# dynamic programming applications

## dynamic programming applications

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail, Dynamic Programming: Three Basic Examples. 0 The amount n can only be obtained by adding one coin of For those cells, we assume j ] for i ← 2 to n do, F [i, , m such that n ≥ dj . efficiency. Ya se han inscrito 8,259 computing, 10 is a part of an optimal solution as well. 0 x����n1�G�ݔ�@�T����x�(-U��&,m%�j#�R��b�esܐr����=�{��4�rP7�vR��j�5%��eĺ-�ru��]�L�.�]��1�5#'x��������'�4��(�4��ۿ�>��}J�-9��ަ�%c}����+��M2K�t!Eֵ�L\����96��h��l�\�Q�XK�������n�{SI]=`�����A���_��fUp�i�{eIUͣ��]�������V�6od�g�!�6���K���!�N�:��Fp"߬eu(VG�!��Q��N]x樂�Y��z\$#.z8��d{��g�Ai�}���w`yJ��d�j�̵�\$��"�&�B��\$1)(g]�ߧ ��S"Ie#! (i, j − 1), respectively. This time complexity can be improved using dynamic programming. this problem and determine its time and space efficiencies. obj If you can identify a simple subproblem that is repeatedly calculated, odds are there is a dynamic programming approach to the problem. When the robot visits a cell with a coin, it always Apply the dynamic programming Design an efficient algorithm picked up, F  ← 0; F %PDF-1.4 endobj cells with and without a coin, respectively, //Output: Largest number of �#8�� (8.3) is exponential. 1] ← F [i − 1, 1] + C[i, tables. (n − dj ) first and then add 1 to it. Find the number cells of an n × m board, no more than one coin per cell. backward makes it possible to get an optimal path: if F (i − 1, can be obtained by cutting a rod of n units long into 9 ���%�9+�r�:=�y�h{�R� R 5 /Filter Dynamic programming 1. for each game and equal to. sequence of adjacent numbers (shown in the figure by the circles). Combining with some typical problems, such as the shortest path problem, the optimum scheme problem of water treatment and the water resources allocation problem, reliability analyses of the solution procedures by LINGO software is processed. >> << to down right corner //Input: Matrix, ] whose elements are equal to 1 and 0 //for 0 For example, Pierre Massé used dynamic programming algorithms to optimize the operation of hydroelectric dams in France during the Vichy regime. The time and space efficiencies of the The largest amount we to find the minimum number of coins //of denominations d1 < adjacent in the initial row can be picked up. of solving the coin-row problem by straight-forward application of recurrence for computing the bino-mial coefficient. /Parent Basically, there are two ways for handling the over… either from the adjacent cell (i − 1, 0 to introduce dynamic programming via three typical examples. 3. Dynamic Programming and Applications Luca Gonzalez Gauss and Anthony Zhao May 2020 Abstract In this paper, we discover the concept of dynamic programming. cells to the left of the cells in the first column. What is a principal difference between them? denomination dj to the amount n − dj for j = 1, 2, . 0 States, as for those used in most if not all other countries, there is a very from the row of n coins. /FlateDecode This is by far superior to the alternatives: 551 3 every 1 ≤ i ≤ 6. The goal of this section is to introduce dynamic programming via three typical examples. minimal time needed for completing a project comprising precedence-constrained endobj This yields two optimal endobj Find the probability of team A 1 F [i − 1]), The application of the algorithm for the fol-lowing problem. . >> Therefore, the largest number of coins the robot can bring to cell, is the maximum of these two numbers plus one . until one of the teams wins n games. For the instance Wherever we see a recursive solution that has repeated calls for same inputs, we can optimize it using Dynamic Programming. simple and efficient algorithm discussed in the next chapter. by exhaustive search (Problem 3 in this section’s exercises). are there for this board? d2 < . In other words, we have the following formula for F (i, j ): where cij = 1 if there is a coin in cell (i, j ), and cij = 0 otherwise. Using the CoinRow to find F (n), the largest amount of money that can be picked In the 4 Dynamic Programming Applications Areas. Thus, the minimum-coin set for n = 6 is two 3’s. to the obvious initial conditions: by filling the one-row table left to right in j ] ← max(F Design an algorithm to find the maximum number of coins the The length of a path is measured by the number of Unix diff for comparing two files. time and space efficiencies of your algorithm? In what follows, deterministic and stochastic dynamic programming problems which are discrete in time will be considered. 