We formulate the bin packing problem as follows: given a set of N file sizes between 0 and 1,000,000 KB (1 GB), find a way to assign them to a minimal number of disks, each of capacity 1 GB. The worst-fit heuristic is a simple rule that considers the file sizes in the order they are presented: if the sound file won't fit on any disk, create a new disk; otherwise place the file on a disk that has the most remaining space. For example, this algorithm would put the sizes 700,000, 800,000, 200,000, 150,000, 150,000 onto three disks: {700,000, 200,000}, {800,000, 150,000}, {150,000}. Note that this does not necessarily lead to the best solution since we could have fit the five files on two disks.
Your main task is to implement the worst-fit heuristic.
Perspective. The bin packing problem is a fundamental problem for minimizing the consumption of a scarce resource, usually space. Applications include: packing the data for Internet phone calls into ATM packets, optimizing file storage on removable media, assigning commercials to station breaks, allocating blocks of computer memory, and minimizing the number of identical processors needed to process a set of tasks by a given deadline. In the latter example, the scarce resource is time instead of space. Cloth, paper, and sheet metal manufacturers use a two-dimensional version of the problem to optimally cut pieces of material (according to customer demand) from standard sized rectangular sheets. Shipping companies use a three-dimensional version to optimally pack containers or trucks.
Disk data type. First, implement a data type Disk.java that represents a 1GB disk. In addition to storing a list of the file sizes that it stores, this data type should also implement the Comparable interface for use with a priority queue.
Priority queue ADT. You will need to develop an efficient data structure to support all of the basic operations to implement the heuristic. For the worst-fit heuristic, you will certainly need insert and delete the maximum, so a priority queue seems like a judicious starting point. Create an ADT PQ.java that implements these operations and any others you need to perform the worst-fit heuristic. As a starting point, you may use this priority queue implementation. It is an adaptation of the heap code in Sedgewick Program 9.2. You are free to add functionality to the PQ.java, but all operations should be meaningful for any Comparable object, not just disks.
Input and output.
Your client program will read in the set of file sizes (guaranteed to be
between zero and a million) from standard input.
Your program should output the number of disks used by the heuristic and
the sum of all file sizes divided by 1 million (a lower bound on
the number of disks required).
If the number of files is less than 100, you should also print out the
disks in decreasing order of remaining space.
For each disk, print out the amount of remaining space and the file sizes that
it includes (in the order that they were inserted). For debugging, the output
below also includes a unique id number (optional) for each disk that represents
the order in which the disk was created.
% worstfit < input20.txt
file sizes sum = 6.58 GB
total disks = 8
5 325754: 347661 326585
0 227744: 420713 351543
7 224009: 383972 392019
4 190811: 324387 484802
6 142051: 340190 263485 254274
3 116563: 347560 204065 331812
2 109806: 396295 230973 262926
1 82266: 311011 286309 320414
Worst-fit decreasing. Experienced travelers know that if small items are packed last, they can fit snugly in the odd gaps in nearly filled luggage. This motivates a smarter strategy: process the items from biggest to smallest. The worst-fit decreasing heuristic is to do worst-fit, but first preprocess the file sizes so that they are in descending order. A modular way to implement this heuristic is to write a separate program IntegerSorter.java that reads in a sequence of integers and prints them out in descending order. Then you can pipe the results through your worst-fit heuristic. You can accomplish this in Unix with:
or in Windows with:% java IntegerSorter < input20.txt | java WorstFit
C:\> java IntegerSorter < input20.txt | java WorstFit
Analysis. Run your heuristics on a variety of inputs, with N = 100, 1,000, 10,000, 100,000, and 1,000,000. Do a few trials for each value of N using random weights between zero and one million. In your writeup, comment on the relative effectiveness (number of disks used) of the two heuristics (worst-fit and worst-fit decreasing).
Submission. Your main programs should be named WorstFit.java and IntegerSorter.java. Submit all of the other files needed by your program (e.g., PQ.java and Disk.java) except for StdIn.java, which we will provide.
Extra Credit.
Design and implement a heuristic that consistently beats worst-fit decreasing.
Your algorithm should work for large N so it must take time proportional
to N log N on average as with worst-fit.
One idea is first-fit decreasing where you insert the next file
in the first disk that has enough room to store it. Use a heap-like structure
called a tournament tree.
Another idea is best-fit decreasing where you insert the next file
in the disk that has the least remaining space among disks capable of storing
the file.
Add a method ceil to a (balanced) binary search tree that takes
a disk x and returns the disk whose remaining capacity is closest to x
without going under.