For the hash function : h(k) = k mod m;
I understand that m=2^n will always give the last n LSB digits. I also understand that m=2^p-1 when K is a string converted to integers using radix 2^p will give same hash value for every permutation of characters in K. But why exactly "a prime not too close to an exact power of 2" is a good choice? What if I choose 2^p - 2 or 2^p-3? Why are these choices considered bad?
Following is the text from CLRS:
"A prime not too close to an exact power of 2 is often a good choice for m. For
example, suppose we wish to allocate a hash table, with collisions resolved by
chaining, to hold roughly n D 2000 character strings, where a character has 8 bits.
We don’t mind examining an average of 3 elements in an unsuccessful search, and
so we allocate a hash table of size m D 701. We could choose m D 701 because
it is a prime near 2000=3 but not near any power of 2."
Suppose we work with radix 2p.
2p-1 case:
Why that is a bad idea to use 2p-1? Let us see,
k = ∑ai2ip
and if we divide by 2p-1 we just get
k = ∑ai2ip = ∑ai mod 2p-1
so, as addition is commutative, we can permute digits and get the same result.
2p-b case:
Quote from CLRS:
A prime not too close to an exact power of 2 is often a good choice for m.
k = ∑ai2ip = ∑aibi mod 2p-b
So changing least significant digit by one will change hash by one. Changing second least significant bit by one will change hash by two. To really change hash we would need to change digits with bigger significance. So, in case of small b we face problem similar to the case then m is power of 2, namely we depend on distribution of least significant digits.
I have a few simple equations that I want to pipe through matlab. But I would like to get exact answers, because these values are expected to be used and simplified later on.
Right now Matlab shows sqrt(2.0) as 1.1414 instead of something like 2^(1/2) as I would like.
I tried turning on format rat but this is dangerous becasue it shows sqrt(2) as 1393/985 without any sort of warning.
There is "symbolic math" but it seems like overkill.
All I want is that 2 + sqrt(50) would return something like 2 + 5 * (2)^(1/2) and even my 5 years old CASIO calculator can do this!
So what can I do to get 2 + sqrt(50) evaluate to 2 + 5 * (2)^(1/2) in matlab?
As per #Oleg's comment use symbolic math.
x=sym('2')+sqrt(sym('50'))
x =
5*2^(1/2) + 2
The average time on ten thousand iterations through this expression is 1.2 milliseconds, whilst the time for the numeric expression (x=2+sqrt(50)) is only 0.4 micro seconds, i.e. a factor of ten thousand faster.
I did pre-run the symbolic expression 50 times, because, as Oleg points out in his second comment the symbolic engine needs some warming up. The first run through your expression took my pc almost 2 seconds.
I would therefore recommend using numeric equations due to the huge difference in calculation time. Only use symbolic expressions when you are forced to (e.g. simplifying expressions for a paper) and then use a symbolic computation engine like Maple or Wolfram Alpha.
Matlab main engine is not symbolic but numeric.
Symbolic toolbox. Create expression in x and subs x = 50
syms x
f = 2+sqrt(x)
subs(f,50)
ans =
50^(1/2) + 2
Say I give something like AB+AB+BA to matlab (or mupad), and ask it to simplify it. the answer should be: 2AB+BA. Can this be done in matlab or mupad?
Edit:
Ok, this is feeling rediculous. I'm trying to do this in either matlab or mulab, and.. it's frustrating not knowing how to do what should be the simplest things, and not being able to find the answers right away via google.
I want to expand the following, multiplied together, as a taylor series:
eq1 := exp(g*l*B):
eq2 := exp(l*A):
eq3 := exp((1-g)*l*B):
g is gamma, l is lambda (don't know how to represent either of these in matlab or mulab). A and B don't commute. I want to multiply the three exponentials together, expand, select all terms of a given power in lambda, and simplify the result. Is there a simple way to do this? or should I give up and go to another system, like maple?
