- - packing p a - packing algo- packing cking rithms algo- - PDF document

algorithmic composition dForm + IM WS2010 Inst. f. Arch. a. Media packing algorithms packing algorithms packing algorithms p a c k i n g packing algorithms packing algorithms algorithms packing packing algorithm 1 algorithms - - packing p a - packing algo- packing cking rithms algo- a l g o - theorems rithms rithms rithms packing algorithms rithms p a - algorithms cking a l g o - rithms packing algo- packing packing algo- rithms packing rithms algorithms packing algo- a l g o - packing algorithms packing algorithms rithms r i t h m s explication the bin packing problem is a combinatorial NP-hard problem in computational complexi- ty theory. 1) different objects must be packed into a number of bins. 2) there are variations of this problem: • 2D packing • linear packing • packing by weight • packing by cost • etc. application • filling up containers • loading trucks with weight capacity • creating file backup in removable media • technology mapping in field-programmab- le gate array semiconductor chip design. variation • cutting stock problem • knapsack problem large instances can be solved with sophis- ticated algorithms, but also heuristics have been developed

algorithmic composition dForm + IM WS2010 Inst. f. Arch. a. Media packing algorithms packing algorithms packing algorithms p a c k i n g packing algorithms packing algorithms algorithms packing packing algorithm 2 algorithms - - packing p a - packing algo- packing cking rithms algo- a l g o - rithms rithms rithms packing algorithms rithms p a - algorithms cking a l g o - rithms packing algo- problems packing packing algo- rithms packing rithms algorithms packing algo- a l g o - packing algorithms packing algorithms rithms r i t h m s knapsack problem optimization: a container C must be filled to its maximum capacity W with a certain amount N of items I, of which a weight Iw and and a value Iv are given. e.g. a thief, robbing a jewelery store variation: • 0-1 knapsack problem: the number of co- pies x of I is RESTRICTED to either 0 or 1. • bound knapsack problem: the number of copies x of each I is RESTRICTED to a number Ci. • unbound knapsack problem: the number of copies x of I in infinite. solution: „greedy approximation algorithm” - sort I in decreasing order of value/weight and pack x copies until C is reached, continue with next I in the array that fits. note: if the next item in the profit density ar- ray has a much higher w than the first one, there might be a loss of v! So, it is “only” guaranteed to achieve at least a value of m/2 and, of course, not recom- mended for use with a bounded knapsack problem.

algorithmic composition dForm + IM WS2010 Inst. f. Arch. a. Media packing algorithms packing algorithms packing algorithms p a c k i n g packing algorithms packing algorithms algorithms packing packing algorithm 3 algorithms - - packing p a - packing algo- packing cking rithms algo- a l g o - rithms rithms rithms packing algorithms rithms p a - algorithms cking a l g o - rithms packing algo- heuristics packing packing algo- rithms packing rithms algorithms packing algo- a l g o - packing algorithms packing algorithms rithms r i t h m s first fit algorithm: puts item into the first container with enough space, opens up a new container, if there is no room in existing containers. ? worst fit algorithm produces results better than the first fit ? algorithm. it picks the least filled container, or creates a new one, instead of just picking ? the first available. first fit decreasing algorithm: if all data of the items is known in advance (offline algorithm), they can be sorted in de- creasing order by criterium and therefore a much higher efficiency is reached. next fit algorithm fills up each container without looking back, until no more items are left. it is the least efficient of all algorithms, but has a practical use, e.g. box filling on a con- veyor belt.