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The term was originally used in the 1940s by Richard Bellman to describe the process of solving problems where one needs to find the best decisions one after another. By 1953 , he had refined this to the modern meaning. The field was founded as a Systems Analysis and Engineering topic which is recognized by the IEEE . Bellman's contribution is remembered in the name of the Bellman Equation , a central result of dynamic programming which restates the optimization problem in Recursive form. The word "programming" in "dynamic programming" has no particular connection to computer Programming at all, and instead comes from the term " Mathematical Programming ", a synonym for Optimization . Thus, the "program" is the optimal plan for action that is produced. For instance, a finalized schedule of events at an exhibition is sometimes called a program. Programming, in this sense, means finding an acceptable plan of action. OVERVIEW ''Optimal substructure'' means that optimal solutions of subproblems can be used to find the optimal solutions of the overall problem. For example, the Shortest Path to a goal from a vertex in a Graph can be found by first computing the shortest path to the goal from all adjacent vertices, and then using this to pick the best overall path, as shown in Figure 1. In general, we can solve a problem with optimal substructure using a three-step process: # Break the problem into smaller subproblems. # Solve these problems optimally using this three-step process recursively. # Use these optimal solutions to construct an optimal solution for the original problem. The subproblems are, themselves, solved by dividing them into sub-subproblems, and so on, until we reach some simple case that is easy to solve. but a DAG indicates overlapping subproblems.]] To say that a problem has ''overlapping subproblems'' is to say that the same subproblems are used to solve many different larger problems. For example, in the Fibonacci Sequence , F3 = F1 + F2 and F4 = F2 + F3 — computing each number involves computing F2. Because both F3 and F4 are needed to compute F5, a naïve approach to computing F5 may end up computing F2 twice or more. This applies whenever overlapping subproblems are present: a naïve approach may waste time recomputing optimal solutions to subproblems it has already solved. In order to avoid this, we instead save the solutions to problems we have already solved. Then, if we need to solve the same problem later, we can retrieve and reuse our already-computed solution. This approach is called '' Memoization '' (not ''memorization'', although this term also fits). If we are sure we won't need a particular solution anymore, we can throw it away to save space. In some cases, we can even compute the solutions to subproblems we know that we'll need in advance. In summary, dynamic programming makes use of: Dynamic programming usually takes one of two approaches:
Some ) have automatic memoization built in. In any case, this is only possible for a Referentially Transparent function. EXAMPLES A naive implementation of a function finding the ''n''th member of the Fibonacci Sequence , based directly on the mathematical definition: function fib(n) if n = 0 '''or''' n = 1 return 1 else return fib(n − 1) + fib(n − 2) Notice that if we call, say, fib(5), we produce a call tree that calls the function on the same value many different times:# fib(5)# fib(4) + fib(3)# (fib(3) + fib(2)) + (fib(2) + fib(1))# ((fib(2) + fib(1)) + (fib(1) + fib(0))) + ((fib(1) + fib(0)) + fib(1))# (((fib(1) + fib(0)) + fib(1)) + (fib(1) + fib(0))) + ((fib(1) + fib(0)) + fib(1))In particular, fib(2) was calculated twice from scratch. In larger examples, many more values of fib, or ''subproblems'', are recalculated, leading to an exponential time algorithm.Now, suppose we have a simple Map object, ''m'', which maps each value of fib that has already been calculated to its result, and we modify our function to use it and update it. The resulting function requires only O (''n'') time instead of exponential time:var m := '''map'''(0 → 1, 1 → 1) function fib(n) if '''map''' m '''does not contain key''' n m {Link without Title} := fib(n − 1) + fib(n − 2) return m {Link without Title} This technique of saving values that have already been calculated is called '' Memoization ''; this is the top-down approach, since we first break the problem into subproblems and then calculate and store values. In the bottom-up approach