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How do I find the time complexity (Big O) of while loop?
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I am just starting with algorithms and I am trying to find out the running time in terms of 'n' for the while loop below.
int k=1;
while(k<n-k){
k+=k;
}
Here n>2. I understand that the value of k doubles everytime and the loop runs only once, once k value becomes greater than n/2. But I am having difficulty in expressing the same in terms of 'n'.
It's worth to list the important points:
k doubles on every loop iteration
your loop condition can be rewritten as: while(2*k < n)*
The essential question is: how many times I have to double the k, untill k doubled will be equal or greater than n?
This is fairly easy. This is exactly how logarithms work. Take a number 2, for example. How many times do I have to double it to reach, let's say, 1000? The answer is log21000 rounded up.
Essentially, your algorithm is log_2(n) - 1, which means that your algorithm runs in logarithmic time complexity.
*As François Andrieux correctly stated in his comment, while mathematically this statement is true, this is not always the case in programming, due to the representation limits of data types. For large ks, the expression 2*k might cause an overflow and invalidate the whole expression, while with the same input the expression k < n-k will behave correctly.
Replace
while(k<n-k)
k+=k;
with equivalent
while(2*k<n)
k*=2
the last is definitely O(log(n)) - it makes log2(n)-1 steps
Expression k < n-k simplifies to k < n/2.
Time Complexity should be O(log(n)) with base being 2
K = 1 -> 2 -> 4 -> 8 -> ... -> m iterations
2^(m-1) < n/2
m-1 < log2(n/2)
m ~ log2(n)
My question's header is similar to this link, however that one wasn't answered to my expectations.
I have an array of integers (1 000 000 entries), and need to mask exactly 30% of elements.
My approach is to loop over elements and roll a dice for each one. Doing it in a non-interrupted manner is good for cache coherency.
As soon as I notice that exactly 300 000 of elements were indeed masked, I need to stop. However, I might reach the end of an array and have only 200 000 elements masked, forcing me to loop a second time, maybe even a third, etc.
What's the most efficient way to ensure I won't have to loop a second time, and not being biased towards picking some elements?
Edit:
//I need to preserve the order of elements.
//For instance, I might have:
[12, 14, 1, 24, 5, 8]
//Masking away 30% might give me:
[0, 14, 1, 24, 0, 8]
The result of masking must be the original array, with some elements set to zero
Just do a fisher-yates shuffle but stop at only 300000 iterations. The last 300000 elements will be the randomly chosen ones.
std::size_t size = 1000000;
for(std::size_t i = 0; i < 300000; ++i)
{
std::size_t r = std::rand() % size;
std::swap(array[r], array[size-1]);
--size;
}
I'm using std::rand for brevity. Obviously you want to use something better.
The other way is this:
for(std::size_t i = 0; i < 300000;)
{
std::size_t r = rand() % 1000000;
if(array[r] != 0)
{
array[r] = 0;
++i;
}
}
Which has no bias and does not reorder elements, but is inferior to fisher yates, especially for high percentages.
When I see a massive list, my mind always goes first to divide-and-conquer.
I won't be writing out a fully-fleshed algorithm here, just a skeleton. You seem like you have enough of a clue to take decent idea and run with it. I think I only need to point you in the right direction. With that said...
We'd need an RNG that can return a suitably-distributed value for how many masked values could potentially be below a given cut point in the list. I'll use the halfway point of the list for said cut. Some statistician can probably set you up with the right RNG function. (Anyone?) I don't want to assume it's just uniformly random [0..mask_count), but it might be.
Given that, you might do something like this:
// the magic RNG your stats homework will provide
int random_split_sub_count_lo( int count, int sub_count, int split_point );
void mask_random_sublist( int *list, int list_count, int sub_count )
{
if (list_count > SOME_SMALL_THRESHOLD)
{
int list_count_lo = list_count / 2; // arbitrary
int list_count_hi = list_count - list_count_lo;
int sub_count_lo = random_split_sub_count_lo( list_count, mask_count, list_count_lo );
int sub_count_hi = list_count - sub_count_lo;
mask( list, list_count_lo, sub_count_lo );
mask( list + sub_count_lo, list_count_hi, sub_count_hi );
}
else
{
// insert here some simple/obvious/naive implementation that
// would be ludicrous to use on a massive list due to complexity,
// but which works great on very small lists. I'm assuming you
// can do this part yourself.
