使用单独的链计算探针的平均数量

Question

Suppose we are given the hash table below:

where the collision resolution method is Separate chaining. I am trying to calculate the average number of probes to find an empty slot. According to my lecture slides the average is 1.77, but I keep getting a different answer. Any assistance would be much appreciated.

Answer 1

So, it works like this: The idea is if you hit an empty bucket, then you are done. But if you hit a non-empty bucket, you need to traverse the chain and find a empty slot at the end of the chain.

So the resulting average probe hits is:

(8 empty slots * (1 probe/empty slot)) / 13 total slots
+
(2 slots with one element *(2 probes for slot with one element) /13 total slots
+
(2 slots with two elements *(3 probes for slot with two elements) /13 total slots
+
(1 slots with four elements *(5 probes for slot with four elements) /13 total slots

= 8/13 + 4/13 + 6/13 + 5/13

= 23/13

= 1.77

Answer 2

Not sure why you want to find out average probes for empty slot in Separate Chaining method. It would be of concern in Open Addressing method. And it would depend on nature of your data ie what is probability of getting hash-code%12 among already occupied index locations.

If you are just interested in metrics for given array:

There are 10 elements in array of size 13. So:

Minimum probability of finding empty location is 3/13 - when each 10 elements lands in different bucket

Maximum probability of finding empty location is 12/13 (if all 10 elements land in one bucket)

For given data it is 8/13