We summarize the performance characteristics of classic algorithms anddata structures for sorting, priority queues, symbol tables, and graph processing.
We also summarize some of the mathematics useful in the analysis of algorithms, including commonly encountered functions;useful formulas and appoximations; properties of logarithms;asymptotic notations; and solutions to divide-and-conquer recurrences.
ALGORITHM CODE TIME SPACE; path: DFS: DepthFirstPaths.java: E + V: V: shortest path (fewest edges) BFS: BreadthFirstPaths.java: E + V: V: cycle: DFS: Cycle.java: E + V: V: directed path: DFS: DepthFirstDirectedPaths.java: E + V: V: shortest directed path (fewest edges) BFS: BreadthFirstDirectedPaths.java: E + V: V: directed cycle: DFS: DirectedCycle.java: E + V: V: topological sort: DFS: Topological.java: E + V: V. In computer science, the time complexity of an algorithm quantifies the amount of time taken by an algorithm to run as a function of the length of the string representing the input. The time complexity of an algorithm is commonly expressed using big O notation, which excludes coefficients and lower order terms. Algorithmic complexity is a measure of how long an algorithm would take to complete given an input of size n. If an algorithm has to scale, it should compute the result within a finite and practical time bound even for large values of n. For this reason, complexity is calculated asymptotically as n approaches infinity. While complexity is usually in terms of time, sometimes complexity is also.
Jai ho song in english lyrics. Wartune 2013.
Sorting.
The table below summarizes the number of compares for a variety of sortingalgorithms, as implemented in this textbook.It includes leading constants but ignores lower-order terms.| ALGORITHM | CODE | STABLE | BEST | AVERAGE | WORST | REMARKS | |
|---|---|---|---|---|---|---|---|
| selection sort | Selection.java | ✔ | ½ n 2 | ½ n 2 | ½ n 2 | n exchanges; quadratic in best case | |
| insertion sort | Insertion.java | ✔ | ✔ | n | ¼ n 2 | ½ n 2 | use for small or partially-sorted arrays |
| bubble sort | Bubble.java | ✔ | ✔ | n | ½ n 2 | ½ n 2 | rarely useful; use insertion sort instead |
| shellsort | Shell.java | ✔ | n log3n | unknown | c n 3/2 | tight code; subquadratic | |
| mergesort | Merge.java | ✔ | ½ n lg n | n lg n | n lg n | n log n guarantee; stable | |
| quicksort | Quick.java | ✔ | n lg n | 2 n ln n | ½ n 2 | n log n probabilistic guarantee; fastest in practice | |
| heapsort | Heap.java | ✔ | n† | 2 n lg n | 2 n lg n | n log n guarantee; in place | |
| †n lg n if all keys are distinct | |||||||
Clean master free official site.
Priority queues.
The table below summarizes the order of growth of the running time ofoperations for a variety of priority queues, as implemented in this textbook.It ignores leading constants and lower-order terms.Except as noted, all running times are worst-case running times.| DATA STRUCTURE | CODE | INSERT | MIN | DELETE | MERGE | ||
|---|---|---|---|---|---|---|---|
| array | BruteIndexMinPQ.java | 1 | n | n | 1 | 1 | n |
| binary heap | IndexMinPQ.java | log n | log n | 1 | log n | log n | n |
| d-way heap | IndexMultiwayMinPQ.java | logdn | d logdn | 1 | logdn | d logdn | n |
| binomial heap | IndexBinomialMinPQ.java | 1 | log n | 1 | log n | log n | log n |
| Fibonacci heap | IndexFibonacciMinPQ.java | 1 | log n† | 1 | 1 † | log n† | 1 |
| † amortized guarantee | |||||||
Symbol tables.
