Dynamic Programming Algorithms: Dynamic Solution Tables, Study notes of Computer Science

Download Dynamic Programming Algorithms: Dynamic Solution Tables and more Study notes Computer Science in PDF only on Docsity! CS1510, Design and Analysis of Algorithms, Fall 2006 Patchrawat Uthaisombut Dynamic Programming general coin change problem Let CHvL be the minimum number of coins of denominations 4, 3, 1 needed to make change for v dollars CHvL = min 8 CHv - 4L + 1, CHv - 3L + 1, CHv - 1L + 1 < knapsack problem Let PHi, wL be the maximum profit that obtainable, with weight at most w, from items in the set 81, 2, …, i < PHi, wL = max 8 PHi - 1, wL, PHi - 1, w - wiL + pi< chained matrix multiplication Let M Hi, jL be the minimum number of multiplications needed to compute Ai Ai+1 … A j M Hi, jL = mini§k§ j-1 8 M Hi, kL + M Hk + 1, jL + pi-1 pk p j < optimal binary search tree Let CHi, jL be the cost of an optimal binary search tree containing real keys i, …, j and dummy keys i - 1, …, j CHi, jL = mini§r§ j 8 CHi, r - 1L + cHr + 1, jL + wHi, jL < longest common subsequence Let LHi, jL be the length of a longest common subsequence of x1 x2 … xi and y1 y2 … y j L Hi, jL = 9 LHi - 1, j - 1L + 1 if xi = y j max 8 LHi, j - 1L, LHi - 1, jL < if xi ∫ y j sequence alignment Let CHi, jL be the cost of an optimal alignment of x1 x2 … xi and y1 y2 … y j C Hi, jL = max 9 LHi - 1, j - 1L + axi y j LHi, j - 1L + d LHi - 1, jL + d = Floyd's shortest path algorithm Let LHi, j, kL be the length of a shortest path from i to j where the internal nodes are from the set 8 1, 2, …, k< We want LHi, j, nL for all i, j LHi, j, kL = min 8 LHi, j, k - 1L, LHi, k, k - 1L + LHk, j, k - 1L < bitonic euclidean TSP Label the vertices using positive integer from left to right For i < j, let LHi, jL be the length of a shortest bitonic spanning path for vertices in the set 81, 2, …, i< where one path ends at vertex i and the other path ends at vertex j We want min1§ j§i-1 8 LHn, jL + d j,n< LHi, jL = min 8 LHi - 1, jL + di-1,i, min1§ k§i-2 LHi - 1, kL + dk,i < maximum contiguous sublist DP-summary.nb 1