In this course we study many well defined algorithms, including (so far) those for ADTs, sorting and searching, and others to come to operate on graphs. Quality open source implementations exist: you often don't need to implement them.
But we also study problem solving methods that guide the design of algorithms for your specific problem. Quality open source implementations may not exist for your specific problem: you may need to:
Such problem solving methods include divide & conquer, dynamic programming, and greedy algorithms.
An optimization problem requires finding a/the "best" of a set of alternatives (alternative approaches or solutions) under some quality metric (which we wish to maximize) or cost metric (which we wish to minimize).
Dynamic Programming is one of several methods we will examine. (Greedy algorithms and linear programming can also apply to optimization problems.)
Dynamic programming solves optimization problems by combining solutions to subproblems.
This sounds familiar: divide and conquer also combines solutions to subproblems, but applies when the subproblems are disjoint. For example, here is the recursion tree for merge sort on an array A[1..8]. Notice that the indices at each level do not overlap):
Dynamic programming applies when the subproblems overlap. For example, here is the recursion tree for a "rod cutting" problem to be discussed in the next section (numbers indicate lengths of rods). Notice that not only do lengths repeat, but also that there are entire subtrees repeating. It would be redundant to redo the computations in these subtrees.
Dynamic programming solves each subproblem just once, and saves its answer in a table, to avoid the recomputation. It uses additional memory to save computation time: an example of a time-memory tradeoff.
There are many examples of computations that require exponential time without dynamic programming but become polynomial with dynamic programming.
This example nicely introduces key points about dynamic programming.
Suppose you get different prices for steel rods of different lengths. Your supplier provides long rods; you want to know how to cut the rods into pieces in order to maximize revenue. Each cut is free. Rod lengths are always an integral number of length units (let's say they are centimeters).
Input: A length n and a table of prices pi for i = 1, 2, ..., n.
Output: The maximum revenue obtainable for rods whose lengths sum to n, computed as the sum of the prices for the individual rods.
We can choose to cut or not cut at each of the n-1 units of measurement. Therefore one can cut a rod in 2n-1 ways.
If pn is large enough, an optimal solution might require no cuts.
Suppose we have a rod of length 4. There are 2n-1 = 23 = 8 ways to cut it up (the numbers show the price we get for each length, from the chart above):
Having enumerated all the solutions, we can see that for a rod of length 4 we get the most revenue by dividing it into two units of length 2 each: p2 + p2 = 5 + 5 = 10.
Any optimal solution (other than the solution that makes no cuts) for a rod of length > 2 results in at least one subproblem: a piece of length > 1 remaining after the cut.
Claim: The optimal solution for the overall problem must include an optimal solution for this subproblem.
Proof: The proof is a "cut and paste" proof by contradiction: if the overall solution did not include an optimal solution for this problem, we could cut out the nonoptimal subproblem solution, paste in the optimal subproblem solution (which must have greater value), and thereby get a better overall solution, contradicting the assumption that the original cut was part of an optimal solution.
Therefore, rod cutting exhibits optimal substructure: The optimal solution to the original problem incorporates optimal solutions to the subproblems. This is a hallmark of problems amenable to dynamic programming. (Not all problems have this property.)
Here is a table of ri, the maximum revenue for a rod of length i, for this problem instance.
To solve a problem of size 7, find the best solution for subproblems of size 7; 1 and 6; 2 and 5; or 3 and 4. Each of these subproblems also exhibits optimal substructure.
One of the optimal solutions makes a cut at 3cm, giving two subproblems of lengths 3cm and 4cm. We need to solve both optimally. The optimal solution for a 3cm rod is no cuts. As we saw above, the optimal solution for a 4cm rod involves cutting into 2 pieces, each of length 2cm. These subproblem optimal solutions are then used in the solution to the problem of a 7cm rod.
The next thing we want to do is write a general expression for the value of an optimal solution that captures its recursive structure.
For any rod length n, we can determine the optimal revenues rn by taking the maximum of:
So, rn = max (pn, r1 + rn-1, r2 + rn-2, .... rn-1 + r1).
There is redundancy in this equation: if we have solved for ri and rn-i, we don't also have to solve for rn-i and ri.
Rather than considering all ways to divide the rod in half, leaving two subproblems, consider all ways to cut off the first piece of length i, leaving only one subproblem of length n - i:
We don't know in advance what the first piece of length i should be, but we do know that one of them must be the optimal choice, so we try all of them.
The above equation leads immediately to a direct recursive implementation (p is the price vector, so p[i] is the price you get for a rod of length i; n the problem size or total length of the rod given):
This works but is inefficient. It calls itself repeatedly on subproblems it has already solved (circled). Here is the recursion tree for n = 4:
In fact we can show that the growth is exponential. Let T(n) be the number of calls to Cut-Rod with the second parameter = n.