The table below summarizes the order of growth of the running time ofoperations for a variety of symbol tables, as implemented in this textbook.It ignores leading constants and lower-order terms.| worst case | average case | ||||||
|---|---|---|---|---|---|---|---|
| DATA STRUCTURE | CODE | SEARCH | INSERT | DELETE | SEARCH | INSERT | DELETE |
| sequential search (in an unordered list) | SequentialSearchST.java | n | n | n | n | n | n |
| binary search (in a sorted array) | BinarySearchST.java | log n | n | n | log n | n | n |
| binary search tree (unbalanced) | BST.java | n | n | n | log n | log n | sqrt(n) |
| red-black BST (left-leaning) | RedBlackBST.java | log n | log n | log n | log n | log n | log n |
| AVL | AVLTreeST.java | log n | log n | log n | log n | log n | log n |
| hash table (separate-chaining) | SeparateChainingHashST.java | n | n | n | 1 † | 1 † | 1 † |
| hash table (linear-probing) | LinearProbingHashST.java | n | n | n | 1 † | 1 † | 1 † |
| † uniform hashing assumption | |||||||
Graph processing.
The table below summarizes the order of growth of the worst-case running time and memory usage (beyond the memory for the graph itself)for a variety of graph-processing problems, as implemented in this textbook.It ignores leading constants and lower-order terms.All running times are worst-case running times.Time Complexity Of Graph Algorithms
| PROBLEM | ALGORITHM | CODE | TIME | SPACE |
|---|---|---|---|---|
| path | DFS | DepthFirstPaths.java | E + V | V |
| shortest path (fewest edges) | BFS | BreadthFirstPaths.java | E + V | V |
| cycle | DFS | Cycle.java | E + V | V |
| directed path | DFS | DepthFirstDirectedPaths.java | E + V | V |
| shortest directed path (fewest edges) | BFS | BreadthFirstDirectedPaths.java | E + V | V |
| directed cycle | DFS | DirectedCycle.java | E + V | V |
| topological sort | DFS | Topological.java | E + V | V |
| bipartiteness / odd cycle | DFS | Bipartite.java | E + V | V |
| connected components | DFS | CC.java | E + V | V |
| strong components | Kosaraju–Sharir | KosarajuSharirSCC.java | E + V | V |
| strong components | Tarjan | TarjanSCC.java | E + V | V |
| strong components | Gabow | GabowSCC.java | E + V | V |
| Eulerian cycle | DFS | EulerianCycle.java | E + V | E + V |
| directed Eulerian cycle | DFS | DirectedEulerianCycle.java | E + V | V |
| transitive closure | DFS | TransitiveClosure.java | V (E + V) | V 2 |
| minimum spanning tree | Kruskal | KruskalMST.java | E log E | E + V |
| minimum spanning tree | Prim | PrimMST.java | E log V | V |
| minimum spanning tree | Boruvka | BoruvkaMST.java | E log V | V |
| shortest paths (nonnegative weights) | Dijkstra | DijkstraSP.java | E log V | V |
| shortest paths (no negative cycles) | Bellman–Ford | BellmanFordSP.java | V (V + E) | V |
| shortest paths (no cycles) | topological sort | AcyclicSP.java | V + E | V |
| all-pairs shortest paths | Floyd–Warshall | FloydWarshall.java | V 3 | V 2 |
| maxflow–mincut | Ford–Fulkerson | FordFulkerson.java | EV (E + V) | V |
| bipartite matching | Hopcroft–Karp | HopcroftKarp.java | V ½ (E + V) | V |
| assignment problem | successive shortest paths | AssignmentProblem.java | n 3 log n | n 2 |
Commonly encountered functions.
Here are some functions that are commonly encounteredwhen analyzing algorithms.| FUNCTION | NOTATION | DEFINITION |
|---|---|---|
| floor | ( lfloor x rfloor ) | greatest integer (; le ; x) |
| ceiling | ( lceil x rceil ) | smallest integer (; ge ; x) |
| binary logarithm | ( lg x) or (log_2 x) | (y) such that (2^{,y} = x) |
| natural logarithm | ( ln x) or (log_e x ) | (y) such that (e^{,y} = x) |
| common logarithm | ( log_{10} x ) | (y) such that (10^{,y} = x) |
| iterated binary logarithm | ( lg^* x ) | (0) if (x le 1;; 1 + lg^*(lg x)) otherwise |
| harmonic number | ( H_n ) | (1 + 1/2 + 1/3 + ldots + 1/n) |
| factorial | ( n! ) | (1 times 2 times 3 times ldots times n) |
| binomial coefficient | ( n choose k ) | ( frac{n!}{k! ; (n-k)!}) |
Time Complexity Of Algorithms
Useful formulas and approximations.