This is mupad, not matlab:
operator("x", _vector_product, Binary, 1999):
A x B + A x B + B x A
returns
2 A x B + B x A
The vetor product is used, simply because it matches the described requirements.
The other day I discovered the following bug in a couple of places in my MATLAB code
I wanted to enter the column vector in my MATLAB script
[a-b,
c-d
e-f]
where a,b,c,d,e,f are long expressions in some variables.
and I entered it in as
[ a -b ;
c -d ;
e -f]
Now MATLAB interprets the second matrix as a 3x2 matrix instead of a column vector.
Is there a way/command/function to force MATLAB to use only the comma and NOT any white space characters as a column separator for matrices ?
I don't think there is any way to force matlab to not treat white space this way, since it is interpretive language, and doing so may affect some built-in functions/third-party code.
However, you can use parentheses to group data - i.e. (a -b) will still be a single element of the matrix.
Well your second matrix does look like it's intended as a 3x2. However, if you do it like this it will be a column vector again:
[a - b;
c - d;
e - f]
which to me is a reasonable intuitive distinction between a minus b and a, negative b.
You can also use brackets as Ilya suggested.
Assuming you have a piece of code in which you only want to have column vectors and no matrix, there is a fairly quick solution:
replace {space}+ by +
replace {space}- by -
It is quite safe to do and unless you have complicated expressions in your vector it should do the trick.
Is there an existing subset of the alphanumerics that is easier to read? In particular, is there a subset that has fewer characters that are visually ambiguous, and by removing (or equating) certain characters we reduce human error?
I know "visually ambiguous" is somewhat waffly of an expression, but it is fairly evident that D, O and 0 are all similar, and 1 and I are also similar. I would like to maximize the size of the set of alpha-numerics, but minimize the number of characters that are likely to be misinterpreted.
The only precedent I am aware of for such a set is the Canada Postal code system that removes the letters D, F, I, O, Q, and U, and that subset was created to aid the postal system's OCR process.
My initial thought is to use only capital letters and numbers as follows:
A
B = 8
C = G
D = 0 = O = Q
E = F
H
I = J = L = T = 1 = 7
K = X
M
N
P
R
S = 5
U = V = Y
W
Z = 2
3
4
6
9
This problem may be difficult to separate from the given type face. The distinctiveness of the characters in the chosen typeface could significantly affect the potential visual ambiguity of any two characters, but I expect that in most modern typefaces the above characters that are equated will have a similar enough appearance to warrant equating them.
I would be grateful for thoughts on the above – are the above equations suitable, or perhaps are there more characters that should be equated? Would lowercase characters be more suitable?
I needed a replacement for hexadecimal (base 16) for similar reasons (e.g. for encoding a key, etc.), the best I could come up with is the following set of 16 characters, which can be used as a replacement for hexadecimal:
0 1 2 3 4 5 6 7 8 9 A B C D E F Hexadecimal
H M N 3 4 P 6 7 R 9 T W C X Y F Replacement
In the replacement set, we consider the following:
All characters used have major distinguishing features that would only be omitted in a truly awful font.
Vowels A E I O U omitted to avoid accidentally spelling words.
Sets of characters that could potentially be very similar or identical in some fonts are avoided completely (none of the characters in any set are used at all):
0 O D Q
1 I L J
8 B
5 S
2 Z
By avoiding these characters completely, the hope is that the user will enter the correct characters, rather than trying to correct mis-entered characters.
For sets of less similar but potentially confusing characters, we only use one character in each set, hopefully the most distinctive:
Y U V
Here Y is used, since it always has the lower vertical section, and a serif in serif fonts
C G
Here C is used, since it seems less likely that a C would be entered as G, than vice versa
X K
Here X is used, since it is more consistent in most fonts
F E
Here F is used, since it is not a vowel
In the case of these similar sets, entry of any character in the set could be automatically converted to the one that is actually used (the first one listed in each set). Note that E must not be automatically converted to F if hexadecimal input might be used (see below).