The initial 1 is for the root call, and T(j) counts the recursive calls in line 5. This has solution 2n. (Use the inductive hypothesis that it holds for j < n and then use formula A5 of Cormen et al. for an exponential series.)
Dynamic programming arranges to solve each sub-problem just once by saving the solutions in a table. There are two approaches.
The top-down approach has the advantages that it is easy to write given the recursive structure of the problem, and only those subproblems that are actually needed will be computed. It has the disadvantage of the overhead of recursion (and the algorithm is not tail-recursive, so can't be convertd to iteration by a compiler).
One can also sort the subproblems by "size" (where size is defined according to which problems use which other ones as subproblems), and solve the smaller ones first.
For example, here we again use a table r, but we arrange (via the loop in line 3) to solve the smallest problem first, then successively larger problems so we always know the smaller solutions are available. The array reference r[j−i] ensures that we only reference subproblems smaller than j, the one we are currently working on.
The bottom-up approach requires extra thought to ensure we arrange to solve the subproblems before they are needed. The bottom-up approach can be more efficient due to the iterative implementation (and with careful analysis, unnecessary subproblems can be excluded).
Both the top-down and bottom-up versions run in Θ(n2) time.
The above programs return the value of an optimal solution. To construct the solution itself, we need to record the choices that led to optimal solutions. Use a table s to record the place where the optimal cut was made (compare to Bottom-Up-Cut-Rod):
For our problem, the input data and the tables constructed are:
We then trace the choices made back through the table s with this procedure:
Exercise: Trace the calls made by Print-Cut-Rod-Solution(p, 8)...
In general, we follow these steps when solving a problem with dynamic programming:
The steps are illustrated in the next example.
A subsequence of sequence S leaves out zero or more elements but preserves order.
Z is a common subsequence of X and Y if Z is a subsequence
of both X and Y.
Z is a longest common subsequence if it is a subsequence of maximal length.
Given two sequences X = 〈 x1, ..., xm 〉 and Y = 〈 y1, ..., yn 〉, find a subsequence common to both whose length is longest. Solutions to this problem have applications to DNA analysis in bioinformatics. The analysis of optimal substructure is elegant.
For every subsequence of X = 〈 x1, ..., xm 〉, check whether it is a subsequence of Y = 〈 y1, ..., yn 〉, and record it if it is longer than the longest previously found.
This involves a lot of redundant work.
Many problems to which dynamic programming applies have exponential brute force solutions that can be improved on by exploiting redundancy in subproblem solutions.
The first step is to characterize the structure of an optimal solution, hopefully to show it exhibits optiomal stubstructure.
Often when solving a problem we start with what is known and then figure out how to contruct a solution. The optimal substructure analysis takes the reverse strategy: assume you have found an optional solution (Z below) and figure out what you must have done to get it!
Theorem: Let Z = 〈 z1, ..., zk 〉 be any LCS of X = 〈 x1, ..., xm 〉 and Y = 〈 y1, ..., yn 〉. Then
Sketch of proofs:
(1) can be proven by contradiction: if the last characters of X and Y are not included in Z, then a longer LCS can be constructed by adding this character to Z, a contradiction.
(2) and (3) have symmetric proofs: Suppose there exists a subsequence W of Xm−1 and Y (or of X and Yn−1) with length > k. Then W is a common subsequence of X and Y, contradicting Z being an LCS.
Therefore, an LCS of two sequences contains as prefix an LCS of prefixes of the sequences. We can now use this fact construct a recursive formula for the value of an LCS.
Let c[i, j] be the length of the LCS of prefixes Xi and Yj. The above recursive substructure leads to the definition of c:
We want to find c[m, n].
A recursive algorithm based on this formulation would have lots of repeated subproblems, for example, on strings of length 4 and 3:
Dynamic programming avoids the redundant computations by storing the results in a table. We use c[i,j] for the length of the LCS of prefixes Xi and Yj (hence it must start at 0). Try to find the correspondence betweeen the code below and the recursive definition shown in the box. (b is part of the third step and is explained next section.)
This is a bottom-up solution: Indices i and j increase through the loops, and references to c always involve either i-1 or j-1, so the needed subproblems have already been computed.
It is clearly Θ(mn); much better than Θ(n2m)!
In the process of computing the value of the optimal solution we can also record the choices that led to this solution. Step 4 is to add this latter record of choices and a way of recovering the optimal solution at the end.
Table b[i, j] is updated above to remember whether each entry is
We reconstruct the path by calling Print-LCS(b, X, n, m) and following the arrows, printing out characters of X that correspond to the diagonal arrows (a Θ(n + m) traversal from the lower right of the matrix to the origin):
What do "spanking" and "amputation" have in common? Use Print-LCS to find out!
Two other applications are covered in the Cormen et al. text, and many others in the Problems at the end of the chapter. I omit them to keep this lecture from being too long, and trust that the student will read them in the text.
