Given series of integers having relation where a number is equal to sum of previous 2 numbers and starting integer is 1
Series ->1,2,3,5,8,13,21,34,55
find the number of ways such that sum of k elements equal to p.We can use an element any number of times.
p=8
k=4.
So,number of ways would be 4.Those are,
1,1,1,5
1,1,3,3
1,2,2,3
2,2,2,2
I am able to sove this question through recursion.I sense dynamic programming here but i am not getting how to do it.Can it be done in much lesser time???
EDIT I forgot to mention that the sequence of the numbers does not matter and will be counted once. for ex=3->(1,2)and(2,1).here number of ways would be 1 only.
EDIT: Poster has changed the original problem since this was posted. My algorithm still works, but maybe can be improved upon. Original problem had n arbitrary input numbers (he has now modified it to be a Fibonacci series). To apply my algorithm to the modified post, truncate the series by taking only elements less than p (assume there are n of them).
Here's an n^(k/2) algorithm. (n is the number of elements in the series)
Use a table of length p, such that table[i] contains all combinations of k/2 elements that sum to i. For example, in the example data that you provided, table[4] contains {1,3} and {2,2}.
EDIT: If the space is prohibitive, this same algorithm can be done with an ordered linked lists, where you only store the non-empty table entries. The linked list has to be both directions: forward and backwards, which makes the final step of the algorithm cleaner.
Once this table is computed, then we get all solutions by combining every table[j] with every table[p-j], whenever both are non-empty.
To get the table, initialize the entire thing to empty. Then:
For i_1 = 0 to n-1:
For i_2 = i_1 to n-1:
...
For i_k/2 = i_k/2-1 to n-1:
sum = series[i_1] + ... + series[i_k/2]
if sum <= p:
store {i_1, i_2, ... , i_k/2 } in table[sum]
This "variable number of loops" looks impossible to implement, but actually it can be done with an array of length k/2 that keeps track of where each i_` is.
Let's go back to your data and see how our table would look:
table[2] = {1,1}
table[3] = {1,2}
table[4] = {1,3} and {2,2}
table[5] = {2,3}
table[6] = {1,5}
table[7] = {2,5}
table[8] = {3,5}
Solutions are found by combining table[2] with table[6], table[3] with table[5], and table[4] with table[4]. Thus, solutions are: {1,1,1,5} {1,2,2,3}, {1,1,3,3}, {2,2,2,2}, {1,3,2,2}.
You can use dynamic programming. Let C(p, k) be the number of ways that sum k element equal to p and a be the array of elements. Then
C(p, k) = C(p - a[0], k - 1) + C(p - a[1], k - 1) + .... + C(p - a[n-1], k - 1)
Then, you can use memorization to speed up your code.
Hint:
Your problem is well-known. It is the sum set problem, a variation of knapsack problem. Check this pretty good explanation. sum-set problem
Related
I am trying to find the maximum sum of two elements in an array minus the distance between them.
Specifically I am trying to calculate max{ a[i]+a[j]-|i-j| }
I am currently stuck. I have obviously considered the naive approach (O(n^2)). However ,I am pretty sure there is a better ,more efficient approach (O(nlogn)) or even O(n).
Can someone please help me on how to approach the problem. I would be grateful if anyone threw some hints or a simple idea to have something to start from. Sorting the array first? Maybe using a dynamic programming approach?
Edit:
I think I have found an O(n) solution
Let's assume that our max sum comes from a[i] and a[j] , a[i] contributes to that sum with : a[i]+i . a[j] contributes to that sum with a[j]-j. (Because our sum is a[i]+a[j]-|j-i|= a[i]+a[j]+i-j. )
Approach: for convenience we compute the matrices A_plus_index=a[i]+i and A_minus_index=a[i]-i.
Then we use two helping arrays:
i) The first one has for every i ,the max value of A_plus_index array considering only the elements from 0 to i.
ii) The second has for every i, the max value of A_minus_index array considering only the elements from N to i ,where N is the length of array a.
Now we traverse the arrays once and find the max: A_plus_index[i]+ A_minus_index[i+1].
Total complexity O(n).
#JeffersonWhite your idea works and you could post it as an answer and accept it.
But I am going to improve upon your idea a little bit:
You could build only one array instead of 2, which contains the maximum of A[j] - j so far for each j from N-1 to 1.
