Bounded Space On-Line Bin Packing: Best Is Better than First

Abstract.

We present a sequence of new linear-time, bounded-space, on-line bin packing algorithms, the K -Bounded Best Fit algorithms (BBF _K ). They are based on the Θ(n log  n) Best Fit algorithm in much the same way as the Next-K Fit algorithms are based on the Θ(n log  n) First Fit algorithm. Unlike the Next-K Fit algorithms, whose asymptotic worst-case ratios approach the limiting value of \frac 17 10 from above as K \rightarrow ∈fty but never reach it, these new algorithms have worst-case ratio \frac 17 10 for all K \geq 2 . They also have substantially better average performance than their bounded-space competition, as we have determined based on extensive experimental results summarized here for instances with item sizes drawn independently and uniformly from intervals of the form (0, u] , 0 < u ≤ 1 . Indeed, for each u < 1 , it appears that there exists a fixed memory bound K(u) such that BBF _K(u) obtains significantly better packings on average than does the First Fit algorithm, even though the latter requires unbounded storage and has a significantly greater running time. For u = 1 , BBF _K can still outperform First Fit (and essentially equal Best Fit) if K is allowed to grow slowly. We provide both theoretical and experimental results concerning the growth rates required.