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Randomized Algorithms
Introduction
• Algorithm uses a random number to make
at least one decision
• Running time depends on input and
random numbers generated
• Worst-case running time often same as
nonrandom algorithm, but no inputs are
bad (only random numbers)
Example
• Quicksort using randomly generated pivot
– running time will vary on same input
– choosing first element as pivot will perform
poorly for almost-sorted input
– almost-sorted input is often common
Generating Random Numbers
• Actually pseudorandom
• Need to be able to generate a sequence of
random numbers
• xi+1 = AximodM
• x0 – seed
• M needs to be large to make period large –
sequence will repeat
Skip Lists
• Linked list that supports insertion and sorting in
O(logN) expected time
• Basic idea – each node may link to several other
nodes ahead of it in the list
• Running time of search if every other node has
an additional link to the node two ahead of it in
the list?
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Skip Lists
• Running time if every 2ith node has a link
to node 2i ahead of it in the list
– essentially, a balanced tree
• Problems?
– how would we insert and delete?
Skip Lists
• A level k node has k links to next node
with at least i levels
• Randomly choose level for a new node
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Skip Lists
• Find algorithm http://www.sable.mcgill.ca/~dbelan2/cs251/skip_lists.html
1) Start at the highest level of the list.
2) Move forward following the pointers at the same level until the
next key is greater than the searched key
3) If the current level is not the lowest, go down one level and
repeat the search at that level from the current node
4) Stop when the level is 1 and the next key is greater than the
searched key
5) If the current key is the searched key return the value of that
node. Otherwise, return a failure.
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Skip Lists
• Insert algorithm
– similar to find algorithm
– keep track of points where you switch to lower
level
– set new node pointers
– update pointers to new node
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Primality Testing
• Application: cryptography
• No polynomial-time algorithm for testing
primality of d-digit number
Primality Testing
• Fermat’s Lesser Theorem: If P is prime,
and 0 < A < P, then AP-1≡1(modP)
– AP-1 % P = 1
• Algorithm
– Given N, compute 2N-1%N
• if result is not 1, N is not prime
• if result is 1, N is robably prime – but not
necessarily
– Randomly choose A – but does not always
work
Primality Testing
• Add additional test to improve result
• Theorem: if P is prime and 0 < X < P, the
only solutions to X2 ≡ 1(modP) are
X=1, P-1
• Test as we calculate AN-1