Data Structures and Algorithms

1. Data Structures and Algorithms Dr. Ken Cosh Recursion

2. Review • Stacks • LIFO • Queues • FIFO • Priority Queues

3. This weeks Topic • Recursion • Tail Recursion • Non-tail Recursion • Indirect Recursion • Nested Recursion • Excessive Recursion

4. Defining new things • A basic rule for designing new things, is only to include terms which have already been defined. • One wouldn’t say “Mix 3 flugglepips with a hippyfrick” • To define an object in terms of itself is a serious violation of this rule – a vicious circle. • However, definitions such as this are called recursive definitions.

5. Infinite Sets • When defining an infinite set, a complete list of the set is impossible, so recursion is often used to define them. • With large finite sets, it can be more efficient to also define them using recursion.

6. Consider defining Natural Numbers 0 ∈ if n ∈ then (n+1) ∈ there are no other objects in set According to these rules, the set of natural numbers contains; 0, 0+1, 0+1+1… etc.

7. Recursion vs Formula • When using a recursive definition, it is necessary to calculate every value; • for instance to calculate 6!, first we need to know 5!, and then 4! etc. • Therefore, sometimes being able to use a formula can be less computationally demanding. • If n=0, g(n) = 1 • if n>0, g(n) = 2*g(n-1) • This is a recursive function, which can be converted to a simple formula – What is it?

8. Function Calls • What happens when a function is called? • If there are formal parameters, they are initialised to the values passed. • The return address – for where the program needs to resume after the function completes needs to be stored. • This return address is key, and for efficiency, the memory for the address is allocated dynamically – using the runtime stack. • The runtime stack contains an activation record (or frame) for each calling function.

9. Activation Records • Activation Records contain; • Values for all parameters to the function (either addresses for call by reference, or copies for call by value) • Local variables can be stored elsewhere, but the activation record will store pointers to their locations. • The return address, where to resume control in the calling function – the address of the next instruction in the calling function • A dynamic link, or pointer, to the caller’s activation record. • The returned value for a non-void function.

10. Runtime stack Parameters and Local Variables Activation Record for f3() Dynamic Link Return Address Return Value Parameters and Local Variables Activation Record for f2() Dynamic Link Return Address Return Value Parameters and Local Variables Activation Record for f1() Dynamic Link Return Address Return Value Activation Record for main()

11. Recursion • An activation record is created whenever a function is called. • This means that each recursive call is not really a function calling itself, but instead, an instance of a function calling another instance of a function.

12. !Factorial! • Consider the factorial problem; int fact(int n) { if (n==0) return 1; else return n* fact(n-1); } • What happens when this function is called? answer = fact(6)

13. Tail Recursion • Tail recursion occurs when the final instruction in a function is the recursive call (and there have been no prior recursive calls). • An example is the factorial function on the previous slide.

14. Non-Tail Recursion • Conversely, in nontail recursion, the recursive call is not the final instruction in the function. Consider this function. void reverse() { char ch; cin.get(ch); if (ch != ‘\n’) { reverse(); cout.put(ch); } }

15. Indirect Recursion • Direct recursion occurs when a function directly calls itself; indirect recursion is when a function calls a function which calls itself – or similar. • f() -> g() -> f() • Consider a buffer, which receives data, decodes it and stores it. This has 3 functions; • receive(), decode(), store() • While there is data to be received, they will repeatedly call each other.

16. Nested Recursion • Nested recursion occurs when a function is not only defined in terms of itself, but is also a parameter to the function. If n=0, A(n,m) = m+1 If n>0, m=0 A(n,m) = A(n-1,1) Else A(n-1, A(n,m-1)) • N.B. this function (known as the Ackermann function) grows very fast • A(4,1) is 265536-3, • The recursive definition is simple, but as it grows faster than addition, multiplication of exponentiation, defining it arithmetically is not practical.

17. Why Recurse? • Logical Simplicity • Some mathematical formulas are naturally recursive – an arithmetic / iterative solution may not be simple. • Readability • Recursive functions are often easier to understand and work through.

18. Why not recurse? • Excessive use of the runtime stack • With the data being stored on the runtime stack, there is danger of it running out of space. • Speed • Some recursive functions can be slow and inefficient.

19. Fibonacci Int Fib(int n) { if (n<2) return n; else return Fib(n-2) + Fib(n-1); } How efficient is this recursive function?

20. Fib(5) • To calculate Fib(5), first we need Fib(4) and Fib(3). • To calculate Fib(4), we need Fib(3) and Fib(2). • To calculate Fib(3), we need Fib(2) and Fib(1). • To calculate Fib(2), we need Fib(1) and Fib(0). • To calculate Fib(2), we need Fib(1) and Fib(0). • To calculate Fib(3), we need Fib(2) and Fib(1). • To calculate Fib(2), we need Fib(1) and Fib(0).

21. Repetition • Notice that on the previous slide (which isn’t complete), we make calls to calculate Fib(2) 3 times. Each time the call is made the value is forgotten by the PC, because of the way unstacked data is thrown away. • To calculate Fib(6), there are 25 calls to the Fib function, and 12 addition operations.

22. Fib(6) Fib(4) Fib(5) Fib(2) Fib(3) Fib(3) Fib(4) Fib(0) Fib(1) Fib(1) Fib(2) Fib(1) Fib(2) Fib(2) Fib(3) Fib(0) Fib(1) Fib(0) Fib(1) Fib(0) Fib(1) Fib(1) Fib(2) Fib(0) Fib(1)

23. Fib Growth • The number of additions required for a recursive definition is; • Fib(n + 1) - 1 • For each addition required, there are 2 function calls; • 2*Fib(n + 1) - 1

24. Fib Growth

25. Recursive Fib • The recursive Fib algorithm is simple. But the number of calls, and run time grows exponentially with n. • Recursive Fib is only really practical with small n.

26. Alternative Fib int iterativeFib(int n) { int previous = -1, result=1; for(int i=0; i<=n; ++i) { int const sum=result+previous; previous = result; result = sum; } return result; }

27. Iterative Fib • The iterative version of fib, is clearly more efficient. • There is however an even more efficient, mathematical approach to approximating Fib(n); int deMoivreFib(int n) { return ceil(exp(n*log(1.6180339897)-log(2.2360679775))-.5); }