I think he means the K-Median selection algorithm (which is what Hopcroft et al call it). This is O(n), where n is the number of elements, but here n = N^2. We can do better than O(N^2) as we have discussed.
On Nov 8, 4:00 pm, Anup Ghatage <ghat...@gmail.com> wrote: > I googled around and couldn't find a good source for the K-Median Theorem. > So, could you provide a good link? > > On Tue, Nov 8, 2011 at 2:55 PM, sagar sindwani > <sindwani.sa...@gmail.com>wrote: > > > > > > > Use K-Median theorem with k=N > > > On Mon, Nov 7, 2011 at 11:33 PM, Gene <gene.ress...@gmail.com> wrote: > > >> Can you explain how a heap helps get the answer? Just to put the > >> elements in a heap requires O ( N^2 log (N) ) time. > > >> On Nov 7, 4:12 pm, vikas <vikas.rastogi2...@gmail.com> wrote: > >> > I think the best we can have is nlogn solution with the heap approach. > > >> > On Nov 6, 10:27 pm, Dave <dave_and_da...@juno.com> wrote: > > >> > > @Mohit: Here is a counterexample: > > >> > > 10 11 52 53 54 > >> > > 20 21 112 113 114 > >> > > 30 31 122 123 124 > >> > > 40 41 132 133 134 > >> > > 50 91 142 143 144 > > >> > > The median is 91, and it is not on the anti-diagonal. > > >> > > Dave > > >> > > On Nov 6, 3:11 am, mohit verma <mohit89m...@gmail.com> wrote: > > >> > > > @Gene > >> > > > As i said in my earlier post right to left diagonal partitions the > >> martix > >> > > > into 2 equal number of elements. So now the median must be in this > >> > > > diagonal. Now our focus is on finding median of this diagonal only. > >> > > > I think this works fine. Can u give some test case for which it > >> fails? > > >> > > > On Sun, Nov 6, 2011 at 3:02 AM, Gene <gene.ress...@gmail.com> > >> wrote: > >> > > > > Unfortunately this isn't true. See the example I gave earlier: > > >> > > > > 1 2 3 > >> > > > > 2 4 5 > >> > > > > 3 4 6 > > >> > > > > Thje median is 3. > > >> > > > > 1 2 2 3 >3< 4 4 5 6 > > >> > > > > Niether one of the 3's lies on the diagonal. > > >> > > > > When you pick any element P on the diagonal, all you know is that > >> > > > > anything to the right and downward is no less than P and > >> everything to > >> > > > > the left and upward is no greater. This leaves the upper right > >> and > >> > > > > lower left rectangles of the matrix unrelated to P. > > >> > > > > On Nov 5, 3:51 am, ankit agarwal <ankit.agarwal.n...@gmail.com> > >> wrote: > >> > > > > > Hi, > > >> > > > > > I think the median will always lie on the diagonal > >> > > > > > a[n][1] ---- a[1][n] > >> > > > > > because the elements on the LHS making the upper triangle will > >> > > > > > always be less than or equal to the elements on the diagonal > >> > > > > > and the RHS, elements in the lower triangle will be greater > >> than or > >> > > > > > equal to them. > > >> > > > > > so sort the diagonal and find the middle element, that will be > >> the > >> > > > > > median. > > >> > > > > > Thanks > >> > > > > > Ankit Agarwal > > >> > > > > > On Nov 5, 1:29 am, Gene <gene.ress...@gmail.com> wrote: > > >> > > > > > > Here's an idea. Say we pick any element P in the 2D array A > >> and use > >> > > > > > > it to fill in an N element array X as follows. > > >> > > > > > > j = N; > >> > > > > > > for i = 1 to N do > >> > > > > > > while A(i, j) > P do > >> > > > > > > j = j - 1; > >> > > > > > > end; > >> > > > > > > X(i) = j; > >> > > > > > > end; > > >> > > > > > > This algorithm needs O(N) time. > > >> > > > > > > The elements of X split each row with respect to P. That is, > >> for each > >> > > > > > > i = 1 to N, > > >> > > > > > > A(i, j) <= P if 0 < j <= X(i), A(i,j) > P if X(i) < j <= > >> N. > > >> > > > > > > Now the strategy is to create two length N arrays a = > >> [0,0,...0]; and > >> > > > > > > b = [N,N,...]