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Algorithm characteristic in computations
In computational complexity theory, the average-case complexity of an algorithm is the amount of some computational resource (typically time) used by the
Average-case_complexity
Measures of how efficiently algorithms use resources
The worst-case analysis is related to the worst-case complexity. Many algorithms with bad worst-case performance have good average-case performance
Best,_worst_and_average_case
Amount of resources to perform an algorithm
worst-case complexity is the maximum of the complexity over all inputs of size n, and the average-case complexity is the average of the complexity over
Computational_complexity
Estimate of time taken for running an algorithm
Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes
Time_complexity
Upper bound on resources required by an algorithm
complexity of an algorithm should be contrasted with its average-case complexity, which is an average measure of the amount of resources the algorithm uses
Worst-case_complexity
Inherent difficulty of computational problems
following complexities: Best-case complexity: This is the complexity of solving the problem for the best input of size n {\displaystyle n} . Average-case complexity:
Computational complexity theory
Computational_complexity_theory
American computer scientist
17. Impagliazzo, Russell (April 17, 1995). "A Personal View of Average-Case Complexity". Klarreich, Erica (April 18, 2022). "Which Computational Universe
Russell_Impagliazzo
probabilistic complexity estimates are the average-case complexity and the almost-always complexity. To obtain the average-case complexity, given an input
Probabilistic analysis of algorithms
Probabilistic_analysis_of_algorithms
Generic-case complexity is a subfield of computational complexity theory that studies the complexity of computational problems on "most inputs". Generic-case
Generic-case_complexity
Soviet-American mathematician
his work in randomness in computing, algorithmic complexity and intractability, average-case complexity, foundations of mathematics and computer science
Leonid_Levin
Hypothesis in computational complexity theory
Computational Complexity: A Modern Approach. Cambridge University Press. pp. 362–363. ISBN 9780521424264.. Feige, Uriel (2002). "Relations between average case complexity
Computational hardness assumption
Computational_hardness_assumption
Subfield of computational complexity theory
direction studies the fine-grained complexity of average-case inputs of random instances rather than adversarial worst-case ones. Ball, Rosen, Sabin, and Vasudevan
Fine_grained_complexity
Algorithm analysis method
worst-case complexity. Average-case analysis was first introduced to overcome the limitations of worst-case analysis. However, the resulting average-case complexity
Smoothed_analysis
Unsolved problem in computer science
JSTOR 2580891. R. Impagliazzo, "A personal view of average-case complexity", p. 134, 10th Annual Structure in Complexity Theory Conference (SCT'95), 1995. "Tentative
P_versus_NP_problem
Equivalence of average-case and expected complexity
be obtained by a deterministic algorithm on a random input (its average-case complexity), for a probability distribution on inputs chosen to be as hard
Yao's_principle
Sorting algorithm which uses multiple comparison intervals
unknown whether Shellsort can reach this asymptotic order of average-case complexity, which is optimal for comparison sorts. The lower bound was improved
Shellsort
Method of searching a sorted array
On average, Fibonacci search performs about 4% more comparisons than binary search, and it has an average-case complexity and worst-case complexity of
Fibonacci_search_technique
Computational problem used in cryptography
applications, worst case complexity is not sufficient, and we need to guarantee cryptographic construction are hard based on average case complexity. A full rank
Short integer solution problem
Short_integer_solution_problem
Software size estimation technique
complexity of the use cases for the system. To find the UUCW for a system, each of the use cases must be identified and classified as Simple, Average
Use_case_points
Variant of heap data structure
worst-case time complexity of O(log n). For a random heap, and for repeated insertions, the insertion operation has an average-case complexity of O(1)
Binary_heap
String search algorithm
for time in order to obtain an average-case complexity of O(n) on random text, although it has O(nm) in the worst case, where the length of the pattern
Boyer–Moore–Horspool algorithm
Boyer–Moore–Horspool_algorithm
Unproven computational hardness assumption
In computational complexity theory, the exponential time hypothesis (ETH) states that satisfiability of 3-CNF Boolean formulas (3-SAT) cannot be solved
Exponential_time_hypothesis
American computer scientist
