Algorithm Design and Analysis Greedy Algorithms (2)

Greedy Algorithms (2)

Outline

• Greedy Algorithms

• Greedy #1: Activity-Selection / Interval Scheduling

• Greedy #2: Coin Changing

• Greedy #3: Fractional Knapsack Problem

• Greedy #4: Breakpoint Selection

• Greedy #5: Huffman Codes

• Greedy #6: Task-Scheduling

• Greedy #7: Scheduling to Minimize Lateness

Algorithm Design Strategy

• Do not focus on “specific algorithms”

• But “some strategies” to “design” algorithms

• First Skill: Divide-and-Conquer (各個擊破/分治)

• Second Skill: Dynamic Programming (動態規劃)

• Third Skill: Greedy (貪婪法則)

Greedy #6: Task-Scheduling

Textbook Exercise 16.2-2

Task-Scheduling Problem

• Input: a finite set 𝑆 = 𝑎

, 𝑎

, … , 𝑎

of 𝑛 unit-time tasks, their

corresponding integer deadlines 𝑑

, 𝑑

, … , 𝑑

(1 ≤ 𝑑

≤ 𝑛), and nonnegative penalties 𝑤

, 𝑤

, … , 𝑤

if 𝑎

is not finished by time 𝑑

• Output: a schedule that minimizes the total penalty

Job 1 2 3 4 5 6 7

Deadline (𝑑_𝑖) 1 2 3 4 4 4 6

Penalty (w_𝑖) 30 60 40 20 50 70 10

𝑎₂ 𝑎₃ 𝑎₆ 𝑎₅

Penalty 30

𝑎₇ 𝑎₁ 𝑎₄

20

Task-Scheduling Problem

• Let a schedule 𝐻 is the OPT

• A task 𝑎

is late in 𝐻 if 𝑓 𝐻, 𝑖 > 𝑑

• A task 𝑎

is early in 𝐻 if 𝑓 𝐻, 𝑖 ≤ 𝑑

• We can have an early-first schedule 𝐻′ with the same total penalty (OPT)

Task-Scheduling Problem

Input: 𝑛 tasks with their deadlines 𝑑₁, 𝑑₂, … , 𝑑_𝑛 and penalties 𝑤₁, 𝑤₂, … , 𝑤_𝑛 Output: the schedule that minimizes the total penalty

𝑎₂ 𝑎₃ 𝑎₆ 𝑎₅

0 n

Penalty 2

𝑎₇ 𝑎0₄

Task 1 2 3 4 5 6 7

𝑑_𝑖 1 2 3 4 4 4 6

w_𝑖 30 60 40 20 50 70 10

𝑎₁ 30 𝑎₂ 𝑎₃ 𝑎₆ 𝑎₅

0 n

Penalty 3

𝑎₇ 𝑎0₁ 𝑎₄

20

𝐻′

𝐻

If the late task proceeds the early task, switching them makes the early one earlier and late one still late

Possible Greedy Choices

• Rethink the problem: “maximize the total penalty for the set of early tasks”

• Greedy idea

• Largest-penalty-first w/o idle time?

• Earliest-deadline-first w/o idle time?

𝑎₂ 𝑎₃ 𝑎₆ 𝑎₅

0 n

Penalty 2

𝑎₇ 𝑎0₄ 𝑎₁ 30

Task 1 2 3 4 5 6 7

𝑑_𝑖 1 2 3 4 4 4 6

w_𝑖 30 60 40 20 50 70 10 60 40 70 50 10

Task-Scheduling Problem

Input: 𝑛 tasks with their deadlines 𝑑₁, 𝑑₂, … , 𝑑_𝑛 and penalties 𝑤₁, 𝑤₂, … , 𝑤_𝑛 Output: the schedule that minimizes the total penalty

Prove Correctness

• Greedy choice: select the largest-penalty task into the early set if feasible

• Proof via contradiction

• Assume that there is no OPT including this greedy choice

• If OPT processes 𝑎_𝑖 after 𝑑_𝑖, we can switch 𝑎_𝑗 and 𝑎_𝑖 into OPT’

