1. CS223 Advanced Data Structures and Algorithms 1 Greedy Algorithms Neil Tang 4/8/2010

2. CS223 Advanced Data Structures and Algorithms 2 Class Overview  Basic idea  Examples: “Greed is good”  The bin packing problem and the algorithms  The path scheduling problem and the algorithms: Greed is no good.

3. CS223 Advanced Data Structures and Algorithms 3 Basic Idea  In each phase, it makes the best choice based on the current partial solution and the performance metric.  No future consequences are considered when making decisions.  Usually it will not go back to change the previous choices.

4. CS223 Advanced Data Structures and Algorithms 4 Examples: “Greed is good”  Dijkstra’s algorithm  Prim’s algorithm  Kruskal’s algorithm

5. CS223 Advanced Data Structures and Algorithms 5 Bin Packing  Bin packing: Given N items with sizes s 1, s 2,…, s N, where 0  s i  1. The bin packing is to pack these items in the fewest bins, given that each bin has unit capacity.  Online bin packing (dynamic case): Each item must be placed in a bin before the size of the next item is given.  Offline bin packing (static case): You cannot make decision until all the input has been read.

6. CS223 Advanced Data Structures and Algorithms 6 An Example  Pack: 0.2,0.5,0.4,0.7,0.1,0.3,0.8

7. CS223 Advanced Data Structures and Algorithms 7 The Next Fit Algorithm  For each new item, check to see if it fits in the same bin as the last one. If it does, place it there. Otherwise, create a new bin.  Time complexity: O(N).

8. CS223 Advanced Data Structures and Algorithms 8 The Next Fit Algorithm  Pack: 0.2,0.5,0.4,0.7,0.1,0.3,0.8

9. CS223 Advanced Data Structures and Algorithms 9 The Next Fit Algorithm  Theorem: Let M be the optimal solution. The next fit algorithm never uses more than 2M bins. There exist sequences such that it uses 2M-2 bins.

10. CS223 Advanced Data Structures and Algorithms 10 The First Fit Algorithm  For each new item, scan the existing bins in order and place it in the first bin that can hold it. Create a new bin if none of them can hold it.  Time complexity: O(N 2 )

11. CS223 Advanced Data Structures and Algorithms 11 The First Fit Algorithm  Pack: 0.2,0.5,0.4,0.7,0.1,0.3,0.8

12. CS223 Advanced Data Structures and Algorithms 12 The First Fit Algorithm  Theorem: Let M be the optimal solution. The first fit algorithm never uses more than  1.7M  bins. There exist sequences such that it uses  1.7(M-1)  bins.

13. CS223 Advanced Data Structures and Algorithms 13 The Best Fit Algorithm  For each new item, scan the existing bins in order and place it in the tightest spot among all bins. Create a new bin if none of them can hold it.  Time complexity: O(N 2 )

14. CS223 Advanced Data Structures and Algorithms 14 The Best Fit Algorithm  Pack: 0.2,0.5,0.4,0.7,0.1,0.3,0.8

15. CS223 Advanced Data Structures and Algorithms 15 The Offline Algorithms  The first/best fit decreasing algorithm: Sort the items in the descending order of their sizes, then use the first/best fit algorithm.  Time complexity: O(N 2 )

16. CS223 Advanced Data Structures and Algorithms 16 The Offline Algorithms  First fit for 0.8, 0.7, 0.5, 0.4, 0.3, 0.2, 0.1

17. CS223 Advanced Data Structures and Algorithms 17 The Offline Algorithms  Theorem: Let M be the optimal solution. The first fit descending algorithm never uses more than  11/9M+4  bins. There exists sequences such that it uses  11/9M  bins.

18. CS440 Computer Networks 18 The Path Scheduling Problem Given a path, and free timeslots of every nodes in the path, find a collision-free transmission schedule. J. Tang, G. Xue and C. Chandler, Interference-aware routing and bandwidth allocation for QoS provisioning in multihop wireless networks, Wireless Communications and Mobile Computing (WCMC), Vol. 5, No. 8, 2005, pp. 933-944.

19. CS440 Computer Networks 19 The First Fit Algorithm

20. CS440 Computer Networks 20 The Optimal Algorithm