This is my first post here so please be patient with me.
I am stuck on a problem and don't know what to do. I am given a
non-overlapping substring of length 'm' that appears twice in a string of length 'n'. What is the probability of finding the substring of length 'm' in both that string and another string of length '2n'?
For argument's sake let's say that m = 4 and n = 33. I have tried to use Independent Event probability as well as Markov Chain Models, but my answer never seems to be correct.
What would be the chance that the same 2 non-overlapping substrings of length 4 that are found in the string of length 33 will be found in a string of length 66?
This is a question I encountered in a Test and I am not able to solve it. Every time I think of an algorithm, a new corner case comes that fails it. Can someone please explain me, how to move through the problem ?
Problem Statement
The Cytes Lottery is the biggest lottery in the world. On each ticket, there is a string of a-z letters. The company produces a draw string S. A person wins if his/her ticket string is a special substring of the draw string. A special substring is a substring which can be formed by ignoring at most K characters from drawString. For example, if draw string = "xyzabc" and tickets are [ac zb yhja] with K=1 then the winning tickets will be 2 i.e ac (won by ignoring "b" in drawstring) and zb (won by ignoring "a" in drawstring).
Now some people change their ticket strings in order to win the lottery. To avoid any kind of suspicion, they can make the following changes in their strings.
They can change character 'o' to character 'a' and vice versa
They can change character 't' to character 'l' and vice versa
They can erase a character from anywhere in the string
Note that they can ignore at most 'K' characters from the draw string to get a match with the ticket string.
Write an algorithm to find the number of people who win the lottery (either honestly or by cheating).
Input:
The first line of the input consists of an integer - numTickets, representing the number of tickets (N).
The second line consists of a string - drawString, representing the draw string (S).
The third line consists of N space-seperated strings - tickets1, tickets2,........., ticketsN representing the tickets.
The last line consists of an integer-tolerance, representing the maximum number of characters that can be deleted from the drawString(K).
Output:
An integer representing the number of winning tickets (either fairly or by cheating).
Constraints:
0 <= numTickets <= 1000
0 <= length of drawString <= 200
0 <= length of tickets[i] <= 200
0 <= tolerance <= 1000
Note:
The drawString contains lowercase English alphabets
Example:
Input:
3
aabacd
abcde aoc actld
1
Output:
2
Vasya has a string s of length n consisting only of digits 0 and 1. Also he has an array a of length n.
Vasya performs the following operation until the string becomes empty: choose some consecutive substring of equal characters, erase it from the string and glue together the remaining parts (any of them can be empty). For example, if he erases substring 111 from string 111110 he will get the string 110. Vasya gets ax points for erasing substring of length x.
Vasya wants to maximize his total points, so help him with this!
https://codeforces.com/problemset/problem/1107/E
i was trying to get my head around the editorial,but couldn't understand it... can anyone tell an easy way to do it?
input:
7
1101001
3 4 9 100 1 2 3
output:
109
Explanation
the optimal sequence of erasings is: 1101001 → 111001 → 11101 → 1111 → ∅.
Here, we consider removing prefixes instead of substrings. Why?
We try to remove a consecutive prefix of a particular state which is actually a substring in the main string. So, our DP states will be start index, end index, prefix length.
Let's consider an example str = "1010110". Here, initially start=0, end=7, and prefix=1(the first '1' will be the only prefix now). we iterate over all the indices in the current state except the starting index and check if str[i]==str[start]. Here, for example, str[4]==str[0]. Now we divide the string into "010" with prefix=1(010) && "110" with prefix=2(1010110). These two are now two individual subproblems. So, when there remains a string with length 1, we return aprefix.
Here is my code.
Starting with a list of integers the task is to convert each integer into a string such that the resulting list of strings will be in numeric order when sorted lexicographically.
This is needed so that a particular system that is only capable of sorting strings will produce an output that is in numeric order.
Example:
Given the integers
1, 23, 3
we could convert the to strings like this:
"01", "23", "03"
so that when sorted they become:
"01", "03", "23"
which is correct. A wrong result would be:
"1", "23", "3"
because that list is sorted in "string order", not in numeric order.
I'm looking for something more efficient than the simple zero-padding scheme. In order to cover all possible 32 bit integers we'd need to pad to 10 digits which is inefficient.
