Sum of all prime numbers below 2 million - java

Problem 10 from Project Euler:
The program runs for smaller numbers and slows to a crawl in the hundred thousands.
At 2 million, an answer fails to show up even though the program seems like it is still running.
I'm trying to implement the Sieve of Eratosthenes. It is supposed to be very fast. What's wrong with my approach?
import java.util.ArrayList;
public class p010
{
/**
* The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17
* Find the sum of all the primes below two million.
* #param args
*/
public static void main(String[] args)
{
ArrayList<Integer> primes = new ArrayList<Integer>();
int upper = 2000000;
for (int i = 2; i < upper; i++)
{
primes.add(i);
}
int sum = 0;
for (int i = 0; i < primes.size(); i++)
{
if (isPrime(primes.get(i)))
{
for (int k = 2; k*primes.get(i) < upper; k++)
{
if (primes.contains(k*primes.get(i)))
{
primes.remove(primes.indexOf(k*primes.get(i)));
}
}
}
}
for (int i = 0; i < primes.size(); i++)
{
sum += primes.get(i);
}
System.out.println(sum);
}
public static boolean isPrime(int number)
{
boolean returnVal = true;
for (int i = 2; i <= Math.sqrt(number); i ++)
{
if (number % i == 0)
{
returnVal = false;
}
}
return returnVal;
}
}

You appear to be trying to implement the Sieve of Eratosthenes which should perform better that O(N^2) (In fact, Wikipedia says it is O(N log(log N)) ...).
The fundamental problem is your choice of data structure. You've chosen to represent the set of remaining prime candidates as an ArrayList of primes. This means that your test to see if a number is still in the set takes O(N) comparisons ... where N is the number of remaining primes. Then you are using ArrayList.remove(int) to remove the non-primes ... which is O(N) also.
That all adds up to making your Sieve implementation worse than O(N^2).
The solution is to replace the ArrayList<Integer> with an boolean[] where the positions (indexes) in the boolean array represent the numbers, and the value of the boolean says whether the number is prime / possibly prime, or not prime.
(There were other problems too that I didn't notice ... see the other answers.)

There are a few issues here. First, lets talk about the algorithm. Your isPrime method is actually the very thing that the sieve is designed to avoid. When you get to a number in the sieve, you already know it's prime, you don't need to test it. If it weren't prime, it would already have been eliminated as a factor of a lower number.
So, point 1:
You can eliminate the isPrime method altogether. It should never return false.
Then, there are implementation issues. primes.contains and primes.remove are problems. They run in linear time on an ArrayList, because they require checking each element or rewriting a large portion of the backing array.
Point 2:
Either mark values in place (use boolean[], or use some other more appropriate data structure.)
I typically use something like boolean primes = new boolean[upper+1], and define n to be included if !(primes[n]). (I just ignore elements 0 and 1 so I don't have to subtract indices.) To "remove" an element, I set it to true. You could also use something like TreeSet<Integer>, I suppose. Using boolean[], the method is near-instantaneous.
Point 3:
sum needs to be a long. The answer (roughly 1.429e11) is larger than the maximum value of an integer (2^31-1)
I can post working code if you like, but here's a test output, without spoilers:
public static void main(String[] args) {
long value;
long start;
long finish;
start = System.nanoTime();
value = arrayMethod(2000000);
finish = System.nanoTime();
System.out.printf("Value: %.3e, time: %4d ms\n", (double)value, (finish-start)/1000000);
start = System.nanoTime();
value = treeMethod(2000000);
finish = System.nanoTime();
System.out.printf("Value: %.3e, time: %4d ms\n", (double)value, (finish-start)/1000000);
}
output:
Using boolean[]
Value: 1.429e+11, time: 17 ms
Using TreeSet<Integer>
Value: 1.429e+11, time: 4869 ms
Edit:
Since spoilers are posted, here's my code:
public static long arrayMethod(int upper) {
boolean[] primes = new boolean[upper+1];
long sum = 0;
for (int i = 2; i <=upper; i++) {
if (!primes[i]) {
sum += i;
for (int k = 2*i; k <= upper; k+=i) {
primes[k] = true;
}
}
}
return sum;
}
public static long treeMethod(int upper) {
TreeSet<Integer> primes = new TreeSet<Integer>();
for (int i = 2; i <= upper; i++) {
primes.add(i);
}
long sum = 0;
for (Integer i = 2; i != null; i=primes.higher(i)) {
sum += i;
for (int k = 2*i; k <= upper; k+=i) {
primes.remove(k);
}
}
return sum;
}

Two things:
Your code is hard to follow. You have a list called "primes", that contains non prime numbers!
Also, you should strongly consider whether or not an array list is appropriate. In this case, a LinkedList would be much more efficient.
Why is this? An array list must constantly resize an array by: asking for new memory to create an array, then copying the old memory over in the newly created array. A Linked list would just resize the memory by changing a pointer. This is a lot quicker! However, I do not think that by making this change you can salvage your algorithm.
You should use an array list if you need to access the items non-sequentially, here, (with a suitable algorithm) you need to access the items sequentially.
Also, your algorithm is slow.Take the advice of SJuan76 (or gyrogearless), thanks sjuan76

The key to the efficiency of classic implementation of the sieve of Eratosthenes on modern CPUs is the direct (i.e. non-sequential) memory access. Fortunately, ArrayList<E> does implement RandomAccess.
Another key to the sieve's efficiency is its conflation of index and value, just like in integer sorting. Actually removing any number from the sequence destroys this ability to directly address without any computations. We must mark, not remove, any composite as we find them, so any numbers greater than it will remain in their places in the sequence.
ArrayList<Integer> can be used for that (except taking more memory than is strictly necessary, but for 2 million this is inconsequential).
So your code with a minimal edit fix (also changing sum to be long as others point out too), becomes
import java.util.ArrayList;
public class Main
{
/**
* The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17
* Find the sum of all the primes below two million.
* #param args
*/
public static void main(String[] args)
{
ArrayList<Integer> primes = new ArrayList<Integer>();
int upper = 5000;
primes.ensureCapacity(upper);
for (int i = 0; i < upper; i++) {
primes.add(i);
}
long sum = 0;
for (int i = 2; i <= upper / i; i++) {
if ( primes.get(i) > 0 ) {
for (int k = i*i; k < upper ; k+=i) {
primes.set(k, 0);
}
}
}
for (int i = 2; i < upper; i++) {
sum += primes.get(i);
}
System.out.println(sum);
}
}
Finds the result for 2000000 in half a second on Ideone. The projected run time for your original code there: between 10 and 400 hours (!).
To find rough estimates for the run time when faced with a slow code, you should always try to find out its empirical orders of growth: run it for some small size n1, then a bigger size n2, record the run times t1 and t2. If t ~ n^a, then a = log(t2/t1) / log(n2/n1).
For your original code the empirical orders of growth measured on 10k .. 20k .. 40k range of upper limit value N, are ~ N^1.7 .. N^1.9 .. N^2.1. For the fixed code it's faster than ~ N (in fact, it's ~ N^0.9 in the tested range 0.5 mln .. 1 mln .. 2 mln). The theoretical complexity is O(N log (log N)).

Your program is not the Sieve of Eratosthenes; the modulo operator gives it away. Your program will be O(n^2), where a proper Sieve of Eratosthenes is O(n log log n), which is essentially n. Here's my version; I'll leave it to you to translate to Java with appropriate numeric datatypes:
function sumPrimes(n)
sum := 0
sieve := makeArray(2..n, True)
for p from 2 to n step 1
if sieve[p]
sum := sum + p
for i from p * p to n step p
sieve[i] := False
return sum
If you're interested in programming with prime numbers, I modestly recommend this essay at my blog.

Related

What is the expected runtime of a while loop that relies on a coin flip?

