1. Data Structures and Algorithms
Hard Problems
PLSD210

2. Polynomial multiplication
Lecture 21 - Key points
Polynomial multiplication
Coefficient representation: O(n2)
Point-value representation: O(n)
Conversion (simple): O(n2)
FFT does conversion in O(nlogn)
FFT
Divide and conquer algorithm
Split transform into odd and even components

3. Lecture 21 - Key points FFT Divide and conquer algorithm
Evaluate A(x):
Use nk, k=0,n-1 to evaluate A(x)
Split log2n times, n/2 work in each step
O(nlogn) overall
Aeven(x) = a0 + a2x + a4x an-2 xn/2-1
Aodd(x) = a1 + a3x + a5x an-1 xn/2-1
A(x) = Aeven(x2) + Aodd(x2) x

4. Lecture 21 - Key points FFT Actual realisations
Iterative - for efficiency
log2n “butterfly” stages
Coefficients swapped based on bit reversal of index

5. Polynomial or Intractable?
Polynomial time complexity
O(nk) time algorithm, k constant
Intractable
No O(nk) algorithm is known
O(nn) or O(n!) or O(kn) time required

6. Similar Problems? Euler’s Problem Hamilton’s Problem
Is there a path through a graph which traverses each edge only once?
Hamilton’s Problem
Is there a path which visits each vertex only once?

7. The Bridges of Königsberg
Euler’s Problem
The Bridges of Königsberg
Can I make a tour through the park, crossing each bridge only once?

8. The Bridges of Königsberg
A Sunday Stroll
Can I make a tour through the park, crossing each bridge only once?
Exhaustive search
Start at each point
Traverse all possible paths
n x (n -1) x (n -2) ...  O(n!)
First
choices
Second
choices

9. Eulerian Path An Eulerian Path exists iff
It is possible to go from any vertex to any other by following the edges (graph must be connected)
Every vertex must have an even number of edges connected to it - with at most two exceptions (the start and end points)
Obviously necessary conditions
Sufficiency proof may be found in the literature

10. Transform the Bridges Problem
Nodes = land
Edges = bridges
No solution!
Determined in O(n) time
Count degree of each vertex
Hamiltonian Path?

11. Hamiltonian Path
Although a path may easily found in this case,
no known efficient algorithm for finding a Hamiltonian path
But if we find a path,
we can verify that it is a Hamiltonian path in polynomial - O(|E|) - time
Check that each edge in the path is an edge in the graph (use adjacency matrix)
Check that each vertex is visited once only - O(n2)

12. Classes P and NP Class P Class NP Problems solvable in polynomial time
eg Euler’s problem
Class NP
Non-deterministic Polynomial
eg Hamilton’s problem

13. Non-deterministic Polynomial
Class NP
Non-deterministic Polynomial
At each step, guess which step to take next (eg which vertex to check)
Have you found a correct solution?
Need polynomial time only
Nothing from previous steps guides the next step
You have to try all possibilities
O(n!) algorithms
Non-deterministic
part
You simply guess!

14. Winning the Turing Award*
No proof that any problem  class NP
Problems (eg Hamiltonian path) are believed to be in class NP
Because no polynomial time algorithm is known
Proving that a problem  class NP
or finding a class P algorithm
Instant fame!
*Computing’s equivalent of the Nobel Prize

15. Composite Numbers
Determine whether a number n can be written as the product of two other numbers
If the divisors can be found, then it’s simple to verify that they are divisors,
but no efficient method of finding them is known

16. Assignment Problem You have to assign students to rooms in a college
You are a compatibility map
Students are vertices,
Compatible students are linked by an edge
If rooms hold 2 students,
a class P algorithm
but if rooms hold 3 students,
a class NP problem!

