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1 | | -# load-balancing-problem-approximation-algorithm |
| 1 | +# Load Balancing Problem Approximation Algorithm |
| 2 | +Approximation Algorithm for balancing loads on machines. (Could be applied to processes management on a CPU). Does not guarantee an optimal solution, but instead, the algorithm guarantees that its solution is within a factor of 1.5 of the optimal solution. |
| 3 | + |
| 4 | +## Problem Statement |
| 5 | +- `M` identical machines |
| 6 | +- `N` jobs |
| 7 | +- Each jobs has processing time **T<sub>j</sub>** |
| 8 | +- Let `J(i)` be the subset of jobs assigned to machine `i` |
| 9 | +- The load of machine `i` is **L<sub>i</sub>** = sum of the processing times for the jobs |
| 10 | + |
| 11 | +**Makespan** = the maximum of the loads on the machines (the machine with the largest load) |
| 12 | +***Goal:* Minimize the Makespan** |
| 13 | + |
| 14 | +## Solution |
| 15 | +Assign each job to a machine to minimize makespan |
| 16 | +There are m<sup>n</sup> different assignments of n jobs to m machines, which is **exponential** |
| 17 | + |
| 18 | +**The greedy solution runs in polynomial time and gives a solution no more than 1.5 times the optimal makespan** |
| 19 | +Proof involves complicated sigma notation and can be found in the references |
| 20 | + |
| 21 | +**Sort jobs by largest processing time 1st & keep assigning jobs to the machine with the smallest load** |
| 22 | + |
| 23 | + |
| 24 | +## Runtime |
| 25 | +- O(n log n) for sorting jobs by processing time |
| 26 | +- O(n log m) for Greedy Balance & assigning jobs to machines |
| 27 | +- O(n log n) + O(n log m) |
| 28 | +- **⇒ O(n log n)** |
| 29 | + |
| 30 | +## Input Jobs |
| 31 | +### Jobs1 Inputs |
| 32 | +<img src="images/jobs1.png" > |
| 33 | + |
| 34 | +### Jobs1 Machine Assignments |
| 35 | +<img src="images/jobs1-machine-assignments.png" > |
| 36 | + |
| 37 | +### Jobs2 Inputs |
| 38 | +<img src="images/jobs2.png" > |
| 39 | + |
| 40 | +### Jobs2 Machine Assignments |
| 41 | +<img src="images/jobs2-machine-assignments.png" > |
| 42 | + |
| 43 | +### Jobs3 Inputs |
| 44 | +<img src="images/jobs3.png" > |
| 45 | + |
| 46 | +### Jobs3 Machine Assignments |
| 47 | +<img src="images/jobs3-machine-assignments.png" > |
| 48 | + |
| 49 | + |
| 50 | +## Usage |
| 51 | +**Machine ID's are meaningless since machines are identical, they're created for readability. |
| 52 | +But the algorithm still creates the job assignments on the machines according to the greedy strategy** |
| 53 | +- `int machineCount` parameter for `balanceMachines()` is how many machines exist in the problem |
| 54 | +- `int[][] jobs` is a 2D array containing the jobs |
| 55 | + - `jobs[i]` is an array represents a job |
| 56 | + - `jobs[i][0]` is the **Job Id or name** |
| 57 | + - `jobs[i][1]` is the **Processing time** |
| 58 | + |
| 59 | +## Code Details |
| 60 | +- Unlike the pseudocode, this version keeps the jobs assigned to a machine inside the `Machine` object in the Priority Queue |
| 61 | +- 1st Creates `M` machines with unique Id's |
| 62 | +- Then loop over the jobs and assign the job with the longest processing time to the smallest load machine |
| 63 | + - Java's Priority Queue doesn't really have an `IncreaseKey()` method so the same effect is achieved by removing the machine from the Queue, updating the Machine's `currentLoad` and then adding the Machine back to the Queue |
| 64 | +- `Machine` class represents a machine |
| 65 | + - **Some Important Parts of a Machine** |
| 66 | + - `id` is the **id or name of the machine** |
| 67 | + - `currentLoad` is the sum of the processing times of the jobs currently assigned to the machine |
| 68 | + - `jobs` is a 2D ArrayList containing the jobs currently assigned to the machine |
| 69 | + - `Machine` class **overrides** `compareTo()` from the `Comparable` interface so that the `PriotityQueue is always has the smallest load machine at the top |
| 70 | + |
| 71 | +## References |
| 72 | +- [Load Balancing - Approximation Algorithms - Kevin Wayne](http://www.serc.iisc.ernet.in/~simmhan/SE252-JAN2014/lectures/SE252.Jan2014.Lecture-17.pdf) |
| 73 | +- [Load Balancing - Arash Rafiey](https://www.sfu.ca/~arashr/lecture24.pdf) |
| 74 | +- [Load Balancing - Hantao Zhang](http://homepage.divms.uiowa.edu/~hzhang/c231/ch11.pdf) |
| 75 | +- [Load Balancing - Yogesh Simmhan](https://www.cs.princeton.edu/~wayne/kleinberg-tardos/pdf/11ApproximationAlgorithms.pdf) |
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