Greedy Algorithms Neil Tang 4/8/2010

CS223 Advanced Data Structures and Algorithms 1

Greedy AlgorithmsGreedy Algorithms

Neil TangNeil Tang4/8/20104/8/2010


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.


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.


Examples: “Greed is good”Examples: “Greed is good”

Dijkstra’s algorithm

Prim’s algorithm

Kruskal’s algorithm


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.


An ExampleAn Example

Pack: 0.2,0.5,0.4,0.7,0.1,0.3,0.8


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).


The Next Fit AlgorithmThe Next Fit Algorithm

Pack: 0.2,0.5,0.4,0.7,0.1,0.3,0.8


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.


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)



Pack: 0.2,0.5,0.4,0.7,0.1,0.3,0.8


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.


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.



The Best Fit AlgorithmThe Best Fit Algorithm

Pack: 0.2,0.5,0.4,0.7,0.1,0.3,0.8


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.



The Offline AlgorithmsThe Offline Algorithms

First fit for 0.8, 0.7, 0.5, 0.4, 0.3, 0.2, 0.1


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.


The Optimal AlgorithmThe Optimal Algorithm

Greedy Algorithms Neil Tang 4/8/2010

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