Prove the correctness

 Date: 11/16 (Thursday)

 Time: 14:20-17:20 (3 hours)

 Location: R103 (check the seat assignment before entering the room)

 Content

 Recurrence and Asymptotic Analysis

 Divide and Conquer

 Dynamic Programming

 Greedy

 Based on slides, assignments, and some variations (practice via textbook exercises)

 Format: Yes/No, Multiple-Choice, Short Answer, Prove/Explanation

 Easy: ~60%, Medium: ~30%, Hard: ~10%

 Close book

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1)

Formulate a problem

2)

Develop an algorithm

3)

Prove the correctness

4)

Analyze running time/space requirement

Design Step

Analysis Step



Analysis Skills

 Prove by contradiction

 Induction

 Asymptotic analysis

 Problem instance



Algorithm Complexity

 In the worst case, what is the growth of function an algorithm takes



Problem Complexity

 In the worst case, what is the growth of the function the optimal algorithm of the problem takes

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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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

Solve a problem recursively



Apply three steps at each level of the recursion

1. Divide the problem into a number of subproblems that are smaller instances of the same problem (比較小的 同樣問題)

2. Conquer the subproblems by solving them recursively If the subproblem sizes are small enough

 then solve the subproblems

 else recursively solve itself

3. Combine the solutions to the subproblems into the solution for the original problem

base case recursive case

1. Divide

2. Conquer

3. Combine

1.

Substitution Method (取代法)

 Guess a bound and then prove by induction

2.

Recursion-Tree Method (遞迴樹法)

 Expand the recurrence into a tree and sum up the cost 3.

Master Method (套公式大法/大師法)

 Apply Master Theorem to a specific form of recurrences

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

Analyze the problem about

 Whether the problem with small inputs can be solved directly

 Whether subproblem solutions can be combined into the original solution

 Whether the overall complexity is better than naïve

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

Dynamic programming, like the divide-and-conquer

method, solves problems by combining the solutions to subproblems

 用空間換取時間

 讓走過的留下痕跡



“Dynamic”: time-varying



“Programming”: a tabular method

Dynamic Programming: planning over time

 Divide-and-Conquer

 partition the problem into independent or disjoint subproblems

 repeatedly solving the common subsubproblems

 more work than necessary

 Dynamic Programming

 partition the problem into dependent or overlapping subproblems

 avoid recomputation

Top-down with memoization

Bottom-up method

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

Apply four steps

1. Characterize the structure of an optimal solution

2. Recursively define the value of an optimal solution

3. Compute the value of an optimal solution, typically in a bottom- up fashion

4. Construct an optimal solution from computed information

 Analyze the problem about

 Whether subproblem solutions can combine into the original solution

 When subproblems are overlapping

 Whether the problem has optimal substructure

 Common for optimization problem

 Two ways to avoid recomputation

 Top-down with memoization

 Bottom-up method

 Complexity analysis

 Space for tabular filling

 Size of the subproblem graph

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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

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local maximal

global maximal

local maximal

 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

Optimal Solution

Possible Case 1 Possible

Case 2

max /min

Subproblem Solution Subproblem

Solution

+

= +

Optimal Solution

Greedy Choice

Subproblem Solution

= +

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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 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

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

 Analyze the problem about

 Whether the problem has optimal substructure

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

 Common for optimization problem

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Optimal Solution

Greedy Choice

Subproblem Solution

= +

 True or False: To prove the correctness of a greedy algorithm, we must prove that every optimal solution contains our greedy choice.

 Given the following recurrence relation, provide a valid traversal order to fill the DP table or justify why no valid traversal exists.

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 Input: a sequence of integers 𝑙₀, 𝑙₁, … , 𝑙_𝑛

 𝑙_𝑖−1 is the number of rows of matrix 𝐴_𝑖

 𝑙_𝑖 is the number of columns of matrix 𝐴_𝑖

 Output: a order of performing 𝑛 − 1 matrix multiplications in the maximum number of operations to obtain the product of 𝐴₁𝐴₂… 𝐴_𝑛

𝐴₁ 𝐴₂ 𝐴₃ 𝐴₄ …… 𝐴_𝑛

𝐴₁.cols=𝐴₂.rows

𝑖 ≤ 𝑘 < 𝑗

 In a binary Huffman code, the number of symbols that can be used to form a codeword is 2 (0 and 1). We can use a 𝑛-ary Huffman code, in which the number of symbols that can be used to form a codeword is 𝑛.

1) Please prove that the decoding tree which represents an optimal code for a file must be a full 𝑛-ary tree, i.e., all non-leaf nodes in the tree have 𝑛 children.

2) Given the following information for a file, what are the lengths of the codewords for the characters ‘a’ and ‘c’ respectively, in a 3-ary

Huffman code derived for this file?

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Character a b c d e f g

Frequency 700 400 200 100 1300 2400 100

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