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second argument. The function inserts the second argument in the list so that the list remains

sorted, and does so in logarithmic (O(log(N))) time. Here, we insert the pair

(priority,

data)

. Since pairs (and other tuples, lists, and sequences in general) are compared

lexicographically, this means that data will be placed in increasing order of priority. Therefore, the

pop

function, by getting (and removing, via the lists'

pop

method) the last item in list

self.queue

, is assured to get the item with the highest priority among those currently in the

queue. It then applies indexing

[1]

to throw the priority away and return only the data.

17.16.4 See Also

Documentation on the

bisect

module in the Library Reference.

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17.17 Converting Numbers to Rationals via Farey

Fractions

Credit: Scott David Daniels

17.17.1 Problem

You have a number

(of almost any type) and need to find a rational number (in reduced form)

that is as close to

as possible but with a denominator no larger than a prescribed value.

17.17.2 Solution

Farey fractions, whose crucial properties were studied by Cauchy, are an excellent way to find

rational approximations of floating-point values:

def farey(v, lim):

""" No error checking on args. lim = maximum denominator.

Results are (numerator, denominator); (1, 0) is

"infinity".

"""

if v < 0:

n, d = farey(-v, lim)

return -n, d

z = lim - lim # Get a "0 of right type" for

denominator

lower, upper = (z, z+1), (z+1, z)

while 1:

mediant = (lower[0] + upper[0]), (lower[1] +

upper[1])

if v * mediant[1] > mediant[0]:

if lim < mediant[1]: return upper

lower = mediant

elif v * mediant[1] == mediant[0]:

if lim >= mediant[1]: return mediant

if lower[1] < upper[1]: return lower

return upper

else:

if lim < mediant[1]: return lower

upper = mediant

For example,

farey(math.pi, 100) == (22, 7)

17.17.3 Discussion

The rationals resulting from this algorithm are in reduced form (numerator and denominator

mutually prime), but the proof, which was given by Cauchy, is rather subtle (see http://www.cut-

the-knot.com/blue/Farey.html

Note the trickiness with

. It is a zero of the same type as the

lim

argument. This lets you use

longs as the limit if necessary, without paying a performance price (not even a test) when there's

no such need.

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To print odds, you can use:

n, d = farey(probability, lim)

print "Odds are %d : %d" % (n, d-n)

This algorithm is ideally suited for reimplementation in a lower-level language (e.g., C or

assembly) if you use it heavily. Since the code uses only multiplication and addition, it can play to

hardware strengths.

If you are using this in an environment where you call it with a lot of values near 0.0, 1.0, or 0.5

(or simple fractions), you may find that its convergence is too slow. You can improve its

convergence in a continued fraction style by appending to the first

in the

farey

function:

if v < 0:

...

elif v < 0.5:

n, d = farey((v-v+1)/v, lim) # lim is wrong; decide what

you want

return d, n

elif v > 1:

intpart = floor(v)

n, d = farey(v-intpart)

return n+intpart*d, d

...

James Farey was an English surveyor who wrote a letter to the Journal of Science around the end

of the 18th century. In that letter he observed that, while reading a privately published list of the

decimal equivalents of fractions, he noticed the following: for any three consecutive fractions in

the simplest terms (e.g., A/B, C/D, E/F), the middle one (C/D), called the mediant, is equal to the

ratio (A + E)/(B + F). I enjoy envisioning Mr. Farey sitting up late on a rainy English night,

reading tables of decimal expansions of fractions by an oil lamp. Calculation has come a long way

since his day, and I'm pleased to be able to benefit from his work.

17.17.4 See Also

Recipe 17.18

for another mathematical evaluation recipe.

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17.18 Evaluating a Polynomial

Credit: Luther Blissett

17.18.1 Problem

You need to evaluate a polynomial function, and you know that the obvious way to evaluate a

polynomial wastes effort; therefore, Horner's well-known formula should always be used instead.

17.18.2 Solution

We often need to evaluate a polynomial

f(x)

, defined by its coefficients (

c[0]+c[1]

x+c[2]

x2+

...), at a given point

. There is an obvious (naive) approach to this, applying the

polynomial's definition directly:

def poly_naive(x, coeff):

result = coeff[0]

for i in range(1, len(coeff)):

result = result + coeff[i] * x**i

return result

However, this is a substantial waste of computational effort, since raising to a power is a time-

consuming operation. Here, we're wantonly raising

to successive powers. It's better to use

Horner's well-known formula, based on the observation that the polynomial formula can also be

indifferently written as

c[0]+x

(c[1]+x

(c[2]+

.... In other words, it can be written

with nested parentheses, but without raise-to-power operations, only additions and multiplications.

Coding a loop for it gives us:

def poly_horner(x, coeff):

result = coeff[-1]

for i in range(-2, -len(coeff)-1, -1):

result = result*x + coeff[i]

return result

17.18.3 Discussion

Python programmers generally emphasize simplicity, not speed. However, when equally simple

solutions exist, and one is always faster (even by a little), it seems sensible to use the faster

solution. Polynomial evaluation is a case in point. The naive approach takes an addition, a

multiplication, and an exponentiation for each degree of the polynomial. Horner's formula takes

just a multiplication and an addition for each degree. On my system, evaluating 10,000 integer

(long) polynomials of order 40 takes 3.37 seconds the naive way and 1.07 seconds the Horner way.