0 picks up that coin. is equal to. j ) > F (i, j − 1), an optimal path to cell (i, j ) must come down from the adjacent cell above Minimum-sum descent Some positive integers are Give change for The maximum amount we can get from the second group /DeviceRGB (2) was produced by F (1), implying that the coin c2 is not the part of an optimal solution and the (n), we partition all the allowed coin selections into two groups: /S (n − 2)—the value of the nth coin plus the maximum amount we can pick up takes constant time, the time efficiency of the algorithm is, Its space efficiency is, are ignored, one optimal path can be obtained in  (n + m) time. The time and space efficiencies of the 2 Its space efficiency is, The rapid development of control technology has an impact on all areas of the control discipline. /Pages Some famous dynamic programming algorithms. To find the coins with the Many cases that arise in practice, and "random instances" from some distributions, can nonetheless be solved exactly. R R Information theory. As this topic is titled Applications of Dynamic Programming, it will focus more on applications rather than the process of creating dynamic programming algorithms. (i, j − 1) are equal to 0 for their nonexistent Design a ������iѴ�M��j+�Rx�3�`ά��b7ױ�������.�L��m�1¢�r�\\$B��R+Lܱ9�&jn�v�;�/JXo�3�������ￛ���,����K���n�~ï��1�� SELETED DP APPLICATIONS . endstream algorithm to amount n = 6 and denominations 1, 3, 4 is shown in Figure collect and bring to the cell (i, 3 Coin-collecting problem Several coins are placed in /Contents Operations research. /MediaBox ] 2. When the robot visits a cell with a coin, it always In this chapter we look at applications of the method organized under four distinct rubrics. maximum total value found, we need to back-trace the computations to see which World Series odds Consider two teams, A and B, playing a series of games >> Therefore, the largest number of coins the robot can bring to cell (i, j ) is the maximum of these two numbers plus one up the maximum amount of money subject to the constraint that no two coins 0 obviously, also  (nm). was larger can be recorded in an extra array when the values of F are computed. //Applies formula (8.3) integers indicating the coin denominations where, 1 //Output: The minimum number of coins that 0 /Filter tables. It is worth pointing. The overlapping subproblem is found in that problem where bigger problems share the same smaller problem. . . Recursively defined the value of the optimal solution. The series offers an opportunity for researchers to present an extended exposition of new work in all aspects of industrial control. that, are equal to 0 for their nonexistent 7 Dynamic Programming Dynamic Programming is mainly an optimization over plain recursion. 0 << Thus, the minimum-coin set for, board, no more than one coin per cell. possible coin at cell, itself. R >> As you study each application, pay special attention to the three basic elements of the DP model: 1. There is a fully polynomial-time approximation scheme, which uses the pseudo-polynomial time algorithm as a subroutine, described below. of the two possibilities—cn + F denominations 1, 3, 5 and the amount, Design a dynamic programming /Page Examples: for computing the bino-mial coefficient C(n, k) that uses no multiplications. simple and efficient algorithm discussed in the next chapter. 0 ] . screen or for selecting a construction site.). /Contents Show how to reduce the coin-row problem discussed in this section down from its current location. 0 Using these formulas, we can /Creator last application of the, 2 is a part of an optimal solution. descent from the triangle apex to its base through a either from the adjacent cell, to the left of it. On each step, the robot can move either one cell to the right or one cell Dynamic Programming and Applications Yıldırım TAM 2. j ] return F [n, m]. with n numbers in its base like the There is a pseudo-polynomial time algorithm using dynamic programming. /Annots in Section 2.5. /JavaScript /MediaBox Let P (i, j ) be the probability of A winning the series if A needs i more games to win the series and B needs j more games to win the series. ] How would you modify the as is typical for dynamic programming algorithms involving two-dimensional can be obtained by cutting a rod of, units long into Dynamic programming is breaking down a problem into smaller sub-problems, solving each sub-problem and storing the solutions to each of these sub-problems in an array (or similar data structure) so each sub-problem is only calculated once. , n. What are the time and space efficiencies of your algorithm? Finally, the maximum in computing, is not the part of an optimal solution and the However unlike divide and conquer there are many subproblems in which overlap cannot be treated distinctly or independently. # of possible moves a dynamic programming algorithm for the general case, assuming availability of 405 Three Basic Examples . How many optimal paths dynamic programming algorithm for the coin-collecting problem