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Python has a built in function sum, which is effectively equivalent to:
def sum2(iterable, start=0):
return start + reduce(operator.add, iterable)
for all types of parameters except strings. It works for numbers and lists, for example:
sum([1,2,3], 0) = sum2([1,2,3],0) = 6 #Note: 0 is the default value for start, but I include it for clarity
sum({888:1}, 0) = sum2({888:1},0) = 888
Why were strings specially left out?
sum( ['foo','bar'], '') # TypeError: sum() can't sum strings [use ''.join(seq) instead]
sum2(['foo','bar'], '') = 'foobar'
I seem to remember discussions in the Python list for the reason, so an explanation or a link to a thread explaining it would be fine.
Edit: I am aware that the standard way is to do "".join. My question is why the option of using sum for strings was banned, and no banning was there for, say, lists.
Edit 2: Although I believe this is not needed given all the good answers I got, the question is: Why does sum work on an iterable containing numbers or an iterable containing lists but not an iterable containing strings?
Python tries to discourage you from "summing" strings. You're supposed to join them:
"".join(list_of_strings)
It's a lot faster, and uses much less memory.
A quick benchmark:
$ python -m timeit -s 'import operator; strings = ["a"]*10000' 'r = reduce(operator.add, strings)'
100 loops, best of 3: 8.46 msec per loop
$ python -m timeit -s 'import operator; strings = ["a"]*10000' 'r = "".join(strings)'
1000 loops, best of 3: 296 usec per loop
Edit (to answer OP's edit): As to why strings were apparently "singled out", I believe it's simply a matter of optimizing for a common case, as well as of enforcing best practice: you can join strings much faster with ''.join, so explicitly forbidding strings on sum will point this out to newbies.
BTW, this restriction has been in place "forever", i.e., since the sum was added as a built-in function (rev. 32347)
You can in fact use sum(..) to concatenate strings, if you use the appropriate starting object! Of course, if you go this far you have already understood enough to use "".join(..) anyway..
>>> class ZeroObject(object):
... def __add__(self, other):
... return other
...
>>> sum(["hi", "there"], ZeroObject())
'hithere'
Here's the source: http://svn.python.org/view/python/trunk/Python/bltinmodule.c?revision=81029&view=markup
In the builtin_sum function we have this bit of code:
/* reject string values for 'start' parameter */
if (PyObject_TypeCheck(result, &PyBaseString_Type)) {
PyErr_SetString(PyExc_TypeError,
"sum() can't sum strings [use ''.join(seq) instead]");
Py_DECREF(iter);
return NULL;
}
Py_INCREF(result);
}
So.. that's your answer.
It's explicitly checked in the code and rejected.
From the docs:
The preferred, fast way to concatenate a
sequence of strings is by calling
''.join(sequence).
By making sum refuse to operate on strings, Python has encouraged you to use the correct method.
Short answer: Efficiency.
Long answer: The sum function has to create an object for each partial sum.
Assume that the amount of time required to create an object is directly proportional to the size of its data. Let N denote the number of elements in the sequence to sum.
doubles are always the same size, which makes sum's running time O(1)×N = O(N).
int (formerly known as long) is arbitary-length. Let M denote the absolute value of the largest sequence element. Then sum's worst-case running time is lg(M) + lg(2M) + lg(3M) + ... + lg(NM) = N×lg(M) + lg(N!) = O(N log N).
For str (where M = the length of the longest string), the worst-case running time is M + 2M + 3M + ... + NM = M×(1 + 2 + ... + N) = O(N²).
Thus, summing strings would be much slower than summing numbers.
str.join does not allocate any intermediate objects. It preallocates a buffer large enough to hold the joined strings, and copies the string data. It runs in O(N) time, much faster than sum.
The Reason Why
#dan04 has an excellent explanation for the costs of using sum on large lists of strings.
The missing piece as to why str is not allowed for sum is that many, many people were trying to use sum for strings, and not many use sum for lists and tuples and other O(n**2) data structures. The trap is that sum works just fine for short lists of strings, but then gets put in production where the lists can be huge, and the performance slows to a crawl. This was such a common trap that the decision was made to ignore duck-typing in this instance, and not allow strings to be used with sum.
Edit: Moved the parts about immutability to history.