I've been working on a SkipList implementation in Java for my CS2 class, and after analyzing the runtime of all my methods (for fun), I had a problem identifying the average runtime of the following:
// Returns 1 with 50% probability, 2 with 25% probability,
// 3 with 12.5% probability, and so on, without exceeding maxHeight
private static int generateRandomHeight(int maxHeight)
{
int height = 1;
// At most O(maxHeight) which is O(log(n))
// Best case is O(1) and on average O(log(log(n)))??
while (--maxHeight > 0 && Math.random() < .5)
height++;
return height;
}
I know that the best runtime is O(1) and the worst runtime is O(h), which is guaranteed to be O(log(n)), since h = ⌈log₂(n)⌉ (ceiling of log base two of n).
Question is, what is the average expected runtime of this function?
At first glance, I assumed that it is expected to be O(log(log(n))). I brought this question to my TAs and after discussing with them, we are leaning on an average runtime of O(1), since it's most likely that we iterate only once or not at all (e.g 50%).
Calculating complexity is often complicated when you have probability. But you were right, it was O(1), but not exactly O(1). O(1) is defined as constant, which means that no matter what you input, the run time would be the same. However, in your case, Math.random() isn't predictable. But, nevertheless, its's still O(1) because the run-time stays relatively constant as n increases. I made a test to see the number of times the loop iterates, and my results were really similar.
public static void main(String[] args) {
List<Integer> l = new ArrayList<>();
for(int n = 0; n < 100; n ++) {
for (int i = 0; i < 1000; i++) {
for (int j = 0; j < n; j++) {
if (Math.random() < 0.5) {
l.add(j);
break;
}
}
}
System.out.println(average(l));
l.clear();
}
}
private static double average(List<Integer> l) {
double sum = 0;
for(int i : l) {
sum += i;
}
return sum / l.size();
}
Here are a few results. As you can see, they hover around 1.
1.011
0.986
1.046
0.991
0.953
0.984
1.017
0.973

Java - Trying to avoid duplicates when using random to get lines in a text file [duplicate]

