Consider the example that you have given:
[0,0,1,1,0,0,1,1,0,1,1,0] , here n = 12 and k=3
Now we need to partition the array into 3 contiguous sub arrays such
that :
a) The expected sum value is maximum
b) and the size of each sub array should be between 2 and 6, both
inclusive. In case, this constraint is not satisfied then its not a
valid candidate for the solution even if the partition produces the
max value.
2 = ceil (n / 2k) = ceil (12/6)
6 = floor (3n / 2k) = floor (36/6)
---------------------------------------------
As mentioned above the following equation :
F(n,k) = MAX for all r such that ceil(n/2k) <= r <= floor(3n/2k)
{ (expected value of array elems from A[n] to A[n-r+1]) + F(n-r,
k-1) }
/**
For understanding how partitioning of the array is represented by the
above equation:
Say there is an array denoted by A[i] and it needs to be divided into
3 contiguous parts, one of the ways to do so would be to take the
following steps :
Let K(partition no.) be initialized to 3.
Let array size N be 12.
a) If N is 0, the goto step 'f'
b) If K == 1 then call it as partition K and goto step 'e'.
c) Lets take X no. of elements from the end of array A of size N and
call it partition K.
d) Decrement K by 1 and N by X { --K; and N-=X;}
d) Goto step 'a'
e) Valid partition and End.
f) Not a valid partition and End.
Now if the above set of steps is run for all values of X such that
2<=X<=6 , then it will generate all possible candidates (partitions)
as per the given problem statement. And for all the valid
partitions(the ones that will hit step 'e') we need to calculate the
expected sum value.
**/
can be translated into,
// I am using 1-based array indexing for better clarity
// A[x .. y] means all elements from A[y] to A[x]..
F(12, 3) = MAX
{
ExpVal (A[12 .. 11]) + F(10, 2) ,
ExpVal (A[12 .. 10]) + F(9, 2) ,
ExpVal (A[12 .. 9]) + F(8, 2) , //
this will yield the maximum sum..
ExpVal (A[12 .. 8]) + F(7, 2) ,
ExpVal (A[12 .. 7]) + F(6, 2)
}
which is nothing but,
F(12, 3) = MAX
{
1/2 + F(10, 2) ,
2/3 + F(9, 2) ,
2/4 + F(8, 2) , // this will yield the
maximum sum..
3/5 + F(7, 2) ,
4/6 + F(6, 2)
}
Trace the above equation and you should get it..
On Nov 28, 12:57 am, Ankur Garg <[email protected]> wrote:
> Hey Sourabh
>
> Could you please explain the solution in a bit detail perhaps using an
> example or so..It wud be really helpful ..Just logic not code
>
>
>
>
>
>
>
> On Mon, Nov 28, 2011 at 1:03 AM, sourabh <[email protected]> wrote:
> > Looks like a dynamic programming problem....
>
> > Say F(n,k) denotes the maximum expected sum value for an array of n
> > elements and partition k , then
>
> > F(n,k) = MAX for all r such that ceil(n/2k) <= r <= floor(3n/2k)
> > { (expected value of array elems from A[n] to A[n-r+1]) + F(n-r,
> > k-1) }
>
> > Base condition:
> > 1) F(N, 1) = expected value for array A[n] such that ceil(n/2k) <= N
> > <= floor(3n/2k)
> > 2) If any of the sub problems where the array size is not between
> > ceil(n/2k) and floor(3n/2k) , both inclusive, then its not a valid
> > candidate for the final solution. This is can be handled by giving
> > initial value to all such combination a value of -1.
>
> > To store that the intermediate computations take an array Max[N][K],
> > F(N,K) = Max[N][K]
>
> > On Nov 28, 12:17 am, sourabh <[email protected]> wrote:
> > > Because in the previous example k = 3.
>
> > > On Nov 27, 10:46 pm, Piyush Grover <[email protected]> wrote:
>
> > > > Optimal split: [0,0][1,1][0,0][1,1][0,1][1,0]
> > > > Expected value of optimal split: 0 + 1 + 0 + 1 + 1/2 + 1/2 = 3
> > > > why this is not the optimal split???
>
> > > > On Sun, Nov 27, 2011 at 6:58 PM, Ankur Garg <[email protected]>
> > wrote:
> > > > > You have an array with *n* elements. The elements are either 0 or 1.
> > You
> > > > > want to *split the array into kcontiguous subarrays*. The size of
> > each
> > > > > subarray can vary between ceil(n/2k) and floor(3n/2k). You can
> > assume that
> > > > > k << n. After you split the array into k subarrays. One element of
> > each
> > > > > subarray will be randomly selected.
>
> > > > > Devise an algorithm for maximizing the sum of the randomly selected
> > > > > elements from the k subarrays. Basically means that we will want to
> > split
> > > > > the array in such way such that the sum of all the expected values
> > for the
> > > > > elements selected from each subarray is maximum.
>
> > > > > You can assume that n is a power of 2.
>
> > > > > Example:
>
> > > > > Array: [0,0,1,1,0,0,1,1,0,1,1,0]
> > > > > n = 12
> > > > > k = 3
> > > > > Size of subarrays can be: 2,3,4,5,6
>
> > > > > Possible subarrays [0,0,1] [1,0,0,1] [1,0,1,1,0]
> > > > > Expected Value of the sum of the elements randomly selected from the
> > subarrays: 1/3 + 2/4 + 3/5 = 43/30 ~ 1.4333333
>
> > > > > Optimal split: [0,0,1,1,0,0][1,1][0,1,1,0]
> > > > > Expected value of optimal split: 1/3 + 1 + 1/2 = 11/6 ~ 1.83333333
>
> > > > > Source ->
> >http://stackoverflow.com/questions/8189334/google-combinatorial-optim...
>
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