And then traverse the array forward each time computing the max( A[i] + i + max_so_far-_reverse[i+1])
//Building the reverse array
max_so_far_reverse = array of length N
max_reverse = A[N-1]-(N-1)
max_so_far_reverse[N-1] = max_reverse
for j = N-2 to 1:
max_reverse = max(max_reverse, A[j]-j)
max_so_far_reverse[j] = max_reverse
//Computing maximum value by traversing forward
max = 0
for i = 0 to N-2:
max = max(max, A[i] + i + max_so_far_reverse[i+1])
return max
A friend of mine recently got this interview question, which seems to us to be solvable but not within the asymptotic time bounds that the interviewer thought should be possible. Here is the problem:
You have an array of N integers, xs, sorted but possibly non-distinct. Your goal is to find four array indices(1) (a,b,c,d) such that the following two properties hold:
xs[a] + xs[b] + xs[c] = xs[d]
a < b < c < d
The goal is to do this in O(N2) time.
First, an O(N3log(N)) solution is obvious: for each (a,b,c) ordered triple, use binary search to see if an appropriate d can be found. Now, how to do better?
One interesting suggestion from the interviewer is to rewrite the first condition as:
xs[a] + xs[b] = xs[d] - xs[c]
It's not clear what to do after this, but perhaps we could chose some pivot value P, and search for an (a,b) pair adding up to P, and a (d,c) pair subtracting to it. That search is easy enough to do in O(n) time for a given P, by searching inwards from both ends of the array. However, it seems to me that the problem with this is that there are N2 such values P, not just N of them, so we haven't actually reduced the problem size at all: we're doing O(N) work, O(N2) times.
We found some related problems being discussed online elsewhere: Find 3 numbers in an array adding to a given sum is solvable in N2 time, but requires that the sum be fixed ahead of time; adapting the same algorithm but iterating through each possible sum leaves us at N3 as always.
Another related problem seems to be Find all triplets in array with sum less than or equal to given sum, but I'm not sure how much of the stuff there is relevant here: an inequality rather than an equality mixes things up quite a bit, and of course the target is fixed rather than varying.
So, what are we missing? Is the problem impossible after all, given the performance requirements? Or is there a clever algorithm we're unable to spot?
(1) Actually the problem as posed is to find all such (a,b,c,d) tuples, and return a count of how many there are. But I think even finding a single one of them in the required time constraints is hard enough.
If the algorithm would have to list the solutions (i.e. the sets of a, b, c, and d that satisfy the condition), the worst case time complexity is O(n4):
1. There can be O(n4) solutions
The trivial example is an array with only 0 values in it. Then a, b, c and d have all the freedom as long as they stay in order. This represents O(n4) solutions.
But more generally arrays which follow the following pattern have O(n4) solutions:
w, w, w, ... x, x, x, ..., y, y, y, ... z, z, z, ....
With just as many occurrences of each, and:
w + x + y = z
However, to only produce the number of solutions, an algorithm can have a better time complexity.
2. Algorithm
This is a slight variation of the already posted algorithm, which does not involve the H factor. It also describes how to handle cases where different configurations lead to the same sums.
Retrieve all pairs and store them in an array X, where each element gets the following information:
a: the smallest index of the two
b: the other index
sum: the value of xs[a] + xs[b]
At the same time also store for each such pair in another array Y, the following:
c: the smallest index of the two
d: the other index
sum: the value of xs[d] - xs[c]
The above operation has a time complexity of O(n²)
Sort both arrays by their element's sum attribute. In case of equal sum values, the sort order will be determined as follows: for the X array by increasing b; for the Y array by decreasing c. Sorting can be done in O(n²) O(n²logn) time.
[Edit: I could not prove the earlier claim of O(n²) (unless some assumptions are made that allow for a radix/bucket sorting algorithm, which I will not assume). As noted in comments, in general an array with n² elements can be sorted in O(n²logn²), which is O(n²logn), but not O(n²)]
Go through both arrays in "tandem" to find pairs of sums that are equal. If that is the case, it needs to be checked that X[i].b < Y[j].c. If so it represents a solution. But there could be many of them, and counting those in an acceptable time needs special care.