}
}
Assuming you can find someone more informed on statistical distributions than I to provide you with a lead on the randomizer you need to split the sublist count, this should give you O(n) performance, with 'n' being the number of masked entries. Also, since the recursion is set up to traverse the actual physical array in constantly-ascending-index order, cache usage should be as optimal as it's gonna get.
Caveat: There may be minor distribution issues due to the discrete nature of the list versus the 30% fraction as you recurse down and down to smaller list sizes. In practice, I suspect this may not matter much, but whatever person this solution is meant for may not be satisfied that the random distribution is truly uniform when viewed under the microscope. YMMV, I guess.
Here's one suggestion. One million bits is only 128K which is not an onerous amount.
So create a bit array with all items initialised to zero. Then randomly select 300,000 of them (accounting for duplicates, of course) and mark those bits as one.
Then you can run through the bit array and, any that are set to one (or zero, if your idea of masking means you want to process the other 700,000), do whatever action you wish to the corresponding entry in the original array.
If you want to ensure there's no possibility of duplicates when randomly selecting them, just trade off space for time by using a Fisher-Yates shuffle.
Construct an collection of all the indices and, for each of the 700,000 you want removed (or 300,000 if, as mentioned, masking means you want to process the other ones) you want selected:
pick one at random from the remaining set.
copy the final element over the one selected.
reduce the set size.
This will leave you with a random subset of indices that you can use to process the integers in the main array.
You want reservoir sampling. Sample code courtesy of Wikipedia:
(*
S has items to sample, R will contain the result
*)
ReservoirSample(S[1..n], R[1..k])
// fill the reservoir array
for i = 1 to k
R[i] := S[i]
// replace elements with gradually decreasing probability
for i = k+1 to n
j := random(1, i) // important: inclusive range
if j <= k
R[j] := S[i]
After watching some Terence Tao videos, I wanted to try implementing algorithms into c++ code to find all the prime numbers up to a number n. In my first version, where I simply had every integer from 2 to n tested to see if they were divisible by anything from 2 to sqrt(n), I got the program to find the primes between 1-10,000,000 in ~52 seconds.
Attempting to optimize the program, and implementing what I now know to be the Sieve of Eratosthenes, I assumed the task would be done much faster than 51 seconds, but sadly, that wasn't the case. Even going up to 1,000,000 took a considerable amount of time (didn't time it, though)
#include <iostream>
#include <vector>
using namespace std;
void main()
{
vector<int> tosieve = {};
for (int i = 2; i < 1000001; i++)
{
tosieve.push_back(i);
}
for (int j = 0; j < tosieve.size(); j++)
{
for (int k = j + 1; k < tosieve.size(); k++)
{
if (tosieve[k] % tosieve[j] == 0)
{
tosieve.erase(tosieve.begin() + k);
}
}
}
//for (int f = 0; f < tosieve.size(); f++)
//{
// cout << (tosieve[f]) << endl;
//}
cout << (tosieve.size()) << endl;
system("pause");
}
Is it the repeated referencing of the vectors or something? Why is this so slow? Even if I'm completely overlooking something (could be, complete beginner at this :I) I would think that finding the primes between 2 and 1,000,000 with this horrible inefficient method would be faster than my original way of finding them from 2 to 10,000,000.
Hope someone has a clear answer to this - hopefully I can use whatever knowledge is gleaned in the future when optimizing programs using a lot of recursion.
The problem is that 'erase' moves every element in the vector down one, meaning it is an O(n) operation.
There are three alternative choices:
1) Just mark deleted elements as 'empty' (make them 0, for example). This will mean future passes have to pass over those empty positions, but that isn't that expensive.
2) Make a new vector, and push_back new values into there.
3) Use std::remove_if: This will move the elements down, but do it in a single pass so will be more efficient. If you use std::remove_if, then you will have to remember it doesn't resize the vector itself.
