Some problems are at least partially susceptible to analytic solution. To solve these problems, approaches that entail a degree of mathematical modeling or analysis can be fruitful. But other problems stubbornly resist such approaches. For these problems we can usually devise candidate solutions by other means, but choosing among the candidates requires experimentation. Because modeling the effects of a solution mathematically is just too difficult, we must actually try a candidate to see how well it works — a practice we sometimes call "piloting." And when we try candidate solutions, we encounter a dilemma: do we proceed depth first, or breadth first? The choice we make can account for much of the difference between success and failure.

What do these two terms — *depth first* and *breadth first* — mean in the context of problem solving?

- Depth-first search for problem solutions
- When we search for solutions to a problem by piloting, we begin by implementing one solution in whole or in part. Soon we come to a place where we recognize an obstacle — a sub-problem. If we dive into solving that sub-problem, we're doing a depth-first exploration.
- Breadth-first search for problem solutions
- An alternative to depth-first exploration is breadth-first. In breadth-first exploration, when we encounter a sub-problem, we decide to set aside the sub-problem "for now," and investigate what else might lie in wait up ahead on our original path.

Which approach is better? Easy (but not very helpful) answer: "It depends." We must use our judgment about whether to solve sub-problems as we find them, or whether to set them aside for now to continue work on the larger problem.

Here are Which approach is better?

Easy (but not very helpful)

answer: "It depends."five guidelines for making choices between depth-first and breadth-first approaches to searching for solutions to problems. In what follows I assume that we have two or more candidate solutions and that mathematical modeling doesn't provide much guidance for choosing among them.

- Consider irrevocability
- Fair comparison of different solutions requires assurance that we compare solutions using similar conditions. Some problem solutions alter the environments in which we evaluate those solutions. Solutions that alter the environment irrevocably are obviously troublesome to evaluate, because they make fair evaluation of other solutions difficult. But even more troublesome are solutions that alter the environment without revealing that they have done so.
- Be certain that you understand how each solution alters the evaluation environment.
- Consider the burden of comparing candidate solutions
- A strategy for comparing candidates might seem simple until we consider resource and schedule constraints. Complexity enters because studying the suitability of a candidate solution might have effects on budget or schedule or both. Or making the comparison might consume the time of people with special knowledge who are needed elsewhere.
- Reducing the field of candidate solutions is advantageous because it reduces the risk of irrevocable or hidden alteration of the evaluation environment. More important though, eliminating candidates limits the need to compare them one to another.
- Keep the immediate goal in mind
- While the ultimate goal is finding a solution that meets our needs, when evaluating a particular solution, the immediate goal can differ. For example, if we can eliminate a candidate solution rapidly or with minimal resource expenditure, we can reduce the total cost of evaluating candidates.
- The overall goal of identifying an acceptable solution can sometimes "hijack" the effort to evaluate suitability of a given candidate solution, biasing the evaluator in favor of approving a candidate. Solution evaluation must remain objective. Eliminating a candidate solution can be almost as valuable as finding an acceptable solution.
- Understand the advantages of depth-first searches
- Searching for solutions depth-first elevates the probability of uncovering information that can make the overall search more effective in some circumstances. For example, when investigating an obstacle that one candidate exposes, investigators might discover attributes of candidates that are more likely to render them vulnerable to the same or similar obstacles. And that can make the rest of the search for solutions more effective.
- Likewise, investigating an obstacle for one candidate can uncover information that enables elimination of other candidates. If that seems possible, you might have found a screening test. To determine whether a given candidate fails, apply that test. As you find more kinds of failures, you can add to your catalog of tests. This makes exploring future candidates cheaper.
- Understand the advantages of breadth-first searches
- Searching for solutions breadth-first elevates the probability of uncovering information that can make the overall search more effective, albeit in different circumstances. For example, when investigating an obstacle that one candidate exposes, investigators might be able to predict conditions that expose additional obstacles not yet discovered. Subjecting the candidate solution, or other candidate solutions, to those conditions can expose obstacles earlier in the investigation, speeding discovery of an acceptable solution.
- If it's possible to explore more than one solution at a time, then breadth-first search has clear advantages. In breadth-first search, what we learn from investigating one candidate can probably be applied immediately to the investigation of another. By contrast, in depth-first search, obstacles encountered by one solution are not as likely to apply to other solutions because the path of exploring the solution space can differ dramatically from one candidate to the next.

If only a few candidate solutions are available, the advantages of breadth-first and depth-first investigations probably don't differ by much. But as the number of candidates increases, the value of choosing wisely can be significant. Top Next Issue

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## Related articles

More articles on Problem Solving and Creativity:

- Problem-Solving Preferences
- When people solve problems together, differences in preferred approaches can surface. Some prefer to
emphasize the goal or objective, while others focus on the obstacles. This difference is at once an
asset and annoyance.
- Call in the Right Expert
- When solving a problem is beyond us, we turn to experts, but sometimes we turn to the wrong experts.
That can make the problem even worse. Why? How does this happen? What can we do about it?
- Virtual Brainstorming: I
- When we need to brainstorm, meeting virtually carries a risk that our results might be problematic.
Here's Part I of some steps to take to reduce the risk.
- The Goal Is Not the Path
- Sometimes, when reaching a goal is more difficult than we thought at first, instead of searching for
another way to get there, we adjust the goal. There are alternatives.
- Newtonian Blind Alleys: II
- Some of our decisions don't turn out well. The nature of our errors does vary, but a common class of
errors is due to applying concepts from physics originated by Isaac Newton. One of these is the concept
of spectrum.

See also Problem Solving and Creativity and Project Management for more related articles.

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