In this case, the MAX is only 5, so I could check the duplicates one by one, but how could I do this in a simpler way? For example, what if the MAX has a value of 20?
Thanks.
int MAX = 5;
for (i = 1 , i <= MAX; i++)
{
drawNum[1] = (int)(Math.random()*MAX)+1;
while (drawNum[2] == drawNum[1])
{
drawNum[2] = (int)(Math.random()*MAX)+1;
}
while ((drawNum[3] == drawNum[1]) || (drawNum[3] == drawNum[2]) )
{
drawNum[3] = (int)(Math.random()*MAX)+1;
}
while ((drawNum[4] == drawNum[1]) || (drawNum[4] == drawNum[2]) || (drawNum[4] == drawNum[3]) )
{
drawNum[4] = (int)(Math.random()*MAX)+1;
}
while ((drawNum[5] == drawNum[1]) ||
(drawNum[5] == drawNum[2]) ||
(drawNum[5] == drawNum[3]) ||
(drawNum[5] == drawNum[4]) )
{
drawNum[5] = (int)(Math.random()*MAX)+1;
}
}
The simplest way would be to create a list of the possible numbers (1..20 or whatever) and then shuffle them with Collections.shuffle. Then just take however many elements you want. This is great if your range is equal to the number of elements you need in the end (e.g. for shuffling a deck of cards).
That doesn't work so well if you want (say) 10 random elements in the range 1..10,000 - you'd end up doing a lot of work unnecessarily. At that point, it's probably better to keep a set of values you've generated so far, and just keep generating numbers in a loop until the next one isn't already present:
if (max < numbersNeeded)
{
throw new IllegalArgumentException("Can't ask for more numbers than are available");
}
Random rng = new Random(); // Ideally just create one instance globally
// Note: use LinkedHashSet to maintain insertion order
Set<Integer> generated = new LinkedHashSet<Integer>();
while (generated.size() < numbersNeeded)
{
Integer next = rng.nextInt(max) + 1;
// As we're adding to a set, this will automatically do a containment check
generated.add(next);
}
Be careful with the set choice though - I've very deliberately used LinkedHashSet as it maintains insertion order, which we care about here.
Yet another option is to always make progress, by reducing the range each time and compensating for existing values. So for example, suppose you wanted 3 values in the range 0..9. On the first iteration you'd generate any number in the range 0..9 - let's say you generate a 4.
On the second iteration you'd then generate a number in the range 0..8. If the generated number is less than 4, you'd keep it as is... otherwise you add one to it. That gets you a result range of 0..9 without 4. Suppose we get 7 that way.
On the third iteration you'd generate a number in the range 0..7. If the generated number is less than 4, you'd keep it as is. If it's 4 or 5, you'd add one. If it's 6 or 7, you'd add two. That way the result range is 0..9 without 4 or 6.
Here's how I'd do it
import java.util.ArrayList;
import java.util.Random;
public class Test {
public static void main(String[] args) {
int size = 20;
ArrayList<Integer> list = new ArrayList<Integer>(size);
for(int i = 1; i <= size; i++) {
list.add(i);
}
Random rand = new Random();
while(list.size() > 0) {
int index = rand.nextInt(list.size());
System.out.println("Selected: "+list.remove(index));
}
}
}
As the esteemed Mr Skeet has pointed out:
If n is the number of randomly selected numbers you wish to choose and N is the total sample space of numbers available for selection:
If n << N, you should just store the numbers that you have picked and check a list to see if the number selected is in it.
If n ~= N, you should probably use my method, by populating a list containing the entire sample space and then removing numbers from it as you select them.
//random numbers are 0,1,2,3
ArrayList<Integer> numbers = new ArrayList<Integer>();
Random randomGenerator = new Random();
while (numbers.size() < 4) {
int random = randomGenerator .nextInt(4);
if (!numbers.contains(random)) {
numbers.add(random);
}
}
This would be a lot simpler in java-8:
Stream.generate(new Random()::ints)
.flatMap(IntStream::boxed)
.distinct()
.limit(16) // whatever limit you might need
.toArray(Integer[]::new);
There is another way of doing "random" ordered numbers with LFSR, take a look at:
http://en.wikipedia.org/wiki/Linear_feedback_shift_register
with this technique you can achieve the ordered random number by index and making sure the values are not duplicated.
But these are not TRUE random numbers because the random generation is deterministic.
But depending your case you can use this technique reducing the amount of processing on random number generation when using shuffling.
Here a LFSR algorithm in java, (I took it somewhere I don't remeber):
public final class LFSR {
private static final int M = 15;
// hard-coded for 15-bits
private static final int[] TAPS = {14, 15};
private final boolean[] bits = new boolean[M + 1];
public LFSR() {
this((int)System.currentTimeMillis());
}
public LFSR(int seed) {
for(int i = 0; i < M; i++) {
bits[i] = (((1 << i) & seed) >>> i) == 1;
}
}
/* generate a random int uniformly on the interval [-2^31 + 1, 2^31 - 1] */
public short nextShort() {
//printBits();
// calculate the integer value from the registers
short next = 0;
for(int i = 0; i < M; i++) {
next |= (bits[i] ? 1 : 0) << i;
}
// allow for zero without allowing for -2^31
if (next < 0) next++;
// calculate the last register from all the preceding
bits[M] = false;
for(int i = 0; i < TAPS.length; i++) {
bits[M] ^= bits[M - TAPS[i]];
}
// shift all the registers
for(int i = 0; i < M; i++) {
bits[i] = bits[i + 1];
}
return next;
}
/** returns random double uniformly over [0, 1) */
public double nextDouble() {
return ((nextShort() / (Integer.MAX_VALUE + 1.0)) + 1.0) / 2.0;
}
/** returns random boolean */
public boolean nextBoolean() {
return nextShort() >= 0;
}
public void printBits() {
System.out.print(bits[M] ? 1 : 0);
System.out.print(" -> ");
for(int i = M - 1; i >= 0; i--) {
System.out.print(bits[i] ? 1 : 0);
}
System.out.println();
}
public static void main(String[] args) {
LFSR rng = new LFSR();
Vector<Short> vec = new Vector<Short>();
for(int i = 0; i <= 32766; i++) {
short next = rng.nextShort();
// just testing/asserting to make
// sure the number doesn't repeat on a given list
if (vec.contains(next))
throw new RuntimeException("Index repeat: " + i);
vec.add(next);
System.out.println(next);
}
}
}
Another approach which allows you to specify how many numbers you want with size and the min and max values of the returned numbers
public static int getRandomInt(int min, int max) {
Random random = new Random();
return random.nextInt((max - min) + 1) + min;
}
public static ArrayList<Integer> getRandomNonRepeatingIntegers(int size, int min,
int max) {
ArrayList<Integer> numbers = new ArrayList<Integer>();
while (numbers.size() < size) {
int random = getRandomInt(min, max);
if (!numbers.contains(random)) {
numbers.add(random);
}
}
return numbers;
}
To use it returning 7 numbers between 0 and 25.
ArrayList<Integer> list = getRandomNonRepeatingIntegers(7, 0, 25);
for (int i = 0; i < list.size(); i++) {
System.out.println("" + list.get(i));
}
The most efficient, basic way to have non-repeating random numbers is explained by this pseudo-code. There is no need to have nested loops or hashed lookups:
// get 5 unique random numbers, possible values 0 - 19
// (assume desired number of selections < number of choices)
const int POOL_SIZE = 20;
const int VAL_COUNT = 5;
declare Array mapping[POOL_SIZE];
declare Array results[VAL_COUNT];
declare i int;
declare r int;
declare max_rand int;
// create mapping array
for (i=0; i<POOL_SIZE; i++) {
mapping[i] = i;
}
max_rand = POOL_SIZE-1; // start loop searching for maximum value (19)
for (i=0; i<VAL_COUNT; i++) {
r = Random(0, max_rand); // get random number
results[i] = mapping[r]; // grab number from map array
mapping[r] = max_rand; // place item past range at selected location
max_rand = max_rand - 1; // reduce random scope by 1
}
Suppose first iteration generated random number 3 to start (from 0 - 19). This would make results[0] = mapping[3], i.e., the value 3. We'd then assign mapping[3] to 19.
In the next iteration, the random number was 5 (from 0 - 18). This would make results[1] = mapping[5], i.e., the value 5. We'd then assign mapping[5] to 18.
Now suppose the next iteration chose 3 again (from 0 - 17). results[2] would be assigned the value of mapping[3], but now, this value is not 3, but 19.
This same protection persists for all numbers, even if you got the same number 5 times in a row. E.g., if the random number generator gave you 0 five times in a row, the results would be: [ 0, 19, 18, 17, 16 ].
You would never get the same number twice.
Generating all the indices of a sequence is generally a bad idea, as it might take a lot of time, especially if the ratio of the numbers to be chosen to MAX is low (the complexity becomes dominated by O(MAX)). This gets worse if the ratio of the numbers to be chosen to MAX approaches one, as then removing the chosen indices from the sequence of all also becomes expensive (we approach O(MAX^2/2)). But for small numbers, this generally works well and is not particularly error-prone.
Filtering the generated indices by using a collection is also a bad idea, as some time is spent in inserting the indices into the sequence, and progress is not guaranteed as the same random number can be drawn several times (but for large enough MAX it is unlikely). This could be close to complexity O(k n log^2(n)/2), ignoring the duplicates and assuming the collection uses a tree for efficient lookup (but with a significant constant cost k of allocating the tree nodes and possibly having to rebalance).
Another option is to generate the random values uniquely from the beginning, guaranteeing progress is being made. That means in the first round, a random index in [0, MAX] is generated:
items i0 i1 i2 i3 i4 i5 i6 (total 7 items)
idx 0 ^^ (index 2)
In the second round, only [0, MAX - 1] is generated (as one item was already selected):
items i0 i1 i3 i4 i5 i6 (total 6 items)
idx 1 ^^ (index 2 out of these 6, but 3 out of the original 7)
The values of the indices then need to be adjusted: if the second index falls in the second half of the sequence (after the first index), it needs to be incremented to account for the gap. We can implement this as a loop, allowing us to select arbitrary number of unique items.
For short sequences, this is quite fast O(n^2/2) algorithm:
void RandomUniqueSequence(std::vector<int> &rand_num,
const size_t n_select_num, const size_t n_item_num)
{
assert(n_select_num <= n_item_num);
rand_num.clear(); // !!
// b1: 3187.000 msec (the fastest)
// b2: 3734.000 msec
for(size_t i = 0; i < n_select_num; ++ i) {
int n = n_Rand(n_item_num - i - 1);
// get a random number
size_t n_where = i;
for(size_t j = 0; j < i; ++ j) {
if(n + j < rand_num[j]) {
n_where = j;
break;
}
}
// see where it should be inserted
rand_num.insert(rand_num.begin() + n_where, 1, n + n_where);
// insert it in the list, maintain a sorted sequence
}
// tier 1 - use comparison with offset instead of increment
}
Where n_select_num is your 5 and n_number_num is your MAX. The n_Rand(x) returns random integers in [0, x] (inclusive). This can be made a bit faster if selecting a lot of items (e.g. not 5 but 500) by using binary search to find the insertion point. To do that, we need to make sure that we meet the requirements.
We will do binary search with the comparison n + j < rand_num[j] which is the same as n < rand_num[j] - j. We need to show that rand_num[j] - j is still a sorted sequence for a sorted sequence rand_num[j]. This is fortunately easily shown, as the lowest distance between two elements of the original rand_num is one (the generated numbers are unique, so there is always difference of at least 1). At the same time, if we subtract the indices j from all the elements rand_num[j], the differences in index are exactly 1. So in the "worst" case, we get a constant sequence - but never decreasing. The binary search can therefore be used, yielding O(n log(n)) algorithm:
struct TNeedle { // in the comparison operator we need to make clear which argument is the needle and which is already in the list; we do that using the type system.
int n;
TNeedle(int _n)
:n(_n)
{}
};
class CCompareWithOffset { // custom comparison "n < rand_num[j] - j"
protected:
std::vector<int>::iterator m_p_begin_it;
public:
CCompareWithOffset(std::vector<int>::iterator p_begin_it)
:m_p_begin_it(p_begin_it)
{}
bool operator ()(const int &r_value, TNeedle n) const
{
size_t n_index = &r_value - &*m_p_begin_it;
// calculate index in the array
return r_value < n.n + n_index; // or r_value - n_index < n.n
}
bool operator ()(TNeedle n, const int &r_value) const
{
size_t n_index = &r_value - &*m_p_begin_it;
// calculate index in the array
return n.n + n_index < r_value; // or n.n < r_value - n_index
}
};
And finally:
void RandomUniqueSequence(std::vector<int> &rand_num,
const size_t n_select_num, const size_t n_item_num)
{
assert(n_select_num <= n_item_num);
rand_num.clear(); // !!
// b1: 3578.000 msec
// b2: 1703.000 msec (the fastest)
for(size_t i = 0; i < n_select_num; ++ i) {
int n = n_Rand(n_item_num - i - 1);
// get a random number
std::vector<int>::iterator p_where_it = std::upper_bound(rand_num.begin(), rand_num.end(),
TNeedle(n), CCompareWithOffset(rand_num.begin()));
// see where it should be inserted