Most of vector operations, including erase() have a O(n) linear time complexity.
Since you have two loops of size 10^6, and a vector of size 10^6, your algorithm executes up to 10^18 operations.
Qubic algorithms for such a big N will take a huge amount of time.
N = 10^6 is even big enough for quadratic algorithms.
Please, read carefully about Sieve of Eratosthenes. The fact that both full search and Sieve of Eratosthenes algorithms took the same time, means that you have done the second one wrong.
I see two performanse issues here:
First of all, push_back() will have to reallocate the dynamic memory block once in a while. Use reserve():
vector<int> tosieve = {};
tosieve.resreve(1000001);
for (int i = 2; i < 1000001; i++)
{
tosieve.push_back(i);
}
Second erase() has to move all Elements behind the one you try to remove. You set the elements to 0 instead and do a run over the vector in the end (untested code):
for (auto& x : tosieve) {
for (auto y = tosieve.begin(); *y < x; ++y) // this check works only in
// the case of an ordered vector
if (y != 0 && x % y == 0) x = 0;
}
{ // this block will make sure, that sieved will be released afterwards
auto sieved = vector<int>{};
for(auto x : tosieve)
sieved.push_back(x);
swap(tosieve, sieved);
} // the large memory block is released now, just keep the sieved elements.
consider to use standard algorithms instead of hand written loops. They help you to state your intent. In this case I see std::transform() for the outer loop of the sieve, std::any_of() for the inner loop, std::generate_n() for filling tosieve at the beginning and std::copy_if() for filling sieved (untested code):
vector<int> tosieve = {};
tosieve.resreve(1000001);
generate_n(back_inserter(tosieve), 1000001, []() -> int {
static int i = 2; return i++;
});
transform(begin(tosieve), end(tosieve), begin(tosieve), [](int i) -> int {
return any_of(begin(tosieve), begin(tosieve) + i - 2,
[&i](int j) -> bool {
return j != 0 && i % j == 0;
}) ? 0 : i;
});
swap(tosieve, [&tosieve]() -> vector<int> {
auto sieved = vector<int>{};
copy_if(begin(tosieve), end(tosieve), back_inserter(sieved),
[](int i) -> bool { return i != 0; });
return sieved;
});
EDIT:
Yet another way to get that done:
vector<int> tosieve = {};
tosieve.resreve(1000001);
generate_n(back_inserter(tosieve), 1000001, []() -> int {
static int i = 2; return i++;
});
swap(tosieve, [&tosieve]() -> vector<int> {
auto sieved = vector<int>{};
copy_if(begin(tosieve), end(tosieve), back_inserter(sieved),
[](int i) -> bool {
return !any_of(begin(tosieve), begin(tosieve) + i - 2,
[&i](int j) -> bool {
return i % j == 0;
});
});
return sieved;
});
Now instead of marking elements, we don't want to copy afterwards, but just directly copy only the elements, we want to copy. This is not only faster than the above suggestion, but also better states the intent.
Very interesting task you have. Thanks!
With pleasure I implemented from scratch my own versions of solving it.
I created 3 separate (independent) functions, all based on Sieve of Eratosthenes. These 3 versions are different in their complexity and speed.
Just a quick note, my simplest (slowest) version finds all primes below your desired limit of 10'000'000 within just 0.025 sec (i.e. 25 milli-seconds).
I also tested all 3 versions to find primes below 2^32 (4'294'967'296), which is solved by "simple" version within 47 seconds, by "intermediate" version within 30 seconds, by "advanced" within 12 seconds. So within just 12 seconds it finds all primes below 4 Billion (there are 203'280'221 such primes below 2^32, see OEIS sequence)!!!
