I was revisiting my notes on Dynamic Programming. Its basically a memoized recursion technique, which stores away the solutions to smaller subproblems for later reuse in computing solutions to relatively larger sub problems.
The question I have is that in order to apply DP to a recursive problem, it must have an optimal substructure. This basically necessitates that an optimal solution to a problem contains optimal solution to subproblems.
Is it possible otherwise ? I mean have you ever seen a case where optimal solution to a problem does not contain an optimal solution to subproblems.
Please share some examples, if you know to deepen my understanding.
In dynamic programming a given problems has Optimal Substructure Property if optimal solution of the given problem can be obtained by using optimal solutions of its sub problems.
For example the shortest path problem has following optimal substructure property: If a node X lies in the shortest path from a source node U to destination node V then the shortest path from U to V is combination of shortest path from U to X and shortest path from X to V.
But Longest path problem doesn’t have the Optimal Substructure property.
i.e the longest path between two nodes doesn't have to be the longest path between the in between nodes.
For example, the longest path q->r->t is not a combination of longest path from q to r and longest path from r to t, because the longest path from q to r is q->s->t->r.
So here: optimal solution to a problem does not contain an optimal solution to the sub problems.
For more details you can read
Longest path problem from wikipedia
Optimal substructure from wikipedia
You're perfectly right that the definitions are imprecise. DP is a technique for getting algorithmic speedups rather than an algorithm in itself. The term "optimal substructure" is a vague concept. (You're right again here!) To wit, every loop can be expressed as a recursive function: each iteration solves a subproblem for the successive one. Is every algorithm with a loop a DP? Clearly not.
What people actually mean by "optimal substructure" and "overlapping subproblems" is that subproblem results are used often enough to decrease the asymptotic complexity of solutions. In other words, the memoization is useful! In most cases the subtle implication is a decrease from exponential to polynomial time, O(n^k) to O(n^p), p<k or similar.
Ex: There is an exponential number of paths between two nodes in a dense graph. DP finds the shortest path looking at only a polynomial number of them because the memos are extremely useful in this case.
On the other hand, Traveling salesman can be expressed as a memoized function (e.g. see this discussion), where the memos cause a factor of O( (1/2)^n ) time to be saved. But, the number of TS paths through n cities, is O(n!). This is so much bigger that the asymptotic run time is still super-exponential: O(n!)/O(2^n) = O(n!). Such an algorithm is generally not called a Dynamic Program even though it's following very much the same pattern as the DP for shortest paths. Apparently it's only a DP if it gives a nice result!
To my understanding, this 'optimal substructure' property is necessary not only for Dynamic Programming, but to obtain a recursive formulation of the solution in the first place. Note that in addition to the Wikipedia article on Dynamic Programming, there is a separate article on the optimal substructure property. To make things even more involved, there is also an article about the Bellman equation.
You could solve the Traveling Salesman Problem, choosing the nearest city at each step, but it's wrong method.
The whole idea is to narrow down the problem into the relatively small set of the candidates for optimal solution and use "brute force" to solve it.
So it better be that solutions of the smaller sub-problem should be sufficient to solve the bigger problem.
This is expressed via a recursion as function of the optimal solution of smaller sub-problems.
answering this question:
Is it possible otherwise ? I mean have you ever seen a case where
optimal solution to a problem does not contain an optimal solution to
subproblems.
no it is not possible, and can even be proven.
You can try to implement dynamic programming on any recursive problem but you will not get any better result if it doesn't have optimal substructure property. In other words dynamic programming methodology is not useful to implement on a problem which doesn't have optimal substructure property.
Related
While reading CLRS section 15.3 about "When I should use dynamic programming." and during the explanation of Optimal substructure they gave 2 examples, they are unweighted longest simple path and unweighted shortest path.
They said,
The subproblems in finding the longest simple path are not independent, whereas for shortest paths they
That's why we can't solve unweighted longest simple path using dynamic programming and I don't have any problem with that but they also said
Both problems examined in Sections 15.1 and 15.2 have independent subproblems ...... In rod cutting, to determine the best way to cut up a rod of length n, we look at the best ways of cutting up rods of length I for I = 0, 1,..., n-1. Because an optimal solution to the length -n problem includes just one of these subproblem solutions (after we have cut off the first piece), independence of subproblems is not an issue.