rand_num.insert(p_where_it, 1, n + p_where_it - rand_num.begin());
// insert it in the list, maintain a sorted sequence
}
// tier 4 - use binary search
}
I have tested this on three benchmarks. First, 3 numbers were chosen out of 7 items, and a histogram of the items chosen was accumulated over 10,000 runs:
4265 4229 4351 4267 4267 4364 4257
This shows that each of the 7 items was chosen approximately the same number of times, and there is no apparent bias caused by the algorithm. All the sequences were also checked for correctness (uniqueness of contents).
The second benchmark involved choosing 7 numbers out of 5000 items. The time of several versions of the algorithm was accumulated over 10,000,000 runs. The results are denoted in comments in the code as b1. The simple version of the algorithm is slightly faster.
The third benchmark involved choosing 700 numbers out of 5000 items. The time of several versions of the algorithm was again accumulated, this time over 10,000 runs. The results are denoted in comments in the code as b2. The binary search version of the algorithm is now more than two times faster than the simple one.
The second method starts being faster for choosing more than cca 75 items on my machine (note that the complexity of either algorithm does not depend on the number of items, MAX).
It is worth mentioning that the above algorithms generate the random numbers in ascending order. But it would be simple to add another array to which the numbers would be saved in the order in which they were generated, and returning that instead (at negligible additional cost O(n)). It is not necessary to shuffle the output: that would be much slower.
Note that the sources are in C++, I don't have Java on my machine, but the concept should be clear.
EDIT:
For amusement, I have also implemented the approach that generates a list with all the indices 0 .. MAX, chooses them randomly and removes them from the list to guarantee uniqueness. Since I've chosen quite high MAX (5000), the performance is catastrophic:
// b1: 519515.000 msec
// b2: 20312.000 msec
std::vector<int> all_numbers(n_item_num);
std::iota(all_numbers.begin(), all_numbers.end(), 0);
// generate all the numbers
for(size_t i = 0; i < n_number_num; ++ i) {
assert(all_numbers.size() == n_item_num - i);
int n = n_Rand(n_item_num - i - 1);
// get a random number
rand_num.push_back(all_numbers[n]); // put it in the output list
all_numbers.erase(all_numbers.begin() + n); // erase it from the input
}
// generate random numbers
I have also implemented the approach with a set (a C++ collection), which actually comes second on benchmark b2, being only about 50% slower than the approach with the binary search. That is understandable, as the set uses a binary tree, where the insertion cost is similar to binary search. The only difference is the chance of getting duplicate items, which slows down the progress.
// b1: 20250.000 msec
// b2: 2296.000 msec
std::set<int> numbers;
while(numbers.size() < n_number_num)
numbers.insert(n_Rand(n_item_num - 1)); // might have duplicates here
// generate unique random numbers
rand_num.resize(numbers.size());
std::copy(numbers.begin(), numbers.end(), rand_num.begin());
// copy the numbers from a set to a vector
Full source code is here.
Your problem seems to reduce to choose k elements at random from a collection of n elements. The Collections.shuffle answer is thus correct, but as pointed out inefficient: its O(n).
Wikipedia: Fisher–Yates shuffle has a O(k) version when the array already exists. In your case, there is no array of elements and creating the array of elements could be very expensive, say if max were 10000000 instead of 20.
The shuffle algorithm involves initializing an array of size n where every element is equal to its index, picking k random numbers each number in a range with the max one less than the previous range, then swapping elements towards the end of the array.
You can do the same operation in O(k) time with a hashmap although I admit its kind of a pain. Note that this is only worthwhile if k is much less than n. (ie k ~ lg(n) or so), otherwise you should use the shuffle directly.
You will use your hashmap as an efficient representation of the backing array in the shuffle algorithm. Any element of the array that is equal to its index need not appear in the map. This allows you to represent an array of size n in constant time, there is no time spent initializing it.
Pick k random numbers: the first is in the range 0 to n-1, the second 0 to n-2, the third 0 to n-3 and so on, thru n-k.
Treat your random numbers as a set of swaps. The first random index swaps to the final position. The second random index swaps to the second to last position. However, instead of working against a backing array, work against your hashmap. Your hashmap will store every item that is out of position.
int getValue(i)
{
if (map.contains(i))
return map[i];
return i;
}
void setValue(i, val)
{
if (i == val)
map.remove(i);
else
map[i] = val;
}
int[] chooseK(int n, int k)
{
for (int i = 0; i < k; i++)
{
int randomIndex = nextRandom(0, n - i); //(n - i is exclusive)
int desiredIndex = n-i-1;
int valAtRandom = getValue(randomIndex);
int valAtDesired = getValue(desiredIndex);
setValue(desiredIndex, valAtRandom);
setValue(randomIndex, valAtDesired);
}
int[] output = new int[k];
for (int i = 0; i < k; i++)
{
output[i] = (getValue(n-i-1));
}
return output;
}
You could use one of the classes implementing the Set interface (API), and then each number you generate, use Set.add() to insert it.
If the return value is false, you know the number has already been generated before.
Instead of doing all this create a LinkedHashSet object and random numbers to it by Math.random() function .... if any duplicated entry occurs the LinkedHashSet object won't add that number to its List ... Since in this Collection Class no duplicate values are allowed .. in the end u get a list of random numbers having no duplicated values .... :D
With Java 8 upwards you can use the ints method from the IntStream interface:
Returns an effectively unlimited stream of pseudorandom int values.
Random r = new Random();
int randomNumberOrigin = 0;
int randomNumberBound = 10;
int size = 5;
int[] unique = r.ints(randomNumberOrigin, randomNumberBound)
.distinct()
.limit(size)
.toArray();
Following code create a sequence random number between [1,m] that was not generated before.
public class NewClass {
public List<Integer> keys = new ArrayList<Integer>();
public int rand(int m) {
int n = (int) (Math.random() * m + 1);
if (!keys.contains(n)) {
keys.add(n);
return n;
} else {
return rand(m);
}
}
public static void main(String[] args) {
int m = 4;
NewClass ne = new NewClass();
for (int i = 0; i < 4; i++) {
System.out.println(ne.rand(m));
}
System.out.println("list: " + ne.keys);
}
}
The most easy way is use nano DateTime as long format.
System.nanoTime();
There is algorithm of card batch: you create ordered array of numbers (the "card batch") and in every iteration you select a number at random position from it (removing the selected number from the "card batch" of course).
Here is an efficient solution for fast creation of a randomized array. After randomization you can simply pick the n-th element e of the array, increment n and return e. This solution has O(1) for getting a random number and O(n) for initialization, but as a tradeoff requires a good amount of memory if n gets large enough.
There is a more efficient and less cumbersome solution for integers than a Collections.shuffle.
The problem is the same as successively picking items from only the un-picked items in a set and setting them in order somewhere else. This is exactly like randomly dealing cards or drawing winning raffle tickets from a hat or bin.
This algorithm works for loading any array and achieving a random order at the end of the load. It also works for adding into a List collection (or any other indexed collection) and achieving a random sequence in the collection at the end of the adds.
It can be done with a single array, created once, or a numerically ordered collectio, such as a List, in place. For an array, the initial array size needs to be the exact size to contain all the intended values. If you don't know how many values might occur in advance, using a numerically orderred collection, such as an ArrayList or List, where the size is not immutable, will also work. It will work universally for an array of any size up to Integer.MAX_VALUE which is just over 2,000,000,000. List objects will have the same index limits. Your machine may run out of memory before you get to an array of that size. It may be more efficient to load an array typed to the object types and convert it to some collection, after loading the array. This is especially true if the target collection is not numerically indexed.
This algorithm, exactly as written, will create a very even distribution where there are no duplicates. One aspect that is VERY IMPORTANT is that it has to be possible for the insertion of the next item to occur up to the current size + 1. Thus, for the second item, it could be possible to store it in location 0 or location 1. For the 20th item, it could be possible to store it in any location, 0 through 19. It is just as possible the first item to stay in location 0 as it is for it to end up in any other location. It is just as possible for the next new item to go anywhere, including the next new location.
The randomness of the sequence will be as random as the randomness of the random number generator.
This algorithm can also be used to load reference types into random locations in an array. Since this works with an array, it can also work with collections. That means you don't have to create the collection and then shuffle it or have it ordered on whatever orders the objects being inserted. The collection need only have the ability to insert an item anywhere in the collection or append it.
// RandomSequence.java
import java.util.Random;
public class RandomSequence {
public static void main(String[] args) {
// create an array of the size and type for which
// you want a random sequence
int[] randomSequence = new int[20];
Random randomNumbers = new Random();
for (int i = 0; i < randomSequence.length; i++ ) {
if (i == 0) { // seed first entry in array with item 0
randomSequence[i] = 0;
} else { // for all other items...
// choose a random pointer to the segment of the
// array already containing items
int pointer = randomNumbers.nextInt(i + 1);
randomSequence[i] = randomSequence[pointer];
randomSequence[pointer] = i;
// note that if pointer & i are equal
// the new value will just go into location i and possibly stay there
// this is VERY IMPORTANT to ensure the sequence is really random
// and not biased
} // end if...else
} // end for
for (int number: randomSequence) {
System.out.printf("%2d ", number);
} // end for
} // end main
} // end class RandomSequence
It really all depends on exactly WHAT you need the random generation for, but here's my take.
First, create a standalone method for generating the random number.
Be sure to allow for limits.
public static int newRandom(int limit){
return generatedRandom.nextInt(limit); }
Next, you will want to create a very simple decision structure that compares values. This can be done in one of two ways. If you have a very limited amount of numbers to verify, a simple IF statement will suffice:
public static int testDuplicates(int int1, int int2, int int3, int int4, int int5){
boolean loopFlag = true;
while(loopFlag == true){
if(int1 == int2 || int1 == int3 || int1 == int4 || int1 == int5 || int1 == 0){
int1 = newRandom(75);
loopFlag = true; }
else{
loopFlag = false; }}
return int1; }
The above compares int1 to int2 through int5, as well as making sure that there are no zeroes in the randoms.
With these two methods in place, we can do the following:
num1 = newRandom(limit1);
num2 = newRandom(limit1);
num3 = newRandom(limit1);
num4 = newRandom(limit1);
num5 = newRandom(limit1);
Followed By:
num1 = testDuplicates(num1, num2, num3, num4, num5);
num2 = testDuplicates(num2, num1, num3, num4, num5);
num3 = testDuplicates(num3, num1, num2, num4, num5);
num4 = testDuplicates(num4, num1, num2, num3, num5);
num5 = testDuplicates(num5, num1, num2, num3, num5);
If you have a longer list to verify, then a more complex method will yield better results both in clarity of code and in processing resources.
Hope this helps. This site has helped me so much, I felt obliged to at least TRY to help as well.
I created a snippet that generates no duplicate random integer. the advantage of this snippet is that you can assign the list of an array to it and generate the random item, too.
No duplication random generator class
With Java 8 using the below code, you can create 10 distinct random Integer Numbers within a range of 1000.
Random random = new Random();
Integer[] input9 = IntStream.range(1, 10).map(i -> random.nextInt(1000)).boxed().distinct()
.toArray(Integer[]::new);
System.out.println(Arrays.toString(input9));
Modify the range to generate more numbers example : range(1,X). It will generate X distinct random numbers.
Modify the nextInt value to select the random number range : random.nextInt(Y)::random number will be generated within the range Y

