Opening your learning space
Starting the app …
Opening your learning space
Starting the app …
Amortized analysis spreads occasional expensive operations across a sequence and gives a worst-case sequence bound without probability.
Free to browse · flashcards or a multiple-choice quiz · import from Essential
Formula
c is cost in an operation unit; potential Φ is nonnegative abstract cost
| Symbol | Meaning | Unit |
|---|---|---|
| Actual cost of operation i | operations or cost | |
| Potential after operation i | potential cost | |
| Amortized cost | cost | |
| Position in the operation sequence | index |
\widehat c_i=c_i+\Phi_i-\Phi_{i-1}For ĉi=ci+Φi−Φi−1, potential terms telescope: Σci=Σĉi+Φ0−Φm. With Φ0=0 and Φm≥0, an upper bound on amortized total cost bounds actual total cost. In a doubling array, some appends copy Θ(n) elements, but m appends require O(m) total copying plus m insertions, so append is O(1) amortized. This is not expected runtime.
For ĉi=ci+Φi−Φi−1, potential terms telescope: Σci=Σĉi+Φ0−Φm. With Φ0=0 and Φm≥0, an upper bound on amortized total cost bounds actual total cost. In a doubling array, some appends copy Θ(n) elements, but m appends require O(m) total copying plus m insertions, so append is O(1) amortized. This is not expected runtime.
Amortized analysis spreads occasional expensive operations across a sequence and gives a worst-case sequence bound without probability.
reale Kosten aus amortisierten Kosten
Use a nonnegative potential and include the initial and final state.
Use the example for Amortized analysis of dynamic arrays to explain which inputs determine the requested output.
The example connects concrete inputs to its result.
Amortized analysis spreads occasional expensive operations across a sequence and gives a worst-case sequence bound without probability.
The relation applies only under its stated assumptions.
Applying the relation to a different case without checking its stated assumptions.
Check the validity range before substituting values: For ĉi=ci+Φi−Φi−1, potential terms telescope: Σci=Σĉi+Φ0−Φm. With Φ0=0 and Φm≥0, an upper bound on amortized total cost bounds actual total cost. In a doubling array, some appends copy Θ(n) elements,.
Formula cluster
Amortized analysis spreads occasional expensive operations across a sequence and gives a worst-case sequence bound without probability.
At capacity 4, append number five copies four values and inserts one, about five units. The next three appends cost one each. Across these four operations the total is eight, or two per append; doubling keeps total copying linear.
Analyze data structures with occasional resizing or rebuilding.
Here is how to work through a typical Amortized analysis of dynamic arrays (amortisierte Kosten = reale Kosten + Potentialänderung) task step by step:
Task
Use the example for Amortized analysis of dynamic arrays to explain which inputs determine the requested output.
Solution path
The example connects concrete inputs to its result.
Amortized analysis spreads occasional expensive operations across a sequence and gives a worst-case sequence bound without probability.
The relation applies only under its stated assumptions.
No. Expected time averages over a probability distribution on inputs or random choices. Amortized analysis bounds total cost for every allowed operation sequence, often using credit or potential. One resize may still cost Θ(n), even though appends are O(1) amortized. This distinction matters when no meaningful probability model exists. Amortized bounds apply to an operation sequence and hold for its worst case. They do not claim every operation is cheap or that inputs are random. An expected value, by contrast, explicitly depends on the chosen probability distribution. No input probability distribution is needed; the bound covers even an adverse operation sequence.
Summing ĉi=ci+Φi−Φi−1 makes intermediate terms cancel: Φ1−Φ0, Φ2−Φ1, and so on. Thus Σĉ=Σc+Φm−Φ0 and Σc=Σĉ+Φ0−Φm. If initial potential is zero and final potential nonnegative, actual total cost is at most amortized total cost. Amortized bounds apply to an operation sequence and hold for its worst case. They do not claim every operation is cheap or that inputs are random. This shows why the reserve held by a state bounds the actual cost of an operation sequence. The telescoping sum tracks credit between consecutive states rather than losing it.
Doubling makes costly copies geometrically less frequent as the array grows. Across m appends, copied capacities form a geometric sum below about 2m, plus m direct writes. Total work is O(m), so amortized append is O(1). Growing by one slot each time copies 1+2+…+m elements and costs Θ(m2). Amortized bounds apply to an operation sequence and hold for its worst case. They do not claim every operation is cheap or that inputs are random. The proof covers every sequence of appends, rather than a single randomly selected run. Regular element writes are included alongside occasional copying costs.
When an array is full, the implementation allocates a larger block and copies existing elements. At size n, that single append can take Θ(n) time and cause a pause. The amortized guarantee concerns sequence total, not per-call latency. Systems with strict latency limits may need incremental migration or another data structure. Amortized bounds apply to an operation sequence and hold for its worst case. They do not claim every operation is cheap or that inputs are random. An amortized bound therefore does not guarantee a strict real-time limit for each operation. Strict latency requirements therefore need a separate bound on the most expensive individual call.
The fifth append meets a full array: four old values are copied and one new value written, about five units. The next three writes cost one each. The four-operation total is eight, averaging two. Future resize costs grow, but occur farther apart; the geometric sum shows constant amortized cost over longer prefixes. Amortized bounds apply to an operation sequence and hold for its worst case. They do not claim every operation is cheap or that inputs are random. The potential function is a proof device and need not exist as extra storage in the program. The chosen potential must cover every allowed state transition, not just the example.
Quanta exam set
Curated exam set for "Amortized analysis of dynamic arrays":
Question (front)
Which formula describes Amortized analysis of dynamic arrays?
Answer in your set
Question (front)
How do you rearrange Amortized analysis of dynamic arrays for reale Kosten aus amortisierten Kosten?
Answer in your set
Question (front)
Which common mistake happens with Amortized analysis of dynamic arrays?
Answer in your set
+ 7 more cards: units, variables, derivation, example, exam task
The formula set contains 10 cards. After importing, review them and start practicing; your ratings determine later review intervals.
This list names subject references and further reading. A page or section number appears when it is part of the reference; these are not verbatim quotations.
Create a curated FSRS exam set for amortisierte Kosten = reale Kosten + Potentialänderung: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
Curated formula set · LaTeX · FSRS spaced repetition · import from Essential