随机算法 (Fall 2011)/Problem set 2
Problem 1
Some known facts:
- Balls-into-bins: [math]\displaystyle{ m }[/math] balls are uniformly and independently thrown to [math]\displaystyle{ n }[/math] bins. If [math]\displaystyle{ m=\Theta(n) }[/math], then the maximum load is [math]\displaystyle{ \Theta\left(\frac{\ln n}{\ln\ln n}\right) }[/math] with high probability.
- Power of two choices: [math]\displaystyle{ m }[/math] balls are sequentially thrown to [math]\displaystyle{ n }[/math] bins. For each ball, uniformly and independently choose random bins [math]\displaystyle{ i }[/math] and [math]\displaystyle{ j }[/math] (might be the same bin), and throw the ball to the currently less loaded bin among bin [math]\displaystyle{ i }[/math] and bin [math]\displaystyle{ j }[/math]. Assuming that [math]\displaystyle{ m=\Theta(n) }[/math], after all [math]\displaystyle{ m }[/math] balls are thrown to bins, the maximum load is [math]\displaystyle{ \Theta\left(\ln \ln n\right) }[/math] with high probability.
Questions: Assume [math]\displaystyle{ n }[/math] balls and [math]\displaystyle{ n }[/math] bins. We throw the [math]\displaystyle{ n }[/math] balls sequentially.
- If we throw the first [math]\displaystyle{ \frac{n}{2} }[/math] balls uniformly and then throw the rest balls as the way of power-of-two-choices, what is the asymptotic maximum load with high probability?
- For the [math]\displaystyle{ k }[/math]th ball, if [math]\displaystyle{ k }[/math] is even, we throw the ball to a uniform bin; and if [math]\displaystyle{ k }[/math] is odd, we throw the ball as the way of power-of-two-choices. What is the asymptotic maximum load with high probability?