Download Data Structures Algorithm Cheatsheet and more Cheat Sheet Computer science in PDF only on Docsity!
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;
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 log
3
n unknown
c n
3/
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 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
log
d
n d log
d
n
1
log
d
n d log
d
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 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 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 greatest integer
ceiling smallest integer
binary logarithm or such that
natural logarithm or such that
common logarithm such that
iterated binary logarithm if otherwise
harmonic number
factorial
⌊ x ⌋ ≤ x
⌈ x ⌉ ≥ x
lg x log x
2
y 2 = x
y
ln x log x
e
y e = x
y
log x
10
y 10 = x
y
lg x
∗
0 x ≤ 1; 1 + lg (lg x )
∗
H
n
1 + 1/2 + 1/3 + … + 1/ n
n! 1 × 2 × 3 × … × n
Common orders of growth.
NAME NOTATION EXAMPLE CODE FRAGMENT
Constant
array access
arithmetic operation
function call
op();
Logarithmic
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
sequential search
grade-school addition
BFPRT median finding
for (int i = 0; i < n; i++)
op();
Linearithmic
mergesort
heapsort
fast Fourier transform
for (int i = 1; i <= n; i++)
for (int j = i; j <= n; j = 2*j)
op();
Quadratic
enumerate all pairs
insertion sort
grade-school multiplication
for (int i = 0; i < n; i++)
for (int j = i+1; j < n; j++)
op();
Cubic
enumerate all triples
Floyd–Warshall
grade-school matrix multiplication
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
ellipsoid algorithm for LP
AKS primality algorithm
Edmond's matching algorithm
Exponential
enumerating all subsets
enumerating all permutations
backtracking search
Asymptotic notations: properties.
Reflexivity: is.
Constants: If is and , then is.
Products: If is and is , then is.
Sums: If is and is , then is.
Transitivity: If is and is , then is.
Polynomials: Let with. Then, is.
Logarithms and polynomials: is for every and every.
Exponentials and polynomials: is for every and every.
Factorials: is.
Limits: If for some constant , then is.
Limits: If , then is but not.
Limits: If , then is but not.
Here are some examples.
FUNCTION
✔ ✔
✔ ✔
✔ ✔ ✔ ✔
O (1)
O (log n )
O ( n )
O ( n log n )
O ( n )
2
O ( n )
3
O ( n )
c
O ( n )
c
f ( n ) O ( f ( n ))
f ( n ) O ( g ( n )) c > 0 c ⋅ f ( n ) O ( g ( n )))
f ( n )
1
O ( g ( n ))
1
f ( n )
2
O ( g ( n )))
2
f ( n ) ⋅ ( n )
1
f
2
O ( g ( n ) ⋅ ( n )))
1
g
2
f ( n )
1
O ( g ( n ))
1
f ( n )
2
O ( g ( n )))
2
f ( n ) + ( n )
1
f
2
O (max{ g ( n ), ( n )})
1
g
2
f ( n ) O ( g ( n )) g ( n ) O ( h ( n )) f ( n ) O ( h ( n ))
f ( n ) = a + n + … +
0
a
1
a
d
n
d
a > 0
d
f ( n ) Θ( n )
d
log n
b
O ( n )
d
b > 0 d > 0
n
d
O ( r )
n
r > 0 d > 0
n! 2
Θ( n log n )
lim = c
n →∞
f ( n )
g ( n )
0 < c < ∞ f ( n ) Θ( g ( n ))
lim = 0
n →∞
f ( n )
g ( n )
f ( n ) O ( g ( n )) Θ( g ( n ))
lim = ∞
n →∞
f ( n )
g ( n )
f ( n ) Ω( g ( n )) O ( g ( n ))
o ( n )
2
O ( n )
2
Θ( n )
2
Ω( n )
2
ω ( n )
2
∼ 2 n
2
∼ 4 n
2
log n
2
10 n + 45
2 n + 45 n + 12
2
✔ ✔ ✔ ✔
✔ ✔
✔ ✔
Divide-and-conquer recurrences.
For each of the following recurrences we assume and that means either or.
RECURRENCE EXAMPLE
binary search
mergesort
insertion sort
tree traversal
towers of Hanoi
Karatsuba multiplication
Strassen multiplication
closest pair
Master theorem.
Let , , and and suppose that is a function on the non-negative integers that satisfies the divide-and-conquer recurrence
with and , where means either or either.
If , then
If , then
If , then
4 n − 2
2
n
3 n
3
n
T (1) = 0 n / 2 ⌊ n / 2⌋ ⌈ n / 2⌉
T ( n )
T ( n ) = T ( n / 2) + 1 ∼ lg n
T ( n ) = 2 T ( n / 2) + n ∼ n lg n
T ( n ) = T ( n − 1) + n ∼
1
2
n
2
T ( n ) = 2 T ( n / 2) + 1 ∼ n
T ( n ) = 2 T ( n − 1) + 1
∼ 2
n
T ( n ) = 3 T ( n / 2) + Θ( n ) Θ( n ) = Θ( )
log 3
2
n
1.58...
T ( n ) = 7 T ( n / 2) + Θ( ) n
2
Θ( n ) = Θ( )
log 7
2
n
2.81...
T ( n ) = 2 T ( n / 2) + Θ( n log n )
Θ( n log n )
2
a ≥ 1 b ≥ 2 c > 0 T ( n )
T ( n ) = a T ( n / b ) + Θ( ) n
c
T (0) = 0 T (1) = Θ(1) n / b ⌊ n / b ⌋ ⌈ n / b ⌉
c < log a
b
T ( n ) = Θ( n )
log a
b
c = log a
b
T ( n ) = Θ( n log n )
c
c > log a
b
T ( n ) = Θ( n )
c