The last sentence
independence of subproblems is not an issue.
Is the independence of subproblems an issue or not? and If not an issue so why they said the first quote or I'm just me misunderstanding something.
The problem is that you cannot simply merge the answers to two dependent subproblems.
If problems are independent, that is fine. If you only look at one subproblem, likewise that is fine. But if you need to combine 2 answers and they are dependent, you have to maintain some additional state.
Where, as in the case of the longest simple path, "some additional state" may blow up on you exponentially. :-)
There are a lot of real-world problems that turn out to be NP-hard. If we assume that P ≠ NP, there aren't any polynomial-time algorithms for these problems.
If you have to solve one of these problems, is there any hope that you'll be able to do so efficiently? Or are you just out of luck?
If a problem is NP-hard, under the assumption that P ≠ NP there is no algorithm that is
deterministic,
exactly correct on all inputs all the time, and
efficient on all possible inputs.
If you absolutely need all of the above guarantees, then you're pretty much out of luck. However, if you're willing to settle for a solution to the problem that relaxes some of these constraints, then there very well still might be hope! Here are a few options to consider.
Option One: Approximation Algorithms
If a problem is NP-hard and P ≠ NP, it means that there's is no algorithm that will always efficiently produce the exactly correct answer on all inputs. But what if you don't need the exact answer? What if you just need answers that are close to correct? In some cases, you may be able to combat NP-hardness by using an approximation algorithm.
For example, a canonical example of an NP-hard problem is the traveling salesman problem. In this problem, you're given as input a complete graph representing a transportation network. Each edge in the graph has an associated weight. The goal is to find a cycle that goes through every node in the graph exactly once and which has minimum total weight. In the case where the edge weights satisfy the triangle inequality (that is, the best route from point A to point B is always to follow the direct link from A to B), then you can get back a cycle whose cost is at most 3/2 optimal by using the Christofides algorithm.
As another example, the 0/1 knapsack problem is known to be NP-hard. In this problem, you're given a bag and a collection of objects with different weights and values. The goal is to pack the maximum value of objects into the bag without exceeding the bag's weight limit. Even though computing an exact answer requires exponential time in the worst case, it's possible to approximate the correct answer to an arbitrary degree of precision in polynomial time. (The algorithm that does this is called a fully polynomial-time approximation scheme or FPTAS).
Unfortunately, we do have some theoretical limits on the approximability of certain NP-hard problems. The Christofides algorithm mentioned earlier gives a 3/2 approximation to TSP where the edges obey the triangle inequality, but interestingly enough it's possible to show that if P ≠ NP, there is no polynomial-time approximation algorithm for TSP that can get within any constant factor of optimal. Usually, you need to do some research to learn more about which problems can be well-approximated and which ones can't, since many NP-hard problems can be approximated well and many can't. There doesn't seem to be a unified theme.
Option Two: Heuristics
In many NP-hard problems, standard approaches like greedy algortihms won't always produce the right answer, but often do reasonably well on "reasonable" inputs. In many cases, it's reasonable to attack NP-hard problems with heuristics. The exact definition of a heuristic varies from context to context, but typically a heuristic is either an approach to a problem that "often" gives back good answers at the cost of sometimes giving back wrong answers, or is a useful rule of thumb that helps speed up searches even if it might not always guide the search the right way.
As an example of the first type of heuristic, let's look at the graph-coloring problem. This NP-hard problem asks, given a graph, to find the minimum number of colors necessary to paint the nodes in the graph such that no edge's endpoints are the same color. This turns out to be a particularly tough problem to solve with many other approaches (the best known approximation algorithms have terrible bounds, and it's not suspected to have a parameterized efficient algorithm). However, there are many heuristics for graph coloring that do quite well in practice. Many greedy coloring heuristics exist for assigning colors to nodes in a reasonable order, and these heuristics often do quite well in practice. Unfortunately, sometimes these heuristics give terrible answers back, but provided that the graph isn't pathologically constructed the heuristics often work just fine.