For simplicity I will describe in details only Simple version out of 3. Here's code:
template <typename T>
std::vector<T> GenPrimes_SieveOfEratosthenes(size_t end) {
// https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
if (end <= 2)
return {};
size_t const cnt = end >> 1;
std::vector<u8> composites((cnt + 7) / 8);
auto Get = [&](size_t i){ return bool((composites[i / 8] >> (i % 8)) & 1); };
auto Set = [&](size_t i){ composites[i / 8] |= u8(1) << (i % 8); };
std::vector<T> primes = {2};
size_t i = 0;
for (i = 1; i < cnt; ++i) {
if (Get(i))
continue;
size_t const p = 2 * i + 1, start = (p * p) >> 1;
primes.push_back(p);
if (start >= cnt)
break;
for (size_t j = start; j < cnt; j += p)
Set(j);
}
for (i = i + 1; i < cnt; ++i)
if (!Get(i))
primes.push_back(2 * i + 1);
return primes;
}
This code implements simplest but fast algorithm of finding primes, called Sieve of Eratosthenes. As a small optimization of speed and memory, I search only over odd numbers. This odd numbers optimization gives me ability to store 2x times less memory and do 2x times less steps, hence improves both speed and memory consumption exactly 2 times.
Algorithm is simple, we allocate array of bits, this array at position K has bit 1 if K is composite, or has 0 if K is probably prime. At the end all 0 bits in array signify Definite primes (that are for sure primes). Also due to odd numbers optimization this bit-array stores only odd numbers, so K-th bit is actually a number 2 * K + 1.
Then left to right we go over this array of bits and if we meet 0 bit at position K then it means we found a prime number P = 2 * K + 1 and now starting from position (P * P) / 2 we mark every P-th bit with 1. It means we mark all numbers bigger than P*P that are composite, because they are divisible by P.
We do this procedure only until P * P becomes greater or equal to our limit End (we're finding all primes < End). This limit guarantees that after reaching it ALL zero bits inside array signify prime numbers.
Second version of code does only one optimization to this Simple version, it makes all multi-core (multi-threaded). But this only optimization makes code much bigger and more complex. Basically it slices whole range of bits into all cores, so that they write bits to memory in parallel.
I'll explain only my third Advanced version, it is most complex of 3 versions. It does not only multi-threaded optimization, but also so-called Primorial optimization.
What is Primorial, it is a product of first smallest primes, for example I take primorial 2 * 3 * 5 * 7 = 210.
We can see that any primorial splits infinite range of integers into wheels by modulus of this primorial. For example primorial 210 splits into ranges [0; 210), [210; 2210), [2210; 3*210), etc.
Now it is easy to mathematically prove that inside All ranges of primorial we can mark same positions of numbers as complex, exactly we can mark all numbers that are multiple of 2 or 3 or 5 or 7 as composite.
We can see that out of 210 remainders there are 162 remainders that are for sure composite, and only 48 remainders are probably prime.
Hence it is enough for us to check primality of only 48/210=22.8% of whole search space. This reduction of search space makes task more than 4x times faster, and 4x times less memory consuming.
One can see that my first Simple version in fact due to odd-only optimization was actually using Primorial equal to 2 optimization. Yes, if we take primorial 2 instead of primorial 210, then we gain exactly first version (Simple) algorithm.
All of my 3 versions are tested for correctness and speed. Although still some tiny bugs can remain. Note. Yet it is recommended not to use my code straight away in production, unless it is tested thoroughly.
All 3 versions are tested for correctness by re-using each other answers. I thoroughly test correctness by feeding all limits (end value) from 0 to 2^18. It takes some time to do this.
See main() function to figure out how to use my functions.
Try it online!
SOURCE CODE GOES HERE. Due to StackOverflow limit of 30K symbols per post, I can't inline source code here, as it is almost 30K in size and together with English post above it takes more than 30K. So I'm providing source code on separate Github Gist server, link below. Note that Try it online! link above also contains full source code, but I reduced search limit of 2^32 to smaller one due to GodBolt limit of running time to 3 seconds.
Github Gist code
Output:
10M time 'Simple' 0.024 sec
Time 2^32 'Simple' 46.924 sec, number of primes 203280221
Time 2^32 'Intermediate' 30.999 sec
Time 2^32 'Advanced' 11.359 sec
All checked till 0
All checked till 5000
All checked till 10000
All checked till 15000
All checked till 20000
All checked till 25000
I'm trying to make two vectors. Where vector1 (total1) is containing some strings and vector2(total2) is containing some random unique numbers(that are between 0 and total1.size() - 1)
I want to make a program that print out total1s strings, but in different order every turn. I don't want to use iterators or something because I want to improve my problem solving capacity.