Let m = n(n-1)/2, i.e. the number of elements in array X (which is also the size of array Y):
i = 0
j = 0
while i < m and j < m:
if X[i].sum < Y[j].sum:
i = i + 1
elif X[i].sum > Y[j].sum:
j = j + 1
else:
# We have a solution. Need to count all others that have same sums in X and Y.
# Find last match in Y and set k as index to it:
countY = 0
while k < m and X[i].sum == Y[j].sum and X[i].b < Y[j].c:
countY = countY + 1
j = j + 1
k = j - 1
# add chunks to `count`:
while i < m and countY >= 0 and X[i].sum == Y[k].sum:
while countY >= 0 and X[i].b >= Y[k].c:
countY = countY - 1
k = k - 1
count = count + countY
i = i + 1
Note that although there are nested loops, the variable i only ever increments, and so does j. The variable k always decrements in the innermost loop. Although it also gets higher values to start from, it can never address the same Y element more than a constant number of times via the k index, because while decrementing this index, it stays within the "same sum" range of Y.
So this means that this last part of the algorithm runs in O(m), which is O(n²). As my latest edit confirmed that the sorting step is not O(n²), that step determines the overall time-complexity: O(n²logn).
So one solution can be :
List all x[a] + x[b] value possible such that a < b and hash them in this fashion
key = (x[a]+x[b]) and value = (a,b).
Complexity of this step - O(n^2)
Now List all x[d] - x[c] values possible such that d > c. Also for each x[d] - x[c] search the entry in your hash map by querying. We have a solution if there exists an entry such that c > b for any hit.
Complexity of this step - O(n^2) * H.
Where H is the search time in your hashmap.
Total complexity - O(n^2)* H. Now H may be O(1). This could done if the range of values in the array is small. Also the choice of hash function would depend on the properties of elements in the array.
I am looking for an algorithm to solve the following problem: We are given an integer array of size n which contains k (0 < k < n) many elements exactly once. Every other integer occurs an even number of times in the array. The output should be any of the k unique numbers. k is a fixed number and not part of the input.
An example would be the input [1, 2, 2, 4, 4, 2, 2, 3] with both 1 and 3 being a correct output.
Most importantly, the algorithm should run in O(n) time and require only O(1) additional space.
edit: There has been some confusion regarding whether there is only one unique integer or multiple. I apologize for this. The correct problem is that there is an arbitrary but fixed amount. I have updated the original question above.
"Dante." gave a good answer for the case that there are at most two such numbers. This link also provides a solution for three. "David Eisenstat" commented that it is also possible to do for any fixed k. I would be grateful for a solution.
There is a standard algorithm to solve such problems using XOR operator:
Time Complexity = O(n)
Space Complexity = O(1)
Suppose your input array contains only one element that occurs odd no of times and rest occur even number of times,we take advantage of the following fact:
Any expression having even number of 0's and 1's in any order will always be = 0 when xor is applied.
That is
0^1^....... = 0 as long as number of 0 is even and number of 1 is even
and 0 and 1 can occur in any order.
Because all numbers that occur even number of times will have their corresponding bits form even number of 1's and 0's and only the number which occurs only once will have its bit left out when we take xor of all elements of array because
0(from no's occuring even times)^1(from no occuring once) = 1
0(from no's occuring even times)^0(from no occuring once) = 0
as you can see the bit of only the number occuring once is preserved.
This means when given such an array and you take xor of all the elements,the result is the number which occurs only once.
So the algorithm for array of length n is:
result = array[0]^array[1]^.....array[n-1]
Different Scenario
As the OP mentioned that input can also be an array which has two numbers occuring only once and rest occur even number of times.
This is solved using the same logic as above but with little difference.
Idea of algorithm:
If you take xor of all the elements then definitely all the bits of elements occuring even number of times will result in 0,which means:
The result will have its bit 1 only at that bit position where the bits of the two numbers occuring only once differ.
We will use the above idea.
Now we focus on the resultant xor bit which is 1(any bit which is 1) and make rest 0.The result is a number which will allow us to differentiate between the two numbers(the required ones).
Because the bit is 1,it means they differ at this position,it means one will have 0 at this position and one will have 1.This means one number when taken AND results in 0 and one does not.
Since it is very easy to set the right most bit,we set it of the result xor as
A = result & ~(result-1)
Now traverse through the array once and if array[i]&A is 0 store the number in variable number_1 as
number_1 = number_1^array[i]
otherwise
number_2 = number_2^array[i]
Because the remaining numbers occur even number of times,their bit will automatically disappear.
So the algorithm is
1.Take xor of all elements,call it xor.
2.Set the rightmost bit of xor and store it in B.
