G. S. Lueker, An Average-Case Analysis of Bin Packing with Uniformly Distributed Item Sizes, Tech. Report 181, Department of Information and Computer Science, University of California, Irvine, CA, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is j ik r 1 for i = r 1) 2, which equals j 1 1)k 2r as claimed above. For the numerical example above, there are two intervals of each type. Intervals of the first type are [22, 44] and [55, 77] they are of length 23 and symmetric about 33 and 66. Intervals of the second type are [12, 21] and [45, 54] they are of length 10 and symmetric about 16.5 and 49.5. In general, intervals of the first type have odd length 2j 1)k r 1 and are symmetric about an even multiple of k 2r. Intervals of the second type have even length k 2j 1 and are symmetric about an odd multiple of ....

....probability space. Our discrete distributions U j, k do not have this failing, and for this reason we can obtain significantly better average case behavior for them. 104 COFFMAN et al. Table 1 Expected waste in the symmetric case. U(0, 1] Ref U j, k , j = k 1, k Ref OPT , FFD, BFD #(n ) [20, 21] #(n ) 10] SS #(n ) 12] FF #(n ) 9] #(n ) k = O(n ) k = ## n BF #(n n) 26] #(n k) 6] Best online #(n n) 26, 27] #(n k) # The upper bound is proved in the reference; the lower bound is conjectured based on experiments. The upper ....

G. S. Lueker, An Average-Case Analysis of Bin Packing with Uniformly Distributed Item Sizes, Tech. Report 181, Department of Information and Computer Science, University of California, Irvine, CA, 1982.

.... (n 1) Coffman, So, Hofri, and Yao [18] E[W FF n ] Theta(n 2=3 ) Shor [55] Coffman, Johnson, Shor, and Weber [14] and E[W BF n ] Theta( p n log 3=4 n) Leighton and Shor [46] Shor [55] Rhee and Talagrand [53] These results can be compared to E[W OPT n ] p n (Lueker [48]) Our discussion of generalizations can be continued to problems of three or more dimensions. In the large, the algorithmic extensions from one to two dimensions can be essentially repeated as extensions from d to d 1 dimensions for d 2; as one might expect, asymptotic ratios tend to worsen as ....

G. S. Lueker. An average-case analysis of bin packing with uniformly distributed item sizes. Technical Report 181, University of California at Irvine, Department of Information and Computer Science, 1982.

....the items in L, observing that s(L) OPT(L) for all L. For all the distributions in question, we have E[s(L n ,u ) Q(n) and n lim E[OPT(L n,u ) s(L n ,u ) 1 [3] Moreover, E[OPT(L n,u ) s(L n ,u ) Q(n . 5 ) O(1) u = 1 0 u 1 where the result for u = 1 was first proved in [16,20] and the results for u 1 are from [3] Based on the above results, it is reasonable to perform one s comparisons between A(L) and s(L ) and this is what we shall do, by means of the following definition. Definition. If A is a bin packing algorithm, ER u [A] n lim E[A(L n,u ) s(L n ,u ....

G. S. LUEKER, "An average-case analysis of bin packing with uniformly distributed item sizes," Report No. 181, Department of Information and Computer Science, University of California, Irvine, CA, 1982.

....where s = 1 p Gamma 1 p 1 ) 1 p 1 Gamma a) 0 b 1. The empty space remains consistently small until b reaches a critical threshold between 0:8 and 0:9 (depending upon n) above which it grows without bound. In particular, the empty space grows roughly as 0:3 p n, for b = 1:0 [1, 7]. Shor [9] provided tighter lower bounds and upper bounds on the expected wasted space for the on line BF and FF algorithms when packing items uniformly distributed on [0,1] When there are many small objects, the empty space seems a good performance measurement. However, if the object sizes tend ....

G. S. Lueker. An average-case analysis of bin packing with uniformly distributed item sizes. Technical Report 181, University of California at Irvine, Feb. 1982.

....If L n has item sizes generated according to U(0; u] for 0 u 1, and A is any on line algorithm, then there exists a constant c 0 such that E[W A (L n ) cn 1=2 for infinitely many n. Table 1: Results for expected waste U(0; 1] Ref Ufj; kg, j = k Gamma 1; k Ref OPT Theta(n 1=2 ) [15], 14] Theta(n 1=2 ) 8] FFD, BFD Theta(n 1=2 ) 15] 14] Theta(n 1=2 ) 8] FF Theta(n 2=3 ) 4] Theta(n 1=2 k 1=2 ) k = O(n 1=3 ) 4] Theta(n 2=3 ) k = Omega Gamma n 1=3 ) 4] BF Theta(n 1=2 log 3=4 n) 17] Theta(n 1=2 log 3=4 k) k = O(n) 7] Theta(n 1=2 ....

....0 u 1, and A is any on line algorithm, then there exists a constant c 0 such that E[W A (L n ) cn 1=2 for infinitely many n. Table 1: Results for expected waste U(0; 1] Ref Ufj; kg, j = k Gamma 1; k Ref OPT Theta(n 1=2 ) 15] 14] Theta(n 1=2 ) 8] FFD, BFD Theta(n 1=2 ) [15], 14] Theta(n 1=2 ) 8] FF Theta(n 2=3 ) 4] Theta(n 1=2 k 1=2 ) k = O(n 1=3 ) 4] Theta(n 2=3 ) k = Omega Gamma n 1=3 ) 4] BF Theta(n 1=2 log 3=4 n) 17] Theta(n 1=2 log 3=4 k) k = O(n) 7] Theta(n 1=2 log 3=4 n) k = Omega Gamma n) 7] best on line ....

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G. S. Lueker. An average-case analysis of bin packing with uniformly distributed item sizes. Technical Report Report No 181, Dept. of Information and Computer Science, University of California, Irvine, CA, 1982.

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