7/31/2019 Greedy Methods

1/14

Greedy Algorithms

7/31/2019 Greedy Methods

2/14

Contents

7/31/2019 Greedy Methods

3/14

Introduction

Suppose that a problem can be solved by asequence of decisions. The greedy method hasthat each decision is locally optimal. Theselocally optimal solutions will finally add up to a

globally optimal solution.

Only a few optimization problems can besolved by the greedy method.

7/31/2019 Greedy Methods

4/14

Greedy-choice property:A globally-optimal solution can always be found

by a series of local improvements from astarting configuration.

AGreedy algorithmworks in phases. At each

phase:You take the best you can get right now,without regard for future consequences

You hope that by choosing a localoptimumat each step, you will end up at a globaloptimum

7/31/2019 Greedy Methods

5/14

Optimization Problem

Constraints

Objective Function

Feasible solution

Optimal Solution

Some Terms

7/31/2019 Greedy Methods

6/14

Problem: Pick k numbers out of n numberssuch that the sum of these k numbers is thelargest.

Algorithm:

FOR i = 1 to k

pick out the largest number anddelete this number from the input.

ENDFOR

A simple example

7/31/2019 Greedy Methods

7/14

Knapsack ProblemGiven:A set S ofnitems, with each item i having

bi - a positive benefit(profit pi )wi - a positive weightSize of Knapsack(capacity)=M

Goal: Choose items with maximum total benefit but withweight at most W.

0-1 Knapsack problem: In this problem each itemmust either be taken or left behind, in this we can not takefractional amount of an item.

Fractional Knapsack Problem: In this problem wecan take fractional amount of items.

7/31/2019 Greedy Methods

8/14

Fractional Knapsack Problem

In this case, we let xi denote the amount wetake of item i.

Objective: Maximize

Constraint:

And 0 1

Siiixb

Siii

Mxw

ix

7/31/2019 Greedy Methods

9/14

AlgorithmfractionalKnapsack(S,W)

Input: Set S of items w/ benefit bi and weight wi; Max.

weight W.Output: Amountxiof each item ito maximize benefit withweight at most W.

foreach item i in Sxi 0

vibi / wi {value}

w 0 {current total weight}

whilew < Wremove item iwith highest viximin {wi / wi , W - w /wi}

w w + min {wi , W - w}

7/31/2019 Greedy Methods

10/14

The greedy algorithm:

Step 1: Sort pi/wi into nonincreasing order.Step 2: Put the objects into the knapsack according

to the sorted sequence as possible as we can.e. g.

n = 3, M = 20,(p1, p2, p3) = (25, 24, 15)(w1, w2, w3) = (18, 15, 10)

Sol: p1/w1 = 25/18 = 1.32

p2/w2 = 24/15 = 1.6p3/w3 = 15/10 = 1.5

Optimal solution: x1 = 0, x2 = 1, x3 = 1/2

total profit = 24 + 7.5 = 31.5

Numerical

7/31/2019 Greedy Methods

11/14

ExampleGiven: A set S of n items, with each item i having

bi - a positive benefitwi - a positive weight

Goal: Choose items with maximum total benefit butwith weight at most W.

Weight:

Benefit:

1 2 3 4 5

4 ml 8 ml 2 ml 6 ml 1 ml

$12 $32 $40 $30 $50

Items:

Value: 3

($ per ml)

4 20 5 50

10 ml

Solution:

1 ml of 5

2 ml of 3 6 ml of 4 1 ml of 2

knapsack

7/31/2019 Greedy Methods

12/14

Job Sequencing With DeadlinesProblem: n jobs, S={1, 2, , n}, each job i has a

deadline di 0 and a profit pi 0. We need one unit oftime to process each job and we can do at most one jobeach time. We can earn the profit pi if job i is completedby its deadline.

i 1 2 3 4 5

pi 20 15 10 5 1

di 2 2 1 3 3

The optimal solution = {1, 2, 4}.

The total profit = 20 + 15 + 5 = 40.

7/31/2019 Greedy Methods

13/14

Algorithm:

Step 1: Sort pi into nonincreasing order. Aftersorting p1 p2 p3 pi.

Step 2:Add the next job i to the solution set if i

can be completed by its deadline. Assign i to timeslot [r-1, r], where r is the largest integer such that 1 r di and [r-1, r] is free.

Step 3: Stop if all jobs are examined. Otherwise, goto step 2.

Time complexity: O(n2)

7/31/2019 Greedy Methods

14/14

e.g.

i pi di1 20 2 assign to [1, 2]

2 15 2 assign to [0, 1]

3 10 1 reject

4 5 3 assign to [2, 3]

5 1 3 reject

solution = {1, 2, 4}total profit = 20 + 15 + 5 = 40