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557 lines
11 KiB
C++
557 lines
11 KiB
C++
//
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// FILE: statHelpers.cpp
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// AUTHOR: Rob Tillaart
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// VERSION: 0.1.7
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// PURPOSE: Arduino library with a number of statistic helper functions.
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// DATE: 2020-07-01
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// URL: https://github.com/RobTillaart/statHelpers
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#include "statHelpers.h"
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///////////////////////////////////////////////////////////////////////////
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//
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// PERMUTATIONS
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//
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uint32_t permutations(uint8_t n, uint8_t k)
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{
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uint32_t rv = 1;
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for (uint8_t i = n; i > (n - k); i--)
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{
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rv *= i;
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}
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return rv;
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}
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uint64_t permutations64(uint8_t n, uint8_t k)
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{
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uint64_t rv = 1;
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for (uint8_t i = n; i > (n - k); i--)
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{
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rv *= i;
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}
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return rv;
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}
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// can be optimized similar to dfactorial
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double dpermutations(uint8_t n, uint8_t k)
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{
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double rv = 1;
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for (uint8_t i = n; i > (n - k); i--)
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{
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rv *= i;
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}
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return rv;
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}
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/*
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http://wordaligned.org/articles/next-permutation snippet
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As an example consider finding the next permutation of:
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8342666411
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The longest monotonically decreasing tail is 666411, and the corresponding head is 8342.
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8342 666411
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666411 is, by definition, reverse-ordered, and cannot be increased by permuting its elements.
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To find the next permutation, we must increase the head; a matter of finding the smallest tail
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element larger than the head’s final 2.
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8342 666411
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Walking back from the end of tail, the first element greater than 2 is 4.
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8342 666411
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Swap the 2 and the 4
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8344 666211
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Since head has increased, we now have a greater permutation. To reduce to the next permutation,
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we reverse tail, putting it into increasing order.
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8344 112666
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Join the head and tail back together. The permutation one greater than 8342666411 is 8344112666.
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*/
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// http://www.nayuki.io/page/next-lexicographical-permutation-algorithm
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// b = nextPermutation<char>(array, 100);
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/*
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template <typename T>
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bool nextPermutation(T * array, uint16_t size)
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{
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// Find longest non-increasing suffix
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int i = size - 1;
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while (i > 0 && array[i - 1] >= array[i]) i--;
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// Now i is the head index of the suffix
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// Are we at the last permutation already?
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if (i <= 0) return false;
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// Let array[i - 1] be the pivot
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// Find rightmost element that exceeds the pivot
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int j = size - 1;
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while (array[j] <= array[i - 1])
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j--;
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// Now the value array[j] will become the new pivot
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// Assertion: j >= i
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// Swap the pivot with j
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T temp = array[i - 1];
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array[i - 1] = array[j];
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array[j] = temp;
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// Reverse the suffix
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j = size - 1;
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while (i < j)
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{
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temp = array[i];
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array[i] = array[j];
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array[j] = temp;
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i++;
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j--;
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}
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return true;
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}
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*/
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///////////////////////////////////////////////////////////////////////////
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//
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// FACTORIAL
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//
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// exact ==> 12!
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uint32_t factorial(uint8_t n)
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{
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uint32_t f = 1;
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while(n > 1) f *= (n--);
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return f;
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}
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// exact ==> 20!
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uint64_t factorial64(uint8_t n)
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{
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// to be tested
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// if ( n <= 12) return factorial(12);
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// uint64_t f = factorial(12);
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// for (uint8_t t = 13; t <= n; t++) f *= t;
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uint64_t f = 1;
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while(n > 1) f *= (n--);
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return f;
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}
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// float (4 byte) => 34!
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// double (8 byte) => 170!
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double dfactorialReference(uint8_t n)
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{
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double f = 1;
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while (n > 1) f *= (n--);
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return f;
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}
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// FASTER VERSION
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// does part of the math with integers.
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// tested on UNO and ESP32, roughly 3x faster
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// numbers differ slightly in the order of IEEE754 precision => acceptable.
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// 10e-7 for 4 bit float
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// 10e-16 for 8 bit double
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double dfactorial(uint8_t n)
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{
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double f = 1;
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while (n > 4)
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{
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uint32_t val = n * (n-1);
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val *= (n-2) * (n-3);
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f *= val;
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n -= 4;
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}
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while (n > 1) f *= (n--); // can be squeezed too.
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return f;
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}
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// stirling is an approximation function for factorial(n).
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// it is slower but constant in time.
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// float => 26!
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// double => 143!
