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22
Recursividad Recursividad Prof. Ing. José María Sola. Introducción a Recursividad Por Ing. José María Sola 1. Factorial Figura 1. Conceptual 2. Fibonacci Figura 2. Definición recursiva Figura 3. Digrafo de pila de llamadas 1

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Page 1: Recursividad - Recursividad - GitHub PagesRecursividad Figura 4. Traza de pila de llamadas Figura 5. Conceptual 3. Listados Completos 3.1. Factorial FactorialDriver.cpp. /* FactorialDriver

RecursividadRecursividad

Prof. Ing. José María Sola.Introducción a Recursividad Por Ing. José María Sola

1. Factorial

Figura 1. Conceptual

2. Fibonacci

Figura 2. Definición recursiva

Figura 3. Digrafo de pila de llamadas

1

Page 2: Recursividad - Recursividad - GitHub PagesRecursividad Figura 4. Traza de pila de llamadas Figura 5. Conceptual 3. Listados Completos 3.1. Factorial FactorialDriver.cpp. /* FactorialDriver

Recursividad

Figura 4. Traza de pila de llamadas

Figura 5. Conceptual

3. Listados Completos

3.1. Factorial

FactorialDriver.cpp.

/* FactorialDriver

* 20130822

* JMS

*/

#include "Factorial.h"

#include <iostream>

#include <iomanip> // setprecision

#include <chrono>

int main(int n, char** argv) {

std::cout << argv[0] << '\n';

auto t = std::chrono::steady_clock::now(); // start

2

Page 3: Recursividad - Recursividad - GitHub PagesRecursividad Figura 4. Traza de pila de llamadas Figura 5. Conceptual 3. Listados Completos 3.1. Factorial FactorialDriver.cpp. /* FactorialDriver

Recursividad

std::cout << "Factorial(17) = " << Factorial(17) << "\n";

std::chrono::duration<double,std::micro> d =

std::chrono::steady_clock::now() - t; // stop

std::cout << std::setprecision(1) << std::fixed << d.count() << "

microseconds\n\n";

}

Makefile.

# Factorial

# JMS

# 2016

.PHONY: all run clean

# S is the Stem

S = Factorial

RM = -rm -f

CXXFLAGS = -std=c++11 -Wall -pedantic-errors

TARGETS = $SIterative $SRecursiveIfElse $SRecursiveIf $SRecursiveSwitch

DRIVER = $SDriver.cpp

RECIPE = $(CXX) $(CPPFLAGS) $(CXXFLAGS) -o $@ $< $(DRIVER)

all: run

run : $(TARGETS)

@($(foreach TARGET,$(TARGETS), ./$(TARGET) && ) echo ) > output.txt

$(TARGETS) : $(DRIVER) $S.h

$SIterative : $SIterative.o

$(RECIPE)

$SRecursiveIf : $SRecursiveIf.o

$(RECIPE)

$SRecursiveIfElse : $SRecursiveIfElse.o

$(RECIPE)

$SRecursiveSwitch : $SRecursiveSwitch.o

$(RECIPE)

clean:

3

Page 4: Recursividad - Recursividad - GitHub PagesRecursividad Figura 4. Traza de pila de llamadas Figura 5. Conceptual 3. Listados Completos 3.1. Factorial FactorialDriver.cpp. /* FactorialDriver

Recursividad

$(RM) $(TARGETS) *.o

FactorialIterative.cpp.

/* FactorialIterative

* 20130822

* JMS

*/

#include "Factorial.h"

unsigned Factorial(unsigned n){

unsigned f=1;

for(unsigned i=1; i<=n; ++i)

f *= i;

return f;

}

FactorialRecursiveIfElse.cpp.

/* FactorialRecursiveIfElse

* 20130822

* JMS

*/

#include "Factorial.h"

unsigned Factorial(unsigned n){

if(n == 1)

return 1;

else

return n * Factorial(n-1);

}

FactorialRecursiveIf.cpp.

