mathematics

Linear search problem

Linear search problem is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Linear search problem rather than just read about it. In short: In computational complexity theory, the linear search problem is an optimal search problem introduced by Richard E. Bellman and independently considered by Anatole Beck.

Key takeaways

Linear search problem belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Linear search problem to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Linear search problem from memory before moving on to harder problems.

Reference excerpt

In computational complexity theory, the linear search problem is an optimal search problem introduced by Richard E. Bellman and independently considered by Anatole Beck.

The problem "An immobile hider is located on the real line according to a known probability distribution. A searcher, whose maximal velocity is one, starts from the origin and wishes to discover the hider in minimal expected time. It is assumed that the searcher can change the direction of his motion without any loss of time. It is also assumed that the searcher cannot see the hider until he actually reaches the point at which the hider is located and the time elapsed until this moment is the duration of the game." The problem is to find the hider in the shortest time possible. Generally, since the hider could be on either side of the searcher and an arbitrary distance away, the searcher has to oscillate back and forth, i.e., the searcher has to go a distance x1 in one direction, return to the origin and go distance x2 in the other direction, etc., (the length of the n-th step being denoted by xn). (However, an optimal solution need not have a first step and could start with an infinite number of small 'oscillations'.) This problem is usually called the linear search problem and a search plan is called a trajectory. The linear search problem for a general probability distribution is unsolved. However, there exists a dynamic programming algorithm that produces a solution for any discrete distribution and also an approximate solution, for any probability distribution, with any desired accuracy. The linear search problem was solved by Anatole Beck and Donald J. Newman (1970) as a two-person zero-sum game. Their minimax trajectory is to double the distance on each step and the optimal strategy is a mixture of trajectories that increase the distance by some fixed constant. This solution gives search strategies that are not sensitive to assumptions concerning the distribution of the target. Thus, it also presents an upper bound for a worst-case scenario. This solution was obtained in the framework of an online algorithm by Shmuel Gal, who also generalized this result to a set of concurrent rays. The best online competitive ratio for the search on the line is 9 but it can be reduced to 4.6 by using a randomized strategy. Demaine et al. gave an online solution with a turn cost. These results were rediscovered in the 1990s by computer scientists as the cow path problem.

Worked examples

Example 1 — a first encounter with Linear search problem

Start with the simplest possible case. Write down what Linear search problem claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Linear search problem before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Linear search problem ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Linear search problem

In research: Linear search problem appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Linear search problem in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Linear search problem is common in secondary-school and first-year university syllabi. It links to neighbouring topics Computational problems, Mathematical optimization, so understanding it makes those chapters shorter.
In everyday life: Look for Linear search problem outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Linear search problem in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Linear search problem means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Linear search problem out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Linear search problem in simple terms?

In computational complexity theory, the linear search problem is an optimal search problem introduced by Richard E. Bellman and independently considered by Anatole Beck.

Why does Linear search problem matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Linear search problem?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Linear search problem.