What is an efficient method for adding thousands of numbers quickly?

I am attempting to solve a challenge, but I have hit a roadblock. I am a beginner programmer attempting to add tens of thousands of numbers. If I wait long enough, my program can easily yield the correct sum, however, I am looking for a more efficient method.
What is an efficient method for adding thousands of numbers quickly?
Side note: I have been reading about modular arithmetic, but I cannot quite wrap my head around it. Not sure if that could be useful for this situation.
I am attempting to get the sum of every prime number below 2 000 000. Here is my code so far:
public class Problem10 {
public static void main (String[] args) {
long sum = 0L;
for(long i = 1L; i < 2000000; i++) {
if(isPrimeNumber((int)i)) {
sum += i;
}
}
System.out.println(sum);
}
public static boolean isPrimeNumber(int i) {
int factors = 0;
int j = 1;
while (j <= i) {
if (i % j == 0) {
factors++;
}
j++;
}
return (factors == 2);
}
}
You can replace your isPrimeNumber() method with this to speed it up substantially.
public static boolean isPrimeNumber(int i) {
if (i==2) return true;
if (i==3) return true;
if (i%2==0) return false;
if (i%3==0) return false;
int j = 5;
int k = 2;
while (j * j <= i) {
if (i % j == 0) return false;
j += k ;
k = 6 - k;
}
return true;
}
This looks like homework, so I'm not going to give you the solution in code. The least you can do is code it yourself.
In your code, isPrimeNumber() is what's taking up most of the time—if I had to guess, I would say 90-99% of it. What you can do to make it faster is implement the Sieve of Eratosthenes.
To start, you create an array that will hold all the prime numbers1. You should start it with a single value: 2. To find more prime numbers, iterate through every integer from 3 to the highest number you want. For each of those numbers, check if that number is divisible by any of the prime numbers in your array. If the next prime number in your array is greater than i / 2, you know that i is prime, and you can add it to your array.
After you have found all the prime numbers from 1 to n, the only way to sum them is by iterating through the array. That part cannot be optimized, but it will not take very long anyways.
1 There are two ways to do this. One is to just use an ArrayList or LinkedList, and add numbers as needed. The other is to create an array that is as large or larger than you need. As mentioned here, the number of primes equal to or less than n is less than (n / log(n)) * (1 + 1.2762 / log(n)), as long as n is greater than 598. If n is less than 598, you can just create an array of length 109.
In regards to the question in the title, "What is an efficient method for adding thousands of numbers quickly?", the only thing I can think of is multithreading. Create an array of all the numbers you want to sum, then have many threads sum different parts of the array. After that, sum all the results from each thread. This method will probably only be noticeably faster when summing huge amount of numbers, e.g. hundreds of thousands or millions.

I'm trying to find the sum of primes below 2 million in Java [duplicate]

This question already has answers here:
Project Euler #10 Java solution not working
(6 answers)
Closed 9 years ago.
I'm trying to find the sum of primes below millions. My code works when I try to find the sum of primes below hundred thousands but when I go large numbers it doesn't work. So I need some help to get this work for big numbers...
import java.util.Scanner;
public class sumPrime {
public static void main (String args []){
long n = 2000000; int i; int j;int sum =0;
for (i=2; i <n; i++){
for (j=2; j<i; j++){
if (i%j==0){
break;
}
}
if (i==j){
sum +=i;
}
}
System.out.print(sum);
}
}
Your code could be improved by making the inner loop stop earlier. If a number N is not prime, then it must have at least one factor (apart from 1) that is less or equal to sqrt(N). In this case, this simple change should make the program roughly 1000 times faster.
For a simple and (more) efficient algorithm, read up on the Sieve of Eratosthenes.
Bug - your sum needs to be a long. An int will probably overflow.
Note that the classic formulation of Sieve of Eratosthenes needs a large array of booleans (or a bitmap) whose size depends on the largest prime candidate you are interested in. In this case that means a 2Mbyte array (or smaller if you use a bitmap) ... which is too small to worry about. Also, you can reduce the memory usage by sieving in stages, though it makes the code more complicated.
Rather than trying to divide by all the numbers below i you could potentially keep the found prime numbers in a list and try to divide by those prime numbers (since any non prime number will be divisible by a prime number less than that).
public static long sumPrime2() {
List<Long> primes = new ArrayList<>();
primes.add(2L);
primes.add(3L);
long primeSum = 5;
for (long primeCandidate = 5; primeCandidate < 2000000; primeCandidate = primeCandidate + 2) {
boolean isCandidatePrime = true;
double sqrt = Math.sqrt(primeCandidate);
for (int i = 0; i < primes.size(); i++) {
Long prime = primes.get(i);
if (primeCandidate % prime == 0) {
isCandidatePrime = false;
break;
}
if (prime > sqrt) {
break;
}
}
if (isCandidatePrime) {
primes.add(primeCandidate);
primeSum += primeCandidate;
}
System.out.println(primeCandidate);
}
System.out.println(primes.size());
return primeSum;
}
This gave the answer in 8 seconds
I suspect integer overflow in i, j, sum - try making them all longs. In the sample code you shouldn't be getting overflows as Java ints are meant to be 32 bit but at some stage you certainly will.
As already mentioned - i only needs to iterate to the square root of n. So I would replace this line:
for (i=2; i <n; i++){
With:
long limit=sqrt(n);
for (i=2; i <limit; i++){
Note that calculating the square root outside the program loops will also speed things up a bit.
Also the sieve algorithm would be faster but requires Java to create an array containing n elements and at some stage that is going to fail with insufficient memory.
The best algorithm for this program uses the Sieve of Eratosthenes:
function sumPrimes(n)
sum, sieve := 0, makeArray(2..n, True)
for p from 2 to n
if sieve[p]
sum := sum + p
for i from p*p to n step p
sieve[i] := False
return sum
Then sumPrimes(2000000) returns the sum of the primes less than two million, in about a second. I'll leave it to you to translate to Java, with an appropriate data type for the sum. If you're interested in programming with prime numbers, I modestly recommend this essay at my blog.