Basically, its a question of preallocation. When you use a statement such as
sum(["a", "b", "c", ..., ])
and expect it to work similar to a reduce statement, the code generated looks something like
v1 = "" + "a" # must allocate v1 and set its size to len("") + len("a")
v2 = v1 + "b" # must allocate v2 and set its size to len("a") + len("b")
...
res = v10000 + "$" # must allocate res and set its size to len(v9999) + len("$")
In each of these steps a new string is created, which for one might give some copying overhead as the strings are getting longer and longer. But that’s maybe not the point here. What’s more important, is that every new string on each line must be allocated to it’s specific size (which. I don’t know it it must allocate in every iteration of the reduce statement, there might be some obvious heuristics to use and Python might allocate a bit more here and there for reuse – but at several points the new string will be large enough that this won’t help anymore and Python must allocate again, which is rather expensive.
A dedicated method like join, however has the job to figure out the real size of the string before it starts and would therefore in theory only allocate once, at the beginning and then just fill that new string, which is much cheaper than the other solution.
I dont know why, but this works!
import operator
def sum_of_strings(list_of_strings):
return reduce(operator.add, list_of_strings)
Second attempt here, I just wanted to know if this is considered a recursive function.
The purpose of the function is to take a string and
if the the first element is equal to the last element
then append the last element to a list and return nothing,
else call istelf and pass the same string from index [1]
finally append the first element to the list
I know that error checking needs to be done on the if statement. However I am only doing this to try and get my head around recursion...Struggling to be honest.
Also I would never write a program like this if it where anything but trivial I just wanted to check if my understanding is correct so far.
def parse(theList):
theList.reverse()
parsedString = ''.join(theList)
return parsedString
def recursiveMessage(theString):
lastElement = theString[len(theString) - 1]
if theString[0] == lastElement:
buildString.append(theString[0])
return None
else:
recursiveMessage(theString[1::])
buildString.append(theString[0])
toPrint = "Hello Everyone!"
buildString = []
recursiveMessage(toPrint)
print(parse(buildString))
Thanks again.
Is this recursive?
If at any point in a function's execution it calls itself, then it is consider recursive. This happens in your example, so recursiveMessage is indeed recursive.
so which is quicker recursion or iteration?
Recursion is usually much slower and consumes more space due to a new stack frame having to be created on the call stack each recursive call. If you know your recursive function will need to be run many times, iteration is the best route.
As an interesting side note, many compilers actually optimize a recursive function by rolling it out into a loop anyways.
I need to find all maximum values among elements of every possible set {a[i], a[i+1],... a[i + k]} (where i is index and k is some given constant). For this I am using.
loop(b, 1, k) {
rloopl(i, b, n) {
if(a[i] < a[i-1])
a[i] = a[i-1];
}
}
but its too slow for large array. Is there any other more efficient way to do this?
I'm very sorry to tell you that, with the requirement as-presented, the answer would be: "no." If "the largest value could be anywhere," you have no choice but to "look ... everywhere."
If you are "doing this 'once and only once,'" for any particular data-set, then you're basically just gonna have to take your lumps. You're stuck with "brute-force."
However, if you're doing this more than once, and/or if you have some influence on the process by which the array in question gets loaded, the situation might start looking a little better.
For instance, if another piece of code is adding elements to this array one-at-a-time, it's trivial for that piece of code to notice the max/min value that it encounters. Code that loads a two-dimensional array might gather statistics about each row (column). And, so on. Such strategies, which are "free, at the time," can be used to eliminate (or, severely curtail) the need to do specific brute-force searches later.
In a program I need to efficiently answer queries of the following form:
Given a set of strings A and a query string q return all s ∈ A such that q is a subsequence of s
For example, given A = {"abcdef", "aaaaaa", "ddca"} and q = "acd" exactly "abcdef" should be returned.
The following is what I have considered considered so far:
For each possible character, make a sorted list of all string/locations where it appears. For querying interleave the lists of the involved characters, and scan through it looking for matches within string boundaries.
This would probably be more efficient for words instead of characters, since the limited number of different characters will make the return lists very dense.
For each n-prefix q might have, store the list of all matching strings. n might realistically be close to 3. For query strings longer than that we brute force the initial list.
