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how to generate a barcode using asp.net c# Knapsack in Software
64 Knapsack Generating QR Code 2d Barcode In None Using Barcode generator for Software Control to generate, create QRCode image in Software applications. QR Code 2d Barcode Decoder In None Using Barcode reader for Software Control to read, scan read, scan image in Software applications. During a robbery, a burglar nds much more loot than he had expected and has to decide what to take His bag (or knapsack ) will hold a total weight of at most W pounds There are n items to pick from, of weight w1 , , wn and dollar value v1 , , vn What s the most valuable combination of items he can t into his bag 1 For instance, take W = 10 and Item 1 2 3 4 Weight 6 3 4 2 Value $30 $14 $16 $9 Paint QR Code JIS X 0510 In C# Using Barcode generator for VS .NET Control to generate, create QR Code 2d barcode image in .NET framework applications. Denso QR Bar Code Printer In .NET Using Barcode drawer for ASP.NET Control to generate, create QR Code JIS X 0510 image in ASP.NET applications. There are two versions of this problem If there are unlimited quantities of each item available, the optimal choice is to pick item 1 and two of item 4 (total: $48) On the other hand, if there is one of each item (the burglar has broken into an art gallery, say), then the optimal knapsack contains items 1 and 3 (total: $46) As we shall see in 8, neither version of this problem is likely to have a polynomialtime algorithm However, using dynamic programming they can both be solved in O(nW ) time, which is reasonable when W is small, but is not polynomial since the input size is proportional to log W rather than W QR Code Drawer In Visual Studio .NET Using Barcode creator for Visual Studio .NET Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications. QR Code Creation In VB.NET Using Barcode generator for .NET Control to generate, create QR image in .NET framework applications. Knapsack with repetition
EAN13 Printer In None Using Barcode encoder for Software Control to generate, create UPC  13 image in Software applications. Printing UCC  12 In None Using Barcode drawer for Software Control to generate, create UPC Code image in Software applications. Let s start with the version that allows repetition As always, the main question in dynamic programming is, what are the subproblems In this case we can shrink the original problem in two ways: we can either look at smaller knapsack capacities w W , or we can look at fewer items (for instance, items 1, 2, , j, for j n) It usually takes a little experimentation to gure out exactly what works The rst restriction calls for smaller capacities Accordingly, de ne K(w) = maximum value achievable with a knapsack of capacity w Can we express this in terms of smaller subproblems Well, if the optimal solution to K(w) includes item i, then removing this item from the knapsack leaves an optimal solution to K(w wi ) In other words, K(w) is simply K(w w i ) + vi , for some i We don t know which i, so we need to try all possibilities K(w) = max {K(w wi ) + vi }, Barcode Drawer In None Using Barcode creation for Software Control to generate, create barcode image in Software applications. Generate Barcode In None Using Barcode creator for Software Control to generate, create bar code image in Software applications. i:wi w
Code39 Drawer In None Using Barcode encoder for Software Control to generate, create Code39 image in Software applications. GS1128 Creator In None Using Barcode creator for Software Control to generate, create GS1 128 image in Software applications. where as usual our convention is that the maximum over an empty set is 0 We re done! The algorithm now writes itself, and it is characteristically simple and elegant Painting USPS OneCode Solution Barcode In None Using Barcode creator for Software Control to generate, create USPS Intelligent Mail image in Software applications. Data Matrix 2d Barcode Drawer In C# Using Barcode generator for .NET Control to generate, create DataMatrix image in Visual Studio .NET applications. If this application seems frivolous, replace weight with CPU time and only W pounds can be taken with only W units of CPU time are available Or use bandwidth in place of CPU time, etc The knapsack problem generalizes a wide variety of resourceconstrained selection tasks Scan Barcode In Visual Studio .NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET framework applications. Code128 Drawer In None Using Barcode encoder for Online Control to generate, create Code 128C image in Online applications. K(0) = 0 for w = 1 to W : K(w) = max{K(w wi ) + vi : wi w} return K(W ) This algorithm lls in a onedimensional table of length W + 1, in lefttoright order Each entry can take up to O(n) time to compute, so the overall running time is O(nW ) As always, there is an underlying dag Try constructing it, and you will be rewarded with a startling insight: this particular variant of knapsack boils down to nding the longest path in a dag! Draw Code128 In Visual C#.NET Using Barcode maker for VS .NET Control to generate, create Code 128B image in .NET framework applications. GS1  12 Generation In ObjectiveC Using Barcode encoder for iPad Control to generate, create UPCA image in iPad applications. Knapsack without repetition
Print EAN13 In None Using Barcode encoder for Font Control to generate, create European Article Number 13 image in Font applications. Encode Code 39 Full ASCII In None Using Barcode drawer for Microsoft Word Control to generate, create Code 3 of 9 image in Word applications. On to the second variant: what if repetitions are not allowed Our earlier subproblems now become completely useless For instance, knowing that the value K(w w n ) is very high doesn t help us, because we don t know whether or not item n already got used up in this partial solution We must therefore re ne our concept of a subproblem to carry additional information about the items being used We add a second parameter, 0 j n: K(w, j) = maximum value achievable using a knapsack of capacity w and items 1, , j The answer we seek is K(W, n) How can we express a subproblem K(w, j) in terms of smaller subproblems Quite simple: either item j is needed to achieve the optimal value, or it isn t needed: K(w, j) = max{K(w wj , j 1) + vj , K(w, j 1)} (The rst case is invoked only if wj w) In other words, we can express K(w, j) in terms of subproblems K( , j 1) The algorithm then consists of lling out a twodimensional table, with W + 1 rows and n + 1 columns Each table entry takes just constant time, so even though the table is much larger than in the previous case, the running time remains the same, O(nW ) Here s the code Initialize all K(0, j) = 0 and all K(w, 0) = 0 for j = 1 to n: for w = 1 to W : if wj > w: K(w, j) = K(w, j 1) else: K(w, j) = max{K(w, j 1), K(w w j , j 1) + vj } return K(W, n)