Creating random numbers with no duplicates

In this case, the MAX is only 5, so I could check the duplicates one by one, but how could I do this in a simpler way? For example, what if the MAX has a value of 20?
Thanks.
int MAX = 5;
for (i = 1 , i <= MAX; i++)
{
drawNum[1] = (int)(Math.random()*MAX)+1;
while (drawNum[2] == drawNum[1])
{
drawNum[2] = (int)(Math.random()*MAX)+1;
}
while ((drawNum[3] == drawNum[1]) || (drawNum[3] == drawNum[2]) )
{
drawNum[3] = (int)(Math.random()*MAX)+1;
}
while ((drawNum[4] == drawNum[1]) || (drawNum[4] == drawNum[2]) || (drawNum[4] == drawNum[3]) )
{
drawNum[4] = (int)(Math.random()*MAX)+1;
}
while ((drawNum[5] == drawNum[1]) ||
(drawNum[5] == drawNum[2]) ||
(drawNum[5] == drawNum[3]) ||
(drawNum[5] == drawNum[4]) )
{
drawNum[5] = (int)(Math.random()*MAX)+1;
}
}
The simplest way would be to create a list of the possible numbers (1..20 or whatever) and then shuffle them with Collections.shuffle. Then just take however many elements you want. This is great if your range is equal to the number of elements you need in the end (e.g. for shuffling a deck of cards).
That doesn't work so well if you want (say) 10 random elements in the range 1..10,000 - you'd end up doing a lot of work unnecessarily. At that point, it's probably better to keep a set of values you've generated so far, and just keep generating numbers in a loop until the next one isn't already present:
if (max < numbersNeeded)
{
throw new IllegalArgumentException("Can't ask for more numbers than are available");
}
Random rng = new Random(); // Ideally just create one instance globally
// Note: use LinkedHashSet to maintain insertion order
Set<Integer> generated = new LinkedHashSet<Integer>();
while (generated.size() < numbersNeeded)
{
Integer next = rng.nextInt(max) + 1;
// As we're adding to a set, this will automatically do a containment check
generated.add(next);
}
Be careful with the set choice though - I've very deliberately used LinkedHashSet as it maintains insertion order, which we care about here.
Yet another option is to always make progress, by reducing the range each time and compensating for existing values. So for example, suppose you wanted 3 values in the range 0..9. On the first iteration you'd generate any number in the range 0..9 - let's say you generate a 4.
On the second iteration you'd then generate a number in the range 0..8. If the generated number is less than 4, you'd keep it as is... otherwise you add one to it. That gets you a result range of 0..9 without 4. Suppose we get 7 that way.
On the third iteration you'd generate a number in the range 0..7. If the generated number is less than 4, you'd keep it as is. If it's 4 or 5, you'd add one. If it's 6 or 7, you'd add two. That way the result range is 0..9 without 4 or 6.
Here's how I'd do it
import java.util.ArrayList;
import java.util.Random;
public class Test {
public static void main(String[] args) {
int size = 20;
ArrayList<Integer> list = new ArrayList<Integer>(size);
for(int i = 1; i <= size; i++) {
list.add(i);
}
Random rand = new Random();
while(list.size() > 0) {
int index = rand.nextInt(list.size());
System.out.println("Selected: "+list.remove(index));
}
}
}
As the esteemed Mr Skeet has pointed out:
If n is the number of randomly selected numbers you wish to choose and N is the total sample space of numbers available for selection:
If n << N, you should just store the numbers that you have picked and check a list to see if the number selected is in it.
If n ~= N, you should probably use my method, by populating a list containing the entire sample space and then removing numbers from it as you select them.
//random numbers are 0,1,2,3
ArrayList<Integer> numbers = new ArrayList<Integer>();
Random randomGenerator = new Random();
while (numbers.size() < 4) {
int random = randomGenerator .nextInt(4);
if (!numbers.contains(random)) {
numbers.add(random);
}
}
This would be a lot simpler in java-8:
Stream.generate(new Random()::ints)
.flatMap(IntStream::boxed)
.distinct()
.limit(16) // whatever limit you might need
.toArray(Integer[]::new);
There is another way of doing "random" ordered numbers with LFSR, take a look at:
http://en.wikipedia.org/wiki/Linear_feedback_shift_register
with this technique you can achieve the ordered random number by index and making sure the values are not duplicated.
But these are not TRUE random numbers because the random generation is deterministic.
But depending your case you can use this technique reducing the amount of processing on random number generation when using shuffling.
Here a LFSR algorithm in java, (I took it somewhere I don't remeber):
public final class LFSR {
private static final int M = 15;
// hard-coded for 15-bits
private static final int[] TAPS = {14, 15};
private final boolean[] bits = new boolean[M + 1];
public LFSR() {
this((int)System.currentTimeMillis());
}
public LFSR(int seed) {
for(int i = 0; i < M; i++) {
bits[i] = (((1 << i) & seed) >>> i) == 1;
}
}
/* generate a random int uniformly on the interval [-2^31 + 1, 2^31 - 1] */
public short nextShort() {
//printBits();
// calculate the integer value from the registers
short next = 0;
for(int i = 0; i < M; i++) {
next |= (bits[i] ? 1 : 0) << i;
}
// allow for zero without allowing for -2^31
if (next < 0) next++;
// calculate the last register from all the preceding
bits[M] = false;
for(int i = 0; i < TAPS.length; i++) {
bits[M] ^= bits[M - TAPS[i]];
}
// shift all the registers
for(int i = 0; i < M; i++) {
bits[i] = bits[i + 1];
}
return next;
}
/** returns random double uniformly over [0, 1) */
public double nextDouble() {
return ((nextShort() / (Integer.MAX_VALUE + 1.0)) + 1.0) / 2.0;
}
/** returns random boolean */
public boolean nextBoolean() {
return nextShort() >= 0;
}
public void printBits() {
System.out.print(bits[M] ? 1 : 0);
System.out.print(" -> ");
for(int i = M - 1; i >= 0; i--) {
System.out.print(bits[i] ? 1 : 0);
}
System.out.println();
}
public static void main(String[] args) {
LFSR rng = new LFSR();
Vector<Short> vec = new Vector<Short>();
for(int i = 0; i <= 32766; i++) {
short next = rng.nextShort();
// just testing/asserting to make
// sure the number doesn't repeat on a given list
if (vec.contains(next))
throw new RuntimeException("Index repeat: " + i);
vec.add(next);
System.out.println(next);
}
}
}
Another approach which allows you to specify how many numbers you want with size and the min and max values of the returned numbers
public static int getRandomInt(int min, int max) {
Random random = new Random();
return random.nextInt((max - min) + 1) + min;
}
public static ArrayList<Integer> getRandomNonRepeatingIntegers(int size, int min,
int max) {
ArrayList<Integer> numbers = new ArrayList<Integer>();
while (numbers.size() < size) {
int random = getRandomInt(min, max);
if (!numbers.contains(random)) {
numbers.add(random);
}
}
return numbers;
}
To use it returning 7 numbers between 0 and 25.
ArrayList<Integer> list = getRandomNonRepeatingIntegers(7, 0, 25);
for (int i = 0; i < list.size(); i++) {
System.out.println("" + list.get(i));
}
The most efficient, basic way to have non-repeating random numbers is explained by this pseudo-code. There is no need to have nested loops or hashed lookups:
// get 5 unique random numbers, possible values 0 - 19
// (assume desired number of selections < number of choices)
const int POOL_SIZE = 20;
const int VAL_COUNT = 5;
declare Array mapping[POOL_SIZE];
declare Array results[VAL_COUNT];
declare i int;
declare r int;
declare max_rand int;
// create mapping array
for (i=0; i<POOL_SIZE; i++) {
mapping[i] = i;
}
max_rand = POOL_SIZE-1; // start loop searching for maximum value (19)
for (i=0; i<VAL_COUNT; i++) {
r = Random(0, max_rand); // get random number
results[i] = mapping[r]; // grab number from map array
mapping[r] = max_rand; // place item past range at selected location
max_rand = max_rand - 1; // reduce random scope by 1
}
Suppose first iteration generated random number 3 to start (from 0 - 19). This would make results[0] = mapping[3], i.e., the value 3. We'd then assign mapping[3] to 19.
In the next iteration, the random number was 5 (from 0 - 18). This would make results[1] = mapping[5], i.e., the value 5. We'd then assign mapping[5] to 18.
Now suppose the next iteration chose 3 again (from 0 - 17). results[2] would be assigned the value of mapping[3], but now, this value is not 3, but 19.
This same protection persists for all numbers, even if you got the same number 5 times in a row. E.g., if the random number generator gave you 0 five times in a row, the results would be: [ 0, 19, 18, 17, 16 ].
You would never get the same number twice.
Generating all the indices of a sequence is generally a bad idea, as it might take a lot of time, especially if the ratio of the numbers to be chosen to MAX is low (the complexity becomes dominated by O(MAX)). This gets worse if the ratio of the numbers to be chosen to MAX approaches one, as then removing the chosen indices from the sequence of all also becomes expensive (we approach O(MAX^2/2)). But for small numbers, this generally works well and is not particularly error-prone.
