Chart Packing Heuristic Xun Shi 208395360

Problem Statement Finding the least area to pack all the texture charts for a 3D models. Optimal packing Extract texture 2006-11

Outline Bin packing and its heuristics Algs Chart packing Algs Non-overlapping bounding rectangle Heuristic Bruno’s Heuristic APTAS Heuristic 2006-11

Bin packing problem Objects of different volumes must be packed into a finite number of bins with capacity V in a way that minimizes the number of bins used. NP-hard; Several Heuristic Algs; Applications: chart packing, linear packing, packing by weight, etc. 2006-11

Heuristic Algs First Fit Decreasing (FFD) Best Fit Decreasing first sort the items by volume, then insert into the first bin with sufficient space; With sorting: 11/9 OPT + 1 bins; Without sorting: 17/10 OPT + 2 bins; A More efficient Alg: 71/60 OPT + 1 bins; Best Fit Decreasing 2006-11

Chart Packing Problem given a set of polygons, find a non-overlapping placement of the polygons that the enclosing rectangle is of minimum area. Tangram 2006-11

Chart Packing Problem Ancient China 2006-11

Chart Packing Problem given a set of polygons, find a non-overlapping placement of the polygons that the enclosing rectangle is of minimum area. Tangram HOW?? 2006-11

ALG 1: Packing by non-overlapping bounding rectangle However, due to the arbitrary shapes… 2006-11

ALG 2: Bruno’s Heuristic Rotate the charts: maximum diameter of the charts are oriented vertically Sort the charts in decreasing order 2006-11

ALG 2: Bruno’s Heuristic cont. Insert the charts one by one 2006-11

ALG 2: Bruno’s Heuristic cont. Using horizontal function to approximate chart position; Rely on experience Not proved !! 2006-11

ALG 2: Bruno’s Heuristic cont. Save space Waste space 2006-11

APTAS for charts packing APTAS: asymptotic polynomial time approximation scheme de la Vega and Leuker, 1980 For every fixed ε> 0 algorithm outputs a solution of size (1+ε)OPT + 1 in time polynomial in n. 2006-11

APTAS for charts packing cont. Instance: Multiset of real numbers Solution: A partition of I into subsets Optimal Solution: A solution with a minimum number of subsets k. 2006-11

APTAS for charts packing cont. Two constructive reductions Reduce BP into Large Set BP[ ], with the threshold Also have Small Set Reduce Large Set BP[ ] into BP[ , m], all are chosen from a set of at most m different values 2006-11

APTAS for charts packing cont. Large Set: BP[ , m] can be solved by dynamic programming in polynomial of n. k ≤(1+ ) OPT Small Set: Greedy insert 2006-11

APTAS for charts packing cont. Running time: dominated for packing Large Set k≤(1+2 ) OPT + 1 for sufficiently small ( ≤½) 2006-11

Reference B. Lévy, S. Petitjean, N. Ray, J. MaillotLeast, “Squares Conformal Maps for Automatic Texture Atlas Generation”, ACM Press, 2002 P. Sander, J. Snyder, S. Gortler, H. Hoppe, “Texture mapping progressive meshes” In SIGGRAPH 01 Conf. Proc. 409-416. ACM Press, 2001. U. Zwick, E. Reshef, “Bin Packing”, course note, 1997/8 del la Vega, G. S. Lueker, “Bin packing can be solved within 1+ε in linear time”, Combinatiorica, 1(4):349-355, 1981 2006-11

THANK YOU VERY MUCH!! 2006-11