As an example of the second type of heuristic, it's helpful to look at SAT solvers. SAT, the Boolean satisfiability problem, was the first problem proven to be NP-hard. The problem asks, given a propositional formula (often written in conjunctive normal form), to determine whether there is a way to assign values to the variables such that the overall formula evaluates to true. Modern SAT solvers are getting quite good at solving SAT in many cases by using heuristics to guide their search over possible variable assignments. One famous SAT-solving algorithm, DPLL, essentially tries all possible assignments to see if the formula is satisfiable, using heuristics to speed up the search. For example, if it finds that a variable is either always true or always false, DPLL will try assigning that variable its forced value before trying other variables. DPLL also finds unit clauses (clauses with just one literal) and sets those variables' values before trying other variables. The net effect of these heuristics is that DPLL ends up being very fast in practice, even though it's known to have exponential worst-case behavior.
Option Three: Pseudopolynomial-Time Algorithms
If P ≠ NP, then no NP-hard problem can be solved in polynomial time. However, in some cases, the definition of "polynomial time" doesn't necessarily match the standard intuition of polynomial time. Formally speaking, polynomial time means polynomial in the number of bits necessary to specify the input, which doesn't always sync up with what we consider the input to be.
As an example, consider the set partition problem. In this problem, you're given a set of numbers and need to determine whether there's a way to split the set into two smaller sets, each of which has the same sum. The naive solution to this problem runs in time O(2n) and works by just brute-force testing all subsets. With dynamic programming, though, it's possible to solve this problem in time O(nN), where n is the number of elements in the set and N is the maximum value in the set. Technically speaking, the runtime O(nN) is not polynomial time because the numeric value N is written out in only log2 N bits, but assuming that the numeric value of N isn't too large, this is a perfectly reasonable runtime.
This algorithm is called a pseudopolynomial-time algorithm because the runtime O(nN) "looks" like a polynomial, but technically speaking is exponential in the size of the input. Many NP-hard problems, especially ones involving numeric values, admit pseudopolynomial-time algorithms and are therefore easy to solve assuming that the numeric values aren't too large.
For more information on pseudopolynomial time, check out this earlier Stack Overflow question about pseudopolynomial time.
Option Four: Randomized Algorithms
If a problem is NP-hard and P ≠ NP, then there is no deterministic algorithm that can solve that problem in worst-case polynomial time. But what happens if we allow for algorithms that introduce randomness? If we're willing to settle for an algorithm that gives a good answer on expectation, then we can often get relatively good answers to NP-hard problems in not much time.
As an example, consider the maximum cut problem. In this problem, you're given an undirected graph and want to find a way to split the nodes in the graph into two nonempty groups A and B with the maximum number of edges running between the groups. This has some interesting applications in computational physics (unfortunately, I don't understand them at all, but you can peruse this paper for some details about this). This problem is known to be NP-hard, but there's a simple randomized approximation algorithm for it. If you just toss each node into one of the two groups completely at random, you end up with a cut that, on expectation, is within 50% of the optimal solution.
Returning to SAT, many modern SAT solvers use some degree of randomness to guide the search for a satisfying assignment. The WalkSAT and GSAT algorithms, for example, work by picking a random clause that isn't currently satisfied and trying to satisfy it by flipping some variable's truth value. This often guides the search toward a satisfying assignment, causing these algorithms to work well in practice.
It turns out there's a lot of open theoretical problems about the ability to solve NP-hard problems using randomized algorithms. If you're curious, check out the complexity class BPP and the open problem of its relation to NP.
Option Five: Parameterized Algorithms
Some NP-hard problems take in multiple different inputs. For example, the long path problem takes as input a graph and a length k, then asks whether there's a simple path of length k in the graph. The subset sum problem takes in as input a set of numbers and a target number k, then asks whether there's a subset of the numbers that dds up to exactly k.