Filtering the generated indices by using a collection is also a bad idea, as some time is spent in inserting the indices into the sequence, and progress is not guaranteed as the same random number can be drawn several times (but for large enough MAX it is unlikely). This could be close to complexity O(k n log^2(n)/2), ignoring the duplicates and assuming the collection uses a tree for efficient lookup (but with a significant constant cost k of allocating the tree nodes and possibly having to rebalance).
Another option is to generate the random values uniquely from the beginning, guaranteeing progress is being made. That means in the first round, a random index in [0, MAX] is generated:
items i0 i1 i2 i3 i4 i5 i6 (total 7 items)
idx 0 ^^ (index 2)
In the second round, only [0, MAX - 1] is generated (as one item was already selected):
items i0 i1 i3 i4 i5 i6 (total 6 items)
idx 1 ^^ (index 2 out of these 6, but 3 out of the original 7)
The values of the indices then need to be adjusted: if the second index falls in the second half of the sequence (after the first index), it needs to be incremented to account for the gap. We can implement this as a loop, allowing us to select arbitrary number of unique items.
For short sequences, this is quite fast O(n^2/2) algorithm:
void RandomUniqueSequence(std::vector<int> &rand_num,
const size_t n_select_num, const size_t n_item_num)
{
assert(n_select_num <= n_item_num);
rand_num.clear(); // !!
// b1: 3187.000 msec (the fastest)
// b2: 3734.000 msec
for(size_t i = 0; i < n_select_num; ++ i) {
int n = n_Rand(n_item_num - i - 1);
// get a random number
size_t n_where = i;
for(size_t j = 0; j < i; ++ j) {
if(n + j < rand_num[j]) {
n_where = j;
break;
}
}
// see where it should be inserted
rand_num.insert(rand_num.begin() + n_where, 1, n + n_where);
// insert it in the list, maintain a sorted sequence
}
// tier 1 - use comparison with offset instead of increment
}
Where n_select_num is your 5 and n_number_num is your MAX. The n_Rand(x) returns random integers in [0, x] (inclusive). This can be made a bit faster if selecting a lot of items (e.g. not 5 but 500) by using binary search to find the insertion point. To do that, we need to make sure that we meet the requirements.
We will do binary search with the comparison n + j < rand_num[j] which is the same as n < rand_num[j] - j. We need to show that rand_num[j] - j is still a sorted sequence for a sorted sequence rand_num[j]. This is fortunately easily shown, as the lowest distance between two elements of the original rand_num is one (the generated numbers are unique, so there is always difference of at least 1). At the same time, if we subtract the indices j from all the elements rand_num[j], the differences in index are exactly 1. So in the "worst" case, we get a constant sequence - but never decreasing. The binary search can therefore be used, yielding O(n log(n)) algorithm:
struct TNeedle { // in the comparison operator we need to make clear which argument is the needle and which is already in the list; we do that using the type system.
int n;
TNeedle(int _n)
:n(_n)
{}
};
class CCompareWithOffset { // custom comparison "n < rand_num[j] - j"
protected:
std::vector<int>::iterator m_p_begin_it;
public:
CCompareWithOffset(std::vector<int>::iterator p_begin_it)
:m_p_begin_it(p_begin_it)
{}
bool operator ()(const int &r_value, TNeedle n) const
{
size_t n_index = &r_value - &*m_p_begin_it;
// calculate index in the array
return r_value < n.n + n_index; // or r_value - n_index < n.n
}
bool operator ()(TNeedle n, const int &r_value) const
{
size_t n_index = &r_value - &*m_p_begin_it;
// calculate index in the array
return n.n + n_index < r_value; // or n.n < r_value - n_index
}
};
And finally:
void RandomUniqueSequence(std::vector<int> &rand_num,
const size_t n_select_num, const size_t n_item_num)
{
assert(n_select_num <= n_item_num);
rand_num.clear(); // !!
// b1: 3578.000 msec
// b2: 1703.000 msec (the fastest)
for(size_t i = 0; i < n_select_num; ++ i) {
int n = n_Rand(n_item_num - i - 1);
// get a random number
std::vector<int>::iterator p_where_it = std::upper_bound(rand_num.begin(), rand_num.end(),
TNeedle(n), CCompareWithOffset(rand_num.begin()));
// see where it should be inserted
rand_num.insert(p_where_it, 1, n + p_where_it - rand_num.begin());
// insert it in the list, maintain a sorted sequence
}
// tier 4 - use binary search
}
I have tested this on three benchmarks. First, 3 numbers were chosen out of 7 items, and a histogram of the items chosen was accumulated over 10,000 runs:
4265 4229 4351 4267 4267 4364 4257
This shows that each of the 7 items was chosen approximately the same number of times, and there is no apparent bias caused by the algorithm. All the sequences were also checked for correctness (uniqueness of contents).
The second benchmark involved choosing 7 numbers out of 5000 items. The time of several versions of the algorithm was accumulated over 10,000,000 runs. The results are denoted in comments in the code as b1. The simple version of the algorithm is slightly faster.
The third benchmark involved choosing 700 numbers out of 5000 items. The time of several versions of the algorithm was again accumulated, this time over 10,000 runs. The results are denoted in comments in the code as b2. The binary search version of the algorithm is now more than two times faster than the simple one.
The second method starts being faster for choosing more than cca 75 items on my machine (note that the complexity of either algorithm does not depend on the number of items, MAX).
It is worth mentioning that the above algorithms generate the random numbers in ascending order. But it would be simple to add another array to which the numbers would be saved in the order in which they were generated, and returning that instead (at negligible additional cost O(n)). It is not necessary to shuffle the output: that would be much slower.
Note that the sources are in C++, I don't have Java on my machine, but the concept should be clear.
EDIT:
For amusement, I have also implemented the approach that generates a list with all the indices 0 .. MAX, chooses them randomly and removes them from the list to guarantee uniqueness. Since I've chosen quite high MAX (5000), the performance is catastrophic:
// b1: 519515.000 msec
// b2: 20312.000 msec
std::vector<int> all_numbers(n_item_num);
std::iota(all_numbers.begin(), all_numbers.end(), 0);
// generate all the numbers
for(size_t i = 0; i < n_number_num; ++ i) {
assert(all_numbers.size() == n_item_num - i);
int n = n_Rand(n_item_num - i - 1);
// get a random number
rand_num.push_back(all_numbers[n]); // put it in the output list
all_numbers.erase(all_numbers.begin() + n); // erase it from the input
}
// generate random numbers
I have also implemented the approach with a set (a C++ collection), which actually comes second on benchmark b2, being only about 50% slower than the approach with the binary search. That is understandable, as the set uses a binary tree, where the insertion cost is similar to binary search. The only difference is the chance of getting duplicate items, which slows down the progress.
// b1: 20250.000 msec
// b2: 2296.000 msec
std::set<int> numbers;
while(numbers.size() < n_number_num)
numbers.insert(n_Rand(n_item_num - 1)); // might have duplicates here
// generate unique random numbers
rand_num.resize(numbers.size());
std::copy(numbers.begin(), numbers.end(), rand_num.begin());
// copy the numbers from a set to a vector
Full source code is here.
Your problem seems to reduce to choose k elements at random from a collection of n elements. The Collections.shuffle answer is thus correct, but as pointed out inefficient: its O(n).
Wikipedia: Fisher–Yates shuffle has a O(k) version when the array already exists. In your case, there is no array of elements and creating the array of elements could be very expensive, say if max were 10000000 instead of 20.
The shuffle algorithm involves initializing an array of size n where every element is equal to its index, picking k random numbers each number in a range with the max one less than the previous range, then swapping elements towards the end of the array.
You can do the same operation in O(k) time with a hashmap although I admit its kind of a pain. Note that this is only worthwhile if k is much less than n. (ie k ~ lg(n) or so), otherwise you should use the shuffle directly.
You will use your hashmap as an efficient representation of the backing array in the shuffle algorithm. Any element of the array that is equal to its index need not appear in the map. This allows you to represent an array of size n in constant time, there is no time spent initializing it.