/* FactorialRecursiveIf

* 20130822

* JMS

*/

#include "Factorial.h"

4

Page 5: Recursividad - Recursividad - GitHub PagesRecursividad Figura 4. Traza de pila de llamadas Figura 5. Conceptual 3. Listados Completos 3.1. Factorial FactorialDriver.cpp. /* FactorialDriver

Recursividad

unsigned Factorial(unsigned n){

if(n == 1)

return 1;

return n * Factorial(n-1);

}

FactorialRecursiveSwitch.cpp.

/* FactorialRecursiveSwitch

* 20130822

* JMS

*/

#include "Factorial.h"

unsigned Factorial(unsigned n){

switch(n){

case 0 : return 1;

default: return n * Factorial(n-1);

}

}

Fatorial output.

./FactorialIterative

Factorial(17) = 4006445056

17.2 microseconds

./FactorialRecursiveIfElse

Factorial(17) = 4006445056

15.2 microseconds

./FactorialRecursiveIf

Factorial(17) = 4006445056

11.2 microseconds

./FactorialRecursiveSwitch

Factorial(17) = 4006445056

11.9 microseconds

3.2. Fibonacci

FibonacciDriver.cpp.

/* FibonacciDriver

5

Page 6: Recursividad - Recursividad - GitHub PagesRecursividad Figura 4. Traza de pila de llamadas Figura 5. Conceptual 3. Listados Completos 3.1. Factorial FactorialDriver.cpp. /* FactorialDriver

Recursividad

* 20130822

* JMS

*/

#include "Fibonacci.h"

#include <iostream>

#include <iomanip> // setprecision

#include <chrono>

int main(int n, char** argv) {

std::cout << argv[0] << '\n';

auto t = std::chrono::steady_clock::now(); // start

for(unsigned i=0; i<27; ++i)

std::cout << "F" << i << " = " << Fibonacci(i) << '\n';

std::chrono::duration<double,std::micro> d =

std::chrono::steady_clock::now() - t; // stop

std::cout << std::setprecision(3) << std::fixed << d.count() << "

microseconds.\n";

std::cout << '\n';

}

FibonacciBenchmark.cpp.

/* FibonacciDriver

* 20130822

* JMS

*/

#include "Fibonacci.h"

#include <iostream>

#include <iomanip> // setprecision

#include <chrono>

int main(int n, char** argv) {

std::cout << argv[0] << '\n';

auto t = std::chrono::steady_clock::now(); // start

std::cout << "F17 = " << Fibonacci(17);

std::chrono::duration<double,std::micro> d =

std::chrono::steady_clock::now() - t; // stop

std::cout << "\t\t" << std::setprecision(3) << std::fixed << d.count()

<< " microseconds.\n";

6

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Recursividad

t = std::chrono::steady_clock::now(); // start

std::cout << "F27 = " << Fibonacci(27);

d = std::chrono::steady_clock::now() - t; // stop

std::cout << '\t' << std::setprecision(3) << std::fixed << d.count()

<< " microseconds.\n";

t = std::chrono::steady_clock::now(); // start

std::cout << "F32 = " << Fibonacci(32);

d = std::chrono::steady_clock::now() - t; // stop

std::cout << '\t' << std::setprecision(3) << std::fixed << d.count()

<< " microseconds.\n";

t = std::chrono::steady_clock::now(); // start

std::cout << "F35 = " << Fibonacci(35);

d = std::chrono::steady_clock::now() - t; // stop

std::cout << '\t' << std::setprecision(3) << std::fixed << d.count()

<< " microseconds.\n\n";

}

Makefile.

