Sometimes it Pays to be Greedy:
Greedy Algorithms in Economic
Epidemiology
Fred Roberts, DIMACS
1
Optimization Problems in Economic
Epidemiology
Many problems in Economic Epi can be
formulated as optimization problems: Find a
solution that maximizes or minimizes some value.
•Find the optimal location for a hospital.
•Find the optimal assignment of health care
workers to jobs.
•Optimize investment in health care supplies.
•Minimize the total cost of a series of medical tests
or public health interventions.
•Control an outbreak with as small an investment
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in vaccines as possible.
Greedy Algorithms
Often, the simplest approach to an optimization
problem is a greedy algorithm:
Choose the best (cheapest, highest-rated,…)
available alternative at each step.
In general, greedy algorithms will find locally
optimal solutions, but not globally optimal ones.
Global optimum
Local optimum
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Greedy Algorithms
•We give examples from Economic Epi:
– Some where a greedy solution achieves a
global optimum
– Others where it doesn’t, but we can either
make modifications or get a bound on how far
from optimal we are.
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Outline
1. Four Applications of Classical Operations
Research Methods
2. Vaccination Strategies for Control of a Highly
Infectious Disease Spreading through a Social
Network
3. Algorithms for Sequential Public Health or
Medical Decision Making
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Classic Example I: Assigning Health
Care Workers to Jobs
•n workers W1, W2, …, Wn
•m jobs J1, J2, …, Jm
•We know which workers are qualified to do which
jobs and the cost of using each worker.
•Goal: assign workers to jobs they are qualified for,
each to at most one job, filling as many jobs as
possible, and among all ways of filling as many
jobs as possible, find the way to do it with
minimum total cost.
•This is known as the Minimum Cost Assignment
Problem
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Assigning Health Care Workers to Jobs
Greedy Algorithm:
•At each stage, add the least expensive worker to
those getting job assignments if there is an
acceptable (feasible) assignment using that worker
and all those who have previously been assigned
jobs, switching job assignments if necessary.
•The greedy algorithm always gives an optimal
job assignment.
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Classic Example II: Investing in Health
Care Options
•Suppose we are faced with a selection of health care
options in which to invest.
•Option i has an estimated cost ci and an estimated value
vi.
AIDS Prevention
– Alternative health care facilities
Options
– Alternative supplies for a clinic Option 1: Condoms
Option 2: Educational
– Alternative research programs
Posters
•Problem: Determine which ones to
Option 3: Clean Needles
invest in so that the total cost is
to Distribute
within budget and the total value is
Option 4: Testing
Option 5: Funded
as large as possible.
Researchers
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Investing in Health Care Options
Knapsack Problem
Maximize i vixi
Subject to i cixi ≤ B
where xi = number of items i chosen
Variants
xi = 0 or 1
xi  {0, 1, …, bi} Bounded Knapsack Problem
xi is any integer Unbounded Knapsack Problem
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Investing in Health Care Options
Greedy Algorithm
•Due to George Dantzig 1957
•Sort items in decreasing order of value
per unit cost: vi/ci
•Pick as many copies of the first item as possible until no
more are possible or until one more would violate i cixi ≤
B.
•Continue in the same way with the second item, then the
third, etc.
•For the unbounded knapsack problem, this algorithm
always achieves at least half of the value obtained by the
optimal solution.
•Is this acceptable?
•It depends on the application: do you need a fast decision?
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Classic Example III: Locating Health
Care Facilities
•We have a number of users of a planned set of
health care facilities. Where do we put the facilities
and how do we assign a user to a facility?
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Locating Health Care Facilities
•There are two costs:
fi = cost of opening a facility at i
cij = cost of sending user at j to facility at i
•Let F = sum of fi over all opened facilities.
•Let C = sum of costs cij over all users j.
•We want to minimize F+ C.
•Assume that there is no limit to the number of facilities
we might open.
