and Outline Reading The Greedy Method
The Greedy Method Technique (5.1) Fractional Knapsack Problem (5.1.1) Task Scheduling (5.1.2) Minimum Spanning Trees (7.3) [future lecture]
The Greedy Method
1
The Greedy Method
2
The Greedy Method Technique
The greedy method is a general algorithm design paradigm, built on the following elements:
configurations: different choices, collections, or values to find objective function: a score assigned to configurations, which we want to either maximize or minimize
Making Change
Problem: A dollar amount to reach and a collection of coin amounts to use to get there. Configuration: A dollar amount yet to return to a customer plus the coins already returned Objective function: Minimize number of coins returned. Greedy solution: Always return the largest coin you can Example 1: Coins are valued $.32, $.08, $.01
Has the greedy-choice property, since no amount over $.32 can be made with a minimum number of coins by omitting a $.32 coin (similarly for amounts over $.08, but under $.32).
It works best when applied to problems with the greedy-choice property:
a globally-optimal solution can always be found by a series of local improvements from a starting configuration.
The Greedy Method 3
Example 2: Coins are valued $.30, $.20, $.05, $.01
Does not have greedy-choice property, since $.40 is best made with two $.20s, but the greedy solution will pick three coins (which ones?)
The Greedy Method 4
The Fractional Knapsack Problem
Given: A set S of n items, with each item i having Goal: Choose items with maximum total benefit but with weight at most W. If we are allowed to take fractional amounts, then this is the fractional knapsack problem.
In this case, we let xi denote the amount we take of item i Objective: maximize bi - a positive benefit wi - a positive weight
Example
Given: A set S of n items, with each item i having Goal: Choose items with maximum total benefit but with weight at most W. knapsack Solution: Items: Weight: Benefit: Value:
5 1 2 3 4 5
bi - a positive benefit wi - a positive weight
b (x / w )
iS i i i
The Greedy Method
4 ml $12 3
8 ml $32 4
2 ml $40 20
6 ml $30 5
1 ml $50 50 10 ml
Constraint:
xi W
iS
2
1 6 1
ml ml ml ml
of of of of
5 3 4 2
($ per ml)
The Greedy Method
6
The Fractional Knapsack Algorithm
Greedy choice: Keep taking item with highest value (benefit to weight ratio)
Since bi ( xi / wi ) = (bi / wi ) xi iS iS Run time: O(n log n). Why?
Algorithm fractionalKnapsack(S, W) Input: set S of items w/ benefit bi and weight wi; max. weight W Output: amount xi of each item i to maximize benefit w/ weight at most W for each item i in S xi 0 {value} vi bi / wi {total weight} w0 while w < W remove item i w/ highest vi xi min{wi , W - w} w w + min{wi , W - w}
7
Task Scheduling
Given: a set T of n tasks, each having: Goal: Perform all the tasks using a minimum number of machines.
Machine 3 Machine 2 Machine 1
A start time, si A finish time, fi (where si < fi)
Correctness: Suppose there is a better solution
there is an item i with higher value than a chosen item j, but xi<wi, xj>0 and vi<vj If we substitute some i with j, we get a better solution How much of i: min{wi-xi, xj} Thus, there is no better solution than the greedy one
1
2
3
4
5
6
7
8
9
The Greedy Method
The Greedy Method
8
Task Scheduling Algorithm
Greedy choice: consider tasks by their start time and use as few machines as possible with Algorithm taskSchedule(T) this order. Input: s...