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COMP3600/6466 – Algorithms Algorithm Design Techniques

[Lev Intro of ch. 2-6, 8-10]

Hanna Kurniawati

https://cs.anu.edu.au/courses/comp3600/comp_3600_6466@anu.edu.au

So far …üThe Problemü Algorithm definition + Model of Computation

üAnalysis of Algorithmsü Asymptotic Notationsü Recurrence Analysis (Divide-and-Conquer)ü Probabilistic Analysis (Avg. case + Randomized Algorithms) ü Correctnessü Empirical Analysis

üAbstract data structuresü Binary Search Treeü Heapsü AVL Treeü Red Black Treeü Hash Tables

Algorithm Design Techniques• Brute Force• Divide & Conquer• Randomized Algorithm• Decrease & Conquer• Transform & Conquer• Dynamic Programming• Greedy• Iterative Improvement

Algorithm Design Techniques• Brute Force: When stuck, start here and improve• Divide & Conquer: Discussed a bit in recurrence analysis• Randomized Algorithm: Discussed a bit in prob. analysis • Decrease & Conquer: Very similar to divide & conquer, but

reduce to 1 smaller sub-problem (e.g., Insertion Sort)• Transform & Conquer: Simplify the problem. For instance, by

pre-sorting and changing data representation (use suitable abstract data structures)

• Dynamic Programming• Greedy• Iterative Improvement

COMP3600/6466 – Algorithms Dynamic Programming

[CLRS ch. 15 + 22.4]

Hanna Kurniawati

https://cs.anu.edu.au/courses/comp3600/comp_3600_6466@anu.edu.au

Topics• What is it?• Example: Fibonacci Sequence• How to develop DP algorithms?• Example: Shortest Path• Example: Chain matrix multiplication• Example: Longest Common Subsequence• Example: Decision-making under uncertainty

What is Dynamic Programming (DP)• DP is an approach to solve problems by trying all

possibilities while avoiding recomputation• Key idea: Remember solutions that have been

computed and reuse as much as possible

DP Requirements• Not all problems can be solved using DP. Can

only be applied to problems whose solutions can be constructed from solutions to its sub-problems• Specifically, two requirements:• Optimal substructure: Optimal solution to the

problem is formed by optimal solutions to sub-problems, which can be solved separately.• Overlapping sub-problems

DP – Trivia • Invented by Richard Bellman in the ‘50s• ”Programming” here has somewhat different

meaning than computer programming• Many stories as to what the word “Programming”

actually means and why it is used• Regardless, it is a powerful algorithm design

technique• Often used for solving optimization problem

Types of Dynamic Programming• Two types:• Top-down (memoization)• Start from the problem we want to solve• Solve sub-problems “reachable” (based on a

decomposition of the problem) from the problem and store the results (aka., made into a memo, memoization)• Reuse the stored solutions whenever possible• Bottom-up• Compute the sub-problems from small to large, starting from

the base, keep the results • Reuse whenever possible

TopicsüWhat is it?• Example: Fibonacci Sequence• How to develop DP algorithms?• Example: Shortest Path• Example: Chain matrix multiplication• Example: Longest Common Subsequence• Example: Decision-making under uncertainty

Example: Fibonacci Sequence• The nth Fibonacci number is

F(n) = F(n-1) + F(n-2)• Algorithm to output the nth Fibonacci number

NaiveFibonacci(n)If n ≤ 2

Return 1Else

Res = NaiveFibonacci(n-1) + NaiveFibonacci(n-2)Return Res

• Time complexity?• 𝑇 𝑛 = 𝑇 𝑛 − 1 + 𝑇 𝑛 − 2 + Θ 1 = Θ(2!)

Upper bound:𝑇 𝑛 ≤ 2𝑇 𝑛 − 1 + 𝑐

= 2 2𝑇 𝑛 − 2 + 𝑐 + 𝑐= 2.2(2𝑇 𝑛 − 3 + 𝑐) + 2𝑐 + 𝑐= 2.2.2…2. 𝑇 2 + 𝑐 ∑!"#$%& 2!

= 2$%' + 𝑐 2$%' − 1= O(2$)

𝑇 𝑛 = 𝑇 𝑛 − 1 + 𝑇 𝑛 − 2 + Θ 1

𝑛 − 2

Lower bound:𝑇 𝑛 ≥ 2𝑇 𝑛 − 2 + 𝑐

= 2 2𝑇 𝑛 − 4 + 𝑐 + 𝑐= 2.2(2𝑇 𝑛 − 6 + 𝑐) + 2𝑐 + 𝑐

= 2.2.2…2. 1 + 𝑐 ∑!"#!" %( 2!

= 2!" + 𝑐 2

!" − 1

= Ω(2$)

𝑛2

𝑇 𝑛 = 𝑇 𝑛 − 1 + 𝑇 𝑛 − 2 + Θ 1 = Θ 2,

Can we do better?• Yes, using DP• The problem satisfies the two requirements

for DP:• Optimal sub-structure: In this case, solution to

Fibonacci(n) is formed by solutions to Fibonacci(n-1) and Fibonacci(n-2), and the sub-problems can be solved separately • Overlapping sub-problems

Overlapping Sub-problems

NaiveFibonacci performs too many repeated calculations

Example: Memoization• Top-down DP (sometimes called memoization)

TopDownDP-Fibonacci(n)Fibo[1..n] = -1Return RecurseFibo(n, Fibo)

RecurseFibo(n, Fibo)If (Fibo[n] != -1)

Return Fibo[n]If n ≤ 2

F = 1Else

F = RecurseFibo(n-1, Fibo) + RecurseFibo(n-2, Fibo)Fibo[n] = FReturn F

Time complexity? Θ(𝑛)

Example: Bottom-Up• Bottom-up DP

BottomUp-Fibonacci(n)Fibo[1] = 1 Fibo[2] = 1 For i = 3 to n

Fibo[i] = Fibo[i-1] + Fibo[i-2] Return Fibo[n]

• Time complexity? Θ(𝑛)

Bottom-Up DP• No recursion• Although asymptotic time complexity is the same as top-down, in

practice bottom-up is faster because it reduces the number of function calls

• How to decide which sub-problem to compute first? • Reverse topological order of the sub-problem graph• Sub-problem graph encodes dependencies of the sub-problems• Vertices: Sub-problems• An edge from u to v: Solving the sub-problem represented by u

requires direct solving of the sub-problem represented by v• For DP to work this graph must be a directed acyclic graph (DAG)

Topological Sort of DAG [CLRS 22.4]• Suppose G(V, E) is a DAG• Topological sort of G is a linear ordering of all its

vertices such that if G contains a directed edge (u, v) then u appears before v in the ordering • Use DFS that maintain time stamp• Discovery time of vertex v: The time v is first visited• Finishing time of vertex v: The time all vertices in E

adjacent to v have been visited

[CLRS p.604]

Example

• Start from u

Topological Sort of DAG [CLRS 22.4]

• Example:

Time complexity: Θ( 𝑉 + 𝐸 )

Now, back to Fibonacci DP Example• Sub-problem graph:

What if sub-problem graph has cycle?• Cause the DP to have infinite loop•What can be done?

TopicsüWhat is it?üExample: Fibonacci Sequence• How to develop DP algorithms?• Example: Shortest Path• Example: Chain matrix multiplication• Example: Longest Common Subsequence• Example: Decision-making under uncertainty

Next: How to develop DP algorithms &

More examples