Here is the specific function that crash the program.
for (unsigned i = 0; i < total1.size();)
{
v1 = rand() % total1.size();
for (unsigned s = 0; s < total1.size(); ++s)
{
if (v1 == total2[s])
;
else
{
total2.push_back(v1);
++i;
}
}
}
I'm very grateful for any help that I can get!
Can I suggest you change of algorithm?. Because, even if your current one is correctly implemented ("s", in your code, must go from 0 to total2.size not total1.size and if element is found, break and generate a new random), it has the following drawback: assume vectors of 1.000.000 elements and you are trying the last random number. You have one probability in 1.000.000 of find a random number not previously used. That is a very small amount.Last but one number has a probability of 2 in 1.000.000 also small. In conclusion, your program will loop and expend lots of CPU resources.
Your best alternative is follow #NathanOliver suggestion and look for function std::shuffle. The manual page shows the implementation algorithm, that is what you are looking for.
Another simple algorithm, with some pros and cons, is:
init total2 with sequence 0,1,2,...,n where n is the size total1 - 1
choice two random numbers, i1 and i2, in range [0,n-1].
Swap elements i1 and i2 in total2.
repeat from (2) a fixed number of times "R".
This method allows to known a priori the necessary steps and to control the level of "randomness" of the final vector (bigger R is more random). However, it is far to be good in its randomness quality.
Another method, better in the probabilistic distribution:
fill a list L with number 0,1,2,...size total1-1.
choice a random number i between 0 and the size of list L - 1 .
Store in total2 the i-th element in list L.
Remove this element from L.
repeat from (2) until L is empty.
If you just want to shuffle vector<string> total1, you can do this without using helping vector<int> total2. Here is an implementation based on Fisher–Yates shuffle.
for(int i=n-1; i>=1; i--) {
int j=rand()%(i+1);
swap(total1[j], total1[i]); // your prof might not allow use of swap:)
}
If you must use vector<int> total2 then shuffle it using above algorithm. Next you can use it to create a new vector<string> result from total1 where result[i]=total1[total2[i]].
What/where are the practical uses of the partial_sum algorithm in STL?
What are some other interesting/non-trivial examples or use-cases?
I used it to reduce memory usage of a simple mark-sweep garbage collector in my toy lambda calculus interpreter.
The GC pool is an array of objects of identical size. The goal is to eliminate objects that aren't linked to other objects, and condense the remaining objects into the beginning of the array. Since the objects are moved in memory, each link needs to be updated. This necessitates an object remapping table.
partial_sum allows the table to be stored in compressed format (as little as one bit per object) until the sweep is complete and memory has been freed. Since the objects are small, this significantly reduces memory use.
Recursively mark used objects and populate the Boolean array.
Use remove_if to condense the marked objects to the beginning of the pool.
Use partial_sum over the Boolean values to generate a table of pointers/indexes into the new pool.
This works because the Nth marked object has N preceding 1's in the array and acquires pool index N.
Sweep over the pool again and replace each link using the remap table.
It's especially friendly to the data cache to put the remap table in the just-freed, thus still hot, memory.
One thing to note about partial sum is that it is the operation that undoes adjacent difference much like - undoes +. Or better yet if you remember calculus the way differentiation undoes integration. Better because adjacent difference is essentially differentiation and partial sum is integration.
Let's say you have simulation of a car and at each time step you need to know the position, velocity, and acceleration. You only need to store one of those values as you can compute the other two. Say you store the position at each time step you can take the adjacent difference of the position to give the velocity and the adjacent difference of the velocity to give the acceleration. Alternatively, if you store the acceleration you can take the partial sum to give the velocity and the partial sum of the velocity gives the position.
Partial sum is one of those functions that doesn't come up too often for most people but is enormously useful when you find the right situation. A lot like calculus.