Interestingly, in the case of the long path problem, there's an algorithm (the color-coding algorithm) whose runtime is O((n3 log n) · bk), where n is the number of nodes, k is the length of the requested path, and b is some constant. This runtime is exponential in k, but is only polynomial in n, the number of nodes. This means that if k is fixed and known in advance, the runtime of the algorithm as a function of the number of nodes is only O(n3 log n), which is quite a nice polynomial. Similarly, in the case of the subset sum problem, there's a dynamic programming algorithm whose runtime is O(nW), where n is the number of elements of the set and W is the maximum weight of those elements. If W is fixed in advance as some constant, then this algorithm will run in time O(n), meaning that it will be possible to exactly solve subset sum in linear time.
Both of these algorithms are examples of parameterized algorithms, algorithms for solving NP-hard problems that split the hardness of the problem into two pieces - a "hard" piece that depends on some input parameter to the problem, and an "easy" piece that scales gracefully with the size of the input. These algorithms can be useful for finding exact solutions to NP-hard problems when the parameter in question is small. The color-coding algorithm mentioned above, for example, has proven quite useful in practice in computational biology.
However, some problems are conjectured to not have any nice parameterized algorithms. Graph coloring, for example, is suspected to not have any efficient parameterized algorithms. In the cases where parameterized algorithms exist, they're often quite efficient, but you can't rely on them for all problems.
For more information on parameterized algorithms, check out this earlier Stack Overflow question.
Option Six: Fast Exponential-Time Algorithms
Exponential-time algorithms don't scale well - their runtimes approach the lifetime of the universe for inputs as small as 100 or 200 elements.
What if you need to solve an NP-hard problem, but you know the input is reasonably small - say, perhaps its size is somewhere between 50 and 70. Standard exponential-time algorithms are probably not going to be fast enough to solve these problems. What if you really do need an exact solution to the problem and the other approaches here won't cut it?
In some cases, there are "optimized" exponential-time algorithms for NP-hard problems. These are algorithms whose runtime is exponential, but not as bad an exponential as the naive solution. For example, a simple exponential-time algorithm for the 3-coloring problem (given a graph, determine if you can color the nodes one of three colors each so that no edge's endpoints are the same color) might work checking each possible way of coloring the nodes in the graph, testing if any of them are 3-colorings. There are 3n possible ways to do this, so in the worst case the runtime of this algorithm will be O(3n · poly(n)) for some small polynomial poly(n). However, using more clever tricks and techniques, it's possible to develop an algorithm for 3-colorability that runs in time O(1.3289n). This is still an exponential-time algorithm, but it's a much faster exponential-time algorithm. For example, 319 is about 109, so if a computer can do one billion operations per second, it can use our initial brute-force algorithm to (roughly speaking) solve 3-colorability in graphs with up to 19 nodes in one second. Using the O((1.3289n)-time exponential algorithm, we could solve instances of up to about 73 nodes in about a second. That's a huge improvement - we've grown the size we can handle in one second by more than a factor of three!
As another famous example, consider the traveling salesman problem. There's an obvious O(n! · poly(n))-time solution to TSP that works by enumerating all permutations of the nodes and testing the paths resulting from those permutations. However, by using a dynamic programming algorithm similar to that used by the color-coding algorithm, it's possible to improve the runtime to "only" O(n2 2n). Given that 13! is about one billion, the naive solution would let you solve TSP for 13-node graphs in roughly a second. For comparison, the DP solution lets you solve TSP on 28-node graphs in about one second.
These fast exponential-time algorithms are often useful for boosting the size of the inputs that can be exactly solved in practice. Of course, they still run in exponential time, so these approaches are typically not useful for solving very large problem instances.
Option Seven: Solve an Easy Special Case
Many problems that are NP-hard in general have restricted special cases that are known to be solvable efficiently. For example, while in general it’s NP-hard to determine whether a graph has a k-coloring, in the specific case of k = 2 this is equivalent to checking whether a graph is bipartite, which can be checked in linear time using a modified depth-first search. Boolean satisfiability is, generally speaking, NP-hard, but it can be solved in polynomial time if you have an input formula with at most two literals per clause, or where the formula is formed from clauses using XOR rather than inclusive-OR, etc. Finding the largest independent set in a graph is generally speaking NP-hard, but if the graph is bipartite this can be done efficiently due to König’s theorem.
