The document discusses greedy algorithms and how they can be applied to solve optimization problems like the knapsack problem, activity selection problem, and job sequencing problem. It provides examples of greedy algorithms for each problem. A greedy algorithm works by making locally optimal choices at each step in the hope of finding a globally optimal solution. For the knapsack problem, the greedy approach sorts items by profit/weight ratio and fills the knapsack accordingly. For activity selection, it schedules activities based on earliest finish time. For job sequencing, it sorts jobs by profit and schedules the most profitable jobs that meet deadlines.

1. GREEDY APPROACH
Shashikant V. Athawale
Assistant Professor ,Computer Engineering Department
AISSMS College of Engineering,
Kennedy Road, Pune , MS, India - 411001
1

2. GREEDY METHOD
 A greedy algorithm always makes the
choice that looks best at the moment
The hope: a locally optimal choice will lead to a
globally optimal solution
For some problems, it works
 greedy algorithms tend to be easier to
code
2

3. KNAPSACK PROBLEM
 Given a knapsack with maximum capacity W,
and a set S consisting of n items
 Each item i has some weight wi and benefit value
bi (all wi and W are integer values)
 Problem: How to pack the knapsack to achieve
maximum total value of packed items?
3

4. KNAPSACK PROBLEM
 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 = 1.39
p2
/w2
= 24/15 = 1.6
p3
/w3
= 15/10 = 1.5
Optimal solution: x1
= 0, x2
= 1, x3
= 1/2
total profit = 24 + 7.5 = 31.5
4

5. ACTIVITY SELECTION
 Problem: Schedule an exclusive resource in
competition with other entities. For example,
scheduling the use of a room (only one entity can
use it at a time) when several groups want to use
it. Or, renting out some piece of equipment to
different people.
 Definition: Set S={1,2, … n} of activities. Each
activity has a start time si and a finish time fj,
where si <fj. Activities i and j are compatible if
they do not overlap. The activity selection
problem is to select a maximum-size set of
mutually compatible activities.
5

6. ACTIVITY SELECTION
 Just march through each activity by finish time
and schedule it if possible:
6

7. JOB SEQUENCING
 Problem: n jobs, S={1, 2, …, n}, each job i has a
deadline di ≥ 0 and a profit pi ≥ 0. We need one
unit of time to process each job and we can do at
most one job each time. We can earn the profit pi
if job i is completed by its deadline.
The optimal solution = {1, 2, 4}.
The total profit = 20 + 15 + 5 = 40. 7

8. JOB SEQUENCING
Algorithm:
Step 1: Sort pi into nonincreasing order. After
sorting p1 ≥ p2 ≥ p3 ≥ … ≥ pn.
Step 2: Add the next job i to the solution set if i can
be completed by its deadline. Assign i to time slot
[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, go
to step 2.
Time complexity: O(n2
) 8

9. Thank You
9

10. Thank You
9