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

5.2 An Application of Dynamic Programming IMPLEMENTATION
5.2.3 Finding the Optimal Solution in Polynomial Time
The analysis of the recursive algorithm reveals that the local costs for each node
are calculated up to three times. Also, the graph in figure 42 has a rectangular
structure, suggesting that the results of the cost calculations could be cached in a
regular matrix. This is precisely what the dynamic programming technique describes
(see Fig. 44): A matrix is filled with cumulated minimal costs and then the optimal
solution is reconstructed from the matrix. To make the reconstruction possible, not
only costs must be stored in the matrix, but also the locally optimal decisions.
The respective algorithm is presented in figure 45. The iterative implementation
of the MergeGridlines() function has the same interface (input and output) as
the recursive variant. However, it runs in polynomial time: The loop body is
executed m + n(m + 1) times, where m and n are the numbers of destination and
source gridlines in the input. Again, I assume the cost function to be of polynomial
time complexity. Thus, the asymptotic running time for the dynamic programming
algorithm is Θ(nm) or Θ(n 2 ), if we assume n and m to be of the same order.
In this case, the cost function is called CostPrev(), because it calculates the costs
for a decision that has already been made to reach the current state. The algorithm
then stores only the decision that has minimal cumulated cost and disregards the
other ones.
Once the matrix is filled, the bottom right corner reflects the total costs of the
optimal path, which can now be reconstructed by following the matching decision
annotations. The reconstruction of the optimal path is trivial and takes linear
time 15 , thus not contributing to the over-all asymptotic running time of the algo-
rithm.
Dest.
Source
1
2
3
4
5
Solution
1 2 3 4 5 6
0
SOURCE
7
SOURCE
11
SOURCE
26
SOURCE
58
DEST
20
SOURCE DEST IDENTIFY IDENTIFY DEST DEST
96 131 81 97 105 139
Solution
DEST DEST DEST DEST DEST
23 77 165 317 428
SOURCE IDENTIFY DEST DEST IDENTIFY DEST
29 31 46 59 178 210
DEST SOURCE DEST SOURCE IDENTIFY IDENTIFY
18 35 41 61 74 199
SOURCE DEST IDENTIFY SOURCE SOURCE DEST
24 29 55 79 115 171
SOURCE SOURCE DEST IDENTIFY DEST DEST
70 89 133 83 109 147
IDENTIFY
134
Figure 44: Dynamic programming cost matrix with optimal path reconstruction,
comp. Figs. 41 and 42
15 The asymptotic running time for optimal path reconstruction is O(n + m), because in each
run of the while loop either i or j or both are decreased.
80