Randomized Dictionary Structures:Structural Properties of Skip Lists
Structural Properties of Skip Lists
Recall that r = 1 + maxi{Zi}. Notice that Zi ≥ k iff the coin tossed for ki gets a run of at least k successes right from the beginning and this happens with probability pk . Since r ≥ k iff at least one of Zi ≥ k − 1 , we easily get the following fact from Boole’s inequality
Choosing k = 4 log1/p n + 1, we immediately obtain a high confidence bound for the number of levels. In fact,
We obtain an estimation for the expected value of r, using the formula stated in theorem ( 13.3) as follows:
THEOREM 13.11 The expected number of levels in a skip list of n elements is O(log n). In fact, the number is O(log n) with high probability.
Recall that the space complexity, | SL | is given by
Since Zi is a geometric random variable with parameter p, Z is a negative binomial random variable by theorem 13.6.
This implies that 4n is in fact a very high confidence bound for Z. Since | SL |= Z+n+2r+2, we easily conclude that
THEOREM 13.12 The space complexity of a skip list for a set of size n is O(n) with very high probability.
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