science

Monotone priority queue

Monotone priority queue is a science 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 Monotone priority queue rather than just read about it. In short: In computer science, a monotone priority queue is a variant of the priority queue abstract data type in which the priorities of extracted items are required to form a monotonic sequence. That is, for a priority queue in which each successively extracted item is the one with the minimum priority (a min-heap), the minimum priority should be monotonically increasing.

Key takeaways

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

Reference excerpt

In computer science, a monotone priority queue is a variant of the priority queue abstract data type in which the priorities of extracted items are required to form a monotonic sequence. That is, for a priority queue in which each successively extracted item is the one with the minimum priority (a min-heap), the minimum priority should be monotonically increasing. Conversely for a max-heap the maximum priority should be monotonically decreasing. The assumption of monotonicity arises naturally in several applications of priority queues, and can be used as a simplifying assumption to speed up certain types of priority queues. A necessary and sufficient condition on a monotone priority queue is that one never attempts to add an element with lower priority than the most recently extracted one.

Applications Monotone priority queues arise naturally when arranging events in order of time, such as network timeouts or discrete event simulation. An event can cause some action to be scheduled at some time in the future, but (real or simulated) causality makes attempts to schedule actions in the past meaningless. In Dijkstra's algorithm for the shortest path problem, vertices of a given weighted graph are extracted in increasing order by their distance from the starting vertex, and a priority queue is used to determine the closest remaining vertex to the starting vertex. Therefore, in this application, the priority queue operations are monotonic. Similarly, in sweep line algorithms in computational geometry, events at which the sweep line crosses a point of interest are prioritized by the coordinate of the crossed point, and these events are extracted in monotonic ordering. A monotonic extraction order also occurs in the best-first version of branch and bound.

Data structures Any priority queue that can handle non-monotone extraction operations can also handle monotone extractions, but some priority queues are specialized to work only for monotone extractions or work better when the extractions are monotone. For instance, the bucket queue is a simple priority queue data structure consisting of an array indexed by priority, where each array cell contains a bucket of items with that priority. An extract-min operation performs a sequential search for the first non-empty bucket and chooses an arbitrary item in that bucket. For non-monotone extractions, each extract-min operation takes time (in the worst case) proportional to the array length (the number of distinct priorities). However, when used as a monotone priority queue, the search for the next non-empty bucket can begin at the priority of the most recent previous extract-min operation rather than at the start of the array. This optimization causes the total time for a sequence of operations to be proportional to the sum of the number of operations and the length of the array, rather than (as in the non-monotonic case) the product of these two quantities. Cherkassky, Goldberg & Silverstein (1999) describe a more complicated scheme called a Heap-on-top (HOT) queue for monotone priority queues with integer priorities, based on multilevel bucketing together with a conventional heap priority queue. Using this method they obtain a structure that can maintain items with integer priorities in a range from 0 to a parameter C. The hot queue uses constant time per insertion or decrease-priority operation and amortized time O ( ( log ⁡ C ) 1 / 3 ( log ⁡ log ⁡ C ) 1 / 2 ) {\displaystyle O((\log C)^{1/3}(\log \log C)^{1/2})} per extract-min operation. Another related structure of Raman (1996) allows the priorities to be machine integers, and again allows constant-time insertion and decrease-priority operations, with extract-min operations on a priority queue of n items taking amortized time O ( log ⁡ n log ⁡ log ⁡ n ) {\displaystyle O({\sqrt {\log n\log \log n}})} . These results lead to a corresponding speedup in Dijkstra's algorithm for graphs with integer edge weights.

References

Worked examples

Example 1 — a first encounter with Monotone priority queue

Start with the simplest possible case. Write down what Monotone priority queue claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, 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 Monotone priority queue 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 Monotone priority queue 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 Monotone priority queue

In research: Monotone priority queue appears in science 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 Monotone priority queue 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: Monotone priority queue is common in secondary-school and first-year university syllabi. It links to neighbouring topics Priority queues, so understanding it makes those chapters shorter.
In everyday life: Look for Monotone priority queue 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 Monotone priority queue in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Monotone priority queue 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 Monotone priority queue out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Monotone priority queue in simple terms?

Why does Monotone priority queue matter?

Because it connects several science 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 Monotone priority queue?

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 Monotone priority queue.