. We'll maintain the invariant that a[i] < > >> Median <= b[i] > >> > > > > > > for some i. I.e, they "bracket" the median. > > >> > > > > > > We define functions L(a) = sum_i( a(i) ) and R(b) = sum_i( N - > >> > > > > > > b(i) ). These tell us how many elements there are left and > >> right of > >> > > > > > > the bracket. > > >> > > > > > > Now reduce the bracket as in binary search: Guess a value P, > >> compute > >> > > > > > > X. If L(X) >= R(X), set b = X else set a = X. > > >> > > > > > > Keep guessing new P values in a way that ensures we reduce > >> the number > >> > > > > > > of elements between a and b by some fixed fraction. If we > >> can do > >> > > > > > > that, we'll get to 1 element in O(N log N) time. > > >> > > > > > > The remaining problem is picking good P's. Certainly the > >> first time is > >> > > > > > > easy. Just take A(N/2, N/2). This has approximately (at > >> least) N^2/4 > >> > > > > > > elements larger than it and N^2/4 smaller due to the sorted > >> rows and > >> > > > > > > columns. This is what we need to get O(N log N) performance. > > >> > > > > > > But after the first split, things get trickier. The area > >> between a and > >> > > > > > > b takes on the shape of a slash / /, so you can't just pick a > >> P that > >> > > > > > > moves a and b together by a fixed fraction of remaining > >> elements. > > >> > > > > > > Not to worry! You can quickly look up the (at most) N row > >> medians in > >> > > > > > > the bracket, i.e. > > >> > > > > > > { A(i, (a[i] + b[i] + 1) / 2) | a[i]<b[i] , i = 1 to N } > > >> > > > > > > and use the well known O(N) median selection algorithm to get > >> a median > >> > > > > > > of this. This has the quality we want of being somewhere > >> roughly in > >> > > > > > > the middle half of the remaining elements. The logic is the > >> same as > >> > > > > > > the selection algorithm itself, but in our case the rows are > >> pre- > >> > > > > > > sorted. > > >> > > > > > > In all, each partitioning step requires O(N), and a fixed > >> fraction > >> > > > > > > (about 1/2) of the elements will be eliminated from the > >> bracket with > >> > > > > > > each step. Thus O(log n) steps will be needed to bring the > >> bracket to > >> > > > > > > size 1 for an overall cost of O(N log N). > > >> > > > > > > I don't doubt that there's a simpler way, but this one seems > >> to work. > >> > > > > > > Anyone see problems? > > >> > > > > > > On Nov 3, 3:41 pm, sravanreddy001 <sravanreddy...@gmail.com> > >> wrote: > > >> > > > > > > > any better solution than O(N^2) in worst case? > >> > > > > > > > How do we take advantage of sorting and find in O(N lg N)- > >> Hide > >> > > > > quoted text - > > >> > > > > > - Show quoted text - > > >> > > > > -- > >> > > > > You received this message because you are subscribed to the > >> Google Groups > >> > > > > "Algorithm Geeks" group. > >> > > > > To post to this group, send email to algogeeks@googlegroups.com. > >> > > > > To unsubscribe from this group, send email to > >> > > > > algogeeks+unsubscr...@googlegroups.com. > >> > > > > For more options, visit this group at > >> > > > >http://groups.google.com/group/algogeeks?hl=en. > > >> > > > -- > >> > > > Mohit- Hide quoted text - > > >> > - Show quoted text - > > >> -- > >> You received this message because you are subscribed to the Google Groups > >> "Algorithm Geeks" group. > >> To post to this group, send email to algogeeks@googlegroups.com. > >> To unsubscribe from this group, send email to > >> algogeeks+unsubscr...@googlegroups.com. > >> For more options, visit this group at > >>http://groups.google.com/group/algogeeks?hl=en. > > > -- > > You received this message because you are subscribed to the Google Groups > > "Algorithm Geeks" group. > > To post to this group, send email to algogeeks@googlegroups.com. > > To unsubscribe from this group, send email to > > algogeeks+unsubscr...@googlegroups.com. > > For more options, visit this group at > >http://groups.google.com/group/algogeeks?hl=en. > > -- > Anup Ghatage- Hide quoted text - > > - Show quoted text - -- You received this message because you are subscribed to the Google Groups "Algorithm Geeks" group. 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