# Algorithms and Data Structures Cheatsheet

We summarize the performance characteristics of classic algorithms and data 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 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 sorting algorithms, 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;
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;
dominated by insertion sort
shellsort Shell.java n log3 n unknown c n 3/2 tight code;
mergesort Merge.java ½ n log2 n n log2 n n log2 n n log n guarantee;
stable
quicksort Quick.java n log2 n 2 n ln n ½ n 2 n log n probabilistic guarantee;
fastest in practice
heapsort Heap.java n 2 n log2 n 2 n log2 n n log n guarantee;
in place
n log2 n if all keys are distinct

## Priority queues.

The table below summarizes the order of growth of the running time of operations 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 logd n d logd n 1 logd n d logd n 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 of operations 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 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 E V (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 encountered when 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:   $$\log_2 (n!) = \log_2 1 + \log_2 2 + \log_2 3 + \ldots + \log_2 n \sim n \log_2 n$$

• 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:   $$\log_b (x^y) = y \log_b x$$

## 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 $$0 \le f(n) \le c \cdot g(n)$$ for all $$n \ge n_0$$
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
op();
Logarithmic $$O(\log n)$$ binary search in a sorted array
insert in a binary heap
search in a red–black tree
for (int i = 1; i <= n; i = 2*i)   op();
Linear $$O(n)$$ sequential search
BFPRT median finding
for (int i = 0; i < n; i++)   op();
Linearithmic $$O(n \log n)$$ mergesort
heapsort
fast Fourier transform
for (int i = 1; i <= n; i++)   for (int j = i; j <= n; j = 2*j)      op();
Quadratic $$O(n^2)$$ enumerate all pairs
insertion sort
for (int i = 0; i < n; i++)   for (int j = i+1; j < n; j++)      op();
Cubic $$O(n^3)$$ enumerate all triples
Floyd–Warshall
for (int i = 0; i < n; i++)   for (int j = i+1; j < n; j++)      for (int k = j+1; k < n; k++)         op();
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

## 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
$$T(n) = T(n\,/\,2) + 1$$ $$\sim \log_2 n$$ binary search
$$T(n) = 2 T(n\,/\,2) + n$$ $$\sim n \log_2 n$$ 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 satisfies the divide-and-conquer recurrence $$T(n) = a \; T(n\,/\,b) + \Theta(n^c)$$ with $$T(0) = 0$$ and $$T(1) = \Theta(1)$$, where $$n\,/\,b$$ means either $$\lfloor n\,/\,b \rfloor$$ 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 $$T(n) = \Theta(n^c)$$
Remark: there are many different versions of the master theorem. The Akra–Bazzi theorem is among the most powerful.