# greedy - Greedy Methods Greedy Methods Problems whose...

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reedy Methods Greedy Methods

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roblems whose solutions can be “ranked” Problems whose solutions can be ranked Travel Investment Course lection selection Feasible solutions stay on highway, finish in x days don’t spend more than one has finish in 4 years Optimal solutions shortest distance, minimum time maximum returns, minimum best combination risk so f depth and breadth ecisions hich highways vest or not in a ke a course Decisions which highways to take invest or not in a portfolio take a course or not Data Structures and Algorithms II
ecisions can be made Decisions can be made one at a time, without backtracking Greedy method Which decisions to make next ? How to guarantee optimality? Try many (all) possible combinations and yy ( ) p choose one which is the best ynamic programming Dynamic programming How to test multiple solutions efficiently? Data Structures and Algorithms II

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he Greedy Method The Greedy Method Input n elements stored in an array A ( 1:n ) Procedure Greedy Solution = NULL for i=1 to n do x = SELECT( A ) if FEASIBLE(Solution, x) then Solution = UNION(Solution, x) endif enddo Data Structures and Algorithms II return (Solution)
A sequence of n decisions w.r.t n inputs q p SELECT: select one of the remaining ecisions to make according to some decisions to make according to some optimization measure nce a decision is made it will ot ecome once a decision is made, it will not become invalid at a later time ptimization ould be based on the partial optimization should be based on the partial solutions built so far EASIBLE: hether the partial solution FEASIBLE: whether the partial solution satisfies some preset constraints Data Structures and Algorithms II

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trategy construct feasible solutions one Strategy : construct feasible solutions one step at a time which optimize (minimize or maximize) a certain objective function Make the obvious decisions first ! Then try to show it is indeed optimal! Data Structures and Algorithms II
napsack problem Knapsack problem Input: a set of n objects a knapsack of capacity M (, ) , . . . , PW i n ii 1 Output: fill the knapsack to maximize the total profit earned p Feasibility constraint: WX M i n 1 Objective function: max PX X i n i  1 01 Data Structures and Algorithms II

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Example p n M  32 0 , PPP WWW 25 2415 181510 123 (, , ) ( , , ) , )(, ,) XX X W X P X ii i n i n 2 11 , ) t i if i t 1 15 0 20 28 2 0 2 12 0 3 1 ( ,, ) . (, ,) largest increase in profit smallest increase in weight 3 01 1 2 315 (,, ) . largest increase in profit to weight ratio Data Structures and Algorithms II ( . , . , . P W P W P W 1 1 2 2 3 3 1391615)
For all three algorithms decisions are made one object at a time the ordering is determined by some ti i ti optimization measure Largest increase in profit clude the remaining object of the largest profit Include the remaining object of the largest profit Smallest increase in weight

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## This note was uploaded on 02/19/2012 for the course ENGR 361 taught by Professor Drexel during the Spring '12 term at Bloomsburg.

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greedy - Greedy Methods Greedy Methods Problems whose...

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