This might speed things up a bit, but one could easily imagine some n-subsequences being present close to all strings in A, which means worst case is the same as just brute forcing the entire set.
Do you know of any data structures, algorithms or preprocessing tricks which might be helpful for performing the above task efficiently for large As? (My ss will be around 100 characters)
Update: Some people have suggested using LCS to check if q is a subsequence of s. I just want to remind that this can be done using a simple function such as:
def isSub(q,s):
i, j = 0, 0
while i != len(q) and j != len(s):
if q[i] == s[j]:
i += 1
j += 1
else:
j += 1
return i == len(q)
Update 2: I've been asked to give more details on the nature of q, A and its elements. While I'd prefer something that works as generally as possible, I assume A will have length around 10^6 and will need to support insertion. The elements s will be shorter with an average length of 64. The queries q will only be 1 to 20 characters and be used for a live search, so the query "ab" will be sent just before the query "abc". Again, I'd much prefer the solution to use the above as little as possible.
Update 3: It has occurred to me, that a data-structure with O(n^{1-epsilon}) lookups, would allow you to solve OVP / disprove the SETH conjecture. That is probably the reason for our suffering. The only options are then to disprove the conjecture, use approximation, or take advantage of the dataset. I imagine quadlets and tries would do the last in different settings.
It could done by building an automaton. You can start with NFA (nondeterministic finite automaton which is like an indeterministic directed graph) which allows edges labeled with an epsilon character, which means that during processing you can jump from one node to another without consuming any character. I'll try to reduce your A. Let's say you A is:
A = {'ab, 'bc'}
If you build NFA for ab string you should get something like this:
+--(1)--+
e | a| |e
(S)--+--(2)--+--(F)
| b| |
+--(3)--+
Above drawing is not the best looking automaton. But there are a few points to consider:
S state is the starting state and F is the ending state.
If you are at F state it means your string qualifies as a subsequence.
The rule of propagating within an autmaton is that you can consume e (epsilon) to jump forward, therefore you can be at more then one state at each point in time. This is called e closure.
Now if given b, starting at state S I can jump one epsilon, reach 2, and consume b and reach 3. Now given end string I consume epsilon and reach F, thus b qualifies as a sub-sequence of ab. So does a or ab you can try yourself using above automata.
The good thing about NFA is that they have one start state and one final state. Two NFA could be easily connected using epsilons. There are various algorithms that could help you to convert NFA to DFA. DFA is a directed graph which can follow precise path given a character -- in particular, it is always in exactly one state at any point in time. (For any NFA, there is a corresponding DFA whose states correspond to sets of states in the NFA.)
So, for A = {'ab, 'bc'}, we would need to build NFA for ab then NFA for bc then join the two NFAs and build the DFA of the entire big NFA.
EDIT
NFA of subsequence of abc would be a?b?c?, so you can build your NFA as:
Now, consider the input acd. To query if ab is subsequence of {'abc', 'acd'}, you can use this NFA: (a?b?c?)|(a?c?d). Once you have NFA you can convert it to DFA where each state will contain whether it is a subsequence of abc or acd or maybe both.
I used link below to make NFA graphic from regular expression:
http://hackingoff.com/images/re2nfa/2013-08-04_21-56-03_-0700-nfa.svg
EDIT 2
You're right! In case if you've 10,000 unique characters in the A. By unique I mean A is something like this: {'abc', 'def'} i.e. intersection of each element of A is empty set. Then your DFA would be worst case in terms of states i.e. 2^10000. But I'm not sure when would that be possible given that there can never be 10,000 unique characters. Even if you have 10,000 characters in A still there will be repetitions and that might reduce states alot since e-closure might eventually merge. I cannot really estimate how much it might reduce. But even having 10 million states, you will only consume less then 10 mb worth of space to construct a DFA. You can even use NFA and find e-closures at run-time but that would add to run-time complexity. You can search different papers on how large regex are converted to DFAs.
EDIT 3
For regex (a?b?c?)|(e?d?a?)|(a?b?m?)
If you convert above NFA to DFA you get:
It actually lot less states then NFA.
Reference:
http://hackingoff.com/compilers/regular-expression-to-nfa-dfa
EDIT 4
After fiddling with that website more. I found that worst case would be something like this A = {'aaaa', 'bbbbb', 'cccc' ....}. But even in this case states are lesser than NFA states.