3.Do the following:
number_1=0,number_2=0;
for(i = 0 to n-1)
{
if(array[i] & B)
number_1 = number_1^array[i];
else
number_2 = number_2^array[i];
}
The number_1 and number_2 are the required numbers.
Here's a Las Vegas algorithm that, given k, the exact number of elements that occur an odd number of times, reports all of them in expected time O(n k) (read: linear-time when k is O(1)) and space O(1) words, assuming that "give me a uniform random word" and "give me the number of 1 bits set in this word (popcount)" are constant-time operations. I'm pretty sure that I'm not the first person to come up with this algorithm (and I'm not even sure that I'm remembering all of the refinements), but I've reached the limits of my patience trying to find it.
The central technique is called random restrictions. Essentially what we do is to filter the input randomly by value, in the hope that we retain exactly one odd-count element. We apply the classic XOR algorithm to the filtered array and check the result; if it succeeded, then we pretend to add it to the array, to make it even-count. Repeat until all k elements are found.
The filtration process goes like this. Treat each input word x as a binary vector of length w (doesn't matter what w is). Compute a random binary matrix A of size w by ceil(1 + lg k) and a random binary vector b of length ceil(1 + lg k). We filter the input by retaining those x such that Ax = b, where the left-hand side is a matrix multiplication mod 2. In implementation, A is represented as ceil(1 + lg k) vectors a1, a2, .... We compute the bits of Ax as popcount(a1 ^ x), popcount(a2 ^ x), .... (This is convenient because we can short-circuit the comparison with b, which shaves a factor lg k from the running time.)
The analysis is to show that, in a given pass, we manage with constant probability to single out one of the odd-count elements. First note that, for some fixed x, the probability that Ax = b is 2-ceil(1 + lg k) = Θ(1/k). Given that Ax = b, for all y ≠ x, the probability that Ay = b is less than 2-ceil(1 + lg k). Thus, the expected number of elements that accompany x is less than 1/2, so with probability more than 1/2, x is unique in the filtered input. Sum over all k odd-count elements (these events are disjoint), and the probability is Θ(1).
Here's a deterministic linear-time algorithm for k = 3. Let the odd-count elements be a, b, c. Accumulate the XOR of the array, which is s = a ^ b ^ c. For each bit i, observe that, if a[i] == b[i] == c[i], then s[i] == a[i] == b[i] == c[i]. Make another pass through the array, accumulate the XOR of the lowest bit set in s ^ x. The even-count elements contribute nothing again. Two of the odd-count elements contribute the same bit and cancel each other out. Thus, the lowest bit set in the XOR is where exactly one of the odd-count elements differs from s. We can use the restriction method above to find it, then the k = 2 method to find the others.
The question title says "the unique integer", but the question body says there can be more than one unique element.
If there is in fact only one non-duplicate: XOR all the elements together. The duplicates all cancel, because they come in pairs (or higher multiples of 2), so the result is the unique integer.
See Dante's answer for an extension of this idea that can handle two unique elements. It can't be generalized to more than that.
Perhaps for k unique elements, we could use k accumulators to track sum(a[i]**k). i.e. a[i], a[i]2, etc. This probably only works for Faster algorithm to find unique element between two arrays?, not this case where the duplicates are all in one array. IDK if an xor of squares, cubes, etc. would be any use for resolving things.
Track the counts for each element and only return the elements with a count of 1. This can be done with a hash map. The below example tracks the result using a hash set while it's still building the counts map. Still O(n) but less efficient, but I think it's slightly more instructive.
Javascript with jsfiddle http://jsfiddle.net/nmckchsa/
function findUnique(arr) {
var uniq = new Map();
var result = new Set();
// iterate through array
for(var i=0; i<arr.length; i++) {
var v = arr[i];
// add value to map that contains counts
if(uniq.has(v)) {
uniq.set(v, uniq.get(v) + 1);
// count is greater than 1 remove from set
result.delete(v);
} else {
uniq.set(v, 1);
// add a possibly uniq value to the set
result.add(v);
}
}
// set to array O(n)
var a = [], x = 0;
result.forEach(function(v) { a[x++] = v; });
return a;
}
alert(findUnique([1,2,3,0,1,2,3,1,2,3,5,4,4]));
EDIT Since the non-uniq numbers appear an even number of times #PeterCordes suggested a more elegant set toggle.
