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The recursive maximum contiguous subsequence sum algorithm works by performing linear work to compute a sum that spans the center border and then performing two recursive calls These calls collectively compute a sum that spans the center border, do further recursive calls, and so on The total work performed by the algorithm is then proportional to the scanning done over all the recursive calls Figure 821 graphically illustrates how the algorithm works for N = 8 elements Each rectangle represents a call to maxSubSum,and the length of the rectangle is proportional to the size of the subarray (and hence the cost of the scanning of the subarray) being operated 011by the invocation The initial call is shown on the first line: The size of the subarray is N,which represents the cost of the scanning for the third case The initial call then makes two recursive calls, yielding two subarrays of size N l 2 The cost of each scan in case 3 is half the original cost, but as there are two such recursive calls, the combined cost of these recursive calls is also N Each of those two recursive instances themselves make two recursive calls, yielding four subproblems that are a quarter of the original size Thus the total of all case 3 costs is also N Eventually, we reach the base case Each base case has size I , and there are N of them Of course, there are no case 3 costs in this instance, but we charge 1 unit for performing the check that determines whether the sole element is positive or negative The total cost then, as illustrated in Figure 821, is N per level of recursion Each level halves the size of the basic problem, so the halving principle tells us that there are approximately log N levels In fact, the number of levels is 1 + [log N] (which is 4 when N equals 8) Thus we expect that the total running time is O(N log N).



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<%@ Page Language="C#" AutoEventWireup="true" CodeFile="ProgrammaticGridaspxcs" Inherits="ProgrammaticGrid" %> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 11//EN" "http://wwww3org/TR/xhtml11/DTD/xhtml11dtd"> <html xmlns="http://wwww3org/1999/xhtml" > <head runat="server"> <title>Untitled Page</title> </head> <body> <form id="form1" runat="server"> <div> <asp:GridView ID="m_CustomersGrid" runat="server"> </asp:GridView> <asp:SqlDataSource ID="m_CustomersDataSource" runat="server"></asp:SqlDataSource> </div> </form> </body> </html>

Trace of recursive calls for recursive maximum contiguous subsequence sum algorithm for N = 8 elements

Figure 616: Defining a relationship between the child and parent tables Some relationships are automatically defined for you

Note that the more formal analysis for all classes of algorithms that recursively solve two halves and use linear additional work

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source InfoPath tries to be helpful by automatically adding relationships between elds (ie, table columns) with the same names (In a moment, we ll see how to manually de ne and edit relationships) In this case, the CustomerID elds exist in both the Customers and Orders tables, so that relationship is already de ned You will want to add other relationships as you see necessary to ensure the integrity of data between tables New relationships are added as the last step before a new table is added The nal result, after adding both the Customers and Orders tables, is shown with a collapsed structure in Figure 614 Using the Show table columns checkbox is similar to expanding the groups in the data source All elds (ie, table columns) of the tables are shown under their respective tables Similar to the Edit Default Values dialog discussed in 3, the checkboxes adjacent to each eld let you decide whether or not you want to include the eld in this database query connection Some checkboxes may be checked and disabled This means the associated eld(s) is either a primary key in the database table or is participating in a relationship with another table and must be included in the query

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This analysis gives an intuitive explanation of why the running time is O ( N log N) In general, however, expanding a recursive algorithm to examine behavior is a bad idea; it violates the third rule of recursion We next consider a more formal mathematical treatment Let T ( N ) represent the time required to solve a maximum contiguous subsequence sum problem of size N If N = 1 the program takes some constant amount of time to execute lines 10 to I I , which we call I unit Thus T ( l ) = 1 Otherwise, the program must perform two recursive calls and the linear work involved in computing the maximum sum for case 3 The constant overhead is absorbed by the O ( N ) term How long do the two recursive calls take Because they solve problems of size N / 2 , we know that they must each require T ( N 1 2 ) units of time; consequently, the total recursive work is 2T(N / 2) This analysis gives the equations

Of course, for the second equation to make sense, N must be a power of 2 Otherwise, at some point N 1 2 will not be even A more precise equation is

Some table columns may contain a large amount of data This is especially true with SQL data types such as binary, image, nvarchar, text, and varbinary If querying the database connection takes a prohibitively long time when you re lling out the form, consider analyzing the data that is queried Deselecting a large eld or two (when Show table columns is enabled) in Figure 614 can make a signi cant difference in performance when lling out the form

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