1. Compsci 201, O-Notation and Maps (Interfaces too)
Owen Astrachan
February 8, 2019
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

2. Compsci 201, Spring 2019: O-Notation + Maps
I is for …
Interface
LinkedList implements List
Inheritance
LinkedList extends AbstractSequentialList
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

3. Compsci 201, Spring 2019: O-Notation + Maps
PFmMaBBD
Interfaces: List, Set, and Map
When it makes sense to use general type
Empirical and Analytical measures of efficiency
Maps: API and Problem Solving
Keys and Values
Big-Oh and O-Notation
Building a mathematical formalism with intuition
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

4. Problems and Solutions
String that occurs most in a list of strings?
CountingStringsBenchmark.java, two ideas
See also CountingStringsFile for same ideas
Parallel arrays: word[k] occurs count[k] times
Use ArrayLists: 2 “the”, 3 “fat”, 4 “fox”
the
fox
cried
fat
tears 2 4 1 3 5
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

5. Compsci 201, Spring 2019: O-Notation + Maps
How does the code work?
Process each string s
First time words.add(s),counter.add(1)
Otherwise, increment count corresponding to s
c[x] += 1 ?
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

6. What is complexity of this code?
Search for each word and … if occurs at k
+1 to counter.get(k), else add at end
Search complexity? O(M) when M different words
One search is O(M) – what about all searches?
Tracking all words. First time zero, then one, …
Avoid analyzing duplicates for the moment
Will take longer if we have multiple occurrences of some of M words
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

7. Compsci 201, Spring 2019: O-Notation + Maps
Tracking N strings
Complexity of search? O(M) for M different words
Complexity of words.indexOf(..) is O(M)
what about all calls? … N is N(N+1)/2
O(n2)
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

8. Should we be more precise?
Adding M different words will be O(M2)
… + M = M(M+1)/2
Adding duplicates: we need to be precise about adding N total words.
Sometimes word will be found, still O(M) for M different words
We have both M and N here, but treat M == N for easier analysis.
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

9. Understanding O-notation
This is an upper bound and in the limit
Coefficients don’t matter, order of growth
N + N + N + N = 4N is O(N) --- why?
N*N is O(N2) – why?
O(1) means independent of N, constant time
In analyzing code and code fragments
Account for each statement
How many teams is each statement executed?
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

10. Running times in seconds machine: 109 instructions/sec
O(log N)
O(N)
O(N log N)
O(N2) 10
3E-9
1E-8
3.3E-8
100
7E-9
1E-7
6.64E-7
0.0001
1,000
1E-6
0.001
10,000
1.3E-8
0.102
100,000
1.7E-8
10.008
1,000,000
0.0199
16.7 min
1,000,000,000
1.002
65.8
31.8 years
This slide has errors, replaced by new version in slides 10
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

11. Compsci 201, Spring 2019: O-Notation + Maps
Just Say No.. When you can
O(n2)
Quadratic is too slow. We can do better. Our goal is linear or better. We cannot always do this, but we can with HashSet
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

12. Example: Analyze using big-Oh
What is runtime of stuff(N)
How to reason about this
What is return value of stuff(N)
What if code changes to sum += k
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

13. Counting for O-notation
Why is O(1) complexity of sum += n
Is this O(1) for any x += y ?
Loop executes N times, doing O(1) per iteration
Total runtime for method? O(n)
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

14. Practice with O-Notation
What is big-Oh of runtime of call calc(N) ?
Number of statements executed, O(1) for 146?
Use calc(16) and generalize
What is big-Oh of value returned by calc(N) ?
Table? N = 1, 2, 4, 8, 16
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

15. Compsci 201, Spring 2019: O-Notation + Maps
WOTO
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps

16. Compsci 201, Spring 2019: O-Notation + Maps
Luis von Ahn
Duke 2000, Math
Duke Honorary Degree 2017
CEO Duolingo
Macarthur Award, 2006
MIT-Lemelson Prize, 2018
Shafi is a serious theoretical computer scientist. Not much to do here but read the bullets and the quote
“It’s amazing how motivating it is to sit with somebody and say, ‘What you’re doing is really important.’ I use that a lot.”
2/8/19
Compsci 201, Spring 2019: O-Notation + Maps