Here's how that would look.
function findUnique(arr) {
var result = new Set();
// iterate through array
for(var i=0; i<arr.length; i++) {
var v = arr[i];
if(result.has(v)) { // even occurances
result.delete(v);
} else { // odd occurances
result.add(v);
}
}
// set to array O(n)
var a = [], x = 0;
result.forEach(function(v) { a[x++] = v; });
return a;
}
JSFiddle http://jsfiddle.net/hepsyqyw/
Assuming you have an input array: [2,3,4,2,4]
Output: 3
In Ruby, you can do something as simple as this:
[2,3,4,2,4].inject(0) {|xor, v| xor ^ v}
Create an array counts that has INT_MAX slots, with each element initialized to zero.
For each element in the input list, increment counts[element] by one. (edit: actually, you will need to do counts[element] = (counts_element+1)%2, or else you might overflow the value for really ridiculously large values of N. It's acceptable to do this kind of modulus counting because all duplicate items appear an even number of times)
Iterate through counts until you find a slot that contains "1". Return the index of that slot.
Step 2 is O(N) time. Steps 1 and 3 take up a lot of memory and a lot of time, but neither one is proportional to the size of the input list, so they're still technically O(1).
(note: this assumes that integers have a minimum and maximum value, as is the case for many programming languages.)
The famous Fisher-Yates shuffle algorithm can be used to randomly permute an array A of length N:
For k = 1 to N
Pick a random integer j from k to N
Swap A[k] and A[j]
A common mistake that I've been told over and over again not to make is this:
For k = 1 to N
Pick a random integer j from 1 to N
Swap A[k] and A[j]
That is, instead of picking a random integer from k to N, you pick a random integer from 1 to N.
What happens if you make this mistake? I know that the resulting permutation isn't uniformly distributed, but I don't know what guarantees there are on what the resulting distribution will be. In particular, does anyone have an expression for the probability distributions over the final positions of the elements?
An Empirical Approach.
Let's implement the erroneous algorithm in Mathematica:
p = 10; (* Range *)
s = {}
For[l = 1, l <= 30000, l++, (*Iterations*)
a = Range[p];
For[k = 1, k <= p, k++,
i = RandomInteger[{1, p}];
temp = a[[k]];
a[[k]] = a[[i]];
a[[i]] = temp
];
AppendTo[s, a];
]
Now get the number of times each integer is in each position:
r = SortBy[#, #[[1]] &] & /# Tally /# Transpose[s]
Let's take three positions in the resulting arrays and plot the frequency distribution for each integer in that position:
For position 1 the freq distribution is:
For position 5 (middle)
And for position 10 (last):
and here you have the distribution for all positions plotted together:
Here you have a better statistics over 8 positions:
Some observations:
For all positions the probability of
"1" is the same (1/n).
The probability matrix is symmetrical
with respect to the big anti-diagonal
So, the probability for any number in the last
position is also uniform (1/n)
You may visualize those properties looking at the starting of all lines from the same point (first property) and the last horizontal line (third property).
The second property can be seen from the following matrix representation example, where the rows are the positions, the columns are the occupant number, and the color represents the experimental probability:
For a 100x100 matrix:
Edit
Just for fun, I calculated the exact formula for the second diagonal element (the first is 1/n). The rest can be done, but it's a lot of work.
h[n_] := (n-1)/n^2 + (n-1)^(n-2) n^(-n)
Values verified from n=3 to 6 ( {8/27, 57/256, 564/3125, 7105/46656} )
Edit
Working out a little the general explicit calculation in #wnoise answer, we can get a little more info.
Replacing 1/n by p[n], so the calculations are hold unevaluated, we get for example for the first part of the matrix with n=7 (click to see a bigger image):
Which, after comparing with results for other values of n, let us identify some known integer sequences in the matrix:
{{ 1/n, 1/n , ...},
{... .., A007318, ....},
{... .., ... ..., ..},
... ....,
{A129687, ... ... ... ... ... ... ..},
{A131084, A028326 ... ... ... ... ..},
{A028326, A131084 , A129687 ... ....}}
You may find those sequences (in some cases with different signs) in the wonderful http://oeis.org/
Solving the general problem is more difficult, but I hope this is a start
The "common mistake" you mention is shuffling by random transpositions. This problem was studied in full detail by Diaconis and Shahshahani in Generating a random permutation with random transpositions (1981). They do a complete analysis of stopping times and convergence to uniformity. If you cannot get a link to the paper, then please send me an e-mail and I can forward you a copy. It's actually a fun read (as are most of Persi Diaconis's papers).
