The functional call returns a hash value of its argument: A hash value is a value that depends solely on its argument, returning always the same value for the same argument (for a given execution of a program).A string a e XX' is said to have a brother if there is a string ye XX' such that. It takes in an input (often a string of.Unary function object class that defines the default hash function used by the standard library. A cryptographic hash function is a one way conversion procedure that takes an arbitrary block of data and returns a fixed-size bit string, the (cryptographic) hash value, such that an accidental or intentional change to the data will change the hash value.Lecture 21: Hash functions CS 3110 Lecture 21Thats a pretty abstract description, so instead I like to imagine a hash function as a fingerprinting machine. The hash algorithms provided by Convert String are common cryptographic hash functions.The algorithm goes.Unfortunately, they are also one of the most misused. Lets have a look at how the Java String class actually computes its hash codes. We’ll be overriding the hash() method to call hash() on the relevant attributes.Writing an adequate hash function for strings. Let’s create a class Student now. Since the default Python hash() implementation works by overriding the hash() method, we can create our own hash() method for our custom objects, by overriding hash(), provided that the relevant attributes are immutable. Hash tables are one of the most useful data structures ever invented.Using hash() on a Custom Object.If we imagineWriting the bucket index as a binary number, a small change to the key shouldRandomly flip the bits in the bucket index. While hash tables are extremely effective when used well, all too often poor hash functions are usedRecall that hash tables work well when the hash function satisfies theSimple uniform hashing assumption - that the hash function should look random.If it is to look random, this means that any change to a key, even a small one,Should change the bucket index in an apparently random way. We want our hash function to use all of the information in the key. For example,If we're mapping names to phone numbers, then hashing each name to itsLength would be a very poor function, as would a hash function that used onlyThe first name, or only the last name. A lot of obvious hash function choices are bad. Clients choose poor hash functions that do not act like random numberGenerators, invalidating the simple uniform hashing assumption.Hash table abstractions do not adequately specify what is required of theHash function, or make it difficult to provide a good hash function.Clearly, a bad hash function can destroy our attempts at a constantRunning time.With anyHash function, it is possible to generate data that cause it to behave poorly,But a good hash function will make this unlikely.To determine whether your hash function is working well is to measureClustering. It's a good idea to test yourFunction to make sure it does not exhibit clustering with the data. This is also the usual implementation-side choice.But memory addresses are typically equal to zero modulo 16, so at most1/16 of the buckets will be used, and the performance of the hash table willBe 16 times slower than one might expect.When the distribution of keys into buckets is not random, we say that the hashTable exhibits clustering. AndSuppose that our implementation hash function is like the one in SML/NJ itTakes the hash code modulo the number of buckets, where the number of bucketsIs always a power of two. The actualHash function is the composition of these two functions,To see what goes wrong, suppose our hash code function on objects is theMemory address of the objects, as in Java.
This means the client can't directly tell whetherThe hash function is performing well or not. If the clustering measure is less than 1.0, the hashFunction is spreading elements out more evenly than a random hash functionUnfortunately most hash table implementations do not give the client aWay to measure clustering. If the hash function isPerfect and every element lands in its own bucket, the clustering measureWill be 0. A clustering measure of C > 1Greater than one means that the performance of the hash table is slowed down byClustering by approximately a factor of C.For example, if m= n and all elements are hashed into one bucket, theClustering measure evaluates to n. If we assume that the e j are independentVar( x i) = n Var( e j) = α - α/ m = ⟨ x i 2⟩ - ⟨ x i⟩ 2Now, if we sum up all m of the variables x i, and divide by n, as in the formula, we should effectively divide this by α:(1/ n) ⟨∑ x i 2⟩ = (1/α)⟨ x i 2⟩ = 1 - 1/ m + αThe clustering measure multiplies this by its reciprocal to get 1.Suppose instead we had a hash function that hit only one of everyC buckets, but was random among those buckets.In this case, for the non-empty buckets, we'd haveTherefore, the clustering measure evaluates in this case to c.In other words, if the clustering measure gives a value significantly greaterThan one, it is like having a hash function that doesn't hitFor a hash table to work well, we want the hash function to have twoThe hash function should give different results h(k 1) ≠No information about h(k 2). For each of the nValue is 1 if the element lands in bucket i (with probabilityThe bucket size x i is a random variable that is the sum of all these random variables:The variance of the sum of independent random variables is the sum of theirVariances. So there will beA wider range of bucket sizes than one would expect from a random hashFor those who have taken some probability theory:Consider bucket i containing x i elements. If clustering is occurring, some buckets willHave more elements than they should, and some will have fewer. Note that it'sNot necessary to compute the sum of squares of allBucket lengths picking a few at random is cheaper and usually good enough.The reason the clustering measure works is because it isBased on an estimate of the variance of theDistribution of bucket sizes. In SML/NJ hash tables, the implementationProvide only the injection property. For example, Java hash tables provide (somewhat weak)Information diffusion, allowing the client hashcode computation toJust aim for the injection property. Knowing the bits of h(k 1) does not give anyAs a hash table designer, you need to figure out which of theClient hash function and the implementation hash function is going toProvide diffusion. There are a number ofGood off-the-shelf ways to accomplish this, with a tradeoff inPerformance versus randomness (and security).Cheap if m is a power of two, but see the caveatsTherefore the client-side hash function h client( k) is defined as ( h diff ∘ h serial)( k) mod m,There are several different good ways to implement diffusion (step 2):Multiplicative hashing, modular hashing, cyclic redundancy checks,And secure hash functions such as MD5 and SHA-1. Diffusion: Map the stream of bytes into a large integer xIn a way that causes every change in the stream to affect the bitsOf x apparently randomly. If the key is a string,Then the stream of bytes would simply be the characters of the string. Two equal keys must result in the same byte stream.Two byte streams should be equal only if the keys are actually equal.How to do this depends on the form of the key. Serialization: Transform the key into a stream of bytes that contains all of the informationIn the original key. Worms armageddon nintendo 64Instead, the client is expected to implementSteps 1 and 2 to produce an integer hash code, as in Java.The implementation side then uses the hash code and the value ofM (usually not exposed to the client, unfortunately) toSome hash table implementations expect the hash code to look completely random,Because they directly use the low-order bits of the hash code as aBucket index, throwing away the information in the high-order bits.Other hash table implementations take a hash code and put it throughAn additional step of applying an integer hash function thatProvides additional diffusion.
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