Tests
There have been four main proposals in this thread:
Shivam Kalra suggested creating an automaton based on all the strings in A. This approach has been tried slightly in the literature, normally under the name "Directed Acyclic Subsequence Graph" (DASG).
J Random Hacker suggested extending my 'prefix list' idea to all 'n choose 3' triplets in the query string, and merging them all using a heap.
In the note "Efficient Subsequence Search in Databases" Rohit Jain, Mukesh K. Mohania and Sunil Prabhakar suggest using a Trie structure with some optimizations and recursively search the tree for the query. They also have a suggestion similar to the triplet idea.
Finally there is the 'naive' approach, which wanghq suggested optimizing by storing an index for each element of A.
To get a better idea of what's worth putting continued effort into, I have implemented the above four approaches in Python and benchmarked them on two sets of data. The implementations could all be made a couple of magnitudes faster with a well done implementation in C or Java; and I haven't included the optimizations suggested for the 'trie' and 'naive' versions.
Test 1
A consists of random paths from my filesystem. q are 100 random [a-z] strings of average length 7. As the alphabet is large (and Python is slow) I was only able to use duplets for method 3.
Construction times in seconds as a function of A size:
Query times in seconds as a function of A size:
Test 2
A consists of randomly sampled [a-b] strings of length 20. q are 100 random [a-b] strings of average length 7. As the alphabet is small we can use quadlets for method 3.
Construction times in seconds as a function of A size:
Query times in seconds as a function of A size:
Conclusions
The double logarithmic plot is a bit hard to read, but from the data we can draw the following conclusions:
Automatons are very fast at querying (constant time), however they are impossible to create and store for |A| >= 256. It might be possible that a closer analysis could yield a better time/memory balance, or some tricks applicable for the remaining methods.
The dup-/trip-/quadlet method is about twice as fast as my trie implementation and four times as fast as the 'naive' implementation. I used only a linear amount of lists for the merge, instead of n^3 as suggested by j_random_hacker. It might be possible to tune the method better, but in general it was disappointing.
My trie implementation consistently does better than the naive approach by around a factor of two. By incorporating more preprocessing (like "where are the next 'c's in this subtree") or perhaps merging it with the triplet method, this seems like todays winner.
If you can do with a magnitude less performance, the naive method does comparatively just fine for very little cost.
As you point out, it might be that all strings in A contain q as a subsequence, in which case you can't hope to do better than O(|A|). (That said, you might still be able to do better than the time taken to run LCS on (q, A[i]) for each string i in A, but I won't focus on that here.)
TTBOMK there are no magic, fast ways to answer this question (in the way that suffix trees are the magic, fast way to answer the corresponding question involving substrings instead of subsequences). Nevertheless if you expect the set of answers for most queries to be small on average then it's worth looking at ways to speed up these queries (the ones yielding small-size answers).
I suggest filtering based on a generalisation of your heuristic (2): if some database sequence A[i] contains q as a subsequence, then it must also contain every subsequence of q. (The reverse direction is not true unfortunately!) So for some small k, e.g. 3 as you suggest, you can preprocess by building an array of lists telling you, for every length-k string s, the list of database sequences containing s as a subsequence. I.e. c[s] will contain a list of the ID numbers of database sequences containing s as a subsequence. Keep each list in numeric order to enable fast intersections later.
Now the basic idea (which we'll improve in a moment) for each query q is: Find all k-sized subsequences of q, look up each in the array of lists c[], and intersect these lists to find the set of sequences in A that might possibly contain q as a subsequence. Then for each possible sequence A[i] in this (hopefully small) intersection, perform an O(n^2) LCS calculation with q to see whether it really does contain q.
A few observations:
The intersection of 2 sorted lists of size m and n can be found in O(m+n) time. To find the intersection of r lists, perform r-1 pairwise intersections in any order. Since taking intersections can only produce sets that are smaller or of the same size, time can be saved by intersecting the smallest pair of lists first, then the next smallest pair (this will necessarily include the result of the first operation), and so on. In particular: sort lists in increasing size order, then always intersect the next list with the "current" intersection.