If the array has repeated entries, then the problem is slightly different. As a shameless plug, this more general problem is addressed by myself, Diaconis and Soundararajan in Appendix B of A Rule of Thumb for Riffle Shuffling (2011).
Let's say
a = 1/N
b = 1-a
Bi(k) is the probability matrix after i swaps for the kth element. i.e the answer to the question "where is k after i swaps?". For example B0(3) = (0 0 1 0 ... 0) and B1(3) = (a 0 b 0 ... 0). What you want is BN(k) for every k.
Ki is an NxN matrix with 1s in the i-th column and i-th row, zeroes everywhere else, e.g:
Ii is the identity matrix but with the element x=y=i zeroed. E.g for i=2:
Ai is
Then,
But because BN(k=1..N) forms the identity matrix, the probability that any given element i will at the end be at position j is given by the matrix element (i,j) of the matrix:
For example, for N=4:
As a diagram for N = 500 (color levels are 100*probability):
The pattern is the same for all N>2:
The most probable ending position for k-th element is k-1.
The least probable ending position is k for k < N*ln(2), position 1 otherwise
I knew I had seen this question before...
" why does this simple shuffle algorithm produce biased results? what is a simple reason? " has a lot of good stuff in the answers, especially a link to a blog by Jeff Atwood on Coding Horror.
As you may have already guessed, based on the answer by #belisarius, the exact distribution is highly dependent on the number of elements to be shuffled. Here's Atwood's plot for a 6-element deck:
What a lovely question! I wish I had a full answer.
Fisher-Yates is nice to analyze because once it decides on the first element, it leaves it alone. The biased one can repeatedly swap an element in and out of any place.
We can analyze this the same way we would a Markov chain, by describing the actions as stochastic transition matrices acting linearly on probability distributions. Most elements get left alone, the diagonal is usually (n-1)/n. On pass k, when they don't get left alone, they get swapped with element k, (or a random element if they are element k). This is 1/(n-1) in either row or column k. The element in both row and column k is also 1/(n-1). It's easy enough to multiply these matrices together for k going from 1 to n.
We do know that the element in last place will be equally likely to have originally been anywhere because the last pass swaps the last place equally likely with any other. Similarly, the first element will be equally likely to be placed anywhere. This symmetry is because the transpose reverses the order of matrix multiplication. In fact, the matrix is symmetric in the sense that row i is the same as column (n+1 - i). Beyond that, the numbers don't show much apparent pattern. These exact solutions do show agreement with the simulations run by belisarius: In slot i, The probability of getting j decreases as j raises to i, reaching its lowest value at i-1, and then jumping up to its highest value at i, and decreasing until j reaches n.
In Mathematica I generated each step with
step[k_, n_] := Normal[SparseArray[{{k, i_} -> 1/n,
{j_, k} -> 1/n, {i_, i_} -> (n - 1)/n} , {n, n}]]
(I haven't found it documented anywhere, but the first matching rule is used.)
The final transition matrix can be calculated with:
Fold[Dot, IdentityMatrix[n], Table[step[m, n], {m, s}]]
ListDensityPlot is a useful visualization tool.
Edit (by belisarius)
Just a confirmation. The following code gives the same matrix as in #Eelvex's answer:
step[k_, n_] := Normal[SparseArray[{{k, i_} -> (1/n),
{j_, k} -> (1/n), {i_, i_} -> ((n - 1)/n)}, {n, n}]];
r[n_, s_] := Fold[Dot, IdentityMatrix[n], Table[step[m, n], {m, s}]];
Last#Table[r[4, i], {i, 1, 4}] // MatrixForm
Wikipedia's page on the Fisher-Yates shuffle has a description and example of exactly what will happen in that case.
You can compute the distribution using stochastic matrices. Let the matrix A(i,j) describe the probability of the card originally at position i ending up in position j. Then the kth swap has a matrix Ak given by Ak(i,j) = 1/N if i == k or j == k, (the card in position k can end up anywhere and any card can end up at position k with equal probability), Ak(i,i) = (N - 1)/N for all i != k (every other card will stay in the same place with probability (N-1)/N) and all other elements zero.
The result of the complete shuffle is then given by the product of the matrices AN ... A1.
