2D Rectangular Packing with LFF and LFF/T Presented by Y. T. Wu

Overview  Introduction  The Problem  LFF Algorithm  LFF/T Algorithm – An Improvement  Experimental Results  Conclusion  Future Direction

Introduction  2D rectangular packing is to pack a set of rectangles into a bounding rectangle  There are many practical applications:  VLSI floor-planning  VLSI placement  Job-scheduling

Introduction  It is a NP-complete problem  Heuristics are proposed for solving these kind of problems  Existing algorithms:  Simulated Annealing  Genetic algorithms

The Problem  To determine whether a set of rectangles(r 1, r 2, r 3,…, r n ) with given length l and width w can be packed into a bounded rectangular container R  Goal:  Minimize total area for packing these rectangles  Maximize packing density

LFF Algorithm  What is LFF?  A quasi-human (cognitive) based heuristic  A deterministic algorithm inspired by Chinese ancient professionals for polygon- shape stone plates  Based on the concept of flexibility

LFF Algorithm  Two types of flexibility:  Flexibility of empty space follows this order: a corner < a side < a central void area a corner < a side < a central void area  Flexibility of objects to be packed cannot be calculated, but follow this order in general: larger objects < smaller objects larger objects < smaller objects

LFF Algorithm  What is a corner?  It is a point which is bounded in two directions Corner bounded on top and left Corner bounded on bottom and left Corner bounded on top and right  As the packed object are all rectangles, all corners are 90º Corner bounded on bottom and right

LFF Algorithm  Basic rules of LFF:  Objects must be packed to at least one corner  There must not be any overlapping of objects  Due to flexibilities, larger objects should be packed to corners first  If more than one available corner (i.e. a corner to which the packing does not overlap with other objects), use fitness cost function to choose the best one  Fitness cost function = total area of rectangle packed

LFF Algorithm  As mentioned in previous slide, overlapping of objects is not allowed  “Region query” is the process for detecting whether the region for insertion of new object is already occupied  In LFF, region query is operated by K-D tree, with K = 4

LFF Algorithm  What is K-D tree?  A “Multi-dimensional Binary Tree” struct kdtree{ kdkey_t kd_keys[4]; kdkey_t kd_lo_min_bound, kd_hi_max_bound,kd_other_bound; struct kdtree *kd_father; struct kdtree * kd_sons[2]; };  The packing is considered to be performed on a coordinate- plane on which all keys are representing points on that coordinate-plane

LFF Algorithm  The 4 kd_keys are:  K 0 (p) = x 1  K 1 (p) = y 1  K 2 (p) = x 2  K 3 (p) = y 2 (x 1, y 1 ) (x 2, y 2 )

LFF Algorithm  The two sons are LOSON and HISON  To determine whether a child q is LOSON or HISON of its parent p, compare their corresponding kdkey  Discriminator j is introduced to control the use kdkey for comparison. Compare K j (p) and K j (q)  For example, when j=0, if K 0 (p)>K 0 (q), q is LOSON; if K 0 (p)<K 0 (q), q is HISON  j increments as it goes down the tree, and reset to 0 when it reaches maximum, then repeat to increment again

LFF Algorithm  An example Fig. 1a) 16 labelled rectangles on a grid 0 x 13 in size Fig. 1b) Placement of these 16 labelled rectangles in a 4-d binary tree a (6, 1, 7, 3) b (2, 6, 4, 9) c (9, 7, 12, 8) d (1, 4, 3, 7) e (1, 10, 3, 13)

LFF Algorithm  Procedures for manipulating K-D Tree in LFF 2D rectangular packing:  KD-INSERT  KD-REGION-SEARCH  KD-SUCCESSOR  KD-INTERSECT-REGION

LFF Algorithm  KD-INSERT  It takes in a kd-node as parameter  Insert this node into the existing KD-TREE by calling the KD-SUCCESSOR to find out where should the node be inserted  KD-SUCCESSOR  It compares the jth kdkey of q and p to determine whether LOSON or HISON should be the next node

LFF Algorithm  KD-REGION-SEARCH  For each kd_node in the KD-TREE, call KD-INTERSECT- REGION to check whether that kd-node intersects with o if o is packed at the corner being evaluated, report overlapping, if any and quit checking  KD-INTERSECT-REGION  Test, based on a set of conditions, whether p intersects or touches with o if o is packed at the corner being examined  If yes, check whether they are touching  If yes, add p to the touches list to indicate that p touches with o  If no, return TRUE to indicate intersection

LFF Algorithm  The relationship between the KD-TREE operations and LFF  Before packing any objects, call KD-REGION- SEARCH to detect overlapping  If no overlapping is reported, call KD-INSERT to update the KD-TREE, indicating that the object is packed at the specific position and occupy there

