## Theory of Combinatorial Algorithms

Prof. Emo Welzl and Prof. Bernd Gärtner

# Mittagsseminar (in cooperation with M. Ghaffari, A. Steger and B. Sudakov)

 Mittagsseminar Talk Information

Date and Time: Tuesday, November 07, 2006, 12:15 pm

Duration: This information is not available in the database

Location: CAB G51

Speaker: Riko Jacob

## Optimal Randomized Comparison Based Algorithms for Collision

We consider the problem of finding two identical elements in a list of length $n$ by randomized comparison based algorithms. We find a trade-off between the success probability $p$ and the running time $t$, and show that this trade-off is optimal up to a constant factor. For worst-case running time $t$, the optimal success probability is $p=\Theta\left(\min\{\frac{t}{n},1\}\frac{t}{n\log t}\right)$. For expected running time $t$, the success probability is $p=\Theta\left(t/(n\log(n))\right)$.

As part of this, but of independent interest, we determine the complexity of finding collisions in randomly chosen input according to three natural (uniform) distributions. Again, we determine the optimal (up to a constant factor) trade-off between success probability (relying on the input distribution) and running time, both worst-case and expected.

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