•However, there is a tradeoff between increased cost of
more facilities and decreased cost of getting to a nearby
facility.
•This is the Uncapacitated Facility Location Problem
•Uncapacitated since we have no limit on the number of
facilities.
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Locating Health Care Facilities
Cost 0.5 f
1
1
a Cost 5
1
Cost 4 e
Numbers on
edges are costs
of moving
along the edge
b Cost 3
1
1
c Cost 6
d
1
Cost 1
Given users at red circled locations, where do we
locate facilities to minimize F+C?
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Locating Health Care Facilities
Greedy Algorithm
•Due to Charikar and Guha (2004)
•First find a preliminary solution S.
•Order the nodes of the network in order of
increasing cost of locating a facility at the node.
•Choose p so that if S is the set of the first p
facilities, then the cost F + C associated with S is
as small as possible.
•Modify the preliminary solution in a series of
steps by randomly selecting nodes to add to S and
subsets of nodes to remove from S.
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Locating Health Care Facilities
•Charikar and Guha show that, given , the
algorithm is guaranteed to achieve a cost F+C that
is at most
2F* + 3C* + (F* + C*)
in at most O(nlog(n/ ) steps, where
F* and C* are costs associated with an arbitrary
optimal solution.
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Classic Example IV: Rerouting
Emergency Vehicles in Case of Floods
•New initiative in Climate and Health at DIMACS.
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Extreme Events due to Global Warming
•We anticipate an increase in number and severity of
extreme events due to global warming.
•More heat waves.
•More floods, hurricanes.
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Extreme Events due to Global Warming
Areas of Emphasis in DIMACS Climate & Health
Initiative
•Evacuations during extreme heat events
•Rolling power blackouts during extreme heat events
•Pesticide applications after floods
•Emergency vehicle rerouting after floods
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Minimum Spanning Tree
Problem
2
20
•
•
•
10
26
14
15
22
8
28
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A spanning tree is a tree using the edges of the graph and
containing all of the nodes.
It is minimum if the sum of the numbers on the edges used is
as small as possible.
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Red edges define a minimum spanning tree.
Minimum Spanning Tree
Problem
• Minimum spanning trees arise in many applications.
• One example: Given a road network, find usable roads
that allow you to go from any node to any other node,
minimizing the lengths of the roads used.
• This problem arises in the DIMACS Climate and Health
project: Find a usable road network for emergency
vehicles in case extreme events leave flooded roads.
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Minimum Spanning Tree
Problem
• Kruskal’s algorithm (greedy algorithm):
– List the edges in order of increasing weight.
– For each edge, greedily include it if it does not
form a cycle with edges already chosen.
– Stop when no more edges can be included.
• Kruskal’s algorithm gives an optimal solution.
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Vaccination Strategies for Control of
a Highly Infectious Disease Spreading
through a Social Network
Work with Paul Dreyer and
Stephen Hartke
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The Model: Moving From State to
State
Social Network = Graph
Nodes = People
Edges = contact
t=0,1,2, …
SI model
Once in infected state, stay there.
Times are discrete: t = 0, 1, 2, …
= infected
= susceptible
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Disease Process
Highly Infectious Disease: You change your state
from to at time t+1 if at least one of your
neighbors have state at time t. You never leave
state .
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Vaccination Strategies
Let’s say you have a limited amount of vaccine
available each time period, say v doses.
Whom should you vaccinate?
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Vaccination Strategies
More precisely: What vaccination strategy minimizes
number of people ultimately infected if a disease
breaks out with one infection?
Sometimes called the firefighter problem:
alternate fire spread and firefighter placement.
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Some Results on the Firefighter
Problem
Thanks to
Kah Loon Ng
DIMACS
for some of the following slides,
slightly modified by me
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Three doses of vaccine per time period (v = 3)
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v=3
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v=3
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v=3
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v=3
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v=3
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v=3
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v=3
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Some questions that can be asked (but
not necessarily answered!)
