CS 4407
Algorithms
Greedy Algorithms
Prof. Gregory Provan
Department of Computer Science
University College Cork
1
Greedy Algorithms
When are greedy algorithms optimal?
 Must satisfy the optimal substructure
property

– Optimal sub-problem must participate in optimal
problem
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
What makes a greedy algorithm?

Feasible
– Has to satisfy the problem’s constraints

Locally Optimal
– The greedy part
– Has to make the best local choice among all feasible choices available on
that step
• If this local choice results in a global optimum then the problem has optimal
substructure

Irrevocable
– Once a choice is made it can’t be un-done on subsequent steps of the
algorithm

Simple examples:
– Playing chess by making best move without lookahead
– Giving fewest number of coins as change

Simple and appealing, but don’t always give the best solution
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Abstract Greedy Algorithm
Inputs: heuristic H, Loop Invariant LI, Cost
Function CF
 Greedy (list)

– L Sort list by H, CF, R  //R is result//
– Do while L
• R  R  head(L) if addition does not violate LI
• L  L - head(L)
– End do
– Return R
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Sample MST
6
4
9
5
14
2
10
15
3
8
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Spanning Tree

Definition
– A spanning tree of a graph G is a tree (acyclic) that
connects all the vertices of G once
• i.e. the tree “spans” every vertex in G
– A Minimum Spanning Tree (MST) is a spanning tree on a
weighted graph that has the minimum total weight
w(T ) 
 w(u, v ) such that w(T) is minimum
u, v T
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Sample MST

Which links to make this a MST?
6
4
9
5
14
2
10
15
3
8
Optimal substructure: A subtree of the MST must in turn be a
MST of the nodes that it spans.
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
MST Claim

Claim: Say that M is a MST
– If we remove any edge (u,v) from M then this
results in two trees, T1 and T2.
– T1 is a MST of its subgraph while T2 is a MST of
its subgraph.
– Then the MST of the entire graph is T1 + T2 + the
smallest edge between T1 and T2
– If some other edge was used, we wouldn’t have
the minimum spanning tree overall
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Greedy Algorithm

We can use a greedy algorithm to find the
MST.
– Two common algorithms
• Kruskal
• Prim
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Example: Kruskal’s MST Algorithm

Idea: Greedily construct the MST
– Go through the list of edges and make a forest
that is a MST
– At each vertex, sort the edges
– Edges with smallest weights examined and
possibly added to MST before edges with higher
weights
– Edges added must be “safe edges” that do not
ruin the tree property.
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Scheduling Problem Statement
INPUT:
A set S = { I1, I2, ···, In} of n event time-intervals Ik = sk , fk, k =1..n,
where sk = start time of Ik ,
fk = finishing time of Ik ,
( sk < fk ).
OUTPUT: A maximum cardinality subset C  S of mutually compatible intervals
(i.e., no overlapping pairs).
Example:
S = the intervals shown below,
C = the blue intervals,
|C| =4.
C is not the unique optimum.
Can you spot another optimum solution?
time
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Task Scheduling

Given: a set T of n tasks, each having:
– A start time, si
– A finish time, fi (where si < fi)

Goal: Perform all the tasks using a minimum number of
“machines.”
Machine 3
Machine 2
Machine 1
1
2
3
4
5
6
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
7
8
9
12
Task Scheduling Algorithm
Greedy choice: consider tasks by their
start time and use as few machines as
possible with this order.
– Run time: O(n log n).
 Correctness: Suppose there is a better
schedule.
– We can use k-1 machines
– The algorithm uses k
– Let i be first task scheduled on
machine k
– Task i must conflict with k-1 other
tasks
– K mutually conflict tasks
– But that means there is no nonconflicting schedule using k-1
machines

Algorithm taskSchedule(T)
Input: set T of tasks w/ start time si
and finish time fi
Output: non-conflicting schedule
with minimum number of machines
m0
{no. of machines}
while T is not empty
remove task i w/ smallest si
if there’s a machine j for i then
schedule i on machine j
else
mm+1
schedule i on machine m
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
13
Example

Given: a set T of n tasks, each having:
– A start time, si
– A finish time, fi (where si < fi)
– [1,4], [1,3], [2,5], [3,7], [4,7], [6,9], [7,8] (ordered by start)

Goal: Perform all tasks on min. number of machines
Machine 3
Machine 2
Machine 1
1
2
3
4
5
6
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
7
8
9
14
Event Scheduling Example
A banquet hall manager has received a list of reservation requests
for the exclusive use of her hall for specified time intervals
She wishes to grant the maximum number of reservation requests
that have no time overlap conflicts
Help her select the maximum number of conflict free time intervals
Activity-Selection Problem

Problem: get your money’s worth out of a
carnival
– Buy a wristband that lets you onto any ride
– Lots of rides, each starting and ending at
different times
– Your goal: ride as many rides as possible
• Another, alternative goal that we don’t solve here:
maximize time spent on rides

Welcome to the activity selection problem
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Activity-Selection

Formally:
– Given a set S of n activities
si = start time of activity i
fi = finish time of activity i
– Find max-size subset A of compatible activities
3
1

2
4
6
5
Assume (wlog) that f1  f2  …  fn
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Activity Selection Problem

Problem:
– Schedule an exclusive resource in competition with other
entities.

Formal Definition:
– Set S={1,2, … n} of activities.
– Each activity has a start time si and a finish time fj, where
si <fj.
– Activities i and j are compatible if they do not overlap.

The activity selection problem
– select a maximum-size set of mutually compatible
activities.
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Greedy Choice Property

Activity selection problem exhibits the
greedy choice property:
– Locally optimal choice  globally optimal sol’n

Theorem: if S is an activity selection problem
sorted by finish time, then  optimal solution
A  S such that {1}  A
– Sketch of proof: if  optimal solution B that does
not contain {1}, can always replace first activity
in B with {1} (Why?). Same number of activities,
thus optimal.
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Activity Selection: A Greedy Algorithm

So actual algorithm is simple:
– Sort the activities by finish time
– Schedule the first activity
– Then schedule the next activity in sorted list
which starts after previous activity finishes
– Repeat until no more activities

Intuition is even more simple:
– Always pick the shortest ride available at the
time
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Greedy Activity Selection

Just march through each activity by finish
time and schedule it if possible:
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Activity Selection Example
Schedule job 1, then try rest: (end up with 1, 4, 8):
T=1
T=2
T=3
T=4
1
1
1
1
Runtime?
T=5
T=6
T=7
T=8
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
T=9
T=10
T=11
T=12
T=13
Activity Selection: Optimal Substructure

Let k be the minimum activity in A (i.e., the
one with the earliest finish time). Then A - {k}
is an optimal solution to S’ = {i  S: si  fk}
– In words: once activity #1 is selected, the
problem reduces to finding an optimal solution
for activity-selection over activities in S
compatible with #1
– Proof: if we could find optimal solution B’ to S’
with |B| > |A - {k}|,
• Then B U {k} is compatible
• And |B U {k}| > |A|
CS 4407, Algorithms
University College Cork,
Gregory M. Provan
Some Greedy Heuristics
Greedy iteration step:
From among undecided intervals, select the interval I that looks BEST.
Commit to I if it’s conflict-free
(i.e., doesn’t overlap with the committed ones so far).
Reject I
otherwise.
Greedy 1: BEST = earliest start time (min sk).
Greedy 2: BEST = latest finishing time (max fk).
Greedy 3: BEST = shortest interval (min fk – sk).
Greedy 4: BEST = overlaps with fewest # of undecided intervals.
CS 4407, Algorithms
University College Cork,
Gregory M. Provan