As a result, if you find yourself needing to solve what might initially appear to be an NP-hard problem, first check whether the inputs you actually need to solve that problem on have some additional restricted structure. If so, you might be able to find an algorithm that applies to your special case and runs much faster than a solver for the problem in its full generality.
Conclusion
If you need to solve an NP-hard problem, don't despair! There are lots of great options available that might make your intractable problem a lot more approachable. No one of the above techniques works in all cases, but by using some combination of these approaches, it's usually possible to make progress even when confronted with NP-hardness.
I've seen references to cut-and-paste proofs in certain texts on algorithms analysis and design. It is often mentioned within the context of Dynamic Programming when proving optimal substructure for an optimization problem (See Chapter 15.3 CLRS). It also shows up on graphs manipulation.
What is the main idea of such proofs? How do I go about using them to prove the correctness of an algorithm or the convenience of a particular approach?
The term "cut and paste" shows up in algorithms sometimes when doing dynamic programming (and other things too, but that is where I first saw it). The idea is that in order to use dynamic programming, the problem you are trying to solve probably has some kind of underlying redundancy. You use a table or similar technique to avoid solving the same optimization problems over and over again. Of course, before you start trying to use dynamic programming, it would be nice to prove that the problem has this redundancy in it, otherwise you won't gain anything by using a table. This is often called the "optimal subproblem" property (e.g., in CLRS).
The "cut and paste" technique is a way to prove that a problem has this property. In particular, you want to show that when you come up with an optimal solution to a problem, you have necessarily used optimal solutions to the constituent subproblems. The proof is by contradiction. Suppose you came up with an optimal solution to a problem by using suboptimal solutions to subproblems. Then, if you were to replace ("cut") those suboptimal subproblem solutions with optimal subproblem solutions (by "pasting" them in), you would improve your optimal solution. But, since your solution was optimal by assumption, you have a contradiction. There are some other steps involved in such a proof, but that is the "cut and paste" part.
'cut-and-paste' technique can be used in proving both correctness of greedy algorithm (both optimal structure and greedy-choice property' and dynamic programming algorithm correctness.
Greedy Correctness
This lecture notes Correctness of MST from MIT 2005 undergrad algorithm class exhibits 'cut-and-paste' technique to prove both optimal structure and greedy-choice property.
This lecture notes Correctness of MST from MIT 6.046J / 18.410J spring 2015 use 'cut-and-paste' technique to prove greedy-choice property
Dynamic Programming Correctness
A more authentic explanation for 'cut-and-paste' was introduced in CLRS (3rd edtion) Chapter 15.3 Element of dynamic programming at page 379
"4. You show that the solutions to the subproblems used within the optimal solution to the problem must themselves be optimal by using a “cut-and-paste”
technique, You do so by supposing that each of the subproblem solutions is not optimal and then deriving a contradiction. In particular, by “cutting out” the nonoptimal subproblem solution and “pasting in” the optimal one, you show that you can get a better solution to the original problem, thus contradicting your supposition that you already had an optimal solution. If there is more than one subproblem, they are typically so similar that the cut-and-paste argument for one can be modified for the others with little effort."
Cut-and-Paste is a way used in proofing graph theory concepts, Idea is this: Assume you have solution for Problem A, you want to say some edge/node, should be available in solution. You will assume you have solution without specified edge/node, you try to reconstruct a solution by cutting an edge/node and pasting specified edge/node and say new solution benefit is at least as same as previous solution.
One of a most important samples is proving MST attributes (proving that greedy choice is good enough). see presentation on MST from CLRS book.
Proof by contradiction
P is assumed to be false, that is !P is true.
It is shown that !P implies two mutually contradictory assertions, Q and !Q.
Since Q and !Q cannot both be true, the assumption that P is false must be wrong, and P must be true.
It is not a new proof technique per se. It is just a fun way to articulate a proof by contradiction.
An example of cut-and-paste:
Suppose you are solving the shortest path problem in graphs with vertices x_1, x_2... x_n. Suppose you find a shortest path from x_1 to x_n and it goes through x_i and x_j (in that order). Then, clearly, the sub path from x_i to x_j must be a shortest path between x_i and x_j as well. Why? Because Math.
