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. Prabhas Chongstitvatana

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. Prabhas Chongstitvatana

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. Prabhas Chongstitvatana

Prabhas Chongstitvatana 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” Prabhas Chongstitvatana

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. Prabhas Chongstitvatana

Prabhas Chongstitvatana 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 Prabhas Chongstitvatana

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. Prabhas Chongstitvatana