# Fibonacci

# JMS

# 2016

.PHONY: all run clean RunDrivers RunBechmarks

# S is the Stem

S = Fibonacci

RM = -rm -f

CXXFLAGS = -std=c++11 -Wall -pedantic-errors

BASETARGETS = \

$SIterative \

$SRecursiveIfElse \

$SRecursiveIf \

$SRecursiveSwitch \

$SRecursiveSwitchFallthrough \

$SMemoizationCode \

$SMemoizationStaticData \

$SMemoizationDynamicDataArray \

$SMemoizationDynamicDataVector

DRIVERS = $(foreach BASETARGET,$(BASETARGETS),

$(BASETARGET)Driver )

7

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Recursividad

BENCHMARKS = $(foreach BASETARGET,$(BASETARGETS),

$(BASETARGET)Benchmark )

DRIVER = $SDriver.cpp

DRIVERRECIPE = $(CXX) $(CPPFLAGS) $(CXXFLAGS) -o $@ $< $(DRIVER)

BENCHMARK = $SBenchmark.cpp

BENCHMARKRECIPE = $(CXX) $(CPPFLAGS) $(CXXFLAGS) -o $@ $< $(BENCHMARK)

all : $(DRIVERS) $(BENCHMARKS)

run: RunDrivers RunBechmarks

RunDrivers : $(DRIVERS)

@($(foreach DRIVER,$(DRIVERS), ./$(DRIVER) && ) echo ) >

DriverOutput.txt

RunBechmarks : $(BENCHMARKS)

@($(foreach BENCHMARK,$(BENCHMARKS), ./$(BENCHMARK) && ) echo ) >

BenchmarkOutput.txt

#All drivers and becnhmarks depend on the header.

$(DRIVERS) $(BENCHMARKS) : $S.h

#All particular drivers depend on the generic driver.

$(DRIVERS) : $(DRIVER)

#All particular benchmarks depend on the generic benchmark.

$(BENCHMARKS) : $(BENCHMARK)

#Iterative

$SIterativeDriver : $SIterative.o

$(DRIVERRECIPE)

$SIterativeBenchmark : $SIterative.o

$(BENCHMARKRECIPE)

#RecursiveIfElse

$SRecursiveIfElseDriver : $SRecursiveIfElse.o

$(DRIVERRECIPE)

$SRecursiveIfElseBenchmark :$SRecursiveIfElse.o

$(BENCHMARKRECIPE)

#RecursiveIf

$SRecursiveIfDriver : $SRecursiveIf.o

$(DRIVERRECIPE)

8

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Recursividad

$SRecursiveIfBenchmark : $SRecursiveIf.o

$(BENCHMARKRECIPE)

#Switch

$SRecursiveSwitchDriver : $SRecursiveSwitch.o

$(DRIVERRECIPE)

$SRecursiveSwitchBenchmark : $SRecursiveSwitch.o

$(BENCHMARKRECIPE)

#SwitchFallthrough

$SRecursiveSwitchFallthroughDriver : $SRecursiveSwitchFallthrough.o

$(DRIVERRECIPE)

$SRecursiveSwitchFallthroughBenchmark : $SRecursiveSwitchFallthrough.o

$(BENCHMARKRECIPE)

#MemoizationCode

$SMemoizationCodeDriver : $SMemoizationCode.o

$(DRIVERRECIPE)

$SMemoizationCodeBenchmark : $SMemoizationCode.o

$(BENCHMARKRECIPE)

#MemoizationStaticData

$SMemoizationStaticDataDriver : $SMemoizationStaticData.o

$(DRIVERRECIPE)

$SMemoizationStaticDataBenchmark : $SMemoizationStaticData.o

$(BENCHMARKRECIPE)

#MemoizationDynamicDataArray

$SMemoizationDynamicDataArrayDriver : $SMemoizationDynamicDataArray.o

$(DRIVERRECIPE)

$SMemoizationDynamicDataArrayBenchmark : $SMemoizationDynamicDataArray.o

$(BENCHMARKRECIPE)

#MemoizationDynamicDataVector

$SMemoizationDynamicDataVectorDriver : $SMemoizationDynamicDataVector.o

$(DRIVERRECIPE)

$SMemoizationDynamicDataVectorBenchmark :

$SMemoizationDynamicDataVector.o

$(BENCHMARKRECIPE)

9

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Recursividad

clean:

$(RM) $(DRIVERS) $(BENCHMARKS) *.o

FibonacciIterative.cpp.

