1

CPSC 320: Intermediate AlgorithmDesign and Analysis

July 25, 2014

2

Course Outline

• Introduction and basic concepts

• Asymptotic notation

• Greedy algorithms

• Graph theory

• Amortized analysis

• Recursion

• Divide-and-conquer algorithms

• Randomized algorithms

• Dynamic programming algorithms

• NP-completeness

3

Dynamic Programming

4

Dynamic Programming

• Greedy algorithm: combines a choice with result of taking the choice

• What if right choice can’t be found easily?

5

Coin Change Problem

• Assume a set of possible coins, and an amount, find the smallest number of coins that add up to the exact amount

• Example: CoinChange(, )=

• Is a greedy solution always optimal?

• Example: GreedyCoinChange(, )=, optimal is

6

Coin Change Problem

• What is a subproblem that can be solved

• If an optimal solution for contains , then an optimal solution for is

• So the subproblem is solving for

• Let’s define the problem recursively:

• For each coin , calculate CoinChange(, )

• Select coin with smallest number of coins in subproblem

• Attach coin to and return

7

Coin Change Recursive

Algorithm CoinChangeRecursive(, ) – is set of coins, is amount

If Then

Return

,

For Each Do

If Then

CoinChangeRecursive(, )

If Then

Return

8

Recursive

• Is this efficient?

• Value for subproblems is calculated repeatedly

• Example: CoinChange(,), value for 86 is calculated on:

• CoinChange()

• CoinChange()

• CoinChange()

• CoinChange()

• Can we save the value (“memoize”) and reuse it?

• Can we calculate from the bottom-up?

9

Dynamic Programming Approach

Algorithm CoinChangeDP(, ) – is set of coins, is amount

For To Do

(length infinity)

For Each Do

If And Then

Return

10

Dynamic Programming Approach - Faster

Algorithm CoinChangeDP2(, ) – is set of coins, is amount

,

For To Do

,

For Each Do

If And Then

,

While Do

,

Return

11

Rod Cutting Problem

• Problem: given a rod of length and a table of prices for rods of length , determine the maximum revenue obtainable by cutting it in pieces

• Example:

• Full rod: revenue is 10

• Cut in 5 pieces of 1: revenue is 5

• Cut in pieces of 2 and 3: revenue is 13

• Problem can be solved with dynamic programming

• Choose each possible size, then solve for remainder

12

Rod Cutting Problem

Algorithm RodCutting(, ) – is size, is price array

For To Do

,

For To Do

If Then

,

While Do

,

Return

13

Weighted Interval Scheduling

• Problem: same as interval scheduling, but with preferences

• Given a set of intervals , and a weight function , return a compatible subset of with maximum total weight

• Greedy approach doesn’t work anymore

• Define the subproblem

• In optimal solution, if we remove last interval , remainder is optimal ending before

14

Weighted Interval Scheduling

Algorithm WeightedIntervalSched(, ) – is set of intervals, is weight function

Sort by increasing finishing time

For To Do

last event that ends before starts (0 if none)

If Then

, false

Else

, true

…

15

Weighted Interval Scheduling (cont.)

…

For DownTo Do

If Then

Add to

last event that ends before starts (0 if none)

Return

16

Subset Sum Problem

• Problem: given a set of weights, find the largest sum of weights below a limit

• Applications: get a bag as full as possible, fill a server with limited resource time as much as possible

• Extension: each weight has a value associated to it (knapsack problem)

• Now the subproblem is slightly different

• If an optimal solution contains weight for limit , then the remainder is optimal for limit

17

Subset Sum Problem

Algorithm SubsetSum(, ) – is array of weights, is limit

, false for

For To Do

For To Do

If Or Then

,

Else

,

,

While And is not false Do

, , ,

Return

18

Knapsack Problem

Algorithm Knapsack(, , ) – is array of weights, is array of values, is limit

, false for

For To Do

For To Do

If Or Then

,

Else

,

,

While And is not false Do

, , ,

Return