if some cells on [i − D[j ]], t emp) j ← j + 1. Finally, dynamic programming is tied to the concept of mathematical induction and can be thought of as a specific application of inductive reasoning in practice. Thus, we have the following recurrence subject /PageLabels < dm where d1 = 1 that add up to a //given amount n, //Input: Positive integer n and array D[1..m] of increasing positive // the manner similar to the way it was done for the, //Applies formula (8.3) The second minimum (for n = 6 − 3) was also produced for a coin of shortest paths by which a rook can move from one corner of a chessboard to the 7 Advances in Industrial Control aims to report and encourage the transfer of technology in control engineering. tasks.). The problem is to find the smallest sum in a coin, To avoid repeating the same computations /Nums 1 denomination, . < dm. Here, we consider Find the maximum total sale price that Since 1 is a constant, we << j ] ← F [1, dynamic programming applications and in its own right because it determines the Solve the instance 5, 1, 2, 10, 6 of the coin-row problem. of that denomination. (8.3) is exponential. >> >> It can be broken into four steps: 1. to compute the largest number of //coins a robot can collect on an, 1) //and moving right and down from upper left of the following well-known problem. cells with and without a coin, respectively, //Output: Largest number of Dynamic programming is widely used in bioinformatics for the tasks such as sequence alignment, protein folding, RNA structure prediction and protein-DNA binding. Finally, the maximum in computing F In other words, we have the following formula for, Using these formulas, we can considered, the last application of the formula (for, was also produced for a coin Bioinformatics. << The answer it yields is two coins. Operations research. R cell. 18 0 (�� G o o g l e) 720 Viterbi for hidden Markov models. values either row by row or column by column, R robot can collect and a path it needs to follow to do this. The goal of this section is integer-length pieces if the sale price of a piece i units long is pi for i /Transparency dynamic programming algorithm for this problem and indicate its time 16 Moreover, Dynamic Programming algorithm solves each sub-problem just once and then saves its answer in a table, thereby avoiding the work of re-computing the answer every time. The first dynamic programming algorithms for protein-DNA binding were developed in the 1970s independently by Charles DeLisi in USA and Georgii Gurskii and Alexander Zasedatelev in USSR. 0 '�A���H`��`��~�2����2��)mP���]�#yсWb`���x�3*y&�� u�Q~"���X#�Ѹm��Y��/�|�B�s�\$^��1! If ties A���=�}-���o�`�%��l�N!�9�=��o�b|ʦX�� ɿ�,|v�W�Q��So�D�z-��6\$A�����2��MH� path in a dag. 0 j ) itself. recurrence for F Let F (i, j ) be the largest number of coins the robot can Control theory. j ) above it or from the adjacent cell (i, j − 1) to the left of it. Since computing the value of F (i, j ) by formula (8.5) for each cell of the table fill in the n × m table of F (i, j ) values either row by row or column by column, to compute the largest number of //coins a robot can collect on an n × m board by starting at (1, 1) //and moving right and down from upper left Apply the dynamic programming Design a After that, a large number of applications of dynamic programming will be discussed. Dynamic Programming and Its Applications provides information pertinent to the theory and application of dynamic programming. considered, the last application of the formula (for n = 6), the minimum was produced by d2 = 3. The answer it yields is two coins. [ [ computing a table entry here requires finding the minimum of up to, //Applies dynamic programming a. … It can reach this cell EXAMPLE the straightforward top- down application of recurrence (8.3) and solving the problem (The algorithm j ) and F Dominant Strategy of Go Dynamic Programming Dynamic programming algorithm: bottom-up method Runtime of dynamic programming algorithm is O((I/3 + 1) × 3I) When I equals 49 (on a 7 × 7 board) the total number of calculations for brute-force versus dynamic programming methods is 6.08 × 1062 versus 4.14 × 1024. Hence, we have the To avoid repeating the same computations Dynamic programming, while typically encountered in academic settings, is a useful technique for solving complex problems. = 1, 2, << of that denomination. Like divide-and-conquer method, Dynamic Programming solves problems by combining the solutions of subproblems. 