Slides credited from Hsueh-I Lu, Hsu-Chun Hsiao, & Michael Tsai

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 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

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

Do not focus on “specific algorithms”



But “some strategies” to “design” algorithms



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



Second Skill: Dynamic Programming (動態規劃)



Third Skill: Greedy (貪婪法則)

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Textbook Chapter 16 – Greedy Algorithms

Textbook Chapter 16.2 – Elements of the greedy strategy

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 always makes the choice that looks best at the moment

 makes a locally optimal choice in the hope that this choice will lead to a globally optimal solution



not always yield optimal solution; may end up at local optimal

Greedy: move towards max gradient and hope it is global maximum local maximal

global maximal

local maximal

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

Dynamic Programming

 has optimal substructure

 make an informed choice after getting optimal solutions to subproblems

 dependent or overlapping subproblems



Greedy Algorithms

 has optimal substructure

 make a greedy choice before solving the subproblem

 no overlapping subproblems

Each round selects only one subproblem

The subproblem size decreases

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Optimal Solution

Possible Case 1 Possible

Case 2

Possible Case k

max /min

Subproblem Solution Subproblem

Solution

Subproblem Solution

+ +

+

=

Optimal Solution

Greedy Choice

Subproblem Solution

= +

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1. Cast the optimization problem as one in which we make a choice and remain one subproblem to solve

2. Demonstrate the optimal substructure



Combining an optimal solution to the subproblem via greedy can arrive an optimal solution to the original problem

3. Prove that there is always an optimal solution to the

original problem that makes the greedy choice

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To yield an optimal solution, the problem should exhibit

1. Optimal Substructure : an optimal solution to the problem contains within it optimal solutions to subproblems

2. Greedy-Choice Property : making locally optimal (greedy) choices leads to a globally optimal solution

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

Optimal Substructure : an optimal solution to the problem contains within it optimal solutions to subproblems



Greedy-Choice Property : making locally optimal (greedy) choices leads to a globally optimal solution

 Show that it exists an optimal solution that “contains” the greedy choice using exchange argument

 For any optimal solution OPT, the greedy choice 𝑔 has two cases

 𝑔 is in OPT: done

 𝑔 not in OPT: modify OPT into OPT’ s.t. OPT’ contains 𝑔 and is at least as good as OPT

OPT OPT’

𝑔

 If OPT’ is better than OPT, the property is proved by contradiction

 If OPT’ is as good as OPT, then we showed that there exists an optimal solution containing 𝑔 by construction

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Textbook Chapter 16.1 – An activity-selection problem

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

Input: 𝑛 activities with start times 𝑠

_𝑖

and finish times 𝑓

_𝑖

(the activities are sorted in monotonically increasing order of finish time 𝑓

₁

≤ 𝑓

₂

≤ ⋯ ≤ 𝑓

_𝑛

)

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Output: the maximum number of compatible activities

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Without loss of generality: 𝑠

₁

< 𝑠

₂

< ⋯ < 𝑠

_𝑛

and 𝑓

₁

< 𝑓

₂

< ⋯ < 𝑓

_𝑛

 大的包小的則不考慮大的  用小的取代大的一定不會變差

1 2 3 4 5 6

activity index

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

Subproblems

 WIS(i): weighted interval scheduling for the first 𝑖 jobs

 Goal: WIS(n)

 Dynamic programming algorithm

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i 0 1 2 3 4 5 … n

M[i]

Weighted Interval Scheduling Problem

Input: 𝑛 jobs with 𝑠_𝑖, 𝑓_𝑖, 𝑣_𝑖 , 𝑝(𝑗) = largest index 𝑖 < 𝑗 s.t. jobs 𝑖 and 𝑗 are compatible Output: the maximum total value obtainable from compatible

Set 𝑣_𝑖 = 1 for all 𝑖 to formulate it into the activity-selection problem

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 Dynamic programming

 Optimal substructure is already proved

 Greedy algorithm

Activity-Selection Problem

Input: 𝑛 activities with 𝑠_𝑖, 𝑓_𝑖 , 𝑝(𝑗) = largest index 𝑖 < 𝑗 s.t. 𝑖 and 𝑗 are compatible Output: the maximum number of activities

select the 𝑖-th activity

Why does the 𝑖-th activity must appear in an OPT?

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

Goal:



Proof

 Assume there is an OPT solution for the first 𝑖 − 1 activities (𝑀_𝑖−1)

 𝐴_𝑗 is the last activity in the OPT solution 

 Replacing 𝐴_𝑗 with 𝐴_𝑖 does not make the OPT worse

time 14

1 2 : i i - 1

:

activity index

1 2 3 4 5 6 7 8 9

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Act-Select(n, s, f, v, p) M[0] = 0

for i = 1 to n if p[i] >= 0

M[i] = 1 + M[p[i]]

return M[n]

Find-Solution(M, n) if n = 0

return {}

return {n} ∪ Find-Solution(p[n])

Activity-Selection Problem

Input: 𝑛 activities with 𝑠_𝑖, 𝑓_𝑖 , 𝑝(𝑗) = largest index 𝑖 < 𝑗 s.t. 𝑖 and 𝑗 are compatible Output: the maximum number of activities

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Textbook Exercise 16.1

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Input: 𝑛 dollars and unlimited coins with values 𝑣

_𝑖

(1, 5, 10, 50)

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Output: the minimum number of coins with the total value 𝑛

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Cashier’s algorithm: at each iteration, add the coin with the largest value no more than the current total

Does this algorithm return the OPT?

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Subproblems

 C(i): minimal number of coins for the total value 𝑖

 Goal: C(n)

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Coin Changing Problem

Input: 𝑛 dollars and unlimited coins with values 𝑣_𝑖 (1, 5, 10, 50) Output: the minimum number of coins with the total value 𝑛

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Suppose OPT is an optimal solution to C(i), there are 4 cases:

 Case 1: coin 1 in OPT

 OPT\coin1 is an optimal solution of C(i – v₁)

 OPT\coin2 is an optimal solution of C(i – v₂)

 OPT\coin3 is an optimal solution of C(i – v₃)

 OPT\coin4 is an optimal solution of C(i – v₄)

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

Greedy choice: select the coin with the largest value no more than the current total



Proof via contradiction (use the case 10 ≤ 𝑖 < 50 for demo)

 Assume that there is no OPT including this greedy choice (choose 10)

 all OPT use 1, 5, 50 to pay 𝑖

 50 cannot be used

 #coins with value 5 < 2  otherwise we can use a 10 to have a better output

 #coins with value 1 < 5  otherwise we can use a 5 to have a better output

 We cannot pay 𝑖 with the constraints (at most 5 + 4 = 9)

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