/* FibonacciIterative

* 20130822

* JMS

*/

#include "Fibonacci.h"

unsigned Fibonacci(unsigned n) {

if(n<2)

return n;

unsigned f;

for(unsigned p=0, pp=1, i=0; i<n; ++i){

f = p + pp;

pp = p;

p = f;

}

return f;

}

FibonacciRecursiveIfElse.cpp.

/* FibonacciRecursiveIf

* 20130822

* JMS

*/

#include "Fibonacci.h"

unsigned Fibonacci(unsigned n){

if(n<2)

return n;

else

return Fibonacci(n-1) + Fibonacci(n-2);

}

FibonacciRecursiveIf.cpp.

/* FibonacciRecursiveIf

10

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Recursividad

* 20130822

* JMS

*/

#include "Fibonacci.h"

unsigned Fibonacci(unsigned n){

if(n<2)

return n;

return Fibonacci(n-1) + Fibonacci(n-2);

}

FibonacciRecursiveSwitch.cpp.

/* FibonacciRecursiveSwitch

* 20130822

* JMS

*/

#include "Fibonacci.h"

unsigned Fibonacci(unsigned n) {

switch(n){

case 0: return 0;

case 1: return 1;

default: return Fibonacci(n-1) + Fibonacci(n-2);

}

}

FibonacciRecursiveSwitchFallthrough.cpp.

/* FibonacciRecursiveSwitchFallthrough

* 20130822

* JMS

*/

#include "Fibonacci.h"

unsigned Fibonacci(unsigned n){

switch(n){

case 0:

case 1: return n;

default: return Fibonacci(n-1) + Fibonacci(n-2);

}

}

11

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Recursividad

FibonacciMemoizationCode.cpp.

/* FibonacciMemoizationCode

* 20140825

* JMS

*/

#include "Fibonacci.h"

unsigned Fibonacci(unsigned n){

switch(n){

case 0: return 0;

case 1: return 1;

case 2: return 1;

case 3: return 2;

case 4: return 3;

case 5: return 5;

case 6: return 8;

case 7: return 13;

case 8: return 21;

case 9: return 34;

case 10: return 55;

case 11: return 89;

case 12: return 144;

case 13: return 233;

case 14: return 377;

case 15: return 610;

case 16: return 987;

case 17: return 1597;

case 18: return 2584;

case 19: return 4181;

case 20: return 6765;

default: return Fibonacci(n-1) + Fibonacci(n-2);

}

}

FibonacciMemoizationStaticData.cpp.

/* FibonacciMemoizationStaticData

* 20140825

* JMS

*/

#include "Fibonacci.h"

#include <array>

12

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Recursividad

unsigned Fibonacci(unsigned n){

static std::array<unsigned,21> f={{

0,

1,

1,

2,

3,

5,

8,

13,

21,

34,

55,

89,

144,

233,

377,

610,

987,

1597,

2584,

4181,

6765

}};

if(n<21)

return f.at(n);

return Fibonacci(n-1) + Fibonacci(n-2);

}

FibonacciMemoizationDynamicDataArray.cpp.

/* FibonacciMemoizationStaticData

* 20140825

* JMS

*/

#include "Fibonacci.h"

#include <array>

unsigned Fibonacci(unsigned n){

static unsigned size=21;

static const unsigned capacity=30;

static std::array<unsigned,capacity> f={{

0,

13

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Recursividad

1,

1,

2,

3,

5,

8,

13,

21,

34,

55,

89,

144,

233,

377,

610,

987,

1597,

2584,

4181,

6765

}};

if(n >= capacity)

return Fibonacci(n-1) + Fibonacci(n-2);

if(n >= size){

f.at(n) = Fibonacci(n-1) + Fibonacci(n-2);

++size;

}

return f.at(n);

}

FibonacciMemoizationDynamicDataVector.cpp.

/* FibonacciMemoizationStaticData

* 20140825

* JMS

*/

#include "Fibonacci.h"

#include <vector>

unsigned Fibonacci(unsigned n){

static std::vector<unsigned> f={{

0,

1,

1,

14

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Recursividad

2,

3,

5,

8,

13,

21,

34,

55,

89,

144,

233,

377,

610,

987,

1597,

2584,

4181,

6765

}};

if(n >= f.size())

f.push_back(Fibonacci(n-1) + Fibonacci(n-2));

return f.at(n);

}

Fibonacci Diver output.

