The CS 5 TimesPenguin Rescue!Dunedin, New Zealand (Penguin Press):

A daring mission has beenmounted to rescue two adorable penguinswho had been given up as lost after aspaceship crash. Risking her life with anuntested experimental jet pack, a brave Chemistry penguin mixed a witches’ brew of propellant, fueled the pack, and set off across the sky in search of her missing colleagues, who were running out of fish when last heard from.

“We are, like, so grateful for this, like, attempt, and, like, we, like, hope for her, like, success,” stated a jittery CS 5 student. “We like, love our, like, penguins and are, like, so helpless with our, like, homework assignments without, like, their, like, help.”

Further quotes were unavailable due to an unexpected attack from a WRIT 1 instructor.

Mergesort

msort([42, 3, 1, 5, 27, 8, 2, 7])

msort([42, 3, 1, 5]) msort([27, 8, 2, 7])

merge([1, 3, 5, 42], [2, 7, 8, 27])

[

Mergesort

msort([42, 3, 1, 5, 27, 8, 2, 7])

msort([42, 3, 1, 5]) msort([27, 8, 2, 7])

merge([1, 3, 5, 42], [2, 7, 8, 27])

[1,

✓

Mergesort

msort([42, 3, 1, 5, 27, 8, 2, 7])

msort([42, 3, 1, 5]) msort([27, 8, 2, 7])

merge([1, 3, 5, 42], [2, 7, 8, 27])

[1,2

✓ ✓

Mergesort

msort([42, 3, 1, 5, 27, 8, 2, 7])

msort([42, 3, 1, 5]) msort([27, 8, 2, 7])

merge([1, 3, 5, 42], [2, 7, 8, 27])

[1,2,3

✓ ✓✓

Mergesort

msort([42, 3, 1, 5, 27, 8, 2, 7])

msort([42, 3, 1, 5]) msort([27, 8, 2, 7])

merge([1, 3, 5, 42], [2, 7, 8, 27])

[1,2,3,5

✓ ✓✓ ✓

Mergesort

msort([42, 3, 1, 5, 27, 8, 2, 7])

msort([42, 3, 1, 5]) msort([27, 8, 2, 7])

merge([1, 3, 5, 42], [2, 7, 8, 27])

[1,2,3,5,7

✓ ✓✓ ✓ ✓

Mergesort

msort([42, 3, 1, 5, 27, 8, 2, 7])

msort([42, 3, 1, 5]) msort([27, 8, 2, 7])

merge([1, 3, 5, 42], [2, 7, 8, 27])

[1,2,3,5,7,8

✓ ✓✓ ✓ ✓ ✓

Mergesort

msort([42, 3, 1, 5, 27, 8, 2, 7])

msort([42, 3, 1, 5]) msort([27, 8, 2, 7])

merge([1, 3, 5, 42], [2, 7, 8, 27])

[1,2,3,5,7,8,27

✓ ✓✓ ✓ ✓ ✓ ✓

Mergesort

msort([42, 3, 1, 5, 27, 8, 2, 7])

msort([42, 3, 1, 5]) msort([27, 8, 2, 7])

merge([1, 3, 5, 42], [2, 7, 8, 27])

[1,2,3,5,7,8,27,42]

✓ ✓✓ ✓ ✓ ✓ ✓✓

Done!

Let’s try it out - and let’s not even make n a power of 2!

msort([42, 3, 1, 6, 5, 2, 7])

msort([42, 3, 1]) msort([6, 5, 2, 7])

msort([42, 3, 1, 6, 5, 2, 7])

msort([42, 3, 1]) msort([6, 5, 2, 7])

msort([42]) msort([3, 1]) msort([6, 5]) msort([2, 7])

msort([42, 3, 1, 6, 5, 2, 7])

msort([42, 3, 1]) msort([6, 5, 2, 7])

msort([42]) msort([3, 1]) msort([6, 5]) msort([2, 7])

[42]

msort([3])msort([1])

msort([42, 3, 1, 6, 5, 2, 7])

msort([42, 3, 1]) msort([6, 5, 2, 7])

msort([42]) msort([3, 1]) msort([6, 5]) msort([2, 7])

[42]

msort([3])msort([1])

[3] [1]

msort([42, 3, 1, 6, 5, 2, 7])

msort([42, 3, 1]) msort([6, 5, 2, 7])

msort([42]) msort([3, 1]) msort([6, 5]) msort([2, 7])

[42]

msort([3])msort([1])

[3] [1]

[1,3]

msort([42, 3, 1, 6, 5, 2, 7])

msort([42, 3, 1]) msort([6, 5, 2, 7])

msort([42]) msort([3, 1]) msort([6, 5]) msort([2, 7])

