CSS 332 : Algorithms - greedy Algorithms

David Luebke 1
06/12/25
CS 332: Algorithms
Greedy Algorithms

David Luebke 2
06/12/25
Review: Dynamic Programming
● Dynamic programming is another strategy for
designing algorithms
● Use when problem breaks down into recurring
small subproblems

David Luebke 3
06/12/25
Review: Optimal Substructure of
LCS
● Observation 1: Optimal substructure
■ A simple recursive algorithm will suffice
■ Draw sample recursion tree from c[3,4]
■ What will be the depth of the tree?
● Observation 2: Overlapping subproblems
■ Find some places where we solve the same subproblem more
than once
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David Luebke 4
06/12/25
Review: Structure of Subproblems
● For the LCS problem:
■ There are few subproblems in total
■ And many recurring instances of each
(unlike divide & conquer, where subproblems unique)
● How many distinct problems exist for the LCS
of x[1..m] and y[1..n]?
● A: mn

David Luebke 5
06/12/25
Memoization
● Memoization is another way to deal with overlapping
subproblems
■ After computing the solution to a subproblem, store in a
table
■ Subsequent calls just do a table lookup
● Can modify recursive alg to use memoziation:
■ There are mn subproblems
■ How many times is each subproblem wanted?
■ What will be the running time for this algorithm? The
running space?

David Luebke 6
06/12/25
● Dynamic programming: build table bottom-up
■ Same table as memoization, but instead of starting
at (m,n) and recursing down, start at (1,1)
● Least Common Subsequence: LCS easy to
calculate from LCS of prefixes
○ As your homework shows, can actually reduce space to
O(min(m,n))
● Knapsack problem: we’ll review this in a bit

David Luebke 7
06/12/25
● Summary of the basic idea:
■ Optimal substructure: optimal solution to problem
consists of optimal solutions to subproblems
■ Overlapping subproblems: few subproblems in total,
many recurring instances of each
■ Solve bottom-up, building a table of solved
subproblems that are used to solve larger ones
● Variations:
■ “Table” could be 3-dimensional, triangular, a tree, etc.

David Luebke 8
06/12/25
Greedy Algorithms
● A greedy algorithm always makes the choice that
looks best at the moment
■ My everyday examples:
○ Walking to the Corner
○ Playing a bridge hand
■ The hope: a locally optimal choice will lead to a globally
optimal solution
■ For some problems, it works
● Dynamic programming can be overkill; greedy
algorithms tend to be easier to code

David Luebke 9
06/12/25
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

David Luebke 10
06/12/25
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
 Assume (wlog) that f1  f2  …  fn
1
2
3
4
5
6

David Luebke 11
06/12/25
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|

David Luebke 12
06/12/25
Activity Selection:
Repeated Subproblems
● Consider a recursive algorithm that tries all
possible compatible subsets to find a maximal
set, and notice repeated subproblems:
S
1A?
S’
2A?
S-{1}
2A?
S-{1,2}
S’’
S’-{2}
S’’
yes no
no
no
yes yes

David Luebke 13
06/12/25
Greedy Choice Property
● Dynamic programming? Memoize? Yes, but…
● Activity selection problem also exhibits the greedy
choice property:
■ Locally optimal choice  globally optimal sol’n
■ Them 17.1: 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.

David Luebke 14
06/12/25
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

David Luebke 15
06/12/25
Minimum Spanning Tree Revisited
● Recall: MST problem has optimal substructure
■ Prove it
● Is Prim’s algorithm greedy? Why?
● Is Kruskal’s algorithm greedy? Why?

David Luebke 16
06/12/25
Review:
The Knapsack Problem
● The famous knapsack problem:
■ A thief breaks into a museum. Fabulous paintings,
sculptures, and jewels are everywhere. The thief has
a good eye for the value of these objects, and knows
that each will fetch hundreds or thousands of dollars
on the clandestine art collector’s market. But, the
thief has only brought a single knapsack to the scene
of the robbery, and can take away only what he can
carry. What items should the thief take to maximize
the haul?

David Luebke 17
06/12/25
Review: The Knapsack Problem
● More formally, the 0-1 knapsack problem:
■ The thief must choose among n items, where the ith item
worth vi dollars and weighs wi pounds
■ Carrying at most W pounds, maximize value
○ Note: assume vi, wi, and W are all integers
○ “0-1” b/c each item must be taken or left in entirety
● A variation, the fractional knapsack problem:
■ Thief can take fractions of items
■ Think of items in 0-1 problem as gold ingots, in fractional
problem as buckets of gold dust

David Luebke 18
06/12/25
Review: The Knapsack Problem
And Optimal Substructure
● Both variations exhibit optimal substructure
● To show this for the 0-1 problem, consider the
most valuable load weighing at most W pounds
■ If we remove item j from the load, what do we know
about the remaining load?
■ A: remainder must be the most valuable load
weighing at most W - wj that thief could take from
museum, excluding item j

David Luebke 19
06/12/25
Solving The Knapsack Problem
● The optimal solution to the fractional knapsack
problem can be found with a greedy algorithm
■ How?
● The optimal solution to the 0-1 problem cannot be
found with the same greedy strategy
■ Greedy strategy: take in order of dollars/pound
■ Example: 3 items weighing 10, 20, and 30 pounds,
knapsack can hold 50 pounds
○ Suppose item 2 is worth $100. Assign values to the other items
so that the greedy strategy will fail

David Luebke 20
06/12/25
The Knapsack Problem:
Greedy Vs. Dynamic
● The fractional problem can be solved greedily
● The 0-1 problem cannot be solved with a
greedy approach
■ As you have seen, however, it can be solved with
dynamic programming

CSS 332 : Algorithms - greedy Algorithms

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