For integers, prefix each number with the length. To make it more readable, use 'a' for length 1, and 'b' for length 2. Example:
non-encoded encoded
1 "a1"
3 "a3"
23 "b23"
This scheme is a bit simpler than prefixing each digit, but only works with numbers, not numbers mixed with text. It can be made to work for negative numbers as well, and even BigDecimal numbers, using some tricks. I wrote an implementation in Apache Jackrabbit 2.x, to make BigDecimal indexable (sortable) as text. For that, I used a format that only uses the characters '0' to '9' and consists of:
one character for: signum(value) + 2
one character for: signum(exponent) + 2
one character for: length(exponent) - 1
multiple characters for: exponent
multiple characters for: value (-1 if inverted)
Only the signum is encoded if the value is zero. The exponent is not encoded if zero. Negative values are "inverted" character by character (0 => 9, 1 => 8, and so on). The same applies to the exponent.
Examples:
non-encoded encoded
0 "2"
2 "322" (signum 1; exponent 0; value 2)
120 "330212" (signum 1; exponent signum 1, length 1, value 2; value 12)
-1 "179" (signum -1, rest inverted; exponent 0; value 1 (-1, inverted))
Values between BigDecimal(BigInteger.ONE, Integer.MIN_VALUE) and BigDecimal(BigInteger.ONE, Integer.MAX_VALUE) are supported.
TL;DR
Encode digits according to their order of magnitude (OM) and other characters so they sort as desired, relative to numbers: jj-a123 would be encoded zjzjz-zaC1B2A3
Longer explanation
This would depend somewhat upon the sorting algorithm that will finally be used to sort and how one would want any given punctuation characters to be sorted in relation to letters and numbers, but if it's "ascii-betical" or similar, you could encode each digit of a number to represent its order of magnitude (OM) in the number, while encoding other characters such that they would sort according to your desired sort order.
For simplicity, I would suggest beginning with encoding every non-numeric character with a "high" value (e.g. lower case z or even ~ if final value is ASCII), so that it sorts after encoded digits. Then cache each digit encountered until another non-numeric is encountered, then encode each cached digit with a value representing its OM. If the number 12945 was encountered in between non-numerics, you would output an E to encode an OM of 5, then the digit that is that order of magnitude, 1, followed by the next OM of 4 (D) and its associated digit, 2. Continue until all numeric digits have been flushed, then continue with non-numerics.
Non-numerics would be treated individually and ranked relative to the OM of digits. If it is desired for them to sort "above" numbers (perhaps the space character or certain others deemed special) they would be encoded by prepending a low-value character (like the space character, if final value will be treated and sorted as ASCII). When/if another numeric is encountered, begin caching and encode according to OM once all consecutive numerics are cached.
Alternately, processing the string in reverse order would preclude the need to cache numbers except for a single "is it a digit?" test and "is the last character a digit?" test. If the first is not true, then use (one of?) the "non-digit" OM character(s). If the first test is true then use the lowest-OM "digit" character (A in my examples). If both tests are true, then increment your OM character (A -> B or E -> F) before use.
Certain levels of additional filtering - or even translation - could be applied. If one wanted to allow accurate sorting based upon Roman numerals, one could encode them as decimal (or even hexadecimal) numbers with an appropriate OM.
Treating decimal points (either periods or commas, depending) as actual decimal separators, and distinct from other punctuation would probably be beyond the true utility of this encoding scheme, as alphanumeric fields seldom use a period or comma as a decimal separator. If it is desired to use them that way, the algorithm would simply detect a decimal separator (either period or comma as appropriate, in between digits) and not encode the numeric portion after that separator as anything but normal text. Fractional portions are actually sorted correctly during a normal ASCII based sort, because more digits represents greater precision - not greater magnitude.
Examples
non-encoded encoded
----------- -------
12345 E1D2C3B4A5
a100 zaC1B0A0
a20 zaB2A0
a2000 zaD2C0B0A0
x100.5 zxC1B0A0z.A5
x100.23 zxC1B0A0z.B2A3
1, 23, 3 A1z,z B2A1z,z A3
1, 2, 3 A1z,z A2z,z A3
1,2,3 A1z,A2z,A3
Potential advantages
Going somewhat beyond simple numeric sorting, some advantages to this encoding method would be several aspects of flexibility with final effective sort order - you are essentially encoding a category for each character - digits get a category based upon their position within the greater string of digits known as a number, while other characters are simply told to sort in their normal way (e.g. ASCII), but after numbers. Any exceptions that should sort before numbers or in other orders would be in one or more additional categories. ASCII can effectively be re-encoded to sort in a non-ASCII way:
You could encode lower case letters to sort before or along with upper case letters. To switch the lower and upper cases, you encode lower case letters with a y and upper case letters with a z. For a pseudo-case-insensitive sort, categorizing both A and a with the same encoding character would sort both of them before B and b, though A would nonetheless always sort before a
If you want Extended ASCII characters (e.g. with diacritics) to sort along with their ASCII cousins, you encode À, Á, Â, Ã, Ä, Å, and Æ along with A by using an a as the OM character, encode B, C, and Ç with a b, and E, È, É, Ê, and Ë with a c, etc. The same intra-category sort order caveat still applies, and some decisions need to be made on characters like capital Eth, and to a certain extent others like Thorn, and Sharp S (Ð, Þ, and ß respectively) as to whether they will sort based on similarities in appearance or pronunciation, or instead more properly perhaps, alphabetical order.