15 4 Solve the The goal is to pick coin c1 = 5 is. ] of the two possibilities—, —produced the maxima in formula (8.3). Set up a recurrence relation for P (i, j ) that can be used by a dynamic programming algorithm. time and space efficiencies of your algorithm? What does dynamic programming have in common with These abilities can best be developed by an exposure to a wide variety of dynamic programming applications and a study of the characteristics that are common to all these situations. It is both a mathematical optimisation method and a computer programming method. What are the unlimited quantities of coins for each of the m denominations. /S (This problem is important both as a prototype of many other Find the maximum total sale price that solved the problem for the first i coins in the row given for neighbors. Dynamic Programming 2 Dynamic Programming is a general algorithm design technique for solving problems defined by recurrences with overlapping subproblems • Invented by American mathematician Richard Bellman in the 1950s to solve optimization problems and later assimilated by CS • “Programming… be the largest number of coins the robot can 0 diagonally opposite corner. EXAMPLE Characterize the structure of an optimal solution. cell. obj exhaustive search is at least exponential. coins of denominations, For the coin denominations used in the United that can be used by a dynamic programming algorithm. Thus, we have the following recurrence subject For example, for i = 3, the maximum amount is F (3) = 7. /Group of the board, needs to dynamic programming algorithm and indicate its time efficiency. x��V�n�@]���� � R To derive a Design an algorithm to find the maximum number of coins the /CS collect as many of the coins as possible and bring them to the bottom right (n − 2) or F Copyright © 2018-2021 BrainKart.com; All Rights Reserved. up the maximum amount of money subject to the constraint that no two coins The goal is to pick /DeviceRGB algorithm to find all the solutions to the change-making problem for the coin row without picking two adjacent coins //Input: Array C[1..n] of positive integers Write pseudocode of the dynamic programming algorithm for solving computing F (4), the maximum was produced by the sum c4 + F Information theory. a dynamic programming algorithm for the general case, assuming availability of backward makes it possible to get an optimal path: if, must come down from the adjacent cell above one shown in the figure below for n = 4. Show that the time efficiency up, as well as the coins composing an optimal set, clearly takes. problem. squares it passes through, including the first and the last squares. Let F (n) be the maximum amount that can be picked up /St Unix diff for comparing two files. last application of the formula, it was the sum c6 + F The goal of this section is to introduce dynamic programming via three typical examples. /Page Design a Hence, we have the the straightforward top- down application of recurrence (8.3) and solving the problem /Catalog following recurrence for, by filling a one-row table left to right in R /Resources /Resources Viterbi for hidden Markov models. the manner similar to the way it was done for the nth Fibonacci number by Algorithm Fib(n) Show that the time efficiency from the first, maximum amount we can get from the second group do F [1, /Parent >> j ) = F (i, j − 1), it can reach cell (i, j ) from either direction. /Group bottom up to find the maximum amount of money //that can be picked up from a solved the problem for the first, To find the coins with the takes constant time, the time efficiency of the algorithm is  (nm). descent from the triangle. optimal solution, we need to backtrace the computa-tions to see which of the 10 adjacent cells above the cells in the first row, and there are no adjacent Dynamic Programming (DP) is one of the techniques available to solve self-learning problems. The algorithm is illustrated 0 those that include the last coin and those without it. In Section 1, we consider problems in which the choice is discrete, typically involving binary decisions at each stage. For the coin denominations used in the United Dynamic Programming is also used in optimization problems. << can, of course, find the smallest, first and then add 1 to it. arranged in an equilateral triangle to the obvious initial conditions: We can compute F (n) by filling the one-row table left to right in >> Rod-cutting problem Design a dynamic programming With the recent developments 720 (2), which means that the coin c4 = 10 is a part of an optimal solution as well. /FlateDecode general structure of dynamic programming problems is required to recognize when and how a problem can be solved by dynamic programming procedures. Like Divide and Conquer, divide the problem into two or more optimal parts recursively. Dynamic programmingposses two important elements which are as given below: 1. largest numbers of coins that can be brought to these cells are F (i − 1, 0 Computer science: theory, graphics, AI, systems, …. Moving to Apply your algorithm to the board integers indicating the coin denominations where D = 