I expect you're looking for an algebraic description of the probabilities; you can get one by expanding out the above matrix product, but it I imagine it will be fairly complex!
UPDATE: I just spotted wnoise's equivalent answer above! oops...
I've looked into this further, and it turns out that this distribution has been studied at length. The reason it's of interest is because this "broken" algorithm is (or was) used in the RSA chip system.
In Shuffling by semi-random transpositions, Elchanan Mossel, Yuval Peres, and Alistair Sinclair study this and a more general class of shuffles. The upshot of that paper appears to be that it takes log(n) broken shuffles to achieve near random distribution.
In The bias of three pseudorandom shuffles (Aequationes Mathematicae, 22, 1981, 268-292), Ethan Bolker and David Robbins analyze this shuffle and determine that the total variation distance to uniformity after a single pass is 1, indicating that it is not very random at all. They give asympotic analyses as well.
Finally, Laurent Saloff-Coste and Jessica Zuniga found a nice upper bound in their study of inhomogeneous Markov chains.
This question is begging for an interactive visual matrix diagram analysis of the broken shuffle mentioned. Such a tool is on the page Will It Shuffle? - Why random comparators are bad by Mike Bostock.
Bostock has put together an excellent tool that analyzes random comparators. In the dropdown on that page, choose naïve swap (random ↦ random) to see the broken algorithm and the pattern it produces.
His page is informative as it allows one to see the immediate effects a change in logic has on the shuffled data. For example:
This matrix diagram using a non-uniform and very-biased shuffle is produced using a naïve swap (we pick from "1 to N") with code like this:
function shuffle(array) {
var n = array.length, i = -1, j;
while (++i < n) {
j = Math.floor(Math.random() * n);
t = array[j];
array[j] = array[i];
array[i] = t;
}
}
But if we implement a non-biased shuffle, where we pick from "k to N" we should see a diagram like this:
where the distribution is uniform, and is produced from code such as:
function FisherYatesDurstenfeldKnuthshuffle( array ) {
var pickIndex, arrayPosition = array.length;
while( --arrayPosition ) {
pickIndex = Math.floor( Math.random() * ( arrayPosition + 1 ) );
array[ pickIndex ] = [ array[ arrayPosition ], array[ arrayPosition ] = array[ pickIndex ] ][ 0 ];
}
}
The excellent answers given so far are concentrating on the distribution, but you have asked also "What happens if you make this mistake?" - which is what I haven't seen answered yet, so I'll give an explanation on this:
The Knuth-Fisher-Yates shuffle algorithm picks 1 out of n elements, then 1 out of n-1 remaining elements and so forth.
You can implement it with two arrays a1 and a2 where you remove one element from a1 and insert it into a2, but the algorithm does it in place (which means, that it needs only one array), as is explained here (Google: "Shuffling Algorithms Fisher-Yates DataGenetics") very well.
If you don't remove the elements, they can be randomly chosen again which produces the biased randomness. This is exactly what the 2nd example your are describing does. The first example, the Knuth-Fisher-Yates algorithm, uses a cursor variable running from k to N, which remembers which elements have already been taken, hence avoiding to pick elements more than once.
I've came across some similar problems to this one in the past, and I still haven't got good idea how to solve this problem. Problem goes like this:
You are given an positive integer array with size n <= 1000 and k <= n which is the number of contiguous subarrays that you will have to split your array into. You have to output minimum m, where m = max{s[1],..., s[k]}, and s[i] is the sum of the i-th subarray. All integers in the array are between 1 and 100. Example :
Input: Output:
5 3 >> n = 5 k = 3 3
2 1 1 2 3
Splitting array into 2+1 | 1+2 | 3 will minimize the m.
My brute force idea was to make first subarray end at position i (for all possible i) and then try to split the rest of the array in k-1 subarrays in the best way possible. However, this is exponential solution and will never work.
So I'm looking for good ideas to solve it. If you have one please tell me.
Thanks for your help.
You can use dynamic programming to solve this problem, but you can actually solve with greedy and binary search on the answer. This algorithm's complexity is O(n log d), where d is the output answer. (An upper bound would be the sum of all the elements in the array.) (or O( n d ) in the size of the output bits)
The idea is to binary search on what your m would be - and then greedily move forward on the array, adding the current element to the partition unless adding the current element pushes it over the current m -- in that case you start a new partition. The current m is a success (and thus adjust your upper bound) if the numbers of partition used is less than or equal to your given input k. Otherwise, you used too many partitions, and raise your lower bound on m.