With float arithmetic, it takes 0.53 seconds the naive way and 0.30 seconds the Horner way. Waste

not, want not, I say.

17.18.4 See Also

Recipe 17.17

for another mathematical evaluation recipe.

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17.19 Module: Finding the Convex Hull of a Set of 2D

Points

Credit: Dinu C. Gherman

Convex hulls of point sets are an important building block in many computational-geometry

applications. Example 17-1

calculates the convex hull of a set of 2D points and generates an

Encapsulated PostScript (EPS) file to visualize it. Finding convex hulls is a fundamental problem

in computational geometry and is a basic building block for solving many problems. The

algorithm used here can be found in any good textbook on computational geometry, such as

Computational Geometry: Algorithms and Applications, 2nd edition (Springer-Verlag). Note that

the given implementation is not guaranteed to be numerically stable. It might benefit from using

the

Numeric

package for gaining more performance for very large sets of points.

Example 17-1. Finding the convex hull of a set of 2D points

""" convexhull.py

Calculate the convex hull of a set of n 2D points in O(n log

n) time.

Taken from Berg et al., Computational Geometry, Springer-

Verlag, 1997.

Emits output as EPS file.

When run from the command line, it generates a random set of

points

inside a square of given length and finds the convex hull

for those,

emitting the result as an EPS file.

Usage:

convexhull.py <numPoints> <squareLength> <outFile>

Dinu C. Gherman

"""

import sys, string, random

# helpers

def _myDet(p, q, r):

""" Calculate determinant of a special matrix with three

2D points.

The sign, - or +, determines the side (right or left,

respectively) on which

the point r lies when measured against a directed vector

from p to q.

"""

# We use Sarrus' Rule to calculate the determinant

# (could also use the Numeric package...)

sum1 = q[0]*r[1] + p[0]*q[1] + r[0]*p[1]

sum2 = q[0]*p[1] + r[0]*q[1] + p[0]*r[1]

return sum1 - sum2

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def _isRightTurn((p, q, r)):

"Do the vectors pq:qr form a right turn, or not?"

assert p != q and q != r and p != r

return _myDet(p, q, r) < 0

def _isPointInPolygon(r, P):

"Is point r inside a given polygon P?"

# We assume that the polygon is a list of points, listed

clockwise

for i in xrange(len(P)-1):

p, q = P[i], P[i+1]

if not _isRightTurn((p, q, r)):

return 0 # Out!

return 1 # It's within!

def _makeRandomData(numPoints=10, sqrLength=100,

addCornerPoints=0):

"Generate a list of random points within a square (for

test/demo only)"

# Fill a square with N random points

min, max = 0, sqrLength

P = []

for i in xrange(numPoints):

rand = random.randint

x = rand(min+1, max-1)

y = rand(min+1, max-1)

P.append((x, y))

# Add some "outmost" corner points

if addCornerPoints:

P = P + [(min, min), (max, max), (min, max), (max,

min)]

return P

# output

epsHeader = """%%!PS-Adobe-2.0 EPSF-2.0

%%%%BoundingBox: %d %d %d %d

/r 2 def %% radius

/circle %% circle, x, y, r --> -

{

0 360 arc %% draw circle

} def

1 setlinewidth %% thin line

newpath %% open page

0 setgray %% black color

"""

def saveAsEps(P, H, boxSize, path):

"Save some points and their convex hull into an EPS

file."

# Save header

f = open(path, 'w')

f.write(epsHeader % (0, 0, boxSize, boxSize))

format = "%3d %3d"

# Save the convex hull as a connected path

if H:

f.write("%s moveto\n" % format % H[0])

for p in H:

f.write("%s lineto\n" % format % p)

f.write("%s lineto\n" % format % H[0])

f.write("stroke\n\n")

# Save the whole list of points as individual dots

for p in P:

f.write("%s r circle\n" % format % p)

f.write("stroke\n")

# Save footer

f.write("\nshowpage\n")

# public interface

def convexHull(P):

"Calculate the convex hull of a set of points."

# Get a local list copy of the points and sort them

lexically

points = map(None, P)

points.sort( )

# Build upper half of the hull

upper = [points[0], points[1]]

for p in points[2:]:

upper.append(p)

while len(upper) > 2 and not _isRightTurn(upper[-

3:]):

del upper[-2]

# Build lower half of the hull

points.reverse( )

lower = [points[0], points[1]]

for p in points[2:]:

lower.append(p)

while len(lower) > 2 and not _isRightTurn(lower[-

3:]):

del lower[-2]

# Remove duplicates

del lower[0]

del lower[-1]

# Concatenate both halves and return

return tuple(upper + lower)

# Test

def test( ):

a = 200

p = _makeRandomData(30, a, 0)

c = convexHull(p)

saveAsEps(p, c, a, file)

if _ _name_ _ == '_ _main_ _':

try:

numPoints = string.atoi(sys.argv[1])

squareLength = string.atoi(sys.argv[2])

path = sys.argv[3]

except IndexError:

numPoints = 30

squareLength = 200

path = "sample.eps"

p = _makeRandomData(numPoints, squareLength,

addCornerPoints=0)

c = convexHull(p)

saveAsEps(p, c, squareLength, path)

17.19.1 See Also

Computational Geometry: Algorithms and Applications, 2nd edition, by M. de Berg, M. van

Kreveld, M. Overmars, and O. Schwarzkopf (Springer-Verlag).

