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We summarize the performance characteristics of classic algorithms anddata structures for sorting, priority queues, symbol tables, and graph processing.
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We also summarize some of the mathematics useful in the analysis of algorithms, including commonly encountered functions;useful formulas and approximations; properties of logarithms;asymptotic notations; and solutions to divide-and-conquer recurrences.
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 | IN PLACE | 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 |
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 | DEL-MIN | MIN | DEC-KEY | 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.
| 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)!}) |
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)
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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 )
Asymptotic 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))) |
Cheatsheet 1 3 2 3/4
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 backtracking search |
Cheatsheet 1 3 2 3
Asymptotic notations: properties.
Cheatsheet 1 3 2 3 Tile Pattern
- 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) | ✔ | ✔ |
Cheatsheet 1 3 2 3
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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 July 05, 2021.
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