Many scientific and business applications involve multiplication of chains of matrices 〈 A1, A2, A3, ... An 〉. Since matrix multiplication is associative, the matrices can be multiplied with their neighbors in this sequence in any order. The order chosen can have a huge difference in the number of multiplications required. For example suppose you have A, a 2x100 matrix, B (100x100) and C (100x20). To compute A*B*C:
(A*B) requires 2*100*100 = 20000 multiplications, and results in a 2x100 matrix. Then you need to multiply by C: 2*100*20 = 4000 multiplications, for a total of 24,000 multiplications (and a 2x20 result).
(B*C) requires 100x100x20 = 200000 multiplications, and results in a 100x20 matrix. Then you need to multiply by A: 2*100*20 = 4000 multiplications, for a total of 204,000 multiplications (and the same 2x20 result).
The Matrix-Chain Multiplication problem is to determine the optimal order of multiplications (not to actually do the multiplications!!). For three matrices I was able to figure out the best sequence by hand, but some problems in science, business and other areas involve many matrices, and the number of combinations to be checked grows exponentially: a brute force solution is Ω(2n). The CLRS textbook develops a dynamic programming solution that runs in Θ(n3) time, and uses Θ(n2) space.
Planning matrix multiplication is perhaps the most "canonical" example of dynamic programming: it is used in most introductory presentations. I chose to present LCS instead because matrix multiplication optimization will be built into turnkey software, and current students will more likely be interested in bioinformatics applications
We saw in Topic 8 that an unfortunate order of insertions of keys into a binary search tree (BST) can result in poor performance (e.g., linear in n). If we know all the keys in advance and also the probability that they will be searched, we can optimize the construction of the BST to minimize search time in the aggregate over a series of queries. An example application is when we want to construct a dictionary from a set of terms that are known in advance along with their frequency in the language.
There are Ω(4n / n3/2) binary trees with n nodes, so obviously a brute force approach will be exponential in time. The CLRS textbook develops a dynamic programming solution that is also Θ(n3). The student need only try problem 15.5-2 from the Cormen et al. text (manual simulation of the algorithm) to appreciate why we want to leave this tedium to computers!
To use dynamic programming, we must show that any optimal solution involves making a choice that leaves one or more subproblems to solve, and the solutions to the subproblems used within the optimal solution must themselves be optimal.
We may not know what that first choice is. Consequently:
How many subproblems are used in an optimal solution may vary:
How many choices in determining which subproblem(s) to use may vary:
Informally, running time depends on (# of subproblems overall) x (# of choices).
(We'll have a better understanding of "overall" when we cover amortized analysis.)
Since optimal substructure varies across problems no one algorithm will handle them all and you may encounter problems that have not already been solved: you need to learn the problem solving skill of constructing algorithms for problems.
When we study graphs, we'll see that finding the shortest path between two vertices in a graph has optimal substructure: if p = p1 + p2 is a shortest path between u and v then p1 must be a shortest path between u and w (etc.). Proof by cut and paste.
But finding the longest simple path (the longest path not repeating any edges) between two vertices is not likely to have optimal substructure.
For example, q → s → t → r is the longest simple path from q to r, but q → s is not the longest simple path for the supbproblem from q to s, so it does not exhibit optimal substructure. Also, if we try to compose optimal solutions q → s → t → r and r → q → s→ t to get a longest simple path from r to t, the composed path is not even legal: the criterion of simplicity is violated.
Dynamic programming requires overlapping yet independently solveable subproblems.
Longest simple path is NP-complete, a topic we will cover at the end of the semester, so is unlikely to have any efficient solution.
Although we wrote the code both ways, in terms of the order in which solutions are found, dynamic programming first finds optimal solutions to subproblems and then choses which to use in an optimal solution to the problem. It applies when one cannot make the top level choice until subproblem solutions are known.
In Topic 13, we'll see that greedy algorithms work top down: first make a choice that looks best, then solve the resulting subproblem. Greedy algorithms apply when one can make the top level choice without knowing how subproblems will be solved.
Dynamic Programming applies when the problem has these characteristics:
Note: do not make the logial error of thinking that if you have found an optimal solution to a subproblem then it must be part of the optimal solution to the overall problem. It may not be. The implication is in the other direction: if we have an optimal overall solution then the subproblems must be solved optimally as well.
Dynamic programming can be approached top-down or bottom-up:
Both have the same asympotic running time. The top-down procedure has the overhead of recursion, but computes only the subproblems that are actually needed. Bottom-up is used the most in practice.
We problem solve with dynamic programming in four steps:
There is an online presentation focusing on LCS at http://www.csanimated.com/animation.php?t=Dynamic_programming.
In the next Topic 13 we look at a related optimization strategy: greedy algorithms.