./FibonacciIterativeDriver

F0 = 0

F1 = 1

F2 = 1

F3 = 2

F4 = 3

F5 = 5

F6 = 8

F7 = 13

F8 = 21

F9 = 34

F10 = 55

F11 = 89

F12 = 144

F13 = 233

F14 = 377

F15 = 610

F16 = 987

F17 = 1597

15

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Recursividad

F18 = 2584

F19 = 4181

F20 = 6765

F21 = 10946

F22 = 17711

F23 = 28657

F24 = 46368

F25 = 75025

F26 = 121393

46.097 microseconds.

./FibonacciRecursiveIfElseDriver

F0 = 0

F1 = 1

F2 = 1

F3 = 2

F4 = 3

F5 = 5

F6 = 8

F7 = 13

F8 = 21

F9 = 34

F10 = 55

F11 = 89

F12 = 144

F13 = 233

F14 = 377

F15 = 610

F16 = 987

F17 = 1597

F18 = 2584

F19 = 4181

F20 = 6765

F21 = 10946

F22 = 17711

F23 = 28657

F24 = 46368

F25 = 75025

F26 = 121393

4148.803 microseconds.

./FibonacciRecursiveIfDriver

F0 = 0

F1 = 1

F2 = 1

F3 = 2

16

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Recursividad

F4 = 3

F5 = 5

F6 = 8

F7 = 13

F8 = 21

F9 = 34

F10 = 55

F11 = 89

F12 = 144

F13 = 233

F14 = 377

F15 = 610

F16 = 987

F17 = 1597

F18 = 2584

F19 = 4181

F20 = 6765

F21 = 10946

F22 = 17711

F23 = 28657

F24 = 46368

F25 = 75025

F26 = 121393

3366.903 microseconds.

./FibonacciRecursiveSwitchDriver

F0 = 0

F1 = 1

F2 = 1

F3 = 2

F4 = 3

F5 = 5

F6 = 8

F7 = 13

F8 = 21

F9 = 34

F10 = 55

F11 = 89

F12 = 144

F13 = 233

F14 = 377

F15 = 610

F16 = 987

F17 = 1597

F18 = 2584

F19 = 4181

17

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Recursividad

F20 = 6765

F21 = 10946

F22 = 17711

F23 = 28657

F24 = 46368

F25 = 75025

F26 = 121393

4468.692 microseconds.

./FibonacciRecursiveSwitchFallthroughDriver

F0 = 0

F1 = 1

F2 = 1

F3 = 2

F4 = 3

F5 = 5

F6 = 8

F7 = 13

F8 = 21

F9 = 34

F10 = 55

F11 = 89

F12 = 144

F13 = 233

F14 = 377

F15 = 610

F16 = 987

F17 = 1597

F18 = 2584

F19 = 4181

F20 = 6765

F21 = 10946

F22 = 17711

F23 = 28657

F24 = 46368

F25 = 75025

F26 = 121393

3360.468 microseconds.

./FibonacciMemoizationCodeDriver

F0 = 0

F1 = 1

F2 = 1

F3 = 2

F4 = 3

F5 = 5

18

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Recursividad

F6 = 8

F7 = 13

F8 = 21

F9 = 34

F10 = 55

F11 = 89

F12 = 144

F13 = 233

F14 = 377

F15 = 610

F16 = 987

F17 = 1597

F18 = 2584

F19 = 4181

F20 = 6765

F21 = 10946

F22 = 17711

F23 = 28657

F24 = 46368

F25 = 75025

F26 = 121393

77.638 microseconds.