It is actually faster to find the intersection a different way, by adding the first element (sequence number) of each of the r lists into a heap data structure, then repeatedly pulling out the minimum value and replenishing the heap with the next value from the list that the most recent minimum came from. This will produce a list of sequence numbers in nondecreasing order; any value that appears fewer than r times in a row can be discarded, since it cannot be a member of all r sets.
If a k-string s has only a few sequences in c[s], then it is in some sense discriminating. For most datasets, not all k-strings will be equally discriminating, and this can be used to our advantage. After preprocessing, consider throwing away all lists having more than some fixed number (or some fixed fraction of the total) of sequences, for 3 reasons:
They take a lot of space to store
They take a lot of time to intersect during query processing
Intersecting them will usually not shrink the overall intersection much
It is not necessary to consider every k-subsequence of q. Although this will produce the smallest intersection, it involves merging (|q| choose k) lists, and it might well be possible to produce an intersection that is nearly as small using just a fraction of these k-subsequences. E.g. you could limit yourself to trying all (or a few) k-substrings of q. As a further filter, consider just those k-subsequences whose sequence lists in c[s] are below some value. (Note: if your threshold is the same for every query, you might as well delete all such lists from the database instead, since this will have the same effect, and saves space.)
One thought;
if q tends to be short, maybe reducing A and q to a set will help?
So for the example, derive to { (a,b,c,d,e,f), (a), (a,c,d) }. Looking up possible candidates for any q should be faster than the original problem (that's a guess actually, not sure how exactly. maybe sort them and "group" similar ones in bloom filters?), then use bruteforce to weed out false positives.
If A strings are lengthy, you could make the characters unique based on their occurence, so that would be {(a1,b1,c1,d1,e1,f1),(a1,a2,a3,a4,a5,a6),(a1,c1,d1,d2)}. This is fine, because if you search for "ddca" you only want to match the second d to a second d. The size of your alphabet would go up (bad for bloom or bitmap style operations) and would be different ever time you get new A's, but the amount of false positives would go down.
First let me make sure my understanding/abstraction is correct. The following two requirements should be met:
if A is a subsequence of B, then all characters in A should appear in B.
for those characters in B, their positions should be in an ascending order.
Note that, a char in A might appear more than once in B.
To solve 1), a map/set can be used. The key is the character in string B, and the value doesn't matter.
To solve 2), we need to maintain the position of each characters. Since a character might appear more than once, the position should be a collection.
So the structure is like:
Map<Character, List<Integer>)
e.g.
abcdefab
a: [0, 6]
b: [1, 7]
c: [2]
d: [3]
e: [4]
f: [5]
Once we have the structure, how to know if the characters are in the right order as they are in string A? If B is acd, we should check the a at position 0 (but not 6), c at position 2 and d at position 3.
The strategy here is to choose the position that's after and close to the previous chosen position. TreeSet is a good candidate for this operation.
public E higher(E e)
Returns the least element in this set strictly greater than the given element, or null if there is no such element.
The runtime complexity is O(s * (n1 + n2)*log(m))).
s: number of strings in the set
n1: number of chars in string (B)
n2: number of chars in query string (A)
m: number of duplicates in string (B), e.g. there are 5 a.
Below is the implementation with some test data.