started to work on various aspects of computational complexity theory including average case complexity. He became one of the founders of the emerging field
Yuri_Gurevich
Sorting algorithm
having constant size). This allows O ( n ) {\displaystyle O(n)} average time complexity even without uniformly distributed input. The most common variant
Bucket_sort
Problem of inverting exponentiation in groups
algorithm known for the worst case, but the average-case complexity can be shown to be about as hard as the worst case using random self-reducibility
Discrete_logarithm
Optimization problem in computer science
NP-hard and discovered some connections between the worst-case complexity and average-case complexity of some lattice problems. Building on these results,
Lattice_problem
the average with respect to the induced distribution on yi, the average-case complexity of f is the same (within polynomial factors) as the worst-case randomized
Random_self-reducibility
Counterintuitive mathematical object
atypical behavior from the algorithm, such as a violation of its average case complexity, or even its correctness. For example, hash tables generally have
Pathological_(mathematics)
Numerical linear algebra algorithm
O(n) complexity. However, this will happen on average only once per rotation. Thus, each rotation has O(n) and one sweep O(n3) average-case complexity, which
Jacobi_eigenvalue_algorithm
Algorithm for the kth smallest element in an array
This reduces the average complexity from O ( n log n ) {\displaystyle O(n\log n)} to O ( n ) {\displaystyle O(n)} , with a worst case of O ( n 2 ) {\displaystyle
Quickselect
Sorting algorithm
values form a single pile and both phases run in O(n) time. The average-case complexity is still O(n log n): any uniformly random sequence of values will
Patience_sorting
Divergent sum of positive unit fractions
its average-case complexity (with the assumption that all input permutations are equally likely) or in its expected time analysis of worst-case inputs
Harmonic_series_(mathematics)
Associative array for storing key–value pairs
according to a probing sequence. In a well-dimensioned hash table, the average time complexity for each lookup is independent of the number of elements stored
Hash_table
Number taken as representative of a list of numbers
In mathematics, an average of a collection or group is a value that is most central, common, or typical in some sense, and represents its overall position
Average
Type of cryptographic software obfuscation
situations about average-case complexity: Algorithmica: In this case P = NP, but iO exists. Heuristica: In this case NP problems are easy on average; iO does
Indistinguishability obfuscation
Indistinguishability_obfuscation
Algorithm for linear programming
worst-case complexity has led to the development of other measures of complexity. The simplex algorithm has polynomial-time average-case complexity under
Simplex_algorithm
Task of transforming a deterministic finite automaton
solving the problem, and for certain distributions of inputs its average-case complexity is even better, O(n log log n). Once Hopcroft's algorithm has been
DFA_minimization
French computer scientist (1948–2011)
methods for analyzing the computational complexity of algorithms, including the theory of average-case complexity. He introduced the theory of analytic
Philippe_Flajolet
Fast approximate median algorithm
asymptotic worst-case complexity quite well, it is typically outperformed in practice by instead choosing random pivots for its average O ( n ) {\displaystyle
Median_of_medians
Simple sorting algorithm using comparisons
cases where the list is already mostly sorted (having a small number of inversions), despite the fact that it has worse average-case time complexity.
Bubble_sort
Data structure for storing non-overlapping sets
also developed one-pass Find algorithms that retain the same worst-case complexity but are more efficient in practice. These are called path splitting
Disjoint-set_data_structure
Binary tree selected at random
these trees. Random binary trees have been used for analyzing the average-case complexity of data structures based on binary search trees. For this application
Random_binary_tree
Sorting algorithm
sorted are distinct, the expected number of comparisons performed in the average case by randomized bogosort is asymptotically equivalent to ( e − 1 ) n !
Bogosort
Search algorithm used in sorted arrays
the array. Therefore, the space complexity of binary search is O(1) in the word RAM model of computation. The average number of iterations performed by
Binary_search
Relaxed variant of the sorting problem
sequence of calls has average-case complexity O(n + k log k), which is asymptotically equivalent to O(n + k log n). The worst-case time is quadratic, but
Partial_sorting
Hungarian-American computer scientist
Almanach, 1986, Budapest. Ajtai, Miklós (1998). "Worst-case complexity, average-case complexity and lattice problems". Documenta Mathematica: 421–428.