[42]

msort([3])msort([1])

[3] [1]

[1,3]

[1,3,42]

msort([42, 3, 1, 6, 5, 2, 7])

msort([42, 3, 1]) msort([6, 5, 2, 7])

msort([42]) msort([3, 1]) msort([6, 5]) msort([2, 7])

[42]

msort([3])msort([1])

[3] [1]

[1,3]

[1,3,42] [2, 5, 6, 7]

msort([42, 3, 1, 6, 5, 2, 7])

msort([42, 3, 1]) msort([6, 5, 2, 7])

msort([42]) msort([3, 1]) msort([6, 5]) msort([2, 7])

[42]

msort([3])msort([1])

[3] [1]

[1,3]

[1,3,42] [2, 5, 6, 7]

[1, 2, 3, 5, 6, 7, 42]

How “Efficient” Is Mergesort?

How big a deal is this?

The Meder-O-Matic Supercomputer: 100 billion steps/second

n2 algorithm n log2 n algorithm

n = 108 11.5+ days

How big a deal is this?

The Meder-O-Matic Supercomputer: 100 billion steps/second

n2 algorithm n log2 n algorithm

n = 108 11.5+ days 0.27 seconds

“Easy” Problems

Sorting a list of n numbers: [42, 3, 17, 26, … , 100]

Multiplying two n x n matrices:

3 5 2 7 1 6 8 9 2 4 6 10 9 3 2 12

( ) 1 5 5 4 5 12 8 6 7 6 1 5 9 23 5 8

( ) = ( )n

n

n n

n

n log2 n

“Easy” Problems

The Shortest Path Problem (i.e. “Google Maps”)

Edsgar Dijkstra

“Easy” Problems

“Polynomial Time” = “Efficient”

n, n2, n3, n4, n5,…

How about something like n log2 n ?

The “class” P

sorting

matrix multiplication shortest paths

“Hard” Problems

Snowplows of Northern Minnesota

Burrsburg

Frostbite City

Shiversville

Tundratown

FreezeapolisBrute-force? Greed?

“Hard” Problems

The Traveling Salesperson Problem

New York

Moscow

Paris

San Francisco

Claremont

366

5625

1545

4664

5868

Brute Force? Greed?

2417

6060

2566

36275563

The Hamiltonian Path Problem

Rome, GA

Athens, GA

Homer, GA

Damascus, GA

Bethlehem, GAThose are some peachy names!

William Rowan Hamilton

n2 Versus 2n

The Meder-O-Matic performs 109 operations/sec

Meder-O-Matic

n2

2n

n!

n = 10 n = 30 n = 50 n = 70

100 < 1 sec

900 < 1 sec

2500 < 1 sec

1024 < 1 sec

109 1 sec

1015 11.6 days

4900 < 1 sec

1021 31,688 years

< 1 sec 1016 years 1057 years 1093 years

Here’s an Idea!

n n2 n3 n5 2n n!

Size of largest problem solvable with “old” computer in one hour = S

Size of largest problem solvable with “new” twice-as-fast computer in one hour

2S 1.41S 1.26S 1.15S S+1 S

Let’s just buy a computer that’s twice as fast!

Snowplows and Traveling Salesperson Revisited!

So are there polynomial time algorithms for the Snowplow and Travelling Salesperson, and Hamiltonian Path Problems?!

Travelling Salesperson Problem

Snowplow Problem

…

Hamiltonian Path Problem

NP-complete problems

Tens of thousands of other known problems go in this cloud!!

Snowplows and Travelling Salesperson Revisited!

Travelling Salesperson Problem

Snowplow Problem

…

Hamiltonian Path Problem

NP-complete problems

Tens of thousands of other known problems go in this cloud!!

If a problem is NP-complete, it doesn’t necessarily mean that it can’t be solved in polynomial time. It does mean…

“I can’t find an efficient algorithm. I guess I’m too dumb.”

Cartoon courtesy of “Computers and Intractability: A Guide to the Theory of NP-Completeness” by M. Garey and D. Johnson

Cartoon courtesy of “Computers and Intractability: A Guide to the Theory of NP-Completeness” by M. Garey and D. Johnson

“I can’t find an efficient algorithm because no such algorithm is possible!”

Cartoon courtesy of “Computers and Intractability: A Guide to the Theory of NP-Completeness” by M. Garey and D. Johnson

“I can’t find an efficient algorithm, but neither can all these famous people.”

$1 million

Vinay Deolalikar

What Is This?!

Are There Problems That Are Even Harder Than NP-Complete?

Kryptonite problems?