Note that there are still similar-sounding letters in the replacement set, this is pretty much unavoidable. When reading aloud, a phonetic alphabet should be used.
Where characters that are also present in standard hexadecimal are used in the replacement set, they are used for the same base-16 value. In theory mixed input of hexadecimal and replacement characters could be supported, provided E is not automatically converted to F.
Since this is just a character replacement, it should be easy to convert to/from hexadecimal.
Upper case seems best for the "canonical" form for output, although lower case also looks reasonable, except for "h" and "n", which should still be relatively clear in most fonts:
h m n 3 4 p 6 7 r 9 t w c x y f
Input can of course be case-insensitive.
There are several similar systems for base 32, see http://en.wikipedia.org/wiki/Base32 However these obviously need to introduce more similar-looking characters, in return for an additional 25% more information per character.
Apparently the following set was also used for Windows product keys in base 24, but again has more similar-looking characters:
B C D F G H J K M P Q R T V W X Y 2 3 4 6 7 8 9
My set of 23 unambiguous characters is:
c,d,e,f,h,j,k,m,n,p,r,t,v,w,x,y,2,3,4,5,6,8,9
I needed a set of unambiguous characters for user input, and I couldn't find anywhere that others have already produced a character set and set of rules that fit my criteria.
My requirements:
No capitals: this supposed to be used in URIs, and typed by people who might not have a lot of typing experience, for whom even the shift key can slow them down and cause uncertainty. I also want someone to be able to say "all lowercase" so as to reduce uncertainty, so I want to avoid capital letters.
Few or no vowels: an easy way to avoid creating foul language or surprising words is to simply omit most vowels. I think keeping "e" and "y" is ok.
Resolve ambiguity consistently: I'm open to using some ambiguous characters, so long as I only use one character from each group (e.g., out of lowercase s, uppercase S, and five, I might only use five); that way, on the backend, I can just replace any of these ambiguous characters with the one correct character from their group. So, the input string "3Sh" would be replaced with "35h" before I look up its match in my database.
Only needed to create tokens: I don't need to encode information like base64 or base32 do, so the exact number of characters in my set doesn't really matter, besides my wanting to to be as large as possible. It only needs to be useful for producing random UUID-type id tokens.
Strongly prefer non-ambiguity: I think it's much more costly for someone to enter a token and have something go wrong than it is for someone to have to type out a longer token. There's a tradeoff, of course, but I want to strongly prefer non-ambiguity over brevity.
The confusable groups of characters I identified:
A/4
b/6/G
8/B
c/C
f/F
9/g/q
i/I/1/l/7 - just too ambiguous to use; note that european "1" can look a lot like many people's "7"
k/K
o/O/0 - just too ambiguous to use
p/P
s/S/5
v/V
w/W
x/X
y/Y
z/Z/2
Unambiguous characters:
I think this leaves only 9 totally unambiguous lowercase/numeric chars, with no vowels:
d,e,h,j,m,n,r,t,3
Adding back in one character from each of those ambiguous groups (and trying to prefer the character that looks most distinct, while avoiding uppercase), there are 23 characters:
c,d,e,f,h,j,k,m,n,p,r,t,v,w,x,y,2,3,4,5,6,8,9
Analysis:
Using the rule of thumb that a UUID with a numerical equivalent range of N possibilities is sufficient to avoid collisions for sqrt(N) instances:
an 8-digit UUID using this character set should be sufficient to avoid collisions for about 300,000 instances
a 16-digit UUID using this character set should be sufficient to avoid collisions for about 80 billion instances.
Mainly drawing inspiration from this ux thread, mentioned by #rwb,
Several programs use similar things. The list in your post seems to be very similar to those used in these programs, and I think it should be enough for most purposes. You can add always add redundancy (error-correction) to "forgive" minor mistakes; this will require you to space-out your codes (see Hamming distance), though.