• The maximum penalty must be equal or lower, because 𝑤

≥ 𝑤

𝑎_𝑗

0 n

Penalty 𝑤_𝑖

𝑎_𝑖 𝑑_𝑖

𝑎_𝑖

0 n

Penalty

𝑎_𝑗 𝑑_𝑖

𝑤_𝑗

𝑤_𝑖 ≥ 𝑤_𝑘 for all 𝑎_𝑘 in the early set Task-Scheduling Problem

Input: 𝑛 tasks with their deadlines 𝑑₁, 𝑑₂, … , 𝑑_𝑛 and penalties 𝑤₁, 𝑤₂, … , 𝑤_𝑛 Output: the schedule that minimizes the total penalty

Prove Correctness

• Greedy algorithm

Task-Scheduling(n, d[], w[])

sort tasks by penalties s.t. w[1] ≥ w[2] ≥ … ≥ w[n]

for i = 1 to n

find the latest available index j <= d[i]

if j > 0

A = A ∪ {i}

mark index j unavailable

return A // the set of early tasks

Can it be better?

Task-Scheduling Problem

Input: 𝑛 tasks with their deadlines 𝑑₁, 𝑑₂, … , 𝑑_𝑛 and penalties 𝑤₁, 𝑤₂, … , 𝑤_𝑛 Output: the schedule that minimizes the total penalty

Example Illustration

Job 1 2 3 4 5 6 7

Deadline (𝑑

) 4 2 4 3 1 4 6

Penalty (w

) 70 60 50 40 30 20 10

𝑎₁ 𝑎₃

𝑎₄ 𝑎₂

0 1 2 3 4 5 6 7

Total penalty = 30 + 20 = 50

2 𝑎₇ 𝑎₅ 𝑎0₆

30

Practice: how about the greedy algorithm using “earliest-deadline-first”

Greedy #7:

Scheduling to Minimize Lateness

Scheduling to Minimize Lateness

• Input: a finite set 𝑆 = 𝑎

, 𝑎

, … , 𝑎

of 𝑛 tasks, their processing time 𝑡

, 𝑡

, … , 𝑡

, and integer deadlines 𝑑

, 𝑑

, … , 𝑑

• Output: a schedule that minimizes the maximum lateness

Job 1 2 3 4

Processing Time (𝑡_𝑖) 3 5 3 2

Deadline (𝑑_𝑖) 4 6 7 8

𝑎₄ 𝑎₁ 𝑎₃ 𝑎₂

0 2 5 8 13

Lateness 0 1 1 7

Scheduling to Minimize Lateness

• Let a schedule 𝐻 contains 𝑠 𝐻, 𝑗 and 𝑓 𝐻, 𝑗 as the start time and finish time of job 𝑗

• 𝑓 𝐻, 𝑗 − 𝑠 𝐻, 𝑗 = 𝑡

• Lateness of job 𝑗 in 𝐻 is 𝐿 𝐻, 𝑗 = max 0, 𝑓 𝐻, 𝑗 − 𝑑

• The goal is to minimize max

𝑗

𝐿 𝐻, 𝑗 = max

𝑗

0, 𝑓 𝐻, 𝑗 − 𝑑

Scheduling to Minimize Lateness Problem

Input: 𝑛 tasks with their processing time 𝑡₁, 𝑡₂, … , 𝑡_𝑛, and deadlines 𝑑₁, 𝑑₂, … , 𝑑_𝑛 Output: the schedule that minimizes the maximum lateness

Possible Greedy Choices

• Greedy idea

• Shortest-processing-time-first w/o idle time?

• Earliest-deadline-first w/o idle time?

Practice: prove that any schedule w/ idle is not optimal

Scheduling to Minimize Lateness Problem

Input: 𝑛 tasks with their processing time 𝑡₁, 𝑡₂, … , 𝑡_𝑛, and deadlines 𝑑₁, 𝑑₂, … , 𝑑_𝑛 Output: the schedule that minimizes the maximum lateness

Possible Greedy Choices

• Idea

• Shortest-processing-time-first w/o idle time?

Job 1 2

Processing Time (𝑡_𝑖) 1 2

Deadline (𝑑_𝑖) 10 2

𝑎₁ 𝑎₂

0 1 3

Lateness 0 1

𝑎₂ 𝑎₁

0 2 3

Lateness 0 0

Scheduling to Minimize Lateness Problem

Input: 𝑛 tasks with their processing time 𝑡₁, 𝑡₂, … , 𝑡_𝑛, and deadlines 𝑑₁, 𝑑₂, … , 𝑑_𝑛 Output: the schedule that minimizes the maximum lateness

Possible Greedy Choices

• Idea

• Earliest-deadline-first w/o idle time?

