# simple uniform hashing: unclear definition of probability

I am trying to understand the assumption of Simple Uniform Hashing (SUHA) as e.g., in CLRS textbook; or other courses about hashing.

The usual description given to SUHA is (cf. CLRS):

"we shall assume that any given element [i.e., key] is equally likely to hash into any of the m slots, independently of where any other element has hashed to. We call this the assumption of simple uniform hashing."

However this is quite informal, and thus I don't fully understand it. My understanding is that the assumption supposedly means that

$Pr[h(k)=\ell]=1/m\,$, for any given key $k$ and slot in the table $\ell=0,\ldots,m-1$, where $m$ is the number of slots in the table. But since given a key $k$ the deterministic hash function $h$ is fixed on $k$, there is no probability distribution here!

Perhaps it is meant here that $k$ is a random variable distributed uniformly over all keys?

Question: what is the distribution the probability $Pr[h(k)=\ell]=1/m\,$ is defined over in SUHA?

## EDIT:

My understanding is the SUHA is an (idealized) assumption that formally means that the probability $Pr[h(k)]$ is the probability of the event of inserting a key to the table nevertheless the time of insertion (it might be the first key I insert or the last one).

In other words, the sample space of probability $\Omega$ is

$\Omega:=($the set of all [ordered] sequences of insertions to the hash table$)$.

The probability event $h(k)=\ell$ is thus the set of all [ordered] sequences of insertions to the hash table in which the $n+1$th insertion hashes to $\ell$, where $n+1$ is the current'' insertion that I'm analyzing and where $n$ is the current number of keys in my hash table.

In SUHA we shall also assume that $h(k)=\ell$ for every $n$ (the current position of insertion in the sequence); namely, we assume that the event $h(k)=\ell$ is completely independent of the sequence.

To sum up the answer: the probability $Pr[h(k)=\ell]$ is taken over all possible ordered sequences of insertions to the hash table.

• There is no hash table involved in SUHA. Hash functions are used for many things beyond hash tables and the SUHA is often relevant to those uses too. There's no reason to talk about "insertions", and the SUHA is a statement that is independent of the sample space, i.e. it is parametric in it. All it says is for the probability measure $\mu_K$ for a random variable $K:\Omega\to D$ and a hash function $h:D\to\{0,\dots,m-1\}$, the function $\mu_K\circ h^{-1}$ is the constantly $\frac{1}{m}$ function. This statement does not require knowing the specific details of $\Omega$ or $D$. Feb 10, 2018 at 12:50
• There is a hash table in the CLRS context. This is where the SUHA was used, and this is what my question asks for: the analysis in CLRS using the SUHA is unclear (I think SUHA was invented by CLRS).
– Jack
Feb 10, 2018 at 13:09
• Note that indeed, we don't need to speak about "insertions", we can speak about simply "picking/choosing" keys.
– Jack
Feb 10, 2018 at 13:28

First, the outcome of a situation being deterministic doesn't mean we will necessarily assign a probability of $1$ or $0$ to it. If I say I'm going to flip a coin, but it is actually a double-sided coin meaning both sides are heads or both sides are tails, the probability that it comes up heads is still $0.5$. Indeed, with respect to classical mechanics, the outcome of a normal coin flip is a deterministic function of the state of the universe, and even quantum mechanics would at best predict near certainty of the classical outcome. But you are correct, that if we knew $h$ and $k$ and $\ell$, then $h(k)=\ell$ would be either a logical tautology or a logical contradiction, and thus of probability $1$ or $0$ respectively.

We could consider any of $h$, $k$, or $\ell$ to be random variables, but the most natural choice would be $k$. I'll write $K$ for the random variable of the keys and reserve $k$ for some particular key. We have $$P(h(K)=\ell)=P(K\in h^{-1}(\ell)) = \mu_K(h^{-1}(\ell))$$ where $\mu_K$ is the probability measure corresponding to $K$. The simple uniform hashing assumption (SUHA) states that $$\mu_K(h^{-1}(\ell))=\frac{1}{m}$$ for every $\ell\in\{0,\dots,m-1\}$. This is simply a constraint on the probability measure $\mu_K$. There is no assumption that $K$ is "uniformly distributed" in any (other) sense. $K$ could be wildly "non-uniform" and still satisfy SUHA. Indeed, it would have to be for some hash functions, $h$.

For example, let's say $m=2$ and $h^{-1}(0)=\{k_0\}$ and thus $h^{-1}(1)=X\setminus \{k_0\}$ where $X$ is the domain of $h$. Let's further say $|X|=101$. Then for this $h$ to satisfy SUHA, it must be the case that $P(K=k_0)=\frac{1}{2}$. For $k\in X\setminus \{k_0\}$, it could be the case that there's a $k_1\in X\setminus \{k_0\}$ such that $P(K=k_1)=\frac{1}{2}$ and thus if $k\neq k_1$ then $P(K=k)=0$. Or maybe, $P(K=k)=\frac{1}{200}$ for any $k\in X\setminus \{k_0\}$. In fact, any distribution over the elements of $X\setminus \{k_0\}$ that sums to $\frac{1}{2}$ will suffice.