Some pseudocode:
// binary search
binary_search ( array, N, k ) {
lower = max( array ), upper = sum( array )
while lower < upper {
mid = ( lower + upper ) / 2
// if the greedy is good
if partitions( array, mid ) <= k
upper = mid
else
lower = mid
}
}
partitions( array, m ) {
count = 0
running_sum = 0
for x in array {
if running_sum + x > m
running_sum = 0
count++
running_sum += x
}
if running_sum > 0
count++
return count
}
This should be easier to come up with conceptually. Also note that because of the monotonic nature of the partitions function, you can actually skip the binary search and do a linear search, if you are sure that the output d is not too big:
for i = 0 to infinity
if partitions( array, i ) <= k
return i
Dynamic programming. Make an array
int best[k+1][n+1];
where best[i][j] is the best you can achieve splitting the first j elements of the array int i subarrays. best[1][j] is simply the sum of the first j array elements. Having row i, you calculate row i+1 as follows:
for(j = i+1; j <= n; ++j){
temp = min(best[i][i], arraysum[i+1 .. j]);
for(h = i+1; h < j; ++h){
if (min(best[i][h], arraysum[h+1 .. j]) < temp){
temp = min(best[i][h], arraysum[h+1 .. j]);
}
}
best[i+1][j] = temp;
}
best[m][n] will contain the solution. The algorithm is O(n^2*k), probably something better is possible.
Edit: a combination of the ideas of ChingPing, toto2, Coffee on Mars and rds (in the order they appear as I currently see this page).
Set A = ceiling(sum/k). This is a lower bound for the minimum. To find a good upper bound for the minimum, create a good partition by any of the mentioned methods, moving borders until you don't find any simple move that still decreases the maximum subsum. That gives you an upper bound B, not much larger than the lower bound (if it were much larger, you'd find an easy improvement by moving a border, I think).
Now proceed with ChingPing's algorithm, with the known upper bound reducing the number of possible branches. This last phase is O((B-A)*n), finding B unknown, but I guess better than O(n^2).
I have a sucky branch and bound algorithm ( please dont downvote me )
First take the sum of array and dvide by k, which gives you the best case bound for you answer i.e. the average A. Also we will keep a best solution seen so far for any branch GO ( global optimal ).Lets consider we put a divider( logical ) as a partition unit after some array element and we have to put k-1 partitions. Now we will put the partitions greedily this way,
Traverse the array elements summing them up until you see that at the next position we will exceed A, now make two branches one where you put the divider at this position and other where you put at next position, Do this recursiely and set GO = min (GO, answer for a branch ).
If at any point in any branch we have a partition greater then GO or the no of position are less then the partitions left to be put we bound. In the end you should have GO as you answer.
EDIT:
As suggested by Daniel, we could modify the divider placing strategy a little to place it until you reach sum of elements as A or the remaining positions left are less then the dividers.
This is just a sketch of an idea... I'm not sure that it works, but it's very easy (and probably fast too).
You start say by putting the separations evenly distributed (it does not actually matter how you start).
Make the sum of each subarray.
Find the subarray with the largest sum.
Look at the right and left neighbor subarrays and move the separation on the left by one if the subarray on the left has a lower sum than the one on the right (and vice-versa).
Redo for the subarray with the current largest sum.
You'll reach some situation where you'll keep bouncing the separation between the same two positions which will probably mean that you have the solution.
EDIT: see the comment by #rds. You'll have to think harder about bouncing solutions and the end condition.
My idea, which unfortunately does not work:
Split the array in N subarrays
Locate the two contiguous subarrays whose sum is the least
Merge the subarrays found in step 2 to form a new contiguous subarray
If the total number of subarrays is greater than k, iterate from step 2, else finish.
If your array has random numbers, you can hope that a partition where each subarray has n/k is a good starting point.
From there
Evaluate this candidate solution, by computing the sums
Store this candidate solution. For instance with:
an array of the indexes of every sub-arrays
the corresponding maximum of sum over sub-arrays
Reduce the size of the max sub-array: create two new candidates: one with the sub-array starting at index+1 ; one with sub-array ending at index-1
Evaluate the new candidates.
If their maximum is higher, discard
If their maximum is lower, iterate on 2, except if this candidate was already evaluated, in which case it is the solution.