Complexity, Origami, etc. Big Oh Traditional Origami Fold and cut

Analyses of Algorithms What resources does an algorithm take? –runtime –space Calculate as function of size of input –size of set –magnitude of input values Typical problems –sort, search, paths/circuits of graph, Fibonacci, GCD –origami, 'off-line Tetris', piano-movers problem

Time complexity Can write program, run program and record time… Want machine independent analysis in terms of inputs –magnitude of inputs, size of set Count basic operations –For sorting, typically count compares Some judgment required Want bounds, not necessarily details

Example How many steps required to find average of N numbers in an array? Pseudo-code (array is aa, size n sum = aa[0]; set sum to the first number for (i=1;i<n;i++) { go through the array… sum = sum + aa[i]; } adding in the numbers How many steps? initialize sum 1 initialize i1 {compare i to n loop done completely n-1 times add aa[i] to sum 1 compare operation increment i } 3*(n-1) + 3

Big Oh Formal definition of upper bounds A function g(n) is said to be O(f(n)) if and only if the following is true: There exists a number n0 and a number c such that is n>n0 g(n) <= c*f(n) English: beyond a certain value of n, f(n) is an upper bound for g(n). You can use a coefficient c.

Jargon You say –g(n) is O(f(n)) –g(n) = O(f(n)) –g(n) є O(f(n)) (contained in, belongs to, is in) Generally, the f(n) functions are standard –O(log n), log NOTE: base doesn't matter –O(n), linear –O(n 2 ), n-squared, quadratic –O(n k ) for some k, polynomial, also called P –O(2 n ), exponential These may be 'NP' or NP-Complete

Exercise What is the relationship between log 2 (n) and ln(n) by definition: log e (n)

log n Why does base not matter? Because log b (n) = log b (a)*log a (n) this is the coefficient

NP, NP-complete Formal definition: an algorithm is NP if it can be performed by a non-deterministic Turing machine. Less formal: an algorithm is NP if it can be done by a process in which there are a finite number of choices, and assuming the correct choice is made, the process is polynomial. The correctness can be checked in polynomial time. If all choices must be tried, the algorithm is O(2 n ) NP-complete refers to a set of problems for which there are NP algorithms that have been shown to be equally hard –(can convert one to the other on polynomial time). –These include: traveling salesman, satisfying logical formulas, subgraph isomorphism. –BIG question in computer science proving either P = NP or P proper subset of NP.

Examples g(n) = 5n+10 is O(n) Let n0=10, c = 6, then if n>10 5*n+10 <= 5*n+n = 6*n g(n) = 3n 2 +5n+100 is O(n 2 ) Let n0 = 100, c=6, then if n>100 3n 2 +5n+100 <= 5n 2 +5n+100<=5n 2 +6n<= 5n 2 +n*n<= 6n 2

log n log n is O(n)—that is, log n is bounded by O(n) Assume base 2. Let n0=2, c=1 log n <= n Why? Raise 2 to each n <=2 n 2<2 2, 3<2 3, etc.

Big Oh This is an asymptotic, upper bound eventually, this is the n>n0 condition May not be a tight bound.

Bounds If g(n) is O(n), it is also O(n 2 ), O(n k ), O(2 n ) But, n 2 is not O(n)? Why? Suppose there was an n0 and a c, n 2 c and n>n0

Exercises g(n) = 4n 3 +n+10. Show g(n) is O(n 3 ) g(n) = log(n) * n. This is called log linear. Show it is O(n).

Growth rates illustrated n=1n=2n=4n=8n=16n=32 O(1) O(logn) O(n) O(nlogn) O(n2)O(n2) O(n 3 ), O(2 n )

Methods For loops, multiply number of steps in loop by upper bound on number of times loop is run For conditionals, add step for doing condition plus maximum of each of the branches

Lower bounds (omega) To give better performance estimates, we may also want to give lower bounds on growth rates Definition (omega): g(n) = Ω(f(n)) if there exist some constants c and n 0 such that g(n) ≥ cf(n) for all n ≥ n 0 This says, g(n) grows at least this much

“Exact” bounds Definition (Theta): g(n) = Θ(f(n)) if and only if g(n) =O(f(n)) and g(n) = Ω(f(n)). An algorithm is Θ(f(n)) means that f(n) is a tight bound (as good as possible) on its running time. –On all inputs of size n, time is ≤ f(n) –On all inputs of size n, time is ≥ f(n)

Complexity Bounds (such as Big Oh) are for specific algorithms –Tight[er] bounds are better than looser bounds –Sometimes there are reports giving average bounds Problems may have different algorithms with different bounds Sometimes possible to prove result about a problem as opposed to an algorithm –There is no method that is better than …

Good source Algorithm (Run Time) Analysis cs /cs201-algo-analysis.ppt

Computing Fibonacci numbers Recursive program 1 long int fib(n) function returns integer 2if (n <= 1) which case? 3 return 1; 0,1 case return 1 4else return fib(n-1) + fib(n-2) else calculate This is based on definition. It takes a long time [for big numbers]

fib(n) runs in exponential time Let T denote the running time. T(0) = T(1) = c T(n) = T(n-1) + T(n-2) + 2 where 2 accounts for line 2 plus the addition at line 3. It can be shown that the running time is Ω((3/2) n ). –at most and at least exponential

Efficient Fibonacci numbers Avoid recomputation –Work iteratively: working up and saving past results Solution with linear running time int fib(int n) { int fibn=0, fibn1=0, fibn2=1; set up 3 numbers if (n < 2) check case return n low(er) cases return n else { for( int i = 2; i <= n; i++ ) { loop fibn = fibn1 + fibn2; calculate fibn fibn1 = fibn2; reset fibn1 fibn2 = fibn; reset fibn2 } ends the looping return fibn; } ends the else clause } ends the function

Origami: paper folding Associated most with Japan Many oldest models come from China Independent development in Spain, elsewhere Worldwide Conventions all over: NYC in June

Water bomb Traditional model Can ask questions: –what is surface area? –what areas are on outside? –from folding pattern, what is foldable assignment of mountain and valley? What is folding order?

Modular origami 6 preliminary folds unit systems of models Rona Gurkewitz, others

Complexity and origami Many complexity questions have been addressed for origami, including: can a model be folded? Can a folding pattern be folded? Can a folding pattern with mountain and valley assignments be folded flat? 03/introdcgeom/demaine/2/index.html

Fold and cut Examples by Betsy Ross, Houdini, Martin Gardner (Scientific American), others. 5 pointed star

General problem Take a square Draw a polygon Is there a way to fold a flat model and make one cut to produce the polygon? Answer: Yes. –two ways. One way: generate skeleton, add perpendicular folds, then determine how to fold this pattern. –NP-complete

Conclusion Very big topic Active research Questions and comments?