Miklós_Ajtai
Sorting algorithm
efficient for data sets that are already substantially sorted: the time complexity is O(kn) when each element in the input is no more than k places away
Insertion_sort
American mathematician
in 2011 his Ph.D. with advisor Madhu Sudan from MIT with thesis Average-Case Complexity of Detecting Cliques. From 2010 to 2013 Rossman was a postdoc at
Benjamin_Rossman
Impagliazzo, Russell (1995), "A personal view of average-case complexity", Proc. Tenth Annual Structure in Complexity Theory Conference (SCT'95), pp. 134–147,
Glossary of mathematical jargon
Glossary_of_mathematical_jargon
Binary tree derived from a sequence of numbers
In particular, Vuillemin used these structures to analyze the average-case complexity of concatenation and splitting operations on binary search trees
Cartesian_tree
Probabilistic data structure
{\mathcal {O}}(\log n)} average complexity for search as well as O ( log n ) {\displaystyle {\mathcal {O}}(\log n)} average complexity for insertion within
Skip_list
Unit hypercube of variable dimension whose corners have been perturbed
have polynomial-time complexity for linear optimization. The Klee–Minty cube has also inspired research on average-case complexity. When eligible pivots
Klee–Minty_cube
Rooted binary tree data structure
guaranteed worst-case performance. The basic operations include: search, traversal, insert and delete. BSTs with guaranteed worst-case complexities perform better
Binary_search_tree
Data structure
the key, the system concludes that the key doesn't exist. Point lookup complexity is O ( L ) {\displaystyle O(L)} without Bloom filters, as each level must
Log-structured_merge-tree
Multidimensional search tree for points in k dimensional space
O(n\log n)} time in the best case. An algorithm that builds a balanced k-d tree to sort points has a worst-case complexity of O ( k n log ( n ) ) {\displaystyle
K-d_tree
Technique for reducing number of solutions
of Valiant-Vazirani in their search-to-decision reduction for average-case complexity. Avi Wigderson used the isolation lemma in 1994 to give a randomized
Isolation_lemma
Information-based complexity (IBC) studies optimal algorithms and computational complexity for the continuous problems that arise in physical science,
Information-based_complexity
Mathematical game
In game theory, "guess 2/3 of the average" is a game where players simultaneously select a real number between 0 and 100, inclusive. The winner of the
Guess_2/3_of_the_average
Sorting algorithm
which is of complexity O(n2) in terms of number of comparisons. Among quadratic sorting algorithms (sorting algorithms with a simple average-case of Θ(n2))
Selection_sort
Topics referred to by the same term
military armoured vehicle. AVGP may refer to: AvgP (mathematics) in average-case complexity AV-GP (contest), the AV Grand Prix, a Japanese adult video contest
AVGP_(disambiguation)
Data structure used for indexing spatial information
tree, inspired by the concept of balancing a B-tree. Worst case query and delete complexity are thus identical to the R-Tree. The insertion strategy to
R*-tree
Sorting algorithm
The complexity of the cocktail shaker sort in big O notation is O ( n 2 ) {\displaystyle O(n^{2})} for both the worst case and the average case, but
Cocktail_shaker_sort
Tree data structure that partitions a 2D area
in an area may be stored as a quadtree, with each leaf node storing the average temperature over the subregion it represents. The point quadtree is an
Quadtree
Divide and conquer sorting algorithm
lowers the average complexity to linear or O(n) time, which is optimal for selection, but the selection algorithm is still O(n2) in the worst case. A variant
Quicksort
American computer scientist
random variate generation; prediction and machine learning; and average-case complexity. Vitter is a Fellow of the National Academy of Inventors (NAI)
Jeffrey_Vitter
String searching algorithm
the combined length of the pattern and text, although its worst-case time complexity is the product of the two lengths. To find multiple matches, the
Rabin–Karp_algorithm
Relative value assigned to a diagnosis-related group of patients
system. The relative average value assigned to each group is its CMI. The CMI of a hospital reflects the diversity, clinical complexity and the needs for
Case_mix_index
Data structures used in spatial indexing
variant of the R-tree is worst-case optimal, but due to its increased complexity it has remained confined to theoretical study and has not received much
R-tree
Method for algorithm analysis in computer science
analysis is that looking at the worst-case run time can be too pessimistic. Instead, amortized analysis averages the running times of operations in a sequence
Amortized_analysis
hard worst-case or average-case complexity. Worst-case complexity measures the difficulty of solving pathological cases rather than typical cases of the underlying
Security of cryptographic hash functions
Security_of_cryptographic_hash_functions
Data structure
Development of operations it supports were primarily motivated by use in that case.[citation needed] A Fenwick tree is an implicit tree where nodes are consecutively
Fenwick_tree
Abstract data type
two implementations. The first one only achieves O(1) per operation on average. That is, the amortized time is O(1), but individual operations can take
Queue_(abstract_data_type)
often in practice. Because of this, researchers have also examined average-case complexity, approximation methods, and restrictions on voter preferences to
Computational_social_choice
Self-balancing binary search tree