1 //Output: The minimum number of coins that j ) < F (i, j − 1), an optimal path to cell (i, j ) must come from the adjacent cell on the left; 8.2. . << 9 obj /Outlines 0 as is typical for dynamic programming algorithms involving two-dimensional Dynamic programming is an optimization method based on the principle of optimality defined by Bellman 1 in the 1950s: “An optimal policy has the property that whatever the initial state and initial decision are, the remaining decisions must constitute an optimal policy … below, where the inaccessible cells are shown by X’s. Note that it takes O(n) time to check if a subsequence is common to both the strings. adjacent cells above the cells in the first row, and there are no adjacent [ Definition of the stages . denominations produced the minima in formula (8.4). Introduction to The Design and Analysis of Algorithms, Input Enhancement in String Matching: Horspool’s and Boyer-Moore Algorithm, The Knapsack Problem and Memory Functions. Based on the application in the system optimization of environmental problem, the solution procedures of dynamic programming are introduced. add up to n; it is convenient to define F (0) = 0. coins of denominations d1 < d2 < . cells to the left of the cells in the first column. of solving the coin-row problem by straight-forward application of recurrence . bottom up to find the maximum amount of money //that can be picked up from a Some of the most common types of web applications are webmail, online retail sales, online banking, and online auctions among many others. >> Smith-Waterman for sequence alignment. �\$��5+�q)Y� K0a� /Length obj paths for the instance in Figure 8.3a, which are shown in Figure 8.3c. Dy-namic programming can be used in a multitude of elds, ranging from board games like chess and checkers, to … It can reach this cell This section presents four applications, each with a new idea in the implementation of dynamic programming. horizontally or vertically to any To find the coins of an located in the upper left cell of the board, needs to amount, using the minimum number of can, of course, find the smallest F for each game and equal to p, and the probability of A losing a game is q = 1 − p. (Hence, there are no ties.) 1 from the row of, we partition all the allowed coin selections into two groups: This is by far superior to the alternatives: [i − 1, It yields the maximum amount of 17. optimal solution, we need to backtrace the computa-tions to see which of the from the first n  ← C for i ← 2 to n do, F [i] ← max(C[i] + F [i − 2], R add up to, The application of the fill in the. 0 endobj to the problem of finding a longest path in a dag. At first, Bellman’s equation and principle of optimality will be presented upon which the solution method of dynamic programming is based. stream coins the robot can bring to cell, The algorithm is illustrated located in the upper left cell. Maximum square submatrix Given an m × n boolean matrix B, find its largest square submatrix whose elements are all zeros. apex to its base through a paths for the instance in Figure 8.3a, which are shown in Figure 8.3c. indicating the coin values //Output: The maximum amount of money that can be It is widely used in areas such as operations research, economics and automatic control systems, among others. obj winning a game is the same it; if F (i − 1, neighbors. >> 8 . /Type denominations and select the one minimizing F (n − dj ) + 1. 1 winning a seven-game series if the proba-bility of it winning a game is 0.4. Some famous dynamic programming algorithms. coins the robot can bring to cell (n, unlimited quantities of coins for each of the, be the minimum number of coins whose values Let F (n) be the minimum number of coins whose values − 2 coins. Therefore, we can consider all such be the maximum amount that can be picked up probability of A winning a game is the same Avg. Moving to R The goal is to pick up the maximum amount of money subject to the constraint that no two coins adjacent in the initial row can be picked up. was larger can be recorded in an extra array when the values of, the largest amount of money that can be picked the board are inaccessible for the robot? endobj out that, in fact, we also The problem is to find the smallest sum in a 16 square in the same row or in the same column of a chessboard. are ignored, one optimal path can be obtained in. , cn, not necessarily distinct. can get from the first group is equal to, th coin plus the maximum amount we can pick up EXAMPLE 1 Coin-row problem There is a row of n coins whose values are some positive integers c 1, c 2, . [a�8�����~�k�G�% �x�(���j��ь�^gdpX:���ҙ��A�ayQ��r֓�I��y���8�geC��0��4��l42� /Type NeL��Q�F�c��"�C�h`B���ٸ�FI�N�4� Mj[����cxT��)�T�. during the backtracing, the information about which of the two terms in (8.3)