Graph Algorithms Cheat Sheet

01-05-2021 admin

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 appoximations; 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 n²	½ n²	½ n²	n exchanges; quadratic in best case
insertion sort	Insertion.java	✔	✔	n	¼ n²	½ n²	use for small or partially-sorted arrays
bubble sort	Bubble.java	✔	✔	n	½ n²	½ n²	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 n²	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	log_dn	d log_dn	1	log_dn	d log_dn	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)	V²
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	V³	V²
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	n³ log n	n²

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)

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
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))
(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.

Graph Algorithms Book

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).