Complexity Cheat Sheet

Heroic Boss Guides: TL;DR

Welcome to our TL;DR Cheat Sheet! This is meant to give you basic information on the key aspects of each fight, and is over-simplified. We strongly recommend that after you finish reading this, you visit the page for each boss to get more in-depth information on how the abilities work and a detailed strategy.

Wrathion

Back and forth between P1 and P2 till he dies

Phase 1

Nobody stand in front of, or behind, the boss (except for active tank)
- Tanks swap at 1-2 stacks
Red circle under you, get out
Boss does the giant red swirlie, get 35 yards away
- Immediately after, run to the furthest edge of the room and wait for explosions to end

Phase 2

3 people need to soak the red soak circles, run over pillars
Everyone else finds and kills stealthed mobs
Kill any pillars your 3 person team couldn’t

1/22/14 Big-O Algorithm Complexity Cheat Sheet bigocheatsheet.com 1/17 I receive $12.19 / wk on Gittip. Big-O Cheat Sheet Searching Sorting Data Structures Heaps Graphs Chart Comments Tweet 2,710 1.7k Know Thy Complexities! This webpage covers the space and time Big-O complexities of common algorithms used in Computer Science.
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He has also made a cheat sheet useful for talking to colleagues or friends about the technique: Click image for PDF version As this technique is quite popular with software developers there is a lot of software implementations to assist you in using this technique: several iPhone apps and versions of the timer used to measure the 25 minutes.

Maut

Back and forth between P1 and P2 till he dies

Phase 1

Melee stand to the boss’s side, ranged and heals behind him
Don’t get hit by Black Wings
If you get the purple targeting circle, move towards the edge of the room to drop puddle
Don’t stand in giant purple swirlie. After it explodes, add spawns from the middle
Tank 1 gets boss, tank 2 gets add 1 when it spawns
- When add 2 spawns, tank 2 taunts boss. Boss will eat add 1 when it gets close
- Repeat this when add 3 spawns
Stand in the purple puddle during Stygian Annihilation

Phase 2

Everyone stack in melee
Tanks run around and soak blue orbs coming from the side
- Drop your debuff on the raid group
If you get small purple swirlie, move 8 yards out till it goes away

Skitra

Back and forth between normal and intermission phase. Intermission every 20% HP

Normal Phase

Everyone stacks in melee behind the boss except the tank
- Tanks swap at 7-8 stacks
If you get Shred Psyche, move 8 yards away until it falls off
- Wait 5s after add spawns, then nuke it
CC one of the Images to give yourself a safe spot
Lust on pull

Intermission

Boss splits raid in half with debuffs
Each half sees 4 Images of the boss
- Most are fake
- Find real one, kill real one

Xanesh

Soccer boss!

Normal abilities

Tanks swap at 1-2 stacks
Move away from red targeting circle if you see one
Don’t stand in the Obelisks
Stay out of the goop during Torment
- Raid use movement CD here like Stampeding Roar

Soccer!

3 players soak the orange soak circles
Find the glowing portal, ball spawns here
“Kick” ball to the other portal on opposite side of the room
Ball can NOT touch wall, Obelisks, or Azshara
After scoring goal, you can’t play soccer for 3 mins. New team next time.

Keep doing these till she dies

Hivemind

1 phase fight against 2 bosses. They don’t share health pool. Summon adds all fight

Tek’ris in control = stack bosses
Ka’zir in control = spread bosses 20 yards
No tank swaps
Don’t stand in Tek’ris’s purple frontal
Spread out when purple circles go on everyone
When Tek’ris makes a Ravager add, everyone hard swap
When Ka’zir makes an add do Volatile Corruption, hard swap
Dodge bowling ball adds
Interrupt Ka’zir’s cast
Keep adds slowed, AoE fest. Kill them as they keep coming
Lust on pull

Shad’har

3 phase fight against a 2 headed doggo

All Phases

Boss does a frontal cone breath ability at a random player
- Does damage + a debuff. Debuff changes based on phase
- Can always be dodged
Does a 1-2 combo on tanks Crush and Dissolve. Always taunt swap as soon as Crush deals damage
If you get the 4 stack application of Debilitating Spit, use a cooldown
If you get fixated by a Living Miasma, move away from people

Phase 1 – 100%-66%

Dodge purple swirlies, they come out more frequently the longer you’re in p1

Phase 2 – 66%-33%

Soak void orbs when they spawn

Phase 3 – 33%-0%

Burn phase, lust at 30%
Green goop puddle starts in the middle of the room, grows over time
- Racing to kill the boss before goop takes over the entire space
Boss does heavy AoE to the raid the rest of the fight
Enrages at 30%, increasing damage done 25%

