Recursion Damian Gordon. Recursion Factorial Fibonacci Decimal to Binary conversion Travelling Salesman Problem Knight’s Tour.

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Presentation transcript:

Recursion Damian Gordon

Recursion Factorial Fibonacci Decimal to Binary conversion Travelling Salesman Problem Knight’s Tour

Recursion

Recursion is a feature of modularisation, when a module can call itself to complete a solution.

Recursion Recursion is a feature of modularisation, when a module can call itself to complete a solution. Let’s remember factorial: 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1

Recursion Recursion is a feature of modularisation, when a module can call itself to complete a solution. Let’s remember factorial: 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 and 6! = 6 * 5 * 4 * 3 * 2 * 1

Recursion So 7! = 7 * (6 * 5 * 4 * 3 * 2 * 1)

Recursion So 7! = 7 * (6 * 5 * 4 * 3 * 2 * 1) is 7! = 7 * 6!

Recursion So 7! = 7 * (6 * 5 * 4 * 3 * 2 * 1) is 7! = 7 * 6! and 9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 9!= 9 * (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1) 9! = 9 * 8!

Recursion Or, in general:

Recursion Or, in general: N! = N * (N-1)!

Recursion Or, in general: N! = N * (N-1)! or Factorial(N) = N * Factorial(N-1)

Recursion Let’s remember how we did Factorial iteratively:

Recursion PROGRAM Factorial: Get Value; Total <- 1; WHILE (Value != 0) DO Total <- Value * Total; Value <- Value - 1; ENDWHILE; Print Total; END.

Recursion Now this is how we implement the following: Factorial(N) = N * Factorial(N-1)

Recursion PROGRAM Factorial: Get Value; PRINT Fact(Value); END. MODULE Fact(N): IF (N <= 0) THEN RETURN 1; ELSE RETURN N * Fact(N-1); ENDIF; END.

Recursion PROGRAM Factorial: Get Value; PRINT Fact(Value); END. MODULE Fact(N): IF (N <= 0) THEN RETURN 1; ELSE RETURN N * Fact(N-1); ENDIF; END.

Recursion So if N is 5

Recursion So if N is 5 MODULE Fact(5) returns – 5 * Fact(4)

Recursion So if N is 5 MODULE Fact(5) returns – 5 * Fact(4) – 5 * 4 * Fact(3)

Recursion So if N is 5 MODULE Fact(5) returns – 5 * Fact(4) – 5 * 4 * Fact(3) – 5 * 4 * 3 * Fact(2)

Recursion So if N is 5 MODULE Fact(5) returns – 5 * Fact(4) – 5 * 4 * Fact(3) – 5 * 4 * 3 * Fact(2) – 5 * 4 * 3 * 2 * Fact(1)

Recursion So if N is 5 MODULE Fact(5) returns – 5 * Fact(4) – 5 * 4 * Fact(3) – 5 * 4 * 3 * Fact(2) – 5 * 4 * 3 * 2 * Fact(1) – 5 * 4 * 3 * 2 * 1 * Fact(0)

Recursion So if N is 5 MODULE Fact(5) returns – 5 * Fact(4) – 5 * 4 * Fact(3) – 5 * 4 * 3 * Fact(2) – 5 * 4 * 3 * 2 * Fact(1) – 5 * 4 * 3 * 2 * 1 * Fact(0) – 5 * 4 * 3 * 2 * 1 * 1

Recursion So if N is 5 MODULE Fact(5) returns – 5 * Fact(4) – 5 * 4 * Fact(3) – 5 * 4 * 3 * Fact(2) – 5 * 4 * 3 * 2 * Fact(1) – 5 * 4 * 3 * 2 * 1 * Fact(0) – 5 * 4 * 3 * 2 * 1 * 1 – 120

Recursion The Fibonacci numbers are numbers where the next number in the sequence is the sum of the previous two. The sequence starts with 1, 1, And then it’s 2 Then 3 Then 5 Then 8 Then 13

Recursion Let’s remember how we did Fibonacci iteratively:

Recursion PROGRAM FibonacciNumbers: READ A; FirstNum <- 1; SecondNum <- 1; WHILE (A != 2) DO Total <- SecondNum + FirstNum; FirstNum <- SecondNum; SecondNum <- Total; A <- A – 1; ENDWHILE; Print Total; END.

Recursion

etc.