• Can the fire be contained?
• How many time steps are required before fire is
contained?
• How many firefighters per time step are necessary?
• What fraction of all nodes will be saved (burnt)?
• Does where the fire breaks out matter?
• Fire starting at more than 1 node?
• Consider different graphs. Construction of
(connected) graphs to minimize damage.
• Complexity/Algorithmic issues
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Containing Fires in Infinite Grids Ld
Fire starts at only one node:
d= 1: Trivial.
d = 2: Impossible to contain the fire with 1
firefighter per time step
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Containing Fires in Infinite Grids Ld
d = 2: Two firefighters per time step needed to contain the
fire.
8 time steps
18 burnt
nodes
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Containing Fires in Infinite Grids Ld
Wang and Moeller (2002): If d  3, 2d-1 firefighters per
time step are sufficient to contain any outbreak starting
at a single node.
Hartke 2004: If d  3, 2d – 2 firefighters per time step
are not enough to contain an outbreak in Ld.
Thus, 2d – 1 firefighters per time step is the minimum
number required to contain an outbreak in Ld and
containment can be attained in 2 time steps.
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Firefighting on Trees
Epidemic starts at the root. Number doses of vaccine: v = 140
Firefighting on Trees
Greedy algorithm:
For each node x, define
weight (x) = number descendants of x + 1
Algorithm: At each time step, place
firefighter at node that has not been saved
such that weight (x) is maximized.
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Firefighting on Trees
26
22
Firefighting on Trees:
12
8
9
2
6
1
1
3
1 1
7
5
1 3
1
6
11
1
4
1
2 1
2
3
1
1
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Firefighting on Trees
Greedy
=7
Optimal
=9
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Firefighting on Trees
Theorem (Hartnell and Li, 2000): For any tree
with one fire starting at the root and one
firefighter to be deployed per time step, the
greedy algorithm always saves more than ½ of
the nodes that any algorithm saves.
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Algorithms for Sequential Public
Health or Medical Decision Making
•A patient presents with certain symptoms.
•Which test do we do first?
•On the basis of the outcome of the first test,
which test do we do next?
•Tests are expensive.
•So are false positive and false negative results.
•“Cost” is a combination of cost of testing and
cost of false results.
•In what order should we do tests in order to
minimize total “cost”?
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Algorithms for Sequential Public
Health or Medical Decision Making
•We have several potential interventions for a
public health crisis.
•Assume funds limit us to one intervention at a time.
•Which intervention do we invest in first?
•On the basis of the outcome of the first intervention,
which do we launch next?
•Interventions are expensive.
•So are false positive and false negative assessments of
the outcome of our interventions.
•“Cost” is a combination of cost of the intervention and
cost of false results.
•In what order should we launch the interventions in 46
order to minimize total “cost”?
Sequential Diagnosis Problem
•Such sequential diagnosis problems arise in many
areas:
–Communication networks (testing connectivity,
paging cellular customers, sequencing tasks, …)
–Manufacturing (testing machines, fault diagnosis,
routing customer service calls, …)
–Inspecting containers at ports
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Sequential Decision Making Problem
•A physician is looking to determine if a patient has
disease x. The doctor has a variety of tests to choose
from. In the end, the patient is to be classified into one
of several categories.
•Simple case: 0 = “doesn’t have the disease”, 1 =
“does have the disease”
•Testing scheme: specifies which tests are to be made
based on previous observations
Blood test
endoscopy
MRI
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Stress test
Sequential Decision Making Problem
•We are looking to determine if an epidemic can be
controlled. We have a variety of interventions to choose
from. In the end, the epidemic is to be classified into one
of several categories.
•Simple case: 0 = controllable, 1 = not controllable
•Intervention scheme: specifies which interventions are to
be made based on assessments of previous interventions.
•H1N1 Virus.
