think-cell technical report TC2003/01 A GUI-based Interaction ...

5.2 An Application of Dynamic Programming IMPLEMENTATION
I suggest that to a large degree it is possible to foresee the user’s intention and
place the pasted shapes where they are meant to be. Some key indicators that can
help to understand the user’s intention are:
• Spatial proximity of edges
• Similarity of shapes in terms of type and formatting
• Patterns that indicate a common case, such as a table row being inserted
These hints can be encoded in a cost function that – given the environment of
the source shapes and that of the destination slide – calculates the costs of some
source gridline being placed before, on, or after some destination gridline. When
every aspect perfectly fits, the costs will be zero; the more counter indication is
detected for the placement in question, the bigger the costs will be. Given such a
local cost function, the gridline matching problem can be seen as an optimization
problem. The optimal solution is a list of source, dest and identify instructions
that describes the order in which gridlines must be taken to create a matching with
minimal cumulated costs.
I do not present a detailed cost function here. It is plausible that a polynomial-
time cost function is feasible; however, a lot of testing and fine-tuning will be
required to achieve reasonable results. For the current implementation I chose a
trivial cost function that takes the relative distance between a source gridline and a
destination gridline as a measure for the cost, and defines the cost for “very close”
to be zero. This algorithm implements a simple snapping behavior, but a much
more sophisticated matching is desirable for a future version of the software.
5.2.2 A Recursive Algorithm with Exponential Running Time
The graph in figure 42 illustrates the interpretation of the gridline matching problem
as a shortest path problem. A straight forward algorithm to find an optimal solution
to this problem is presented in figure 43. The recursive MergeGridlines() function
takes as input two lists of gridlines and returns the minimal costs along with the
optimal path. The algorithm effectively does a complete enumeration of all possible
paths from source s to target t. Assuming that the number of source gridlines n
and the number of destination gridlines m are of the same order, this leads to an
exponential asymptotic running time of Θ(3 n ): The algorithm does not have any
loops and therefore only the recursive function calls contribute to the asymptotic
running time. There are three recursive calls altogether, and for about half of all
nodes in the shortest-path-graph, all three calls are carried out. I further assume
that the cost function has some polynomial running time characteristics.
The cost function in the recursive algorithm is called CostNext(). This is, be-
cause given a certain state of the calculation, this function calculates the costs for
making a source, dest or identify decision for the next gridlines (i. e., the heads
of the listSource and listDest parameters).
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