6.856 — Randomized Algorithms
David Karger
Handout #5, Mar. 2, 2011 — Homework 4, Due 3/9
1. Improving the two-choice bound.
(a) In class we proved that the two-choices approach improves the maximum load
to O(log log n). A generalization is that choosing the least loaded of dchoices
reduces the maximum load to O(logdlog n). Explain what changes to the proof
are needed to derive this result. Give only the diffs; do not bother writing a
complete proof.
(b) Optional. Suppose that instead of making two choices at random, you divide
the bins into a left and right half and break all ties by putting items in the
left bin. Show that the maximum load improves by a constant factor, from
O(log log n/ log 2) to O(log log n/2 log φ) where φ= (1 + √5)/2 is the golden
ratio. Hint: for the number of height ibins on each side, use different recur-
rences βifor the left side and γifor the right side. Show that βi+1 ≤c1βiγi/n2
while γi+1 ≤c2βi+1γi/n2.
(c) Optional. Generalize to dbins, showing a load of O((log log n)/d).
2. Show that there is no perfect hash family mapping from mto nof size polynomial in
m(i.e., that can be represented in a constant number of log m-bit machine words) if
2n≤m≤2o(n).Hint: use a probabilistic existence proof to show that for any family
of that size, there exists a set that does not get perfectly hashed.
3. Cuckoo hashing is nice, but does cost a factor of two in space. Develop a related
approach that uses less space while still guaranteeing worst-case constant-time lookups.
Use the following ideas:
•Probing more than twice in a table increases the chances of finding an empty cell.
•If after some probes you fail to find an empty cell, move the failed item into an
“overflow” table that uses cukoo hashing
Use this to achieve constant-time lookups using only (1 + ) space for any constant .
Determine the best tradeoffs you can between number of probes required and amount
of space used.
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