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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. - PowerPoint PPT Presentation
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CS223 Advanced Data Structures and Algorithms 1
Greedy AlgorithmsGreedy Algorithms
Neil TangNeil Tang4/8/20104/8/2010
CS223 Advanced Data Structures and Algorithms 2
Class OverviewClass 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 IdeaBasic 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”Examples: “Greed is good”
Dijkstra’s algorithm
Prim’s algorithm
Kruskal’s algorithm
CS223 Advanced Data Structures and Algorithms 5
Bin PackingBin Packing
Bin packing: Given N items with sizes s1, s2,…, sN, where
0 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 ExampleAn 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 AlgorithmThe 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 AlgorithmThe 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 AlgorithmThe 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 AlgorithmThe 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 AlgorithmThe 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 AlgorithmThe 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 AlgorithmThe 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 AlgorithmThe 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 AlgorithmsThe 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 AlgorithmsThe 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 AlgorithmsThe 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.
CS440 Computer Networks 18
The Path Scheduling ProblemThe Path Scheduling ProblemGiven 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 AlgorithmThe First Fit Algorithm
CS440 Computer Networks 20
The Optimal AlgorithmThe Optimal Algorithm