Last time I (would have) used it is when converting a discrete probability distribution (an array of p(X = k)) into a cumulative distribution (an array of p(X <= k)). To select once from the distribution, you can pick a number from [0-1) randomly, then binary search into the cumulative distribution.
That code wasn't in C++, though, so I did the partial sum myself.
You can use it to generate a monotonically increasing sequence of numbers. For example, the following generates a vector containing the numbers 1 through 42:
std::vector<int> v(42, 1);
std::partial_sum(v.begin(), v.end(), v.begin());
Is this an everyday use case? Probably not, though I've found it useful on several occasions.
You can also use std::partial_sum to generate a list of factorials. (This is even less useful, though, since the number of factorials that can be represented by a typical integer data type is quite limited. It is fun, though :-D)
std::vector<int> v(10, 1);
std::partial_sum(v.begin(), v.end(), v.begin());
std::partial_sum(v.begin(), v.end(), v.begin(), std::multiplies<int>());
Personal Use Case: Roulette-Wheel-Selection
I'm using partial_sum in a roulette-wheel-selection algorithm (link text). This algorithm choses randomly elements from a container with a probability which is linear to some value given beforehands.
Because all my elements to choose from bringing a not-necessarily normalized value, I use the partial_sum algorithm for constructing something like a "roulette-wheel", because I sum up all the elements. Then I chose a random variable in this range (the last partial_sum is the sum of all) and use stl::lower_bound for searching "the wheel" where my random search landed. The element returned by the lower_bound algorithm is the chosen one.
Besides the advantage of clear and expressive code with the use of partial_sum, I could also gain some speed when experimenting with the GCC parallel mode which brings parallelized versions for some algorithms and one of them is the partial_sum (link text).
Another use I know of: One of the most important algorithmic primitives in parallel processing (but maybe a little bit away from STL)
If you're interested in heavy optimized algorithms which are using partial_sum (in this case maybe more results under the synonyms "scan" or "prefix_sum"), than go to the parallel algorithms community. They need it all the time. You won't find a parallel sorting algorithm based on quicksort or mergesort without using it. This operation is one of the most important parallel primitives used. I think it is most commonly used for calculating offsets in dynamic algorithms. Think of a partition step in quicksort, which is split and fed to the parallel threads. You don't know the number of elements in each slot of the partition before calculating it. So you need some offsets for all the threads for later access.
Maybe you will find more informatin in the now-hot topic of GPU processing. One short article regarding Nvidia's CUDA and the scan-primitive with a few application examples you will find in Chapter 39. Parallel Prefix Sum (Scan) with CUDA.
Personal Use Case: intermediate step in counting sort from CLRS:
COUNTING_SORT (A, B, k)
for i ← 1 to k do
c[i] ← 0
for j ← 1 to n do
c[A[j]] ← c[A[j]] + 1
//c[i] now contains the number of elements equal to i
// std::partial_sum here
for i ← 2 to k do
c[i] ← c[i] + c[i-1]
// c[i] now contains the number of elements ≤ i
for j ← n downto 1 do
B[c[A[i]]] ← A[j]
c[A[i]] ← c[A[j]] - 1
You could build a "moving sum" (precursor to a moving average):
template <class T>
void moving_sum (const vector<T>& in, int num, vector<T>& out)
{
// cummulative sum
partial_sum (in.begin(), in.end(), out.begin());
// shift and subtract
int j;
for (int i = out.size() - 1; i >= 0; i--) {
j = i - num;
if (j >= 0)
out[i] -= out[j];
}
}
And then call it with:
vector<double> v(10);
// fill in v
vector<double> v2 (v.size());
moving_sum (v, 3, v2);
You know, I actually did use partial_sum() once... It was this interesting little problem that I was asked on a job interview. I enjoyed it so much, I went home and coded it up.
The problem was: Given a sequential sequence of integers, find the shortest sub-sequence with the highest value. E.g. Given:
Value: -1 2 3 -1 4 -2 -4 5
Index: 0 1 2 3 4 5 6 7
We would find the subsequence [1,4]
Now the obvious solution is to run with 3 for loops, iterating over all possible starts & ends, and adding up the value of each possible subsequence in turn. Inefficient, but quick to code up and hard to make mistakes. (Especially when the third for loop is just accumulate(start,end,0).)
