1. Probabilistic algorithms
Sometimes it is preferable to choose a course at random, rather than spend time working out which alternative is best.
Main characteristic
The same algorithm may behave differently when it is applied twice to the same instance.
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2. Probabilistic != Uncertain
Paradox :
The error probability can often be brought down below that of a hardware error during the significantly larger time needed to compute the answer deterministically.
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3. Prabhas Chongstitvatana
There are problems for which no algorithm is know, be deterministic or probabilistic, that can give the answer with certainty within a reasonable amount of time, yet probabilistic algorithm can solve the problem quickly if an arbitrarily small error probability is tolerated.
Example : determine whether a 1000 digit number is prime or composite.
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Type of probabilistic algorithms
Numerical
Confidence interval
More time = more precise
Monte Carlo
Exact but sometimes wrong
More time = less error probability
Las Vegas
Always correct but sometimes no answer
If soln can be verified efficiently
Answer with confidence interval : “with probability 90% the answer is 59 plus or minus 3”
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5. Expected vs average time
deterministic
Average time
The average time taken by an algorithm when each possible instance of a given size is equally likely.
probabilistic
Expected time
The mean time that it would take to solve the same instance over and over.
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Worst-case expected time
Expected time taken by the worst possible instance of a given size.
Example
Las Vegas can be more efficient than deterministic one but only with respect to expected time. (if bad luck, LA takes long time)
Quicksort deterministic worst-case
Quicksort probabilistic
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7. Prabhas Chongstitvatana
QuicksortLV( T[i..j] )
If j-i is sufficiently small then insertsort( T[i..j] )
else
p = T[uniform(i..j) ]
pivotbis(T[i..j],p,k,l)
quicksortLV(T[i..k])
quicksortLV(T[l..j])
Pivotbis(T[i..j], p, k, l) partitions T into three sections, p as pivot. After pivoting the elements in T[i..k] < p, T[k+1.. l-1] = p and T[l..j] > p. Return k,l.
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