Additionally, after finding a node for insertion and deletion, the amortized complexity of the tree restructuring operations is constant. Adding or deleting the
WAVL_tree
Proof method
average-case complexity of Shellsort", Journal of the ACM (JACM), 47:5(2000) 905–911. doi:10.1145/355483.355488 P.M.B. Vitanyi (2018), On the average‐case
Incompressibility_method
conclusions. Woźniakowski's 1991 paper, showing the connection between average case complexity of integration and QMC, led to new interest in QMC. Woźniakowski's
Quasi-Monte Carlo methods in finance
Quasi-Monte_Carlo_methods_in_finance
Professor of operations research at MIT
introduced Overlap Gap Property, a new method for understanding average-case complexity, based on theory of spin glasses. He has served in editorial roles
David_Gamarnik
Divide and conquer sorting algorithm
terms of moves, merge sort's worst case complexity is O(n log n) - the same complexity as quicksort's best case. Merge sort is more efficient than quicksort
Merge_sort
Search tree data structure
has the complexity of O ( m ) {\displaystyle O(m)} , whereas an imperfect hash function may have numerous colliding keys, and the worst-case lookup speed
Trie
Code-breaking game
for the codebreaker gives the minimal number of moves for worst and average case and in the sense of a minimax value of a zero-sum game in game theory
Mastermind_(board_game)
Self-balancing binary search tree data structure
self-balancing. Rebalancing (i.e. color changes) has a worst-case time complexity of O(logn) and average of O(1), though these are very quick in practice. Additionally
Red–black_tree
Data structure
insert or delete an element as half of the array will need to be shifted on average. To overcome this problem, elements inside a node can be organized in a
B+_tree
Largest independent set of paired elements
{\displaystyle k} , and applying Yao's principle relating expected and average-case complexity, one can show that any deterministic or randomized algorithm for
Matroid_parity_problem
Attack against cryptographic algorithms
must be below the chance of discovering it randomly (that is, the average case complexity of 2 n {\displaystyle 2^{n}} of the rotational cryptanalytic attack
Rotational_cryptanalysis
Computational complexity
in computer science In computational complexity theory, NL (Nondeterministic Logarithmic-space) is the complexity class containing decision problems that
NL_(complexity)
Type of sorting algorithm
better worst case complexity when a self-balancing tree is used, but even more overhead. Adding one item to a binary search tree is on average an O(log n)
Tree_sort
Algorithm that arranges lists in order
Sorting algorithms can be classified by: Computational complexity Best, worst and average case behavior in terms of the size of the list. For typical
Sorting_algorithm
Study of resources used by an algorithm
the analysis of algorithms is the process of finding the computational complexity of algorithms—the amount of time, storage, or other resources needed to
Analysis_of_algorithms
Concept in computational complexity theory
time; that is, if they lie in the complexity class P. In modern terms, it identifies tractable problems with the complexity class P. Formally, to say that
Cobham's_thesis
Sequentially looking in an array
likely to be searched, then linear search has an average case of n+1/2 comparisons, but the average case can be affected if the search probabilities for
Linear_search
Data structure in computer science
size can be continually reduced by seeking out a bottom-level node and averaging its bit data up into a leaf node, pruning part of the tree. Once sampling
Octree
Measure of complexity of real-valued functions
Rademacher complexity 0. The set A = { ( 1 , 1 ) , ( 1 , 2 ) } ⊆ R 2 {\displaystyle A=\{(1,1),(1,2)\}\subseteq \mathbb {R} ^{2}} has average width 1 /
Rademacher_complexity
Probabilistic primality test
2307/2008231. JSTOR 2008231. I. Damgård; P. Landrock; C. Pomerance (1993). "Average case error estimates for the strong probable prime test". Mathematics of Computation
Solovay–Strassen primality test
Solovay–Strassen_primality_test
Russian-American mathematician
Myasnikov, Alexei; Schupp, Paul; Shpilrain, Vladimir (2005). "Average-case complexity and decision problems in group theory". Advances in Mathematics
Ilya_Kapovich
Hybrid sorting algorithm based on insertion sort and merge sort
Peter McIlroy's 1993 paper "Optimistic Sorting and Information Theoretic Complexity". Timsort was designed to take advantage of runs of consecutive ordered
Timsort
Computational benchmark
Michael J.; Montanaro, Ashley; Shepherd, Dan J. (2016-08-18). "Average-case complexity versus approximate simulation of commuting quantum computations"
Quantum_supremacy
Number of bits that differ between two strings
strings over an alphabet of size q ≥ 2 the Hamming distance is applied in case of the q-ary symmetric channel, while the Lee distance is used for phase-shift
Hamming_distance
In-place, comparison-based sorting algorithm
Leapfrogging Samplesort: A Class of O(n log22 n) Worst-Case Complexity and O(n log n) Average-Case Complexity Sorting Algorithms". arXiv:1801.09431 [cs.DS]. Chen
Proportion_extend_sort
Function for sorting in C++ standard library
language standard and may vary across implementations, but the worst-case asymptotic complexity of the function is specified: a call to sort must perform no more
Sort_(C++)
Searching algorithm
(2): 130–146. doi:10.1147/rd.12.0130. Simon Yuan. "Understanding The Complexity Of Interpolation Search, Seminar Advanced Algorithms and Data Structures"
Interpolation_search
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