Proof using Cut and Paste:
Suppose there exists a strictly shorter path from x_i to x_j. "Cut" that path, and "Paste" into the overall path from x_1 to x_n. Then, you will have another path from x_1 to x_n that is (strictly) shorter than the previously known path, and contradicts the "shortest" nature of that path.
Plain Old Proof by Contradiction:
Suppose P: The presented path from x_1 to x_n is a shortest path.
Q: The subpath from x_i to x_j is a shortest path.
The, Not Q => not P (using the argument presented above.
Hence, P => Q.
So, "Cut" and "Paste" makes it a bit more visual and easy to explain.
I read this article, it suggests (page 1025 last paragraph) that there is a polynomial time algorithm to find the optimum of a k-tsp problem using binary search.
Using binary search would suggest there exists an algorithm for checking if a solution exists with cost<X and this algorithm is used for the binary search.
I 'googled' around for this and the only algorithm i could find was a non deterministic one (which is pretty trivial), but obviously i'm looking for a deterministic one.
I am interested in this for learning purposes,
Any help/links would be appreciated.
EDIT
I am referring to finding the value of the optimal solution and not about finding the solution itself.
Since TSP is a special case of k-TSP where k = number of nodes in the graph. If you had a solution for "what's the cheapest k-TSP route" in polynomial in relation to graph size, then you'd have a polynomial solution to decision problem version of TSP which would imply that P = NP.
So the answer is no. Deterministic polynomial algorithm for both decision problem and optimization version (they're essentially the same) of k-TSP doesn't exist (yet).
The paper you mentioned proposes a polynomial-time approximation algorithm for the directed k-TSP problem.
Approximation algorithms are those which are guaranteed to yield solutions with a limited deviation from the optimal solution value. There are examples of polynomial-time approximation algorithms for NP-Hard problems: the Christofides Algorithm yields, in time O(n³), solutions to the metric TSP problem whose values are at most 3/2 the value of the optimal solution.
David Karger, in a lecture (link)
mentions a randomized algorithm for efficient k-TSP problem which runs in polynomial time in n (but exponential in k). It is based on the idea of color coding: color each of the node with a random color in [1..k], and find a shortest chromatic path (where each color appears exactly once). With a simple dynamic programming algorithm, this approach gives a runtime of O(n^2 2^k) and it succeeds (in finding the path with minimal cost) with probability e^{-k}. By repeating e^k times, one achieves an algorithm that finds the minimum k-TSP with high probability.
I heard the only difference between dynamic programming and back tracking is DP allows overlapping of sub problems, e.g.
fib(n) = fib(n-1) + fib (n-2)
Is it right? Are there any other differences?
Also, I would like know some common problems solved using these techniques.
There are two typical implementations of Dynamic Programming approach: bottom-to-top and top-to-bottom.
Top-to-bottom Dynamic Programming is nothing else than ordinary recursion, enhanced with memorizing the solutions for intermediate sub-problems. When a given sub-problem arises second (third, fourth...) time, it is not solved from scratch, but instead the previously memorized solution is used right away. This technique is known under the name memoization (no 'r' before 'i').
This is actually what your example with Fibonacci sequence is supposed to illustrate. Just use the recursive formula for Fibonacci sequence, but build the table of fib(i) values along the way, and you get a Top-to-bottom DP algorithm for this problem (so that, for example, if you need to calculate fib(5) second time, you get it from the table instead of calculating it again).
In Bottom-to-top Dynamic Programming the approach is also based on storing sub-solutions in memory, but they are solved in a different order (from smaller to bigger), and the resultant general structure of the algorithm is not recursive. LCS algorithm is a classic Bottom-to-top DP example.
Bottom-to-top DP algorithms are usually more efficient, but they are generally harder (and sometimes impossible) to build, since it is not always easy to predict which primitive sub-problems you are going to need to solve the whole original problem, and which path you have to take from small sub-problems to get to the final solution in the most efficient way.
Dynamic problems also requires "optimal substructure".
According to Wikipedia:
Dynamic programming is a method of
solving complex problems by breaking
them down into simpler steps. It is
applicable to problems that exhibit
the properties of 1) overlapping
subproblems which are only slightly
smaller and 2) optimal substructure.