The correct solution involves a divide-and-conquer / bottom up approach. E.g. Divide the problem space in half, and for each half compute the largest subsequence contained within that section, the largest subsequence including the starting number, the largest subsequence including the ending number, and the entire section's subsequence. Armed with this data we can then combine the two halves together without any further evaluation of either one. Obviously the data for each half can be computed by further breaking each half into halves (quarters), each quarter into halves (eighths), and so on until we have trivial singleton cases. It's all quite efficient.
But all that aside, there's a third (somewhat less efficient) option that I wanted to explore. It's similar to the 3-for-loop case, only we add the adjacent numbers to avoid so much work. The idea is that there's no need to add a+b, a+b+c, and a+b+c+d when we can add t1=a+b, t2=t1+c, and t3=t2+d. It's a space/computation tradeoff thing. It works by transforming the sequence:
Index: 0 1 2 3 4
FROM: 1 2 3 4 5
TO: 1 3 6 10 15
Thereby giving us all possible substrings starting at index=0 and ending at indexes=0,1,2,3,4.
Then we iterate over this set subtracting the successive possible "start" points...
FROM: 1 3 6 10 15
TO: - 2 5 9 14
TO: - - 3 7 12
TO: - - - 4 9
TO: - - - - 5
Thereby giving us the values (sums) of all possible subsequences.
We can find the maximum value of each iteration via max_element().
The first step is most easily accomplished via partial_sum().
The remaining steps via a for loop and transform(data+i,data+size,data+i,bind2nd(minus<TYPE>(),data[i-1])).
Clearly O(N^2). But still interesting and fun...
Partial sums are often useful in parallel algorithms. Consider the code
for (int i=0; N>i; ++i) {
sum += x[i];
do_something(sum);
}
If you want to parallelise this code, you need to know the partial sums. I am using GNUs parallel version of partial_sum for something very similar.
I often use partial sum not to sum but to calculate the current value in the sequence depending on the previous.
For example, if you integrate a function. Each new step is a previous step, vt += dvdt or vt = integrate_step(dvdt, t_prev, t_prev+dt);.
In nonparametric Bayesian methods there is a Metropolis-Hastings step (per observation) that determines to sample a new or an existing cluster. If an existing cluster has to be sampled this needs to be done with different weights. These weighted likelihoods are simulated in the following example code.
#include <random>
#include <iostream>
#include <algorithm>
int main() {
std::default_random_engine generator(std::random_device{}());
std::uniform_real_distribution<double> distribution(0.0,1.0);
int K = 8;
std::vector<double> weighted_likelihood(K);
for (int i = 0; i < K; ++i) {
weighted_likelihood[i] = i*10;
}
std::cout << "Weighted likelihood: ";
for (auto i: weighted_likelihood) std::cout << i << ' ';
std::cout << std::endl;
std::vector<double> cumsum_likelihood(K);
std::partial_sum(weighted_likelihood.begin(), weighted_likelihood.end(), cumsum_likelihood.begin());
std::cout << "Cumulative sum of weighted likelihood: ";
for (auto i: cumsum_likelihood) std::cout << i << ' ';
std::cout << std::endl;
std::vector<int> frequency(K);
int N = 280000;
for (int i = 0; i < N; ++i) {
double pick = distribution(generator) * cumsum_likelihood.back();
auto lower = std::lower_bound(cumsum_likelihood.begin(), cumsum_likelihood.end(), pick);
int index = std::distance(cumsum_likelihood.begin(), lower);
frequency[index]++;
}
std::cout << "Frequencies: ";
for (auto i: frequency) std::cout << i << ' ';
std::cout << std::endl;
}
Note that this is not different from the answer by https://stackoverflow.com/users/13005/steve-jessop. It's added to give a bit more context about a particular situation (nonparametric Bayesian mehods, e.g. the algorithms by Neal using the Dirichlet process as prior) and the actual code which uses partial_sum in combination with lower_bound.