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double stirling(uint8_t n)
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{
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double v = exp(-n) * pow(n, n) * sqrt(TWO_PI * n);
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return v;
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}
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///////////////////////////////////////////////////////////////////////////
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//
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// SEMIFACTORIAL
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//
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// exact ==> 20!!
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uint32_t semiFactorial(uint8_t n)
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{
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uint32_t f = 1;
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while(n > 1)
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{
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f *= n;
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n -= 2;
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}
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return f;
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}
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// exact ==> 33!!
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uint64_t semiFactorial64(uint8_t n)
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{
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uint64_t f = 1;
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while(n > 1)
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{
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f *= n;
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n -= 2;
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}
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return f;
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}
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// float (4 byte) => 56!!
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// double (8 byte) => 300!!
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double dSemiFactorial(uint16_t n)
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{
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double f = 1;
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while (n > 6) // TODO: test performance gain by integer math. (depends on platform ?)
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{
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uint32_t val = n * (n-2) * (n-4);
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f *= val;
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n -= 6;
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}
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while (n > 1)
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{
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f *= n;
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n -= 2;
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}
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return f;
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}
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///////////////////////////////////////////////////////////////////////////
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//
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// COMBINATIONS
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//
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// works for n = 0..30 for all k
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uint32_t combinations(uint16_t n, uint16_t k)
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{
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if ((k == 0) || (k == n)) return 1;
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if (k < (n-k)) k = n - k; // symmetry
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uint32_t rv = n;
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uint8_t p = 2;
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for (uint8_t i = n-1; i > k; i--)
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{
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// if ((0xFFFFFFFF / i) < rv) return 0; // overflow detect...
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rv = (rv * i) / p;
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p++;
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}
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return rv;
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}
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// works for n = 0..61 for all k
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uint64_t combinations64(uint16_t n, uint16_t k)
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{
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if ((k == 0) || (k == n)) return 1;
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if (k < (n-k)) k = n - k; // symmetry
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uint64_t rv = n;
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uint8_t p = 2;
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for (uint8_t i = n-1; i > k; i--)
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{
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// overflow detect here ?
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rv = (rv * i) / p;
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p++;
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}
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return rv;
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}
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// experimental - not exact but allows large values
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// float (4 bits) works till n = 125 for all k
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// double (8 bits) works till n = 1020 for all k
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double dcombinations(uint16_t n, uint16_t k)
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{
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if ((k == 0) || (k == n)) return 1;
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if (k < (n-k)) k = n - k; // symmetry
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double rv = n;
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uint16_t p = 2;
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for (uint16_t i = n-1; i > k; i--)
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{
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rv *= i;
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rv /= p;
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p++;
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}
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return rv;
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}
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// recursive (mind your stack and time)
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// works for n = 0..30 for all k
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// educational purpose
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uint32_t rcombinations(uint16_t n, uint16_t k)
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{
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if (k > (n-k)) k = n - k; // symmetry
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if (k == 0) return 1;
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return (n * rcombinations(n - 1, k - 1)) / k;
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}
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// recursive
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// works for n = 0..61 for all k
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// educational purpose
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uint64_t rcombinations64(uint16_t n, uint16_t k)
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{
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if (k > (n-k)) k = n - k; // symmetry
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if (k == 0) return 1;
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return (n * rcombinations64(n - 1, k - 1)) / k;
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}
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// very slow double recursive way by means of Pascals triangle.
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// works for n = 0..30 for all k (but takes a lot of time)
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// educational purpose
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uint32_t combPascal(uint16_t n, uint16_t k)
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{
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if (k > (n-k)) k = n - k; // symmetry
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if (k > n ) return 0;
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if (k == 0) return 1;
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if (n < 2) return 1;
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uint32_t rv = combPascal(n-1, k-1);
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rv += combPascal(n-1, k);
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return rv;
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}
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/////////////////////////////////////////////////
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//
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// EXPERIMENTAL
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//
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// BIG SECTION
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//
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// keep track of exponent myself in 32 bit unsigned integer
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// - can be extended to a 64 bit integer
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// however it already takes hours to calculate with 32 bits
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/*
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UNO
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n n! millis() from earlier version
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-----------------------------------------
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1 1.00000e0 0
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10 3.62880e6 1
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100 9.33262e157 7
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1000 4.02386e2567 105
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2000 3.31627e5735 231
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4000 1.82880e12673 504
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8000 5.18416e27752 1086
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10000 2.84625e35659 1389
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16000 5.11880e60319 2330
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32000 1.06550e130270 4978
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100000 2.82428e456573 17201
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1000000 8.26379e5565708 206421 (3.5 minutes)
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10000000 an hour? too long...