No references as to particular method used in deriving the lists, except trial and error
with humans (which is great for non-ocr: your users are humans)
It may make sense to use character grouping (say, groups of 5) to increase context ("first character in the second of 5 groups")
Ambiguity can be eliminated by using complete nouns (from a dictionary with few look-alikes; word-edit-distance may be useful here) instead of characters. People may confuse "1" with "i", but few will confuse "one" with "ice".
Another option is to make your code into a (fake) word that can be read out loud. A markov model may help you there.
If you have the option to use only capitals, I created this set based on characters which users commonly mistyped, however this wholly depends on the font they read the text in.
Characters to use: A C D E F G H J K L M N P Q R T U V W X Y 3 4 6 7 9
Characters to avoid:
B similar to 8
I similar to 1
O similar to 0
S similar to 5
Z similar to 2
What you seek is an unambiguous, efficient Human-Computer code. What I recommend is to encode the entire data with literal(meaningful) words, nouns in particular.
I have been developing a software to do just that - and most efficiently. I call it WCode. Technically its just Base-1024 Encoding - wherein you use words instead of symbols.
Here are the links:
Presentation: https://docs.google.com/presentation/d/1sYiXCWIYAWpKAahrGFZ2p5zJX8uMxPccu-oaGOajrGA/edit
Documentation: https://docs.google.com/folder/d/0B0pxLafSqCjKOWhYSFFGOHd1a2c/edit
Project: https://github.com/San13/WCode (Please wait while I get around uploading...)
This would be a general problem in OCR. Thus for end to end solution where in OCR encoding is controlled - specialised fonts have been developed to solve the "visual ambiguity" issue you mention of.
See: http://en.wikipedia.org/wiki/OCR-A_font
as additional information : you may want to know about Base32 Encoding - wherein symbol for digit '1' is not used as it may 'confuse' the users with the symbol for alphabet 'l'.
Unambiguous looking letters for humans are also unambiguous for optical character recognition (OCR). By removing all pairs of letters that are confusing for OCR, one obtains:
!+2345679:BCDEGHKLQSUZadehiopqstu
See https://www.monperrus.net/martin/store-data-paper
It depends how large you want your set to be. For example, just the set {0, 1} will probably work well. Similarly the set of digits only. But probably you want a set that's roughly half the size of the original set of characters.
I have not done this, but here's a suggestion. Pick a font, pick an initial set of characters, and write some code to do the following. Draw each character to fit into an n-by-n square of black and white pixels, for n = 1 through (say) 10. Cut away any all-white rows and columns from the edge, since we're only interested in the black area. That gives you a list of 10 codes for each character. Measure the distance between any two characters by how many of these codes differ. Estimate what distance is acceptable for your application. Then do a brute-force search for a set of characters which are that far apart.
Basically, use a script to simulate squinting at the characters and see which ones you can still tell apart.
Here's some python I wrote to encode and decode integers using the system of characters described above.
def base20encode(i):
"""Convert integer into base20 string of unambiguous characters."""
if not isinstance(i, int):
raise TypeError('This function must be called on an integer.')
chars, s = '012345689ACEHKMNPRUW', ''
while i > 0:
i, remainder = divmod(i, 20)
s = chars[remainder] + s
return s
def base20decode(s):
"""Convert string to unambiguous chars and then return integer from resultant base20"""
if not isinstance(s, str):
raise TypeError('This function must be called on a string.')
s = s.translate(bytes.maketrans(b'BGDOQFIJLT7KSVYZ', b'8C000E11111X5UU2'))
chars, i, exponent = '012345689ACEHKMNPRUW', 0, 1
for number in s[::-1]:
i += chars.index(number) * exponent
exponent *= 20
return i
base20decode(base20encode(10))
base58:123456789ABCDEFGHJKLMNPQRSTUVWXYZabcdefghijkmnopqrstuvwxyz