Point Lookout: a free weekly publication of Chaco Canyon Consulting
Volume 22, Issue 16;   April 27, 2022:

# Depth First or Breadth First?

When investigating candidate solutions to a problem, we tend to focus first on what we believe is the "best bet." But a more systematic approach can sometimes yield dramatic advantages by reducing the cost of the investigation and the time it requires.

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.
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?
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.
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.

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## Related articles

More articles on Problem Solving and Creativity:

Workplace Barn Raisings
Until about 75 years ago, barn raising was a common custom in the rural United States. People came together from all parts of the community to help construct one family's barn. Although the custom has largely disappeared in rural communities, we can still benefit from the barn raising approach in problem-solving organizations.
Breaking the Rules
Many outstanding advances are due to those who broke rules to get things done. And some of those who break rules get fired or disciplined. When is rule breaking a useful tactic?
New Ideas: Generation
When groups work together to solve problems, they employ three processes repeatedly: they generate ideas, they judge those ideas, and they experiment with those ideas. We first examine idea generation.
Design Errors and Group Biases
Design errors can cause unwanted outcomes, but they can also lead to welcome surprises. The causes of many design errors are fundamental attributes of the way groups function. Here is Part II of our exploration.
Cost Concerns: Comparisons
When we assess the costs of different options for solving a problem, we must take care not to commit a variety of errors in approach. These errors can lead to flawed decisions. One activity at risk for error is comparing the costs of two options.

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

## Forthcoming issues of Point Lookout

Coming July 13: What Do We Actually Know?
Precision in both writing and speech can be critical in determining the success of collaborations in the modern workplace. Precision is especially important when we distinguish between what we surmise or assume and what we actually know. Available here and by RSS on July 13.
And on July 20: Overt Verbal Abuse at Work
Verbal abuse in the workplace involves using written or spoken language to disparage, to disadvantage, or to otherwise harm others. Perpetrators tend to favor tactics that they can subsequently deny having used to harm anyone. Available here and by RSS on July 20.

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