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ESP32 240MHz
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n n! millis() from earlier version
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-----------------------------------------
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1 1.00000e0 0
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10 3.62880e6 0
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100 9.33262e157 0
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1000 4.02387e2567 8
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10000 2.84626e35659 110
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100000 2.82423e456573 1390
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1000000 8.26393e5565708 16781
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10000000 1.20242e65657059 196573 (3++ minutes)
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100000000 1.61720e756570556 2253211
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1000000000 9.90463e4270738226 25410726 (7++ hrs to detect overflow! :(
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1 0
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10 0.6
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100 1.57
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1000 2.567
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10000 3.5659
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100000 4.56573
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1000000 5.565708
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10000000 6.5657059
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100000000 7.56570556
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// largest found - exponent is approaching max_uint32_t - 4294967296
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518678058! 4.1873547e4294967283 // break condition was hit...
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next one should fit too
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518678059! 2.1718890e4294967292
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*/
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void bigFactorial(uint32_t n, double &mantissa, uint32_t &exponent)
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{
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exponent = 0;
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double f = 1;
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while (n > 1)
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{
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f *= n--;
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while (f > 1000000000)
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{
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f /= 1000000000;
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exponent += 9;
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}
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}
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while (f > 10) // fix exponent if needed.
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{
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f /= 10;
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exponent++;
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}
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mantissa = f;
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}
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// Should work full range for n = 518678059, k = { 0..n }
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// and n > 518678059 => max k is definitely smaller ==> expect n+1 ==> k-15 at start
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// performance: low. depends on k.
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//
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//
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// for relative small k and large n (multiple orders of magnitude) one can get an estimate
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// P(n,k) ~ raw = log10(n - k/2) * k;
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// exponent = int(raw);
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// mantissa = pow(10, raw - int(raw));
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void bigPermutations(uint32_t n, uint32_t k, double &mantissa, uint32_t &exponent)
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{
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exponent = 0;
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double f = 1;
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for (uint32_t i = n; i > (n - k); i--)
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{
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f *= i;
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while (f > 1000000000)
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{
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f /= 1000000000;
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exponent += 9;
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}
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}
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while (f > 10) // fix exponent if needed.
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{
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f /= 10;
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exponent++;
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}
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mantissa = f;
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}
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void bigCombinations(uint32_t n, uint32_t k, double &mantissa, uint32_t &exponent)
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{
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exponent = 0;
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if ((k == 0) || (k == n)) return;
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if (k < (n-k)) k = n - k; // symmetry
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double f = n;
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uint32_t p = 2;
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for (uint32_t i = n-1; i > k; i--)
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{
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f = (f * i) / p;
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p++;
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while (f > 1000000000)
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{
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f /= 1000000000;
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exponent += 9;
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}
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}
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while (f > 10) // fix exponent if needed.
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{
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f /= 10;
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exponent++;
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}
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mantissa = f;
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}
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////////////////////////////////////////////////////////////
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//
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// EXPERIMENTAL 64 BIT (not tested)
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//
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void bigFactorial64(uint64_t n, double &mantissa, uint64_t &exponent)
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{
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exponent = 0;
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double f = 1;
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while (n > 1)
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{
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f *= n--;
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while (f > 1000000000) // optimize - per 1e9 to save divisions.
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{
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f /= 1000000000;
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exponent += 9;
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}
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}
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while (f > 10) // fix exponent if needed.
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{
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f /= 10;
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exponent++;
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}
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mantissa = f;
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}
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void bigPermutations64(uint64_t n, uint64_t k, double &mantissa, uint64_t &exponent)
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{
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exponent = 0;
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double f = 1;
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for (uint64_t i = n; i > (n - k); i--)
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{
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f *= i;
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while (f > 1000000000)
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{
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f /= 1000000000;
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exponent += 9;
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}
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}
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while (f > 10) // fix exponent if needed.
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{
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f /= 10;
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exponent++;
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}
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mantissa = f;
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}
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void bigCombinations64(uint64_t n, uint64_t k, double &mantissa, uint64_t &exponent)
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{
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exponent = 0;
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mantissa = 0;
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if ((k == 0) || (k == n)) return;
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if (k < (n-k)) k = n - k; // symmetry
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double f = n;
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uint64_t p = 2;
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for (uint64_t i = n-1; i > k; i--)
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{
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f = (f * i) / p;
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p++;
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while (f > 1000000000)
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{
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f /= 1000000000;
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exponent += 9;
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}
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}
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while (f > 10) // fix exponent if needed.
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{
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f /= 10;
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exponent++;
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}
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mantissa = f;
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}
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// -- END OF FILE --
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