Drest’agath

One phase encounter against a giant tentacle and her smaller tentacle adds

Drest’agath Abilities

Heals 11 mil every 5s
Puts big purple circle on the tank. Taunt swap when this happens
- Tank with the circle needs to run to an add and explode on the add
Move away from boss when she does Entropic Crash
If you get purple swirlie, move away from people
Dodge Void Glare lazer beam
At 100 energy, does a large AoE on the raid

Adds

3 types, Tentacles, Eyes, Maws
- Eyes do mind flay, interrupt it. Kill these last.
- Tentacles drop Smoke Bomb. Melee get inside smoke & kill it
  - 1-2 healers need to go in smoke to heal
- Maw does a heal absorb to anyone near. Ranged nuke this
When adds die, they drop purple goop. Run over goop to get Void Infused Ichor
- When you have Ichor, any damage you deal to the boss can NOT be regenerated by her passive healing. Raid should all grab ichor at same time and use CDs

Rotate through killing adds & boss 3-4 times and kill her

Il’gynoth

2 phase encounter. P1 you fight Il’gynoth, P2 you kill organs. Killing 1 organ moves you back to P1. Need to kill all 3 organs to win

Il’gynoth abilities

Beam nukes the tank. Swap after taking 2 of these
Also does standard Eye Beam lasers at a few people. Run away from this. Leaves goop on ground, try to keep goop in a small circle
MC’s random players. Everyone swap and kill MC’d people
In your 2nd and 3rd P1, you’ll have blood goop adds. Slow & kill them
Lust during 3rd Phase 1

P2 abilities

If you get red circle on you, spread out from people
Pick 1 organ to kill each P2, all DPS focus it
Interrupt organs when they cast. If you miss, you’ll get a blood goop add
- Blood goop adds need to be slowed and killed

Vexiona

3 phase fight. Alternates between P1 and P2 until boss reaches 40%, then goes into P3.

Phase 1

Start fight off against boss + one big Ascendant add
- Nuke Ascendant add asap
- Ascendant does a massive lazer beam at it’s tank
- Leaves an orb where it dies, this comes into play later
If you get the purple circle under you, move to the edges of the room
- Drops a puddle. Try to drop new puddles on or near old ones
Boss does a frontal breath on the tank, gives tank stacks of a DoT
- After 3rd breath, tank should get the orb the Ascendant dropped. Resets your stacks
When tank gets Despair, healers need to bomb heal them
Boss also periodically puts down a gateway and small adds come out. Casters, rogues, weak warriors. Just kill these

Phase 2

Boss flies into air, summons new adds, make 1 add turn into Ascendant
- Everyone nuke Ascendant ASAP, then Vexiona tank pick up new orb to reset stacks
Dodge the flyover flame breath from Vexiona
Handle the purple circles like in P1, drop puddles near edges
After 3 flyovers, back to P1

Big O Complexity Cheat Sheet

Repeat P1 and P2 until boss hits 40%

Phase 3

Burn phase. Lust here
Everyone gets DoT stacks now whenever you take damage
- Healers save raid CDs for late in fight, boss sub 20% hp
Move away from boss when she does Heart of Darkness, get out of glowing orange circle
Stand next to a friend all phase
Have 5+ people stack on tank when Desolation goes out

Ra-den

2 phase fight. P2 starts at 40% hp

Phase 1

Everyone but tanks should stack in melee range
- Put 2 markers behind the raid, one left one right
Orbs spawn, one void one lightning. Kill one, let other hit boss
- Alternate which type of orb you kill
Boss explodes for AoE damage when orb hits him
An add spawns based on which orb
- Lightning orb makes a lightning add casting Chain Lightning
- Void orb makes Void add casting a big soak circle. Targeted player make sure you’re with the group to split damage with everyone
Also does an Unstable ability based on which orb hit
- Lightning orb makes a jumping lightning bolt. Affected player runs to one of the markers behind raid. Someone else runs to other marker for bolt to jump to. Keep repeating until bolt stops jumping
- Void orb makes small soak circles. Soak these
- Different people need to soak or be hit by bolt every time
Tanks swap at 2 stacks of the debuff he does

Phase 2

Burn phase, lust here
Heavy AoE to everyone for rest of fight
Dodge purple swirlies
If you get Chained, run away from people you’re connected with
Healers should rotate raid CDs after lust ends
Keep tanks around half health this phase, so the DoT from Decaying Strike doesn’t wreck them

Carapace

Coming Soon. See full guide for now.

N’zoth

Coming soon. See full guide for now.

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.

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.

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).

Complexity Cheat Sheet

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 theoremis among the most powerful.