Small advantage of being basically human-readable, with effort
Caveats
Though this allows many 'categories' of characters to be defined, be sure to remember that each order of magnitude for digits is its own category - you need to know that the data will not contain numbers that are greater in OM than approximately 250, depending upon how many other categories you wish to define (ASCII 0 is reserved for storing strings, and there needs to be at least one other character to indicate "not a digit" - at least for alphanumeric data - making the maximum perhaps 254 orders of magnitude), but that should be plenty for any situation I can imagine. I'm not sure what other issues quantum computing will bring about, but there's probably a quantum solution to it, whatever it is.
Finally, if the hyphen is encoded as a non-numeric character, and all non-numerics are encoded with a higher OM than digits, negative numbers would be encoded as greater than any positive number. The hyphen should be encoded as a lower-than-digit-OM (perhaps only when preceding a digit) if negative numbers need to be sorted correctly according to magnitude.
Since the ASCII code of A is greater than 9, you could encode them as hexadecimal strings.
The integers
1, 23, 3
can be encoded as
00000001, 00000017, 00000003
and 32-bit integers can always be encoded as 8-character strings. (assume unsigned)
This is supposed to be a low-level course, and it is only the third day of class. However, we are asked to "Write the null-terminated string 'R5' in hexadecimal, binary, and octal notations. Assume that ASCII code is used"
I have no idea where to go to learn how to do this. Any suggestions? Thanks.
NULL-terminated ASCII strings are stored with one byte per character, plus one byte for the NULL. You would therefore be printing three bytes - 'R', '5', and 0.
Look up 'R' and '5' on an ASCII chart to see what the numeric values are for those characters in ASCII. Then, write out your three bytes three different ways - one each for hexadecimal, binary and octal.
Hope that helps.
It seems like this just requires you to look up the appropriate entries from the ASCII table, which in most cases lists hex and octal and the characters themselves.
ASCII is a standard way of defining how characters are represented, and most tables will list characters against corresponding hex, decimal, and octal values. The first 128 is standard and the next 128 are the extended characters (those weird characters that don't map to an English keyboard).
If you google "ASCII table" you'll be inundated with different links. The top one I saw at www.asciitable.com appears to have everything you need - except binary.
Most of the times you're not going to see binary listed, but it's fairly academic to translate a hex value into binary - your Windows Calculator will happily do this for you.
To more directly translate your specific string you'll look up each character (including the NULL) separately and translate each individually.
Ultimately to the computer, everything is a number. To represent characters such as letters or symbols, we can agree on an encoding, or a numbering of these characters. For example, we could invent a new encoding where 1 means 'A', 2, means 'B', and so on. ASCII is one commonly used text encoding which maps characters to numbers. In this case, we are concerned with a string of 3 characters: 'R', '5', and null (a null character marks the end of a string. It is represented by the value 0. If you look in an ASCII table, you'll find that the numeric values are 82, 53, and 0.
String: R, 5, <null>
Decimal numbers: 82, 53, 0
Our normal number system is base-10, or decimal. This means that each digit represents a value ten times larger than the next (1, to 10, to 100, to 1000, etc.). Alternate bases include 8 (octal), 16 (hexadecimal), and 2 (binary). There is a straightforward way to convert between bases, although you can also easily find calculators that will do the conversion for you. You may want to review the relevant section of your textbook, or check out the Wikipedia articles. For the example of decimal 82, the hexadecimal value is 52 (this means 5*16 + 2 = 8*10 + 2). Oftentimes you will see a prefix of "0x", this is commonly used to make it clear the following digits are in base 16. (otherwise, you might think "52" refers to the decimal value 52).
Interesting. So would it be correct to say that the null-terminated string "R5" is simply "52, 35, 30" or is there a more correct format to it? Thank you for your patience. –
As I pointed out in another comment, the actual value 0 marks the end of a string, not the value 0x30, which represents a character '0' in the string. Note that the value of zero (0) is the same regardless of which base your numbers are in.
String: R, 5, <null>
Decimal : 82, 53, 0
Hexadecimal: 52, 35, 0