Here are some useful formulas for approximations that are widely used in the analysis of algorithms.- Harmonic sum: (1 + 1/2 + 1/3 + ldots + 1/n sim ln n)
- Triangular sum: (1 + 2 + 3 + ldots + n = n , (n+1) , / , 2 sim n^2 ,/, 2)
- Sum of squares: (1^2 + 2^2 + 3^2 + ldots + n^2 sim n^3 , / , 3)
- Geometric sum: If (r neq 1), then(1 + r + r^2 + r^3 + ldots + r^n = (r^{n+1} - 1) ; /; (r - 1))
- (r = 1/2): (1 + 1/2 + 1/4 + 1/8 + ldots + 1/2^n sim 2)
- (r = 2): (1 + 2 + 4 + 8 + ldots + n/2 + n = 2n - 1 sim 2n), when (n) is a power of 2
- Stirling's approximation: (lg (n!) = lg 1 + lg 2 + lg 3 + ldots + lg n sim n lg n)
- Exponential: ((1 + 1/n)^n sim e; ;;(1 - 1/n)^n sim 1 / e)
- Binomial coefficients: ({n choose k} sim n^k , / , k!) when (k) is a small constant
- Approximate sum by integral: If (f(x)) is a monotonically increasing function, then( displaystyle int_0^n f(x) ; dx ; le ; sum_{i=1}^n ; f(i) ; le ; int_1^{n+1} f(x) ; dx)
Properties of logarithms.
- Definition: (log_b a = c) means (b^c = a).We refer to (b) as the base of the logarithm.
- Special cases: (log_b b = 1,; log_b 1 = 0 )
- Inverse of exponential: (b^{log_b x} = x)
- Product: (log_b (x times y) = log_b x + log_b y )
- Division: (log_b (x div y) = log_b x - log_b y )
- Finite product: (log_b ( x_1 times x_2 times ldots times x_n) ; = ; log_b x_1 + log_b x_2 + ldots + log_b x_n)
- Changing bases: (log_b x = log_c x ; / ; log_c b )
- Rearranging exponents: (x^{log_b y} = y^{log_b x})
- Exponentiation: (log_b (x^y) = y log_b x )
Aymptotic notations: definitions.
| NAME | NOTATION | DESCRIPTION | DEFINITION |
|---|---|---|---|
| Tilde | (f(n) sim g(n); ) | (f(n)) is equal to (g(n)) asymptotically (including constant factors) | ( ; displaystyle lim_{n to infty} frac{f(n)}{g(n)} = 1) |
| Big Oh | (f(n)) is (O(g(n))) | (f(n)) is bounded above by (g(n)) asymptotically (ignoring constant factors) | there exist constants (c > 0) and (n_0 ge 0) such that (0 le f(n) le c cdot g(n)) forall (n ge n_0) |
| Big Omega | (f(n)) is (Omega(g(n))) | (f(n)) is bounded below by (g(n)) asymptotically (ignoring constant factors) | ( g(n) ) is (O(f(n))) |
| Big Theta | (f(n)) is (Theta(g(n))) | (f(n)) is bounded above and below by (g(n)) asymptotically (ignoring constant factors) | ( f(n) ) is both (O(g(n))) and (Omega(g(n))) |
| Little oh | (f(n)) is (o(g(n))) | (f(n)) is dominated by (g(n)) asymptotically (ignoring constant factors) | ( ; displaystyle lim_{n to infty} frac{f(n)}{g(n)} = 0) |
| Little omega | (f(n)) is (omega(g(n))) | (f(n)) dominates (g(n)) asymptotically (ignoring constant factors) | ( g(n) ) is (o(f(n))) |
Common orders of growth.
| NAME | NOTATION | EXAMPLE | CODE FRAGMENT |
|---|---|---|---|
| Constant | (O(1)) | array access arithmetic operation function call | |
| Logarithmic | (O(log n)) | binary search in a sorted array insert in a binary heap search in a red–black tree | |
| Linear | (O(n)) | sequential search grade-school addition BFPRT median finding | |
| Linearithmic | (O(n log n)) | mergesort heapsort fast Fourier transform | |
| Quadratic | (O(n^2)) | enumerate all pairs insertion sort grade-school multiplication | |
| Cubic | (O(n^3)) | enumerate all triples Floyd–Warshall grade-school matrix multiplication | |
| Polynomial | (O(n^c)) | ellipsoid algorithm for LP AKS primality algorithm Edmond's matching algorithm | |
| Exponential | (2^{O(n^c)}) | enumerating all subsets enumerating all permutations backtracing search |
Asymptotic notations: properties.