Backtracking is a general algorithm
for finding all (or some) solutions to
some computational problem, that
incrementally builds candidates to the
solutions, and abandons each partial
candidate c ("backtracks") as soon as
it determines that c cannot possibly
be completed to a valid solution.
For a detailed discussion of "optimal substructure", please read the CLRS book.
Common problems for backtracking I can think of are:
Eight queen puzzle
Map coloring
Sudoku
DP problems:
This website at MIT has a good collection of DP problems with nice animated explanations.
A chapter from a book from a professor at Berkeley.
One more difference could be that Dynamic programming problems usually rely on the principle of optimality. The principle of optimality states that an optimal sequence of decision or choices each sub sequence must also be optimal.
Backtracking problems are usually NOT optimal on their way! They can only be applied to problems which admit the concept of partial candidate solution.
Say that we have a solution tree, whose leaves are the solutions for the original problem, and whose non-leaf nodes are the suboptimal solutions for part of the problem. We try to traverse the solution tree for the solutions.
Dynamic programming is more like BFS: we find all possible suboptimal solutions represented the non-leaf nodes, and only grow the tree by one layer under those non-leaf nodes.
Backtracking is more like DFS: we grow the tree as deep as possible and prune the tree at one node if the solutions under the node are not what we expect.
Then there is one inference derived from the aforementioned theory: Dynamic programming usually takes more space than backtracking, because BFS usually takes more space than DFS (O(N) vs O(log N)). In fact, dynamic programming requires memorizing all the suboptimal solutions in the previous step for later use, while backtracking does not require that.
DP allows for solving a large, computationally intensive problem by breaking it down into subproblems whose solution requires only knowledge of the immediate prior solution. You will get a very good idea by picking up Needleman-Wunsch and solving a sample because it is so easy to see the application.
Backtracking seems to be more complicated where the solution tree is pruned is it is known that a specific path will not yield an optimal result.
Therefore one could say that Backtracking optimizes for memory since DP assumes that all the computations are performed and then the algorithm goes back stepping through the lowest cost nodes.
IMHO, the difference is very subtle since both (DP and BCKT) are used to explore all possibilities to solve a problem.
As for today, I see two subtelties:
BCKT is a brute force solution to a problem. DP is not a brute force solution. Thus, you might say: DP explores the solution space more optimally than BCKT. In practice, when you want to solve a problem using DP strategy, it is recommended to first build a recursive solution. Well, that recursive solution could be considered also the BCKT solution.
There are hundreds of ways to explore a solution space (wellcome to the world of optimization) "more optimally" than a brute force exploration. DP is DP because in its core it is implementing a mathematical recurrence relation, i.e., current value is a combination of past values (bottom-to-top). So, we might say, that DP is DP because the problem space satisfies exploring its solution space by using a recurrence relation. If you explore the solution space based on another idea, then that won't be a DP solution. As in any problem, the problem itself may facilitate to use one optimization technique or another, based on the problem structure itself. The structure of some problems enable to use DP optimization technique. In this sense, BCKT is more general though not all problems allow BCKT too.
Example: Sudoku enables BCKT to explore its whole solution space. However, it does not allow to use DP to explore more efficiently its solution space, since there is no recurrence relation anywhere that can be derived. However, there are other optimization techniques that fit with the problem and improve brute force BCKT.
Example: Just get the minimum of a classic mathematical function. This problem does not allow BCKT to explore the state space of the problem.
Example: Any problem that can be solved using DP can also be solved using BCKT. In this sense, the recursive solution of the problem could be considered the BCKT solution.
Hope this helps a bit.
In a very simple sentence I can say: Dynamic programming is a strategy to solve optimization problem. optimization problem is about minimum or maximum result (a single result). but in, Backtracking we use brute force approach, not for optimization problem. it is for when you have multiple results and you want all or some of them.
Depth first node generation of state space tree with bounding function is called backtracking. Here the current node is dependent on the node that generated it.
Depth first node generation of state space tree with memory function is called top down dynamic programming. Here the current node is dependant on the node it generates.