import java.util.ArrayList;
import java.util.HashMap;
import java.util.List;
import java.util.Map;
import java.util.TreeSet;
public class SubsequenceStr {
public static void main(String[] args) {
String[] testSet = new String[] {
"abcdefgh", //right one
"adcefgh", //has all chars, but not the right order
"bcdefh", //missing one char
"", //empty
"acdh",//exact match
"acd",
"acdehacdeh"
};
List<String> subseqenceStrs = subsequenceStrs(testSet, "acdh");
for (String str : subseqenceStrs) {
System.out.println(str);
}
//duplicates in query
subseqenceStrs = subsequenceStrs(testSet, "aa");
for (String str : subseqenceStrs) {
System.out.println(str);
}
subseqenceStrs = subsequenceStrs(testSet, "aaa");
for (String str : subseqenceStrs) {
System.out.println(str);
}
}
public static List<String> subsequenceStrs(String[] strSet, String q) {
System.out.println("find strings whose subsequence string is " + q);
List<String> results = new ArrayList<String>();
for (String str : strSet) {
char[] chars = str.toCharArray();
Map<Character, TreeSet<Integer>> charPositions = new HashMap<Character, TreeSet<Integer>>();
for (int i = 0; i < chars.length; i++) {
TreeSet<Integer> positions = charPositions.get(chars[i]);
if (positions == null) {
positions = new TreeSet<Integer>();
charPositions.put(chars[i], positions);
}
positions.add(i);
}
char[] qChars = q.toCharArray();
int lowestPosition = -1;
boolean isSubsequence = false;
for (int i = 0; i < qChars.length; i++) {
TreeSet<Integer> positions = charPositions.get(qChars[i]);
if (positions == null || positions.size() == 0) {
break;
} else {
Integer position = positions.higher(lowestPosition);
if (position == null) {
break;
} else {
lowestPosition = position;
if (i == qChars.length - 1) {
isSubsequence = true;
}
}
}
}
if (isSubsequence) {
results.add(str);
}
}
return results;
}
}
Output:
find strings whose subsequence string is acdh
abcdefgh
acdh
acdehacdeh
find strings whose subsequence string is aa
acdehacdeh
find strings whose subsequence string is aaa
As always, I might be totally wrong :)
You might want to have a look into the Book Algorithms on Strings and Sequences by Dan Gusfield. As it turns out part of it is available on the internet. You might also want to read Gusfield's Introduction to Suffix Trees. As it turns out this book covers many approaches for you kind of question. It is considered one of the standard publications in this field.
Get a fast longest common subsequence algorithm implementation. Actually it suffices to determine the length of the LCS. Notice that Gusman's book has very good algorithms and also points to more sources for such algorithms.
Return all s ∈ A with length(LCS(s,q)) == length(q)
I am doing an application where I want to find a specific char in an array of chars. In other words, I have the following char array:
char[] charArray= new char[] {'\uE001', '\uE002', '\uE003', '\uE004', '\uE005', '\uE006', '\uE007', '\uE008', '\uE009'};
At some point, I want to check if the character '\uE002' exists in the charArray. My method was to make a loop on every character in the charArray and find if it exists.
for (int z = 0 ; z < charArray; z ++) {
if (charArray[z] == myChar) {
//Do the work
}
}
Is there any other solution than making a char array and finding the character by looping every single char?
One option is to pre-sort charArray and use Arrays.binarySearch(charArray, myChar). A non-negative return value will indicate that myChar is present in charArray.
char[] charArray = new char[] {'\uE001', '\uE002', '\uE003', '\uE004', '\uE005', '\uE006', '\uE007', '\uE008', '\uE009'};
Arrays.sort(charArray); // can be omitted if you know that the values are already sorted
...
if (Arrays.binarySearch(charArray, myChar) >= 0) {
// Do the work
}
edit An alternative that avoids using the Arrays module is to put the characters into a string (at initialization time) and then use String.indexOf():
String chars = "\uE001...";
...
if (chars.indexOf(myChar) >= 0) {
// Do the work
}
This is not hugely different to what you're doing already, except that it requires less code.
If n is the size of charArray, the first solution is O(log n) whereas the second one is O(n).
You can use hash/map to check existance of the characer. This approach has better time of O(log n) or O(1) depending on hash/map inner structure.
If you don't have access to Arrays, since you are working in JavaME, then you should try to:
Or implement a sorted array and the binary search youself
Or just use a O(n) solution, wich is a good solution anyways.
Your solution is O(n), along with the one stated by aix.
You could try to use a Map, but it would depends on how much elements you are in your array. If you think there won't be more than 1000 elements in the array, just use a O(n) solution. But if you think you could have an unkown number of elements, a Map would be a reasoable choice, providing a better solution.
You can use net.rim.device.api.util.Arrays.getIndex(char[] array, char element)
It depends on what Java ME configuration / profile you are using. If you're on CDC then check for what parts of Java SE 1.3 Collections framework are supported (just find javadocs targeted for your device and look in java.util package). Another thing worth checking is whether your device has some blackberry-specific API extensions to work with collections.
If you are limited to bare minimum of CLDC/MIDP then other solution than you mentioned would be to add chars to Vector and use Vector.contains(Object)
If you don't want to implement it yourself you could use ArrayUtils in apache commons project:
ArrayUtils apache-commons