Intervention 1: Close Schools if 15% absenteeism
Intervention 2: Close Airports
Intervention 3: Tamiflu to health care workers
Intervention 4: Invest in vaccine.
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Sequential Decision Making Problem
•0’s and 1’s suggest binary digits (bits)
•Bit String: A sequence of bits:
0001, 1101, …
•Boolean Function: A function that assigns to
each bit string a 0 or a 1.
Bit String x
00
01
10
11
B(x)
1
0
0
1
B(00) = 1, B(10) = 0
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Sequential Decision Making Problem
•Following in language of medical testing.
•Patients have attributes related to the disease
being tested for, each in a number of states
•Sample attributes:
–White blood cell count
–PSA
–Creatinin clearance
–Fever > 40 degrees Centigrade
–Severe cough
–Severe fatigue
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Sequential Decision Making Problem
•Simplest Case: Attributes are in state 0 or 1
(absent or present, higher than threshold or not)
•Then: Patient corresponds to a bit string like
011001
•So: Classification is a decision function F that
assigns each bit string to a category.
011001
F(011001)
If attributes 2, 3, and 6 are present, assign
patient to category F(011001).
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Sequential Decision Making Problem
•If there are two categories, 0 and 1 (“has disease”
or “doesn’t have disease”), the decision function F
is a Boolean function.
Example:
F(000) = F(111) = 1, F(abc) = 0 otherwise
This classifies a patient as positive (sick with the
disease) iff he has none of the attributes or all of
them.
1=
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Binary Decision Tree Approach
•Tests measure presence/absence of attributes: so 0
or 1
•Use two categories: 0, 1 (has disease or doesn’t)
•Binary Decision Tree:
–Nodes are tests or categories
–Two arcs exit from each test node, labeled left
and right.
–Take the right arc when test says the attribute
is present, left arc otherwise
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Binary Decision Tree Approach
•Reach category 1 from
the root by:
a0 L to a1 R a2 R 1 or
a0 R a2 R1
•Patient classified in
category 1 iff he has
a1 and a2 and not a0 or
a0 and a2 and possibly a1.
•Corresponding Boolean
function:
• F(111) = F(101) = F(011) = 1,
F(abc) = 0 otherwise.
Figure 1
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Binary Decision Tree Approach
•This binary decision
tree corresponds to the
same Boolean function
F(111) = F(101) =
F(011) = 1, F(abc) = 0
otherwise.
However, it has one less
test node ai. So, it is
more efficient if all tests
are equally costly and
equally likely.
Figure 2
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Binary Decision Tree Approach
•The problem of finding the “least cost” binary
decision tree for a given Boolean function is very
hard (NP-complete).
•For small n = number of attributes, can try to
solve it by trying all possible binary decision trees
corresponding to the Boolean function F.
•Even for n = 4, not practical.
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Binary Decision Tree Approach
Promising Approach: Special Assumptions about
Boolean Function F
•Stroud and Saeger (Los Alamos National Lab)
enumerate all “complete, monotone” Boolean
functions and calculate the least expensive
corresponding binary decision trees.
•Their method practical for n up to 4, not n = 5.
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Binary Decision Tree Approach
Monotone Boolean Functions:
•Given two bit strings x1x2…xn, y1y2…yn
•Suppose that xi  yi for all i implies that
F(x1x2…xn)  F(y1y2…yn).
•Then we say that F is monotone.
Incomplete Boolean Functions:
•Boolean function F is incomplete if F can be
calculated by finding at most n-1 attributes and
knowing the value of the input string on those
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attributes
Complete, Monotone Boolean
Functions
Combinatorial Explosion!
2
2
No. BDTs
from CM
Bool. Funs.
4
3
9
60
4
114
11,808
1,079,779,602
5
6,894
63,515,920
5 x 1018
No. of
attributes
No. CM
Bool. Funs.
No. BDTs
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Cost Functions
•Stroud-Saeger method applies to more
sophisticated cost models, not just cost =
number of tests in the BDT.