17.20 Module: Parsing a String into a Date/Time Object

Portably

Credit: Brett Cannon

Python's

time

module supplies the parsing function

strptime

only on some platforms, and

not on Windows. Example 17-2

shows a

strptime

function that is a pure Python

implementation of the

time.strptime

function that comes with Python. It is similar to how

time.strptime

is documented in the standard Python documentation. It accepts two more

optional arguments, as shown in the following signature:

strptime(string, format="%a %b %d %H:%M:%S %Y", option=AS_IS,

locale_setting=ENGLISH)

option

's default value of

AS_IS

gets time information from the string, without any checking

or filling-in. You can pass

option

CHECK

, so that the function makes sure that whatever

information it gets is within reasonable ranges (raising an exception otherwise), or

FILL_IN

(like

CHECK

, but also tries to fill in any missing information that can be computed).

locale_setting

accepts a locale tuple (as created by

LocaleAssembly

) to specify

names of days, months, and so on. Currently,

ENGLISH

and

SWEDISH

locale tuples are built

into this recipe's

strptime

module.

Although this recipe's

strptime

cannot be as fast as the version in the standard Python library,

that's hardly ever a major consideration for typical

strptime

use. This recipe does offer two

substantial advantages. It runs on any platform supporting Python and gives perfectly identical

results on different platforms, while

time.strptime

exists only on some platforms and tends

to have different quirks on each platform that supplies it. The optional checking and filling-in of

information that this recipe provides is also quite handy.

The locale-setting support of this version of

strptime

was inspired by that in Andrew

Markebo's own

strptime

, which you can find at http://www.fukt.hk-

r.se/~flognat/hacks/strptime.py

. However, this recipe has a more complete implementation of

strptime

's specification that is based on regular expressions, rather than relying on whitespace

and miscellaneous characters to split strings. For example, this recipe can correctly parse strings

based on a format such as

"%Y%m%d"

Example 17-2. Parsing a string into a date/time object portably

""" A pure-Python version of strptime.

As close as possible to time.strptime's specs in the

official Python docs.

Locales supported via LocaleAssembly -- examples supplied

for English and

Swedish, follow the examples to add your own locales.

Thanks to Andrew Markebo for his pure Python version of

strptime, which

convinced me to improve locale support -- and, of course, to

Guido van Rossum

and all other contributors to Python, the best language I've

ever used!

"""

import re

from exceptions import Exception

_ _all_ _ = ['strptime', 'AS_IS', 'CHECK', 'FILL_IN',

'LocaleAssembly', 'ENGLISH', 'SWEDISH']

# metadata module

_ _author_ _ = 'Brett Cannon'

_ _email_ _ = 'drifty@bigfoot.com'

_ _version_ _ = '1.5cb'

_ _url_ _ = 'http://www.drifty.org/'

# global settings and parameter constants

CENTURY = 2000

AS_IS = 'AS_IS'

CHECK = 'CHECK'

FILL_IN = 'FILL_IN'

def LocaleAssembly(DirectiveDict, MonthDict, DayDict,

am_pmTuple):

""" Creates locale tuple for use by strptime.

Accepts arguments dictionaries DirectiveDict (locale-

specific regexes for

extracting info from time strings), MonthDict (locale-

specific full and

abbreviated month names), DayDict (locale-specific full

and abbreviated

weekday names), and the am_pmTuple tuple (locale-

specific valid

representations of AM and PM, as a two-item tuple). Look

at how the

ENGLISH dictionary is created for an example; make sure

your dictionary has values

corresponding to each entry in the ENGLISH dictionary.

You can override

any value in the BasicDict with an entry in

DirectiveDict.

"""

BasicDict={'%d':r'(?P<d>[0-3]\d)', # Day of the month

[01,31]

'%H':r'(?P<H>[0-2]\d)', # Hour (24-h) [00,23]

'%I':r'(?P<I>[01]\d)', # Hour (12-h) [01,12]

'%j':r'(?P<j>[0-3]\d\d)', # Day of the year [001,366]

'%m':r'(?P<m>[01]\d)', # Month [01,12]

'%M':r'(?P<M>[0-5]\d)', # Minute [00,59]

'%S':r'(?P<S>[0-6]\d)', # Second [00,61]

'%U':r'(?P<U>[0-5]\d)', # Week in the year, Sunday

first [00,53]

'%w':r'(?P<w>[0-6])', # Weekday [0(Sunday),6]

'%W':r'(?P<W>[0-5]\d)', # Week in the year, Monday

first [00,53]

'%y':r'(?P<y>\d\d)', # Year without century [00,99]

'%Y':r'(?P<Y>\d\d\d\d)', # Year with century

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