• Greedy algorithm

Min-Lateness(n, t[], d[])

sort tasks by deadlines s.t. d[1]≤d[2]≤ ...≤d[n]

ct = 0 // current time for j = 1 to n

assign job j to interval (ct, ct + t[j]) s[j] = ct

f[j] = s[j] + t[j]

ct = ct + t[j]

return s[], f[]

Scheduling to Minimize Lateness Problem

Input: 𝑛 tasks with their processing time 𝑡₁, 𝑡₂, … , 𝑡_𝑛, and deadlines 𝑑₁, 𝑑₂, … , 𝑑_𝑛 Output: the schedule that minimizes the maximum lateness

Prove Correctness – Greedy-Choice Property

• Greedy choice: first select the task with the earliest deadline

• Proof via contradiction

• Assume that there is no OPT including this greedy choice

• If OPT processes 𝑎₁ as the 𝑖-th task (𝑎_𝑘), we can switch 𝑎_𝑘 and 𝑎₁ into OPT’

• The maximum lateness must be equal or lower → 𝐿 OPT′ ≤ 𝐿 OPT exchange argument

Scheduling to Minimize Lateness Problem

Input: 𝑛 tasks with their processing time 𝑡₁, 𝑡₂, … , 𝑡_𝑛, and deadlines 𝑑₁, 𝑑₂, … , 𝑑_𝑛 Output: the schedule that minimizes the maximum lateness

Prove Correctness – Greedy-Choice Property

•

𝑎_𝑘 𝑎₁

L(OPT, k)

𝑎₁ 𝑎_𝑘

L(OPT’, 1) L(OPT’, k) L(OPT, 1) OPT

OPT’

If 𝑎_𝑘 is not late in OPT’: If 𝑎_𝑘 is late in OPT’:

Generalization of this property?

Scheduling to Minimize Lateness Problem

Input: 𝑛 tasks with their processing time 𝑡₁, 𝑡₂, … , 𝑡_𝑛, and deadlines 𝑑₁, 𝑑₂, … , 𝑑_𝑛 Output: the schedule that minimizes the maximum lateness

Prove Correctness – No Inversions

• There is an optimal scheduling w/o inversions given 𝑑

≤ 𝑑

≤ ⋯ ≤ 𝑑

• 𝑎

and 𝑎

are inverted if 𝑑

< 𝑑

but 𝑎

is scheduled before 𝑎

• Proof via contradiction

• Assume that OPT has 𝑎

and 𝑎

that are inverted

• Let OPT’ = OPT but 𝑎

and 𝑎

are swapped

• OPT’ is equal or better than OPT → 𝐿 OPT′ ≤ 𝐿 OPT

Scheduling to Minimize Lateness Problem

Input: 𝑛 tasks with their processing time 𝑡₁, 𝑡₂, … , 𝑡_𝑛, and deadlines 𝑑₁, 𝑑₂, … , 𝑑_𝑛 Output: the schedule that minimizes the maximum lateness

Prove Correctness – No Inversions

•

𝑎_𝑗 𝑎_𝑖

L(OPT, j)

𝑎_𝑖 𝑎_𝑗

L(OPT’, i) L(OPT’, j) L(OPT, i) OPT

OPT’

If 𝑎_𝑗 is not late in OPT’: If 𝑎_𝑗 is late in OPT’:

……

……

The earliest-deadline-first greedy algorithm is optimal

Optimal Solution

Greedy Choice

Subproble m Solution

= +

Scheduling to Minimize Lateness Problem

Input: 𝑛 tasks with their processing time 𝑡₁, 𝑡₂, … , 𝑡_𝑛, and deadlines 𝑑₁, 𝑑₂, … , 𝑑_𝑛 Output: the schedule that minimizes the maximum lateness

Concluding Remarks

• “Greedy”: always makes the choice that looks best at the moment in the hope that this choice will lead to a globally optimal solution

• When to use greedy

• Whether the problem has optimal substructure

• Whether we can make a greedy choice and remain only one subproblem

• Common for optimization problem

• Prove for correctness

• Optimal substructure

• Greedy choice property

Optimal Solution

Greedy Choice

Subproblem Solution

= +

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