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

More documents

Recommendations

Info

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 algorithm. 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
5.2 An Application of Dynamic Programming IMPLEMENTATION Algorithm: MergeGridlines (shortest path by dynamic programming) Input: listSource : Array [1..n] of Gridline listDest : Array [1..m] of Gridline Output: minimalCost : Accumulated cost for the optimal path optimalPath : A list of source, dest, identify Variables: costSource, costDest, costIdentify, minimalCost : Cost optimalPath : List of Choice matrixCost : Array [0..n+1][0..m+1] of Cost matrixChoice : Array [1..n+1][1..m+1] of Choice iSrc, iDst, iDstStart : Index Begin Initialize(matrixCost, ∞) matrixCost[1][1] ← 0 For iSrc ← 1 To n + 1 Do If iSrc = 1 Then iDstStart ← 2 Else iDstStart ← 1 // Skip upper left corner For iDst ← iDstStart To m + 1 Do costSource ← CostPrev(source, . . . ) + matrixCost[iSrc − 1][iDst] costDest ← CostPrev(dest, . . . ) + matrixCost[iSrc][iDst − 1] costIdentify ← CostPrev(identify, . . . ) + matrixCost[iSrc − 1][iDst − 1] If costDest < costSource Then If costDest < costIdentify Then matrixCost[iSrc][iDst] ← costDest matrixChoice[iSrc][iDst] ← dest Else matrixCost[iSrc][iDst] ← costIdentify matrixChoice[iSrc][iDst] ← identify End If Else If costSource < costIdentify Then matrixCost[iSrc][iDst] ← costSource matrixChoice[iSrc][iDst] ← source Else matrixCost[iSrc][iDst] ← costIdentify matrixChoice[iSrc][iDst] ← identify End If End If End For End For // Reconstruct optimal solution i ← n + 1, j ← m + 1 minimalCost ← matrixCost[i][j] optimalP ath ← () While i > 1 Or j > 1 Do optimalP ath ← matrixChoice[i][j].optimalP ath // a.(b, c) ↦→ (a, b, c) If matrixChoice[i][j] = source Or matrixChoice[i][j] = identify Then i ← i − 1 End If If matrixChoice[i][j] = dest Or matrixChoice[i][j] = identify Then j ← j − 1 End If End While Return (minimalCost, optimalP ath) End Figure 45: A dynamic programming algorithm for gridline matching, running in polynomial-time 81
Page 1 and 2:
think-cell technical report TC2003/
Page 3 and 4:
Contents List of Figures 5 1 Introd
Page 5 and 6:
List of Figures 1 The consultant’
Page 7 and 8:
1 Introduction INTRODUCTION “Layo
Page 9 and 10:
2 Field Study FIELD STUDY Jeff Hawk
Page 11 and 12:
2.1 Qualitative Aspects FIELD STUDY
Page 13 and 14:
Page 15 and 16:
Page 17 and 18:
Page 19 and 20:
2.2 Quantitative Aspects FIELD STUD
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
3 State of the Art STATE OF THE ART
Page 27 and 28:
3.1 User Interfaces for Layout Spec
Page 29 and 30: 3.2 Computer Supported Layout STATE
Page 31 and 32: 3.2 Computer Supported Layout STATE
Page 33 and 34: 3.3 State of the Art - Résumé STA
Page 35 and 36: 4.1 A New Approach to Slide Layout
Page 43 and 44: 4.2 Interaction Toolbox INTERACTION
Page 53 and 54: 4.3 Specifying the User Interface I
Page 69 and 70: 5 Implementation IMPLEMENTATION The
Page 71 and 72: 5.1 Program Architecture IMPLEMENTA
Page 77 and 78: 5.2 An Application of Dynamic Progr
Page 79: 5.2 An Application of Dynamic Progr
Page 83 and 84: 6 Evaluation EVALUATION In this cha
Page 85 and 86: 6.2 Case Study EVALUATION presented
Page 87 and 88: 6.2 Case Study EVALUATION AÖGHKÖ
Page 89 and 90: 7.2 Résumé CONCLUSION User Studie
Page 91 and 92: References REFERENCES [Adoa] Adobe
Page 93 and 94: REFERENCES [Hur78] A. Hurlburt. The
show all

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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?