•Cost Complication: How many nodes of the
decision tree are actually visited during average
procedure toward diagnosis? Depends on
“distribution” of the disease.
•Answer can also depend on probability of test
errors and probability a patient has the disease.
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Cost Functions:
Unit Costs
Tree Utilization
•Assume we are given probability of test errors
for different tests and a priori probability a
patient has the disease.
•This allows us to calculate “expected” cost of
utilization of the tree Cutil.
•It also allows us to calculate probability of false
positive and probability of false negative.
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Cost Functions
OTHER COSTS:
•Cost of false positive: Cost of additional
tests.
–If it means beginning a series of
treatments, it could be expensive, not to
mention psychological cost to patient.
•Cost of false negative:
–Complex issue.
–What is cost of patient going untreated?
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Cost Function used for Evaluating
the Decision Trees
CTot = CFalsePositive *PFalsePositive + CFalseNegative *PFalseNegative +
Cutil
CFalsePositive is the cost of false positive (Type I error)
CFalseNegative is the cost of false negative (Type II error)
PFalsePositive is the probability of a false positive occurring
PFalseNegative is the probability of a false negative occurring
Cutil is the expected cost of utilization of the tree.
PFalsePositive and PFalseNegative are calculated from the tree.
Cutil is calculated from tree and probability of disease and
probability of test errors.
CFalsePositive, CFalseNegative are input – given information.
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Stroud Saeger Results
• Using this cost function Ctot, Stroud-Saeger found an
algorithm for enumerating all BDTs coming from
complete, monotone Boolean functions and then ranking
all trees in terms of their total costs.
• The method is feasible for n = 3 or 4 types of tests, not
for n > 4.
D. Madigan, S. Mittal, F. Roberts: A new approach:
Searching through a Generalized Tree Space
• Idea: Sometimes adding more possibilities results in
being able to do more efficient searches.
• We expand the space of trees from those corresponding
to Stroud and Saeger’s “Complete and Monotonic”
Boolean Functions to “Complete and Monotonic”
BDTs.
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CM Trees
•
Monotonic Decision Trees
– A binary decision tree will be called monotonic
if all the left leaves are class “0” and all the
right leaves are class “1”.
•
Complete Decision Trees
– A binary decision tree will be called complete
if every type of test occurs at least once in the
tree and, at any non-leaf node in the tree, its
left and right sub-trees are not identical.
•
CM Tree = complete, monotonic BDT
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The CM Tree Space
complete, monotonic BDTs
No. of
attributes
Distinct BDTs
Trees From CM
Boolean
Functions
Complete,
Monotonic
BDTs
2
74
4
4
3
16,430
60
114
4
1,079,779,602
11,808
66,600
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Tree Neighborhood and Tree Space
• Define tree neighborhood by giving four operations
for moving from one tree in CM Tree Space to
another.
• We have developed an algorithm for finding lowcost BDTs by searching through CM Tree Space
from a tree to one of its neighbors.
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Tree Space Traversal
•
Naïve Idea: Greedy Search
1. Randomly start at any tree in the CM tree space
2. Find its neighboring trees using the four
operations
3. Move to the neighbor with the lowest cost
4. Iterate until we find a minimum
–
Problem: The CM Tree space is highly multimodal (more than one local minimum)!
–
Therefore, we implement a stochastic search
algorithm with simulated annealing to find the best
tree – a variant of the greedy algorithm.
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Results: Searching CM Tree Space
• We were able to perform experiments for 3, 4 and
5 tests, successfully; significantly faster than
existing methods of searching through BDTs
obtained from complete, monotonic Boolean
functions.
• Results show improvement compared to existing
extensive search methods.
– They found the optimal tree almost half the
time
– They often found a less costly tree than the best
tree arising from a complete, monotone
Boolean function.
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Conclusion: Sometimes it Pays to be
Greedy
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