- Reflexivity: (f(n)) is (O(f(n))).
- Constants: If (f(n)) is (O(g(n))) and ( c > 0 ),then (c cdot f(n)) is (O(g(n)))).
- Products: If (f_1(n)) is (O(g_1(n))) and ( f_2(n) ) is (O(g_2(n)))),then (f_1(n) cdot f_2(n)) is (O(g_1(n) cdot g_2(n)))).
- Sums: If (f_1(n)) is (O(g_1(n))) and ( f_2(n) ) is (O(g_2(n)))),then (f_1(n) + f_2(n)) is (O(max { g_1(n) , g_2(n) })).
- Transitivity: If (f(n)) is (O(g(n))) and ( g(n) ) is (O(h(n))),then ( f(n) ) is (O(h(n))).
- Polynomials: Let (f(n) = a_0 + a_1 n + ldots + a_d n^d) with(a_d > 0). Then, ( f(n) ) is (Theta(n^d)).
- Logarithms and polynomials: ( log_b n ) is (O(n^d)) for every ( b > 0) and every ( d > 0 ).
- Exponentials and polynomials: ( n^d ) is (O(r^n)) for every ( r > 0) and every ( d > 0 ).
- Factorials: ( n! ) is ( 2^{Theta(n log n)} ).
- Limits: If ( ; displaystyle lim_{n to infty} frac{f(n)}{g(n)} = c)for some constant ( 0 < c < infty), then(f(n)) is (Theta(g(n))).
- Limits: If ( ; displaystyle lim_{n to infty} frac{f(n)}{g(n)} = 0),then (f(n)) is (O(g(n))) but not (Theta(g(n))).
- Limits: If ( ; displaystyle lim_{n to infty} frac{f(n)}{g(n)} = infty),then (f(n)) is (Omega(g(n))) but not (O(g(n))).
Here are some examples.
| FUNCTION | (o(n^2)) | (O(n^2)) | (Theta(n^2)) | (Omega(n^2)) | (omega(n^2)) | (sim 2 n^2) | (sim 4 n^2) |
|---|---|---|---|---|---|---|---|
| (log_2 n) | ✔ | ✔ | |||||
| (10n + 45) | ✔ | ✔ | |||||
| (2n^2 + 45n + 12) | ✔ | ✔ | ✔ | ✔ | |||
| (4n^2 - 2 sqrt{n}) | ✔ | ✔ | ✔ | ✔ | |||
| (3n^3) | ✔ | ✔ | |||||
| (2^n) | ✔ | ✔ |
Divide-and-conquer recurrences.
For each of the following recurrences we assume (T(1) = 0)and that (n,/,2) means either (lfloor n,/,2 rfloor) or(lceil n,/,2 rceil).| RECURRENCE | (T(n)) | EXAMPLE |
|---|---|---|
| (T(n) = T(n,/,2) + 1) | (sim lg n) | binary search |
| (T(n) = 2 T(n,/,2) + n) | (sim n lg n) | mergesort |
| (T(n) = T(n-1) + n) | (sim frac{1}{2} n^2) | insertion sort |
| (T(n) = 2 T(n,/,2) + 1) | (sim n) | tree traversal |
| (T(n) = 2 T(n-1) + 1) | (sim 2^n) | towers of Hanoi |
| (T(n) = 3 T(n,/,2) + Theta(n)) | (Theta(n^{log_2 3}) = Theta(n^{1.58..})) | Karatsuba multiplication |
| (T(n) = 7 T(n,/,2) + Theta(n^2)) | (Theta(n^{log_2 7}) = Theta(n^{2.81..})) | Strassen multiplication |
| (T(n) = 2 T(n,/,2) + Theta(n log n)) | (Theta(n log^2 n)) | closest pair |
Master theorem.