Typically, we would specify a distribution for $K$ and then SUHA would constrain what $h$ could be. The point is that the hash function is "well matched" to the distribution of keys whatever that is. There is no sense in stating a hash function satisfies SUHA without specifying the distribution of keys, but we can say a pair of a hash function and a distribution of keys satisfy SUHA while giving the details of neither. This is all that is needed to prove many results about hashing. We don't want to limit ourselves to only talking about "uniformly" distributed keys. A hash table will still gave the same performance guarantees for "non-uniformly" distributed keys as long as the hash function is chosen to "compensate" for the "non-uniform" distribution, i.e. to satisfy SUHA.

• Thanks. I don't fully understand the answer. Are you simply claiming that $Pr[h(k)=\ell]$ means that the probability a randomly picked element/key $k$ is hashed to $\ell$ is $1/m$ (where the probability is over the distribution of keys)? I find then the CLRS description quite inadequate.
– Jack
Feb 10, 2018 at 11:46
• I also am not sure you are correct that your condition for SUHA is $Pr[\mu_K(h^{-1}(\ell))=1/m$. This condition is not sufficient for SUHA nor the analysis of SUHA, assuming you mean that $K$ is the random variable of the key where the key is chosen randomly from all possible keys. Indeed, all hash functions from universe $U$ to $0,...,m-1$, defined by using $\mod m$ would then satisfy SUHA by your assumption; but this is not what is meant by SUHA. The question again boils down to what is the sample space and what are the events.
– Jack
Feb 10, 2018 at 12:13
• @Jack You are definitely reading way more into the SUHA than is there. The only inadequacy with the CLRS description (that may well be handled in the omitted context) is that it should explicitly state that $k$ is the relevant random variable. For your second comment, they would only satisfy SUHA with respect to a given key distribution. $\pmod m$ does satisfy the SUHA if $U=\{n\mid n < km\}$ for some natural $k>0$ and a uniform key distribution. If $|U|\neq km$ for some natural $k>0$ then it won't because $|h^{-1}(\ell)|$ will not be the same for all $\ell$. Feb 10, 2018 at 12:51
• Generally speaking, given a finite universe $U$ and a uniform key distribution over it, a hash function $h$ satisfies SUHA with respect to that key distribution if and only if $|h^{-1}(y)|$ is the same for all $y$. Thus generally (under these assumptions) we must have $|U|=km$ for the SUHA to have any chance of holding. The modulus function is a great hash function if your keys are uniformly distributed. A major purpose of a hash function is to "adapt" not-necessarily-uniformly distributed keys to a new uniformly distributed set of values. The SUHA is exactly this statement. Feb 10, 2018 at 13:07
• Indeed, for your first comment: $\mod m$ satisfies SUHA only if we assume further that the keys are given to me (that is, for the insertion; what you call equivalently, "key distribution") uniformly. Formally (at least in the context of insertions to a dictionary as in CLRS), this means that the sample space is the space of all ordered sequences of keys. That is what I said.
– Jack
Feb 10, 2018 at 13:21

The way I think about Simple Uniform Hashing is like this:

Since CLRS assumes the universe of keys $$U$$ fixed, it is also natural to assume that the probability distribution of keys is also fixed by the problem at hand. In fact, this is what CLRS say on page 274 (3rd edition) in the context of open-addressing:

Of course, a given key has a unique fixed probe sequence associated with it; what we mean here is that, considering the probability distribution on the space of keys and the operation of the hash function on the keys, each possible probe sequence is equally likely.

So let $$K: S \to U$$ be the random variable of choosing a key with its own probability distribution (which is not assumed uniform(!): some keys might be chosen much more often than the other ones).

Then if one composes $$K$$ with the hash function (which is a deterministic object) $$h: U \to \{0,1,...,m-1\},$$ one gets a new random variable$$h(K): S \to \{0,1,...,m-1\}$$ that would have its own probability distribution showing how often certain hash-values would be used in our experiment of randomly choosing keys.

Then the hash function $$h$$ satisfies SUHA when $$h(K)$$ has $$Unif \ \{0,1,...,m-1\}$$ distribution.

For example, if $$U = \{k_1, k_2,k_3,k_4 \}$$ consists of only four keys with $$P(K=k_1) = P(K=k_2) = 1/3, \ P(K=k_3) = P(K=k_4) = 1/6,$$ the appropriate hashing function with $$m=2$$ slots would e.g. be $$h(k_1)=h(k_3)=0, \quad h(k_2)=h(k_4)=1,$$ because then $$P(h(K)=0)=1/2, \quad P(h(K)=1)=1/2.$$

Certainly, for some configurations of $$U, K$$ and $$m$$ there might be no hash function $$h$$ satisfying SUHA. Take e.g. $$U=\{k_1, k_2, k_3 \}$$ with $$P(K=k_1)=2/3, P(K=k_2)=P(K=k_3)=1/6,$$ and $$m = 2$$. There is no way to evenly spread out $$k_1,k_2,k_3$$ in just two slots - if you set $$h(k_1)=j$$, then $$P(h(K)=j) \geq 2/3$$no matter what $$h$$ was.