CS 401

Greedy Algorithms / Interval Partition / Minimizing Lateness

Xiaorui Sun

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Homework 1

If the solution involves reducing the problem into a question we solved in class, then the solution should look like:

Reduction Outline (how to use algo learned in class)!"#$%&'()*+,-./0☺23456789:;<=>?@ABCDEF☹HIJKLMNOPQRST

Proof of Runtime/Termination!"#$%&'()*+,-./0☺23456789:;<=>?@ABCDEF☹HIJKLMNOPQRST

Proof of Correctness!"#$%&'()*+,-./0☺23456789:;<=>?@ABCDEF☹HIJKLMNOPQRST

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Homework 1

If the solution involves a new algorithm (usually modified from an algo in class), then the solution should look like:

Algorithm Outline!"#$%&'()*+,-./0☺23456789:;<=>?@ABCDEF☹HIJKLMNOPQRST

Proof of Runtime/Termination!"#$%&'()*+,-./0☺23456789:;<=>?@ABCDEF☹HIJKLMNOPQRST

Proof of Correctness!"#$%&'()*+,-./0☺23456789:;<=>?@ABCDEF☹HIJKLMNOPQRST

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Problem 2

!" = $ !% , Ω !% or Θ(!%)?!" = $ !+ , Ω !+ or Θ(!+)?!" = $ !, , Ω !, or Θ(!,)?

Answer the following 10 questions:

!" = $ !- , Ω !- or Θ(!-)?!% = $ !+ , Ω !+ or Θ(!+)?

!% = $ !, , Ω !, or Θ(!,)?!% = $ !- , Ω !- or Θ(!-)?!+ = $ !, , Ω !, or Θ(!,)?!+ = $ !- , Ω !- or Θ(!-)?!, = $ !- , Ω !- or Θ(!-)?

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Greedy Algorithms

• Hard to define exactly but can give general properties• Solution is built in small steps• Decisions on how to build the solution are made to

maximize some criterion without looking to the future• Want the ‘best’ current partial solution as if the

current step were the last step

Interval Scheduling: Earliest Finish Time

Consider jobs in increasing order of finish time. Take each job provided it’s compatible with the ones already taken.

Implementation. O(n log n).• Remember job !∗ that was added last to A.• Job ! is compatible with A if # ! ≥ %(!∗).

Correctness proof: greedy stays ahead6

Sort jobs by finish times so that f(1) £ f(2) £ ... £ f(n).( ← ∅for j = 1 to n {

if (job j compatible with ()( ← ( ∪ {-}

}return (

Interval PartitioningTechnique: Structural

Interval Partitioning

Lecture ! starts at "(!) and finishes at %(!).Goal: find minimum number of classrooms to schedule all lectures so that no two occur at the same time in the same room.

8Time9 9:30 10 10:30 11 11:30 12 12:30 1 1:30 2 2:30

h

c

b

a

e

d g

f i

j

3 3:30 4 4:30

Room 1

Room 2

Room 3

Room 4

Interval Partitioning

9Time9 9:30 10 10:30 11 11:30 12 12:30 1 1:30 2 2:30

h

c

b

a

e

d g

f i

j

3 3:30 4 4:30

Room 1

Room 2

Room 3

Room 4

C

BA

ED G

F

J

H

I

Note: graph coloring is very hard in general, but graphs corresponding to

interval intersections are simpler.

A Better Schedule

This one uses only 3 classrooms

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Time9 9:30 10 10:30 11 11:30 12 12:30 1 1:30 2 2:30

h

c

a e

f

g i

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3 3:30 4 4:30

d

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A Greedy Algorithm

Greedy algorithm: Consider lectures in increasing order of finish time: assign lecture to any compatible classroom.

Correctness: This is wrong!11

Sort intervals by finish time so that f1 £ f2 £ ... £ fn.d ¬ 0

for j = 1 to n {if (let j is compatible with some classroom k, ! ≤ # ≤ $)

schedule lecture j in classroom kelse

allocate a new classroom d + 1schedule lecture j in classroom d + 1d ¬ d + 1

}

Example

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Time0 1 2 3 4 5 6

d

a

b

c

Time0 1 2 3 4 5 6

d

a

b

c

Greedy by finish time gives: OPT:

Time0 1 2 3 4 5 6

da

b c

A Greedy Algorithm

Greedy algorithm: Consider lectures in increasing order of start time: assign lecture to any compatible classroom.

Implementation: Exercise!13

Sort intervals by starting time so that s1 £ s2 £ ... £ sn.d ¬ 0

for j = 1 to n {if (lect j is compatible with some classroom k, ! ≤ # ≤ $)

schedule lecture j in classroom kelse

allocate a new classroom d + 1schedule lecture j in classroom d + 1d ¬ d + 1

}

A Structural Lower-Bound on OPT

Def. The depth of a set of intervals is the maximum number that contain any given time.

Key observation. Number of classrooms needed ³ depth.

Ex: Depth of schedule below = 3Þ schedule below is optimal.

Q. Does there always exist a schedule equal to depth of intervals?

14Time9 9:30 10 10:30 11 11:30 12 12:30 1 1:30 2 2:30

h

c

a e

f

g i

j

3 3:30 4 4:30

d

b

Correctness

Observation: Greedy algorithm never schedules two incompatible lectures in the same classroom.

Theorem: Greedy algorithm is optimal.Proof (exploit structural property). Let ! = number of classrooms that the greedy algorithm allocates.Classroom ! is opened because we needed to schedule a job, say #, that is incompatible with all ! − 1 previously used classrooms.Since we sorted by start time, all these incompatibilities are caused by lectures that start no later than &(#).Thus, we have ! lectures overlapping at time & # + *, i.e. depth ≥ !�OPT Observation� Þ all schedules use ³ depth classrooms, so ! = depth and greedy is optimal

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Minimizing Lateness

Scheduling to Minimizing Lateness• Similar to interval scheduling.• Instead of start and finish times, request ! has

ØTime Requirement "! which must be scheduled in a contiguous block

ØDeadline #! by which time the request would like to be finished

• Requests are scheduled into time intervals [%!, '!] s.t. "! = '! − %!.

• Lateness for request ! is• If #! < '! then request ! is late by ,! = '! − #- otherwise its

lateness ,! = .• Goal: Find a schedule that minimize the

Maximum lateness , = /01! ,!

23 6

43 3

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8

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9

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9

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14

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15

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6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

d5 = 14d2 = 8 d6 = 15 d1 = 6 d4 = 9d3 = 9

lateness = 0lateness = 2 max lateness = 6lateness = 0lateness = 0lateness = 0

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Minimizing Lateness: Greedy Algorithms

Greedy template. Consider jobs in some order. • [Shortest processing time first]

Consider jobs in ascending order of processing time !".

• [Smallest slack]Consider jobs in ascending order of slack #" − !".

• [Earliest deadline first]Consider jobs in ascending order of deadline #".

counterexampledj

tj

100

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1

10

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counterexampledj

tj2

1

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10

10

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