02-greedy - GreedyAlgorithms CS4102:Algorithms Spring2014...

Greedy Algorithms
CS 4102: Algorithms
Spring 2014
Mark Floryan
1

Overview
2

Inventory of greedy algorithms
3

Coin change

Knapsack algorithm

Interval scheduling

Prim’s MST

Kruskal’s MST

Dijkstra’s shortest path

Huffman Codes

Greedy Method: Overview
4

Optimization problems: terminology

A solution must meet certain constraints: A solution is
feasible

Example: All edges in solution are in graph, form a simple path

Solutions judged on some criteria: Objective function

Example: Sum of edge weights in path is smallest

One (or more) feasible solutions that scores best (by the
objective function) is the optimal solution(s)

Greedy Method: Overview
5

Greedy strategy:

Build solution by stages, adding one item to partial
solution found so far

At each stage, make locally optimal choice based on the
greedy rule (sometimes called the selection function )

Locally optimal, i.e. best given what info we have now

Irrevocable, a choice can’t be un-done

Sequence of locally optimal choices leads to globally
optimal solution (hopefully)

Must prove this for a given problem!

Approximation algorithms, heuristics

Proving them correct
6

Given a greedy algorithm, how do you show it is
optimal?

As opposed to other types of algorithms (divide-and-
conquer , etc.)

One common way is to compare the solution given
with an optimal solution

Making Change
7

Making change: algorithm description
8

The problem:

Give back the right amount of change, and…

Return the fewest number of coins!

Inputs: the dollar-amount to return

Also, the set of possible coins. (Do we have half-dollars?
That affects the answer we give.)

Output: a set of coins

Making change: algorithm solution
9

Problem description: providing coin change of a
given amount in the fewest number of coins

Inputs: the dollar-amount to return. Perhaps the
possible set of coins, if it is non-obvious.

Output: a set of coins that obtains the desired
amount of change in the fewest number of coins

Assumptions: If the coins are not stated, then they
are the standard quarter, dime, nickel, and penny.
All inputs are non-negative, and dollar amounts are
ignored.

Strategy: a greedy algorithm that uses the largest
coins first

Making Change
10

Inputs:

Value N of the change to be returned

An unlimited number of coins of values d1, d2,.., dk

Output: the smallest possible set of coins that sums
to N

Objective function? Smallest set

Constraints on feasible solutions? Must sum to N

Greedy rule: choose coin of largest value that is less
than N - Sum(coins chosen so far)

Always optimal? Depends on set of coin values

Algorithm for making change
11

This algorithm makes change for an amount A
using coins of denominations
denom [1] > denom [2] > ∙∙∙ > denom [ n ] = 1.