./FibonacciMemoizationStaticDataDriver

F0 = 0

F1 = 1

F2 = 1

F3 = 2

F4 = 3

F5 = 5

F6 = 8

F7 = 13

F8 = 21

F9 = 34

F10 = 55

F11 = 89

F12 = 144

F13 = 233

F14 = 377

F15 = 610

F16 = 987

F17 = 1597

F18 = 2584

F19 = 4181

F20 = 6765

F21 = 10946

19

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Recursividad

F22 = 17711

F23 = 28657

F24 = 46368

F25 = 75025

F26 = 121393

44.976 microseconds.

./FibonacciMemoizationDynamicDataArrayDriver

F0 = 0

F1 = 1

F2 = 1

F3 = 2

F4 = 3

F5 = 5

F6 = 8

F7 = 13

F8 = 21

F9 = 34

F10 = 55

F11 = 89

F12 = 144

F13 = 233

F14 = 377

F15 = 610

F16 = 987

F17 = 1597

F18 = 2584

F19 = 4181

F20 = 6765

F21 = 10946

F22 = 17711

F23 = 28657

F24 = 46368

F25 = 75025

F26 = 121393

43.826 microseconds.

./FibonacciMemoizationDynamicDataVectorDriver

F0 = 0

F1 = 1

F2 = 1

F3 = 2

F4 = 3

F5 = 5

F6 = 8

F7 = 13

20

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Recursividad

F8 = 21

F9 = 34

F10 = 55

F11 = 89

F12 = 144

F13 = 233

F14 = 377

F15 = 610

F16 = 987

F17 = 1597

F18 = 2584

F19 = 4181

F20 = 6765

F21 = 10946

F22 = 17711

F23 = 28657

F24 = 46368

F25 = 75025

F26 = 121393

57.489 microseconds.

Fibonacci Bechmark output.

./FibonacciIterativeBenchmark

F17 = 1597 9.034 microseconds.

F27 = 196418 1.765 microseconds.

F32 = 2178309 1.579 microseconds.

F35 = 9227465 1.543 microseconds.

./FibonacciRecursiveIfElseBenchmark

F17 = 1597 41.551 microseconds.

F27 = 196418 2333.753 microseconds.

F32 = 2178309 24162.599 microseconds.

F35 = 9227465 103341.085 microseconds.

./FibonacciRecursiveIfBenchmark

F17 = 1597 25.558 microseconds.

F27 = 196418 1973.196 microseconds.

F32 = 2178309 25276.450 microseconds.

F35 = 9227465 144965.583 microseconds.

./FibonacciRecursiveSwitchBenchmark

F17 = 1597 35.157 microseconds.

F27 = 196418 3175.235 microseconds.

F32 = 2178309 35336.870 microseconds.

21

Page 22: Recursividad - Recursividad - GitHub PagesRecursividad Figura 4. Traza de pila de llamadas Figura 5. Conceptual 3. Listados Completos 3.1. Factorial FactorialDriver.cpp. /* FactorialDriver

Recursividad

F35 = 9227465 151168.537 microseconds.

./FibonacciRecursiveSwitchFallthroughBenchmark

F17 = 1597 32.217 microseconds.

F27 = 196418 2778.870 microseconds.

F32 = 2178309 30693.526 microseconds.

F35 = 9227465 124443.841 microseconds.

./FibonacciMemoizationCodeBenchmark

F17 = 1597 11.449 microseconds.

F27 = 196418 2.338 microseconds.

F32 = 2178309 4.971 microseconds.

F35 = 9227465 15.155 microseconds.

./FibonacciMemoizationStaticDataBenchmark

F17 = 1597 8.708 microseconds.

F27 = 196418 2.482 microseconds.

F32 = 2178309 5.751 microseconds.

F35 = 9227465 17.158 microseconds.

./FibonacciMemoizationDynamicDataArrayBenchmark

F17 = 1597 8.575 microseconds.

F27 = 196418 1.921 microseconds.

F32 = 2178309 1.628 microseconds.

F35 = 9227465 1.806 microseconds.

./FibonacciMemoizationDynamicDataVectorBenchmark

F17 = 1597 16.356 microseconds.

F27 = 196418 6.208 microseconds.

F32 = 2178309 1.799 microseconds.

F35 = 9227465 1.557 microseconds.

22


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