I know that a regular hash table usually has O(1) lookup time, but of course this sometimes degrades due to collisions.

But the original paper on Linear Hashing says that Linear Hashing usually gets O(1) lookup time:

"A new bucket has therefore been appended to the last bucket of the file to which all the records have been moved in one access. Since for all these records the bucket 100 is henceforward the primary bucket, they are all accessible in one access."

This seems to be saying that all the records which reside in buckets which were appended due to splits are accessible in O(1) time. But this doesn't seem to be the case. In order to access data in an appended bucket, first you need to look in the base region, and only then, if the key is not found, perform the second hash function and look in the extended region. But how is this "one access", as the above quote says? Shouldn't all records in the extended region require O(2) access time?

>> But how is this "one access", as the above quote says?

Although you might have to use several hashing functions to calculate the bucket address, you only have to access one bucket. Because when you've found the address, you know the value you're looking for will be at that address.

So, you start with h0. If you need to, you use h1, h2 etc. untill you find the bucket address you need. Then you access the bucket and get the value out of it.

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chsalviaAuthor Commented:

But wouldn't you still need to access the address in the base region to compare your key to see if it's there?

Say you have a table of size 10, and you append 2 buckets after 2 splits occur. So now the split address is 2, and so buckets 0 and 1 are addressed with h1, while buckets 2 through 9 are addressed with h0.

Now say you lookup a key. So you do h0(key). And suppose this gives you bucket 0, which is below the split address. Therefore, the key you're looking for could be in two places. It might be in bucket 0, or it might be in bucket 10. If it's not in bucket 0, you need call h1(key) to get to bucket 10. Isn't this two accesses? You have to access bucket 0 to check for the key, and then access bucket 10 since you don't find it in bucket 0.

>> If it's not in bucket 0, you need call h1(key) to get to bucket 10. Isn't this two accesses?

You don't need to check bucket 0 if you don't want to ... you can immediately apply h1 to know for sure which bucket it will be in. All depends on which is faster, and which is the more plausible :)

btw O(2) doesn't "exist" - it's the same as O(1).

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Okay. Just to clarify, you mentioned that it could take several hash functions to calculate an address. But wouldn't it only ever take 2 hash functions at max?

I thought it worked as follows: You start with a base table of N buckets, and your base hash functions is h0. All appended buckets are addressed with h1.

After N splits, the table size is N * 2 and now you go up another "level". So now your "base" hash function is h1, and all appended buckets are addressed with h2. Then after another N * 2 splits, your table size is N * 4, and your base hash function becomes h2, and all appended buckets are addressed with h3, and so on...

So wouldn't you only need two hash functions max to calculate any address, no matter how big the table got?

That's just a requirement for the hx function. It does not actually involve recursively calling the other hashing functions. In other words :

h0 will generate 0 ... N-1
h1 will generate 0 ... 2N-1
h2 will generate 0 ... 4N-1
h3 will generate 0 ... 8N-1
etc.

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Although you might have to use several hashing functions to calculate the bucket address, you only have to access one bucket. Because when you've found the address, you know the value you're looking for will be at that address.

So, you start with h0. If you need to, you use h1, h2 etc. untill you find the bucket address you need. Then you access the bucket and get the value out of it.