Pick k random numbers: the first is in the range 0 to n-1, the second 0 to n-2, the third 0 to n-3 and so on, thru n-k.
Treat your random numbers as a set of swaps. The first random index swaps to the final position. The second random index swaps to the second to last position. However, instead of working against a backing array, work against your hashmap. Your hashmap will store every item that is out of position.
int getValue(i)
{
if (map.contains(i))
return map[i];
return i;
}
void setValue(i, val)
{
if (i == val)
map.remove(i);
else
map[i] = val;
}
int[] chooseK(int n, int k)
{
for (int i = 0; i < k; i++)
{
int randomIndex = nextRandom(0, n - i); //(n - i is exclusive)
int desiredIndex = n-i-1;
int valAtRandom = getValue(randomIndex);
int valAtDesired = getValue(desiredIndex);
setValue(desiredIndex, valAtRandom);
setValue(randomIndex, valAtDesired);
}
int[] output = new int[k];
for (int i = 0; i < k; i++)
{
output[i] = (getValue(n-i-1));
}
return output;
}
You could use one of the classes implementing the Set interface (API), and then each number you generate, use Set.add() to insert it.
If the return value is false, you know the number has already been generated before.
Instead of doing all this create a LinkedHashSet object and random numbers to it by Math.random() function .... if any duplicated entry occurs the LinkedHashSet object won't add that number to its List ... Since in this Collection Class no duplicate values are allowed .. in the end u get a list of random numbers having no duplicated values .... :D
With Java 8 upwards you can use the ints method from the IntStream interface:
Returns an effectively unlimited stream of pseudorandom int values.
Random r = new Random();
int randomNumberOrigin = 0;
int randomNumberBound = 10;
int size = 5;
int[] unique = r.ints(randomNumberOrigin, randomNumberBound)
.distinct()
.limit(size)
.toArray();
Following code create a sequence random number between [1,m] that was not generated before.
public class NewClass {
public List<Integer> keys = new ArrayList<Integer>();
public int rand(int m) {
int n = (int) (Math.random() * m + 1);
if (!keys.contains(n)) {
keys.add(n);
return n;
} else {
return rand(m);
}
}
public static void main(String[] args) {
int m = 4;
NewClass ne = new NewClass();
for (int i = 0; i < 4; i++) {
System.out.println(ne.rand(m));
}
System.out.println("list: " + ne.keys);
}
}
The most easy way is use nano DateTime as long format.
System.nanoTime();
There is algorithm of card batch: you create ordered array of numbers (the "card batch") and in every iteration you select a number at random position from it (removing the selected number from the "card batch" of course).
Here is an efficient solution for fast creation of a randomized array. After randomization you can simply pick the n-th element e of the array, increment n and return e. This solution has O(1) for getting a random number and O(n) for initialization, but as a tradeoff requires a good amount of memory if n gets large enough.
There is a more efficient and less cumbersome solution for integers than a Collections.shuffle.
The problem is the same as successively picking items from only the un-picked items in a set and setting them in order somewhere else. This is exactly like randomly dealing cards or drawing winning raffle tickets from a hat or bin.
This algorithm works for loading any array and achieving a random order at the end of the load. It also works for adding into a List collection (or any other indexed collection) and achieving a random sequence in the collection at the end of the adds.
It can be done with a single array, created once, or a numerically ordered collectio, such as a List, in place. For an array, the initial array size needs to be the exact size to contain all the intended values. If you don't know how many values might occur in advance, using a numerically orderred collection, such as an ArrayList or List, where the size is not immutable, will also work. It will work universally for an array of any size up to Integer.MAX_VALUE which is just over 2,000,000,000. List objects will have the same index limits. Your machine may run out of memory before you get to an array of that size. It may be more efficient to load an array typed to the object types and convert it to some collection, after loading the array. This is especially true if the target collection is not numerically indexed.
This algorithm, exactly as written, will create a very even distribution where there are no duplicates. One aspect that is VERY IMPORTANT is that it has to be possible for the insertion of the next item to occur up to the current size + 1. Thus, for the second item, it could be possible to store it in location 0 or location 1. For the 20th item, it could be possible to store it in any location, 0 through 19. It is just as possible the first item to stay in location 0 as it is for it to end up in any other location. It is just as possible for the next new item to go anywhere, including the next new location.
The randomness of the sequence will be as random as the randomness of the random number generator.
This algorithm can also be used to load reference types into random locations in an array. Since this works with an array, it can also work with collections. That means you don't have to create the collection and then shuffle it or have it ordered on whatever orders the objects being inserted. The collection need only have the ability to insert an item anywhere in the collection or append it.
// RandomSequence.java
import java.util.Random;
public class RandomSequence {
public static void main(String[] args) {
// create an array of the size and type for which
// you want a random sequence
int[] randomSequence = new int[20];
Random randomNumbers = new Random();
for (int i = 0; i < randomSequence.length; i++ ) {
if (i == 0) { // seed first entry in array with item 0
randomSequence[i] = 0;
} else { // for all other items...
// choose a random pointer to the segment of the
// array already containing items
int pointer = randomNumbers.nextInt(i + 1);
randomSequence[i] = randomSequence[pointer];
randomSequence[pointer] = i;
// note that if pointer & i are equal
// the new value will just go into location i and possibly stay there
// this is VERY IMPORTANT to ensure the sequence is really random
// and not biased
} // end if...else
} // end for
for (int number: randomSequence) {
System.out.printf("%2d ", number);
} // end for
} // end main
} // end class RandomSequence
It really all depends on exactly WHAT you need the random generation for, but here's my take.
First, create a standalone method for generating the random number.
Be sure to allow for limits.
public static int newRandom(int limit){
return generatedRandom.nextInt(limit); }
Next, you will want to create a very simple decision structure that compares values. This can be done in one of two ways. If you have a very limited amount of numbers to verify, a simple IF statement will suffice:
public static int testDuplicates(int int1, int int2, int int3, int int4, int int5){
boolean loopFlag = true;
while(loopFlag == true){
if(int1 == int2 || int1 == int3 || int1 == int4 || int1 == int5 || int1 == 0){
int1 = newRandom(75);
loopFlag = true; }
else{
loopFlag = false; }}
return int1; }
The above compares int1 to int2 through int5, as well as making sure that there are no zeroes in the randoms.
With these two methods in place, we can do the following:
num1 = newRandom(limit1);
num2 = newRandom(limit1);
num3 = newRandom(limit1);
num4 = newRandom(limit1);
num5 = newRandom(limit1);
Followed By:
num1 = testDuplicates(num1, num2, num3, num4, num5);
num2 = testDuplicates(num2, num1, num3, num4, num5);
num3 = testDuplicates(num3, num1, num2, num4, num5);
num4 = testDuplicates(num4, num1, num2, num3, num5);
num5 = testDuplicates(num5, num1, num2, num3, num5);
If you have a longer list to verify, then a more complex method will yield better results both in clarity of code and in processing resources.
Hope this helps. This site has helped me so much, I felt obliged to at least TRY to help as well.
I created a snippet that generates no duplicate random integer. the advantage of this snippet is that you can assign the list of an array to it and generate the random item, too.
No duplication random generator class
With Java 8 using the below code, you can create 10 distinct random Integer Numbers within a range of 1000.
Random random = new Random();
Integer[] input9 = IntStream.range(1, 10).map(i -> random.nextInt(1000)).boxed().distinct()
.toArray(Integer[]::new);
System.out.println(Arrays.toString(input9));
Modify the range to generate more numbers example : range(1,X). It will generate X distinct random numbers.
Modify the nextInt value to select the random number range : random.nextInt(Y)::random number will be generated within the range Y

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