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## Bin Packing: From Theory to Experiment and Back Again

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**Bin Packing: From Theory to Experiment and Back Again**• David S. Johnson • AT&T Labs – Research • http://www.research.att.com/~dsj/**Applications**• Packing commercials into station breaks • Packing files onto floppy disks • Packing MP3 songs onto CDs • Packing IP packets into frames, SONET time slots, etc. • Packing telemetry data into fixed size packets Standard Drawback: Bin Packing is NP-complete**OUTLINE**• Worst-Case Performance • Average-Case Performance • Classical Models • Experiments Theory • Discrete Distributions • Theory Experiments Theory****First Fit (FF):Put each item in the first bin with enough space Best Fit (BF):Put each item in the bin that will hold it with the least space left over First Fit Decreasing, Best Fit Decreasing (FFD,BFD): Start by reordering the items by non-increasing size.**Worst-Case Bounds**• Theorem [Ullman, 1971]. For all lists L, BF(L), FF(L) ≤ (17/10)OPT(L) + 3. • Theorem [Johnson, 1973]. For all lists L, BFD(L), FFD(L) ≤ (11/9)OPT(L) + 4. (Note 1: 11/9 = 1.222222…) (Note 2: These bounds are asymptotically tight.)**Lower Bounds: FF and BF**½ - ½ - ½ - ½ + ½ + OPT: N bins FF, BF: N/2 bins + N bins = 1.5 OPT**Lower Bounds: FF and BF**1/6 - 2 1/6 - 2 1/6 - 2 1/6 - 2 1/6 - 2 1/6 - 2 1/6 - 2 1/3 + 1/3 + 1/3 + ½ + 1/2 + OPT: N bins FF, BF: N/6 bins + N/2 bins + N bins = 5/3 OPT**Lower Bounds: FF and BF**1/43 + , 1/1806 + , etc. 1/3 + 1/7 + FF, BF = N(1 + 1/2 + 1/6 + 1/42 + 1/1805 + … ) (1.691..) OPT 1/2 + OPT: N bins**Proof Idea for FF, BF:View as a 2-Dimensional Matching**Problem**Distributions U[0,u]**Item sizes uniformly distributed in the interval (0,u], 0 < u < 1**FFD on U(0,u]**u = .6 FFD(L) – s(L) u = .5 u = .4 N = Experimental Results from [Bentley, Johnson, Leighton, McGeoch, 1983]**OUTLINE** • Worst-Case Performance • Average-Case Performance • Classical Models • Experiments Theory • Discrete Distributions • Theory Experiments Theory **y**z (0,2,1) (1,0,2) (2,1,1) x (0,0,0)**Theorem [Csirik et al. 2000]**Note: The LP’s for (1) and (3) are both of size polynomial in B, not log(B), and hence “pseudo-polynomial”**U{6,8}**U{12,16} U{3,4} U(0,¾] 1 2/3 1/3 0.00 0.25 0.50 0.75 1.00 Discrete Uniform Distributions**Theorem [Coffman et al. 1997]**(Results analogous to those for the corresponding U(0,u])**Experimental Results for Best Fit0 ≤ u ≤ 1, 1 ≤ j ≤**k = 51 Averages of 25 trials for each distribution, N = 2,048,000**Average Waste under Best Fit(Experimental values for N =**100,000,000 and 200,000,000) [KRS, 1996] Holds for all j = k-2 [GJSW, 1993]**¾β**6 β/24 3 2 3 3 3 3 4 6 2 5 5 2 4 2 β/6 β/2 β/2 2 4 2 β/2 β/2 β/3 β/8 β/24 “Fluid Algorithm” Analysis: U{6,13} Size = 6 5 4 3 2 1 Amount = ββββββ Bin Type = Amount =**Theorem[Coffman, Johnson, McGeoch, Shor, & Weber, 1994-2008]**