we calculate the smaller values of fib first, then build larger values from them. This method also uses linear (O(''n'')) time since it contains a loop that repeats n − 1 times:function fib(n) var previousFib := 0, currentFib := 1 repeat n − 1 '''times''' var newFib := previousFib + currentFib previousFib := currentFib currentFib := newFib return currentFib In both these examples, we only calculate fib(2) one time, and then use it to calculate both fib(4) and fib(3), instead of computing it every time either of them is evaluated.Checkerboard Consider a checkerboard with ''n'' × ''n'' squares and a cost-function ''c''(''i'', ''j'') which returns a cost associated with square ''i'',''j'' (''i'' being the row, ''j'' being the column). For instance (on a 5 × 5 checkerboard), Thus ''c''(1, 3) = 5 Let us say you had a checker that could start at any square on the first rank (i.e., row) and you wanted to know the shortest path (sum of the costs of the visited squares are at a minimum) to get to the last rank, assuming the checker could move only diagonally left forward, diagonally right forward, or straight forward. That is, a checker on (1,3) can move to (2,2), (2,3) or (2,4). This problem exhibits optimal substructure. That is, the solution to the entire problem relies on solutions to subproblems. Let us define a function ''q''(''i'', ''j'') as q If we can find the values of this function for all the squares at rank ''n'', we pick the minimum and follow that path backwards to get the shortest path. It is easy to see that ''q''(''i'', ''j'') is equal to the minimum cost to get to any of the three squares below it (since those are the only squares that can reach it) plus ''c''(''i'', ''j''). For instance: Now, let us define ''q''(''i'', ''j'') in little more general terms: This equation is pretty straightforward. The first line is simply there to make the recursive property simpler (when dealing with the edges, so we need only one recursion). The second line says what happens in the first rank, so we have something to start with. The third line, the recursion, is the important part. It is basically the same as the A,B,C,D example. From this definition we can make a straightforward recursive code for ''q''(''i'', ''j''). In the following pseudocode, ''n'' is the size of the board, c(i, j) is the cost-function, and min() returns the minimum of a number of values: function minCost(i, j) if j < 1 '''or''' j > n return infinity else if i = 1 return c(i, j) else return '''min'''( minCost(i-1, j-1), minCost(i-1, j), minCost(i-1, j+1) ) + c(i, j) It should be noted that this function just computes the path-cost, not the actual path. We will get to the path soon. This, like the Fibonacci-numbers example, is horribly slow since it spends mountains of time recomputing the same shortest paths over and over. However, we can compute it much faster in a bottom up-fashion if we use a two-dimensional array q j instead of a function. Why do we do that? Simply because when using a function we recompute the same path over and over, and we can choose what values to compute first. We also need to know what the actual path is. The path problem we can solve using another array p j , a ''predecessor array''. This array basically says where paths come from. Consider the following code:function computeShortestPathArrays() for x '''from''' 1 '''to''' n q x := c(1, x) for y '''from''' 1 '''to''' n q 0 := infinity q n + 1 := infinity for y '''from''' 2 '''to''' n for x '''from''' 1 '''to''' n m := min(q x-1 , q x , q x+1 ) q x := m + c(y, x) if m = q x-1 p x := -1 else if m = q x p x := 0 else p x := 1 Now the rest is a simple matter of finding the minimum and printing it. function computeShortestPath() computeShortestPathArrays() minIndex := 1 min := q 1 for i '''from''' 2 '''to''' n if q i < min minIndex := i min := q i printPath(n, minIndex) function printPath(y, x) print(x) print("<-") if y = 2 print(x + p x ) else printPath(y-1, x + p x ) Sequence alignment The steps for using dynamic programming to perform sequence alignment are as follows:
Matrix chain multiplication ''Not yet written'' Show how the placement of parentheses affects the number of scalar multiplications required when multiplying a bunch of matrices. Show how to write a dynamic program to calculate the optimal parentheses placement. This is such a long example that it might be better to make it its own article.--> ALGORITHMS THAT USE DYNAMIC PROGRAMMING
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