Recursion: Function and Programming Software Engineering at Azusa Pacific University  Evolutionary Approach  Examples and Algorithms  Programming in C++  Recursion and Iteration  Summary This is an example in regard to programming and algorithm. Here I took part of material from my previous teaching. August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 Evolutionary Approach (Present = Past + Span) Let’s learn from math (break-down / gather-up) Span Solvable Unsolved ==+ Present Past ==+ ==+ Break-down Gather-up Recursion: Function and Programming Software Engineering at Azusa Pacific University August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 With recursively thinking and programming, we do break-down-into-past first, then gather-up-to-present funcT RunBack (T timeTag) { if (bEarlyEnough(timeTag))// termination return Answer; else // recursive approach return RunBack (DeTimeTag) + Span (timeTag); } A function calls to itself (directly or indirectly) Recursion: Function and Programming Software Engineering at Azusa Pacific University  Examples and Algorithms (typical pattern) August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 Examples and Algorithms (Factorial) /*Fact = 1 * 2 * (i-1) * i * … 8 * *Fact (i) = Fact(i-1) * i * * Past Span Assume we need to do factorial. We use the number as time tag, so that i refers to the present while i-1 to the previous (or past). The recursive function calling itself fulfils recursion Recursion: Function and Programming Software Engineering at Azusa Pacific University *==== How the recursion works is described as follows ===== * Fact = Fact(7) * 8 * └  fact(6) * 7 * └  Fact(5) * 6 *… …. * └  fact(0) *1  8! = (7!)*8 =  7! = (6!)*7 = 5040  6! = (5!)*4 =720  1 */ Gather-up Break-down August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 Evolutionary Approach  Examples and Algorithms  Programming in C++  Recursion and Iteration  Summary Recursion: Function and Programming Software Engineering at Azusa Pacific University August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 Programming in C++ (Factorial) /* ==== recursive function ======= * !! What a intuitive mapping !! *F n = F n-1 * n * * (1) (2) (3) */ long Fact (int n) // n is time tag { if (n == 1)// Termination return n; else// recurring back return Fact (n-1) * n; } Recall that the Fibonacci numbers are defined recursively: However, the algorithm used in left program is non-recursive --it is iterative, while in right program we use the definition of Fibonacci numbers to implement directly a recursive algorithm. /*===== iterative implementation === * ?? Where is intuitive definition?? */ int IterativeFact (int n) { int previous = -1, result = 0; for (int i=0; I <= n; i++) { int const sum = result + previous; previous = result; result = sum; } return result; } Recursion: Function and Programming Software Engineering at Azusa Pacific University August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 Evolutionary Approach  Examples and Algorithms  Programming in C++  Recursion and Iteration  Summary Recursion: Function and Programming Software Engineering at Azusa Pacific University August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 Life cycle issues (logic equivalence)  Tree elements for both iteration and recursion Initialization / Termination / Decrease /* ======Recursive Process=====*/ int main () { int ret = Fact(8); // Initialization } long Factint n) // n is time tag { if (i == 1) // Termination return 1; else return Fact(n-1) * n; // Decrease } /* ===== Iterative Process======*/ int main () { long ret= 1; // Initialization while (n>0)// Termination { ret *= n; n--;// Decrease }  Recursion and Iteration Recursion: Function and Programming Software Engineering at Azusa Pacific University August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 Life cycle issues (logic equivalence)  Logic equivalent solution (I/T/D)  Recursion and Iteration Recursion: Function and Programming Software Engineering at Azusa Pacific University /* General recursive function * F n = F n-1 + S n * *========= * Fact = Fact(7) * 8 * └  fact(6) * 7 * └  Fact(5) * 6 * └  Fact(4) * 5 * └  Fact(3) * 4 * └  Fact(2) * 3 * └  Fact(1) * 2 * └  1 * = = 5040 = 720 = 120 = 24 = 6 = 2 = Break down to pastGather up to present August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 Life cycle issues (logic equivalence)  Logic equivalent solution (I/T/D)  Recursion and Iteration Recursion: Function and Programming Software Engineering at Azusa Pacific University /* General recursive function * F n = F n-1 + S n * ========= Long ret = 1 for (i=8; I > 0; i--) { // descending to past push (I, stack) } For (i=8; i > 0; i --) { // ascending to present ret *= pop (stack) } = = 5040 = 720 = 120 = 24 = 6 = 2 = August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 Life cycle issues (logic equivalence)  Logic equivalent solution (I/T/D)  Recursion and Iteration Recursion: Function and Programming Software Engineering at Azusa Pacific University By means of stack, we can always find a iterative equivalence to any recursive algorithm. However some iterative solution will be very difficult to understand, because it loses straightforwardness August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 Recursion and Iteration  Simplified thought (KISS)  Inefficient running (procedural call)  Straightforward (intuitive in math)  Life cycle issues ( Initialization / Termination / Decrease )  Tricky exercise between recursion and iteration  Iteration provides plane algorithm  Complicated thinking thread  Efficient running (just iteratively going through)  Non-straightforward in description (twisty)  Recursion provides solid algorithm Recursion: Function and Programming Software Engineering at Azusa Pacific University August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 Life cycle issues ( Initialization / Termination / Decrease )  Tricky exercise between recursion and iteration  Mechanical conversion from recursion to iteration  Converted by compilers and algorithm designer  Keep it simplified (for man to write)  Auto complicated (for machine to convert)  Efficiency improved (for application)  Go do right thing with recursive design because keeping it simple is really wise, while taking advantage of computer for conversion if necessary  Recursion and Iteration Recursion: Function and Programming Software Engineering at Azusa Pacific University August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 Evolutionary Approach  Examples and Algorithms  Programming in C++  Recursion and Iteration  Summary Recursion: Function and Programming Software Engineering at Azusa Pacific University August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

 Summary of Recursion  Recursive thinking:  Mathematic intuition (straightforwardness)  Be wise: looking back prior to looking ahead  Recursive programming  Function calls itself, directly or indirectly  Remember this: the past is the key to the present  Recurring back:  If complicated, break down to past  Conquer the parts, then gather up to present Recursion: Function and Programming Software Engineering at Azusa Pacific University August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,

Software Engineering at Azusa Pacific University August 10, Azusa Pacific University, Azusa, CA 91702, Tel: (800) Department of Computer Science,