LFF Algorithm  The algorithm: Starting with an empty work space (bounding rectangle) 1.Based on the current packing configuration, find all possible COPMs for each unpacked rectangle, represent each COPM b a quinary-tuple 1.Based on the current packing configuration, find all possible COPMs for each unpacked rectangle, represent each COPM b a quinary-tuple 2.Sort all these quinary-tuples in lexicographical order 3.For each candidate COPM, do a) to c) to find its fitness function value (FFV). a)Pseudo-pack this COPM b)Pseudo-pack all the remaining rectangles based on the current COPM list and with a greedy approach, until no more COPM can be packed c)Calculate FFV of this candidate COPM as the occupied area. Note: Before the pseudo-packing for the next candidate COPM is tried, one needs to remove the previous pseudo-packed COPM 4.Pick the candidate COPM with the highest FFB and really pack the corresponding rectangle according to the COPM 5.Mark the rectangle as packed 6.Return to step 1 until no more packing can be done

LFF Algorithm  A sample case:  Suppose rectangles A(2x5), B(4x4) and C(6x4) are to be packed in a work space (8x8) and a rectangle P(6x3) at (0, 8) is already packed in previous iteration  COPM is represented as  COPM is represented as  As P is packed, one of the container’s corner is occupied, the available corners are: (0, 0), (0, 2), (3, 8), (8, 0), (8, 8)

LFF algorithm  The COPM list generated stores all feasible packing (no overlapping and within the bounding rectangle)  (6, 4, 1, 4, 2) – rectangle C vertically packed at (8, 8)  (6, 4, 1, 3, 2) – rectangle C vertically packed at (8, 0)  (6, 4, 1, 4, 0) – rectangle C vertically packed at (3, 8)  (5, 2, 0, 3, 6) – rectangle A horizontally packed at (8, 8) or (3, 8)  (5, 2, 1, 6, 3) – rectangle A vertically packed at (8, 8)  (5, 2, 0, 3, 0) – rectangle A vertically packed at (8, 0)  (5, 2, 1, 6, 0) – rectangle A vertically packed at (8, 0)  (5, 2, 1, 3, 3) – rectangle A vertically packed at (3, 8)  (5, 2, 0, 0, 0) – rectangle A vertically packed at (0, 2) or (0, 0)  For rectangle B, the generation is by checking all orientation against all corners, if feasible in location, it will shorten itself by deleting not feasible solutions

LFF Algorithm  Consider rectangle C is pseudo-packed to (8, 8) vertically, i.e., COPM (6, 4, 1, 4, 2)  The COPM list for next pseudo-packing step is shortened by removing all entries related to rectangle C as well as all entries packing rectangles to corner (8, 8) since it is already occupied  The list becomes:  (5, 2, 0, 3, 0) – rectangle A vertically packed at (8, 0)  (5, 2, 1, 6, 0) – rectangle A vertically packed at (8, 0)  (5, 2, 1, 3, 3) – rectangle A vertically packed at (3, 8)  (5, 2, 0, 0, 0) – rectangle A vertically packed at (0, 2) or (0, 0)  (4, 4, 0, 4, 0) – rectangle B horizontally or vertically packed at (8, 0)  (4, 4, 0, 3, 4) – rectangle B horizontally or vertically packed at (3, 8)  Packing continues until all rectangles are packed or no more available space for packing. Remove all rectangles and pseudo-pack C to another corner.

LFF/T – An Improvement of LFF  Introduce two new concepts:  Tightness For each candidate rectangle placement location, we look at the points that are immediate adjacent to each corner of the candidate rectangle (There are 8 points for measurement) The 8 corner-adjacent ptsB has a larger tightness value (4) than A(3)  A parameter q  Greedy Search done for the COPMs of the q longest rectangles only AB

LFF/T – An Improvement of LFF The LFF/T algorithm: Starting with an empty work space (bounding rectangle) 1.Based on the current packing configuration, find all possible COPMs for each unpacked rectangle, represent each COPM b a quinary-tuple <longer side, shorter side, orientation, x1, y1) 2.Sort all these quinary-tuples in lexicographical order 3.For each candidate COPM, do a) to c) to find its fitness function value (FFV). a)Pseudo-pack this COPM b)While there are rectangles remaining and there are spaces in the box, If there is a location that the next rectangle in the sorted list can fit, then,  For each legal corner moves of the rectangle, calculate the tightness values as described above  Choose the corner move with the highest tightness values and pseudo-pack this rectangle there  Update the corner lists and move to the next rectangle in the list Else Skip this rectangle Calculate FFV of this candidate COPM as the occupied area. Note: Before the pseudo-packing for the next candidate COPM is tried, one needs to remove the previous pseudo-packed COPM 4.Pick the candidate COPM with the highest FFB and really pack the corresponding rectangle according to the COPM 5.Mark the rectangle as packed 6.Return to step 1 until no more packing can be done

Experimental Results Problem No. of Rects Box Size Original LFF LFF/T (q=n) PackingDensity Packing Density ami * %95.84% ami * %97.12% playout629397* %99.26%

Conclusion  LFF/T can really improve the packing density of LFF  Reasons:  In LFF, there may be several corners result in the same FFV (fitness value). The algorithm will just arbitrary take one  Tightness value ensures that less gaps or spaces exist between any pair of rectangles, which in turn gives smaller dead space  LFF/T takes longer processing time  Both algorithms have their own advantages

Future Direction  Extension of LFF or LFF/T to 3D applications  Experiments will be carried out to see which one is more suitable