MS 101: Algorithms

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MS 101 Algorithms

• Instructor
• Neelima Gupta
• ngupta_at_cs.du.ac.in

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Table of Contents

• Five representative problems

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Five Representative Problems

• Interval Scheduling can be solved by a greedy
  approach.
• Weighted Interval Scheduling Natural greedy
  doesnt work, no other greedy is known, more
  sophisticated technique DP solves the problem.
• Maximum Bipartite Matching
• Independent Set
• Competitive Facility Location

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Interval Scheduling Problem
P(1)10
P(2)3
P(3)4
P(4)20
P(5)2
Time
0
1
7
2
3
4
5
6
8
9
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Examples

• Jobs submitted to an operating system, Resource
  CPU.
• An HR of a company needs to schedule meetings of
  some committees in a meeting room, resource is
  meeting room
• Scheduling classes in a room, resource is
  class-room

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Greedy Approach Increasing Finishing Times
P(1)10
P(2)3
P(3)4
P(4)20
P(5)2
Time
0
1
7
2
3
4
5
6
8
9
.
back
7

Weighted Interval Scheduling
P(1)10
P(2)3
P(3)4
P(4)20
P(5)2
Time
0
1
7
2
3
4
5
6
8
9
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Examples

• Jobs submitted to an operating system, Resource
  CPU.
• Weights profit by executing the job

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Greedy Approach

P(1)10
P(2)3
P(3)4
P(4)20
P(5)2
Time
0
1
7
2
3
4
5
6
8
9
.
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Greedy does not work

P(1)10
P(2)3
Optimal schedule
P(3)4
P(4)20
Schedule chosen by greedy app
P(5)2
Time
0
1
7
2
3
4
5
6
8
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Greedy approach takes job 2, 3 and 5 as best
schedule and makes profit of 7. While optimal
schedule is job 1 and job4 making profit of 30
(1020). Hence greedy will not work
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Example of a Bipartite graph
u1
u1
v1
u2
v2
u3
v3
u2
u4
V2
V1
Edge like this is not acceptable in Bipartite
Graph
Figure 1
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Examples

• There is a set T of teachers with a set C of
  courses. A teacher can teach only some set of
  courses represented by the edges in the bipartite
  graph.
• Thus, bipartite graphs are used to represent
  relationships between two distinct sets of
  objectsteachers and courses here.
• Jobs/Employers and Applicants An employer
  receives several applications but only few of
  them qualify for the interview. Similarly an
  applicant applies for many jobs but qualify only
  for few for them for interview. An edge (a, e) in
  the bipartite graph represents that the applicant
  a qualifies for the interview for job e.

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Maximum Bipartite Matching Problem
Maximum Matching is a matching of maximum
Cardinality.
By Applying the definition of matching, If we
choose the edge (u1,v1) first
u1
And then (u2, v2)
v1
u2
So no more edge can be included, hence matching
in this case is
v2
u3
v3
(u1,v1) , (u2, v2)
u4
V1
V2
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u1
v1

• But instead of picking (u2,v2) , if we pick
• (u2 ,v3) after (u1 , v1) then
• ( u4 ,v2) .

u2
v2
u3
v3
Hence the Maximum Matching is
(u1,v1) , (u2, v3 ) , (u4 , v2)
u4
V1
V2
So, the problem is to find the Matching with
MAXIMUM CARDINALITY in a given Bipartite graph.
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Examples

• There is a set T of teachers with a set C of
  courses. A teacher can teach only some set of
  courses represented by the edges in the bipartite
  graph.
• A teacher needs to be assigned at most one course
  and one course must be taught by only one
  teacher.
• Suppose a number of committee meetings are to be
  scheduled in various meeting rooms during the
  time 3pm - 4pm. A committee meeting can be held
  only in few rooms (may be because other rooms are
  smaller in size than the committee size etc). An
  edge (c, r) represents that committee c can be
  scheduled in room r.
• A committee needs to be assigned one room and one
  room must be assigned to only one committee.

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Maximum Bipartite Matching from Abhishek Aditya
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Maximum Bipartite Matching from Abhishek Aditya
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Maximum Bipartite Matching from Abhishek Aditya
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Independent Set

• Given a graph G (V, E), a subset S of V is said
  to be independent if no two nodes in S are
  joined by an edge in G.

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Maximal Independent set of size 2
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Maximal Independent set of size 3 also Maximum
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More on Independent Set and Comp. FLP from Anurag