Recursion ExampleShow how to develop algorithms based on recursion and review some recursion research.
C#
Recursion. This is a concept—a recursive method calls itself. Recursive methods are used extensively in programming and in compilers.
Recursion, notes. These algorithms help with complex problems. They solve problems and puzzles with brute force. They exhaust all possibilities.
Maze
Example. Here is a recursive method. The method has 2 parameters, including a ref parameter. It checks a condition near the top of its method body, as many recursive algorithms do.
ref
And It calls itself again based on an incremented value of the parameter it receives. The program also has a commented-out exception.
Tip This demonstrates the appearance of the method call stack frames in a recursive algorithm.
Next The primitive example here continues until it sums up a value of 10 by incrementing an integer.
Info The call stack has six method stack frames with the Recursive method signature in them.
using System; class Program { static int Recursive(int value, ref int count) { count++; if (value >= 10) { // throw new Exception("End"); return value; } return Recursive(value + 1, ref count); } static void Main() { // // Call recursive method with two parameters. // int count = 0; int total = Recursive(5, ref count); // // Write the result from the method calls and also the call count. // Console.WriteLine(total); Console.WriteLine(count); } }
10 Total value of 10 was added up. 6 Six method calls.
Unhandled Exception: System.Exception: End at Program.Recursive(Int32 value, Int32& count) ...Program.cs:line 10 at Program.Recursive(Int32 value, Int32& count) ...Program.cs:line 13 at Program.Recursive(Int32 value, Int32& count) ...Program.cs:line 13 at Program.Recursive(Int32 value, Int32& count) ...Program.cs:line 13 at Program.Recursive(Int32 value, Int32& count) ...Program.cs:line 13 at Program.Recursive(Int32 value, Int32& count) ...Program.cs:line 13 at Program.Main() in ...Program.cs:line 22
StackOverflowException. This may be the shortest valid C# program that will crash. The Main method calls Main. The stack will overflow, leading to a memory error.
StackOverflowException
Tip The exception here is a good thing, as it stops a process that would otherwise continue infinitely, wasting time and resources.
class Program { static void Main() { // Do not run this program. Main(); } }
Process is terminated due to StackOverflowException.
Tail recursion. Does the C# compiler optimize tail recursion? Here we show a method called Recurse that could be optimized to iteration using tail recursion.
Result The compiler does not optimize the tail calls in this program. So it causes a stack overflow.
Thus The programmer will need to write tail-recursive methods with iteration to achieve optimal performance in C# programs.
class Program { static void Recurse(int remaining) { // This method could be optimized with tail recursion. if (remaining <= 0) { return; } Recurse(remaining - 1); } static void Main() { // Attempt to call this method. Recurse(1000000); } }
Process is terminated due to StackOverflowException.
Change. This program uses recursion to compute different ways of making change to match a specified amount. Change() will print out all the ways you can make change for a specified amount.
Note If you specify 51 cents, it will tell you can make this out of 36 1-cent coins and three 5-cent coins.
Note 2 The 2 Lists store the currently recorded values, and the amounts or denominations of coins that are possible to be used.
List
int, uint
Note 3 The program emphasizes the recursive Change method. This implements the logic that actually computes the change.
Tip In many recursive methods, it is clearest to add the exit conditions at the start of the function.
using System; using System.Collections.Generic; using System.Linq; class Program { static void Main() { List<int> coins = new List<int>(); List<int> amounts = new List<int>() { 1, 5, 10, 25, 50 }; // // Compute change for 51 cents. // Change(coins, amounts, 0, 0, 51); } static void Change(List<int> coins, List<int> amounts, int highest, int sum, int goal) { // // See if we are done. // if (sum == goal) { Display(coins, amounts); return; } // // See if we have too much. // if (sum > goal) { return; } // // Loop through amounts. // foreach (int value in amounts) { // // Only add higher or equal amounts. // if (value >= highest) { List<int> copy = new List<int>(coins); copy.Add(value); Change(copy, amounts, value, sum + value, goal); } } } static void Display(List<int> coins, List<int> amounts) { foreach (int amount in amounts) { int count = coins.Count(value => value == amount); Console.WriteLine("{0}: {1}", amount, count); } Console.WriteLine(); } }
1: 51 5: 0 10: 0 25: 0 50: 0 1: 46 5: 1 10: 0 25: 0 50: 0 1: 41 5: 2 10: 0 25: 0 50: 0 1: 41 5: 0 10: 1 25: 0 50: 0 1: 36 5: 3 10: 0 25: 0 50: 0
Notes, Change method. Change() first tests for the ideal amount, and if this is reached, it prints it to the screen. It tests to make sure the change collected has not exceeded the target.
if
Info This algorithm adds coins in increasing size. It starts with the 1-cent coin and continues with the 5-cent coin and then larger coins.
Output. Display() prints the frequencies of the coins that can be added together to reach the target value. In the example, all of the output will add up to 51 cents.
Console.WriteLine
SICP. The Structure and Interpretation of Computer Programs book uses the "change" example in its first chapter. It counts ways to compute change for 100 cents. See section 1.2.2 Tree Recursion.
Detail The answer of 292 is computed from interpreting the Lisp program. On the C# program shown here, you will also get 292 answers.
Also If you modify the program to include a counter, it is clearer that the result is 292.
Notes, an end. Every recursive method sequence must be terminated. Often the first part of the recursive method will have a branch that tests for a condition being met.
And In this way, the recursive methods continue until the result is attained (or the algorithm fails).
Notes, ref. The ref keyword is useful when dealing with recursive methods. We have the recursive method return more than one value without using any allocations on the managed heap.
Tip You may want to use a count parameter to make sure you don't enter an infinite recursion series.
Summary. There are tricks to using recursive methods. Reference parameters (with the ref and out keywords) are often useful. We examined the call stack for a recursive algorithm.
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Sam Allen is passionate about computer languages. In the past, his work has been recommended by Apple and Microsoft and he has studied computers at a selective university in the United States.