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Greedy Algorithms Neil Tang 4/8/2010

Greedy Algorithms Neil Tang 4/8/2010. 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. Basic Idea.

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Greedy Algorithms Neil Tang 4/8/2010

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  1. Greedy AlgorithmsNeil Tang4/8/2010 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. 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. CS223 Advanced Data Structures and Algorithms

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

  5. Bin Packing • Bin packing: Given N items with sizes s1, s2,…, sN, where0 si 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. CS223 Advanced Data Structures and Algorithms

  6. An Example • Pack: 0.2,0.5,0.4,0.7,0.1,0.3,0.8 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). 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 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. 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(N2) 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 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. 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(N2) 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 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(N2) 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 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. CS223 Advanced Data Structures and Algorithms

  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. CS440 Computer Networks

  19. The First Fit Algorithm CS440 Computer Networks

  20. The Optimal Algorithm CS440 Computer Networks

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