Let (a ge 1), (b ge 2), and (c > 0) and suppose that(T(n)) is a function on the non-negative integers that satisfiesthe divide-and-conquer recurrence$$T(n) = a ; T(n,/,b) + Theta(n^c)$$with (T(0) = 0) and (T(1) = Theta(1)), where (n,/,b) meanseither (lfloor n,/,b rfloor) or either (lceil n,/,b rceil).- If (c < log_b a), then (T(n) = Theta(n^{log_{,b} a}))
- If (c = log_b a), then (T(n) = Theta(n^c log n))
- If (c > log_b a), then (T(n) = Theta(n^c))
Last modified on September 12, 2020.
Copyright © 2000–2019Robert SedgewickandKevin Wayne.All rights reserved.
| Sorting Algorithms | Space complexity | Time complexity | ||
|---|---|---|---|---|
| Worst case | Best case | Average case | Worst case | |
| Insertion Sort | O(1) | O(n) | O(n2) | O(n2) |
| Selection Sort | O(1) | O(n2) | O(n2) | O(n2) |
| Smooth Sort | O(1) | O(n) | O(n log n) | O(n log n) |
| Bubble Sort | O(1) | O(n) | O(n2) | O(n2) |
| Shell Sort | O(1) | O(n) | O(n log n2) | O(n log n2) |
| Mergesort | O(n) | O(n log n) | O(n log n) | O(n log n) |
| Quicksort | O(log n) | O(n log n) | O(n log n) | O(n log n) |
| Heapsort | O(1) | O(n log n) | O(n log n) | O(n log n) |
| Data Structures | Average Case | Worst Case | ||||
|---|---|---|---|---|---|---|
| Search | Insert | Delete | Search | Insert | Delete | |
| Array | O(n) | N/A | N/A | O(n) | N/A | N/A |
| Sorted Array | O(log n) | O(n) | O(n) | O(log n) | O(n) | O(n) |
| Linked List | O(n) | O(1) | O(1) | O(n) | O(1) | O(1) |
| Doubly Linked List | O(n) | O(1) | O(1) | O(n) | O(1) | O(1) |
| Stack | O(n) | O(1) | O(1) | O(n) | O(1) | O(1) |
| Hash table | O(1) | O(1) | O(1) | O(n) | O(n) | O(n) |
| Binary Search Tree | O(log n) | O(log n) | O(log n) | O(n) | O(n) | O(n) |
| B-Tree | O(log n) | O(log n) | O(log n) | O(log n) | O(log n) | O(log n) |
| Red-Black tree | O(log n) | O(log n) | O(log n) | O(log n) | O(log n) | O(log n) |
| AVL Tree | O(log n) | O(log n) | O(log n) | O(log n) | O(log n) | O(log n) |
| n f(n) | log n | n | n log n | n2 | 2n | n! |
|---|---|---|---|---|---|---|
| 10 | 0.003ns | 0.01ns | 0.033ns | 0.1ns | 1ns | 3.65ms |
| 20 | 0.004ns | 0.02ns | 0.086ns | 0.4ns | 1ms | 77years |
| 30 | 0.005ns | 0.03ns | 0.147ns | 0.9ns | 1sec | 8.4x1015yrs |
| 40 | 0.005ns | 0.04ns | 0.213ns | 1.6ns | 18.3min | -- |
| 50 | 0.006ns | 0.05ns | 0.282ns | 2.5ns | 13days | -- |
| 100 | 0.07 | 0.1ns | 0.644ns | 0.10ns | 4x1013yrs | -- |
| 1,000 | 0.010ns | 1.00ns | 9.966ns | 1ms | -- | -- |
| 10,000 | 0.013ns | 10ns | 130ns | 100ms | -- | -- |
| 100,000 | 0.017ns | 0.10ms | 1.67ms | 10sec | -- | -- |
| 1'000,000 | 0.020ns | 1ms | 19.93ms | 16.7min | -- | -- |
| 10'000,000 | 0.023ns | 0.01sec | 0.23ms | 1.16days | -- | -- |
| 100'000,000 | 0.027ns | 0.10sec | 2.66sec | 115.7days | -- | -- |
| 1,000'000,000 | 0.030ns | 1sec | 29.90sec | 31.7 years | -- | -- |