17. Boolean Satisfiability
Find an assignment of values (true or false) to ai so that
is true
where op are boolean operators: and, or, ...
Equivalent to circuit satisfiability problem:
Find a set of inputs which produce a true at the output of a circuit composed of logic gates
Solution:
Try all 2n possible assignments
a1 op a2 op a3 ... op an

18. Map Colouring
Colour a map using only three colours, so that adjoining countries don’t have the same colour
Easily answered for 2 colours:
point at which an odd number of countries meet
Solvable for 4 colours
proof that 4 colours suffice for any map

19. Map Colouring - Equivalent Problems
Graph equivalent
Vertices = countries
Edges connect countries with common borders
Work scheduling
Vertices = tasks
Edges represent common resources
eg linked tasks require some machine
3 colour marking of the vertices
allocation of tasks to 3 shifts in a day

20. Reducing Problems Reduction Map Colouring  boolean satisfiability
A solution can be constructed by assignment of true or false to a set of variables:
ar = true  A is red
ab = true  A is blue
ag = true  A is green
br = true  B is red

21. Reducing Problems Map colouring solution
If we can find a set of values for ax, bx, ... which make this expression true, we’ve found a map colouring
Special class of problems in NP
The NP-complete problems
All problems in NP are efficiently reducible to them
Efficiently = “in polynomial time”
((ar & ~ ab & ~ag ) & (bb & ~ br. ... ) & (cb ...

22. NP-Complete problems
Many problems have been shown to be efficiently reducible to boolean satisfiability
but an efficient solution to this (and thus any other NP-complete problem!) has eluded many researchers for a long time!
It’s believed that class NP  class P
Key word!

23. Solving NP-Complete problems
Intractible problems are soluble
It’s just a question of patience
(or longevity!)
Non-deterministic algorithms
Alternately
Non-deterministic (guessing) step
Select the next step randomly
No guidelines: any step may be the solution
Deterministic step
Determine (in polynomial time) whether you have found a solution or not
Key word!

24. The Travelling Salesman Problem
A salesman has a number of cities which he must visit, find the minimum cost tour
Optimisation problem
Can we reduce it to boolean satisfiability?
Problem must have a true / false answer
Recast as:
Does a tour exist with cost less than x?
Reduce x until the answer is provably no ...
optimal tour
This problem may be proved equivalent to finding a Hamiltonian circuit

25. The Travelling Salesman Problem
A salesman has a number of cities which he must visit, find the minimum cost tour
Many real world problems map to TSP
Drilling circuit boards
Inspection tours
...
How do we solve it?
Heuristics
Approximate techniques leading to near-optimal solutions
If you find the optimal solution, will you know?

26. Travelling Salesman Heuristics
Start with the Minimum Spanning Tree
Optimal tour cost <= 2 MST cost
Heuristics
Christofides’ algorithm
Prune the tour by taking short-cuts
Tour must be an Eulerian path
Each vertex of even degree
Add edges to odd degree vertices
Christofides’ algorithm produces a tour <= 1.5 optimal tour cost

27. Travelling Salesman Heuristics
“Banding”
Divide the region into bands
Number of cities in each band sufficiently small that the TSP problem can be solved by exhaustion (brute-force or try all solutions)
Greedy algorithm to join the band solutions

28. Travelling Salesman Heuristics
Genetic algorithms
Biological evolution analogy
Reasonably effective
Trivially parallelisable!
Simulated annealing
Model: Annealing of solid
Solution heads towards local minimum
Temperature allows some probability of climbing hills to escape a local minimum

29. Games Search trees can be huge
Chess: ~40 possible choices for each move
10+-move look-ahead needed to win
>4010 positions to explore
Modern computers just “touching” this capability
150MHz ~104 evaluations/second
(EE student programming .. so you can add a factor of 10 if you like  )
Still a rather boring game ... ~1 year/move??

30. Games Board games Assume a “score” can be assigned to a position
Player tries to maximise this score
Opponent tries to minimise this score
Game proceeds in alternating maximise and minimise steps
Minimax algorithm