence s n-2 , s n-1 , s n into the appropriate bin - an approach which also allows us to keep but the two last elements of S in memory.
We still have a big problem, though. Given a label, how to find the bin among the possibly huge number of existing bins? If we maintain, say, an array of bins, then we will have to traverse that entire array until we find the desired bin.
It is not even possible to attempt direct addressing here, for the range B of the numbers that can be drawn is not only unknown but also possibly huge - let alone the number of subsets of three such numbers! Even if we did know B , there would probably not be enough available memory for such an array. We know, however, that the game always ends in reasonable time, owing to an appropriate probability distribution along B enforced by the game dealer.
Thus, the actual number of bins that will ever be used is small, and available memory to store only that number of bins shall not be an issue. We can therefore keep the bins in a linked list, or even in an array without direct addressing. When the sequence has size n , such list or array of bins will have size O n , and the time it takes to traverse it all and find the desired bin will also be O n.
The use of hashing can improve the performance even further. We use a hash map where the keys are the bin labels and the values are the bins themselves each one a list, or bit array etc.
By doing this, the amortized cost of inserting each new bin is also constant, and the algorithm processes each drawn number in overall O 1 expected time. Programming Challenges. By using a clever backtracking approach aided by hashing, we obtain a simpler algorithm which is just as efficient, in the worst case. As a matter of fact, our algorithm may demand significantly less time and space, in practice, than the usual dynamic programming approach. There is a ferry to transport cars across a river.
The ferry comprises two lanes of cars throughout its length. Before boarding on the ferry, the cars wait on a single queue in their order of arrival.
When boarding, each car goes either to the left or to the right lane at the discretion of the ferry operator. Whenever the next car in the queue cannot be boarded on either lane, the ferry departs, and all cars still in the queue should wait for the next ferry. A brute force approach could be as follows. If it is possible to load k cars on the ferry, proceed with the next value of k.
Stop when there is no feasible distribution of cars for some k. Clearly, if the number n of cars loaded in the optimal solution is not ridiculously small, such an exponential-time algorithm is doomed to run for ages.
An algorithm for determining the k-best solutions of the one-dimensional knapsack problem. Pesquisa Operacional, 20 1 : The easiest hard problem. American Scientist, 90 2 :
0コメント