# O() Analysis of Methods and Data Structures

Back to George One More Time Before they invented drawing boards, what did they go back to? If all the world is a stage, where is the audience sitting? If the #2 pencil is the most popular, why is it still #2? If work is so terrific, how come they have to pay you to do it? If you ate pasta and antipasto, would you still be hungry? If you try to fail, and succeed, which have you done? "People who think they know everything are a great annoyance to those of us who do. - Anon

O() Analysis Reasonable vs. Unreasonable Algorithms Using O() Analysis in Design Concurrent Systems Parallelism Recipe for Determining O() Break algorithm down into known pieces Well learn the Big-Os in this section Identify relationships between pieces Sequential is additive Nested (loop / recursion) is multiplicative

Drop constants Keep only dominant factor for each variable LB Comparing Data Structures and Methods Data Structure Unsorted L List Sorted L List Unsorted Array Sorted Array Binary Tree

BST N F&B BST N Traverse N N N N N N Log N

Search N N N Log N N N Log N Insert 1 N 1

N 1 Reasonable vs. Unreasonable Algorithms Algorithmic Performance Thus Far Some examples thus far: O(1) Insert to front of linked list O(N) Simple/Linear Search O(N Log N) MergeSort

O(N2) BubbleSort But it could get worse: O(N5), O(N2000), etc. An O(N5) Example For N = 256 N5 = 2565 = 1,100,000,000,000 If we had a computer that could execute a million instructions per second 1,100,000 seconds = 12.7 days to complete But it could get worse

The Power of Exponents A rich king and a wise peasant The Wise Peasants Pay Day(N) 1 2 3 4 ... 63 64

Pieces of Grain 2 4 8 16 2N 9,223,000,000,000,000,000 18,450,000,000,000,000,000 How Bad is 2N? Imagine being able to grow a billion

(1,000,000,000) pieces of grain a second It would take 585 years to grow enough grain just for the 64th day Over a thousand years to fulfill the peasants request! LB So the King cut off the peasants head. The Towers of

Hanoi A B C Goal: Move stack of rings to another peg Rule 1: May move only 1 ring at a time Rule 2: May never have larger ring on top of smaller ring

The Towers of Hanoi A B C The Towers of Hanoi A

B C The Towers of Hanoi A B C

The Towers of Hanoi A B C The Towers of Hanoi

A B C The Towers of Hanoi A B

C The Towers of Hanoi A B C The Towers of Hanoi

A B C The Towers of Hanoi A B

C The Towers of Hanoi A B C The Towers of Hanoi

A B C The Towers of Hanoi A B

C The Towers of Hanoi A B C The Towers of

Hanoi A B C The Towers of Hanoi A

B C The Towers of Hanoi A B C

Towers of Hanoi Complexity For 1 rings we have 1 operations. For 2 rings we have 3 operations. For 3 rings we have 7 operations. For 4 rings we have 15 operations. In general, the cost is 2N 1 = O(2N) Each time we increment N, we double the amount of work. This grows incredibly fast!

Towers of Hanoi (2N) Runtime For N = 64 2N = 264 = 18,450,000,000,000,000,000 If we had a computer that could execute a million instructions per second It would take 584,000 years to complete But it could get worse The Bounded Tile Problem Match up the patterns in the tiles. Can it be done, yes or no?

The Bounded Tile Problem Matching tiles Tiling a 5x5 Area 25 available tiles remaining Tiling a 5x5 Area 24 available tiles remaining

Tiling a 5x5 Area 23 available tiles remaining Tiling a 5x5 Area 22 available tiles remaining Tiling a 5x5 Area 2 available tiles remaining

Analysis of the Bounded Tiling Problem Tile a 5 by 5 area (N = 25 tiles) 1st location: 25 choices 2nd location: 24 choices And so on Total number of arrangements: 25 * 24 * 23 * 22 * 21 * .... * 3 * 2 * 1 25! (Factorial) = 15,500,000,000,000,000,000,000,000 Bounded Tiling Problem is O(N!)

Tiling (N!) Runtime For N = 25 25! = 15,500,000,000,000,000,000,000,000 If we could place a million tiles per second It would take 470 billion years to complete Why not a faster computer? A Faster Computer If we had a computer that could execute a trillion instructions per second (a million times faster than our MIPS computer)

5x5 tiling problem would take 470,000 years 64-ring Tower of Hanoi problem would take 213 days Why not an even faster computer! The Fastest Computer Possible? What if: Instructions took ZERO time to execute CPU registers could be loaded at the speed of light These algorithms are still unreasonable! The speed of light is only so fast!

Where Does this Leave Us? Clearly algorithms have varying runtimes. Wed like a way to categorize them: Reasonable, so it may be useful Unreasonable, so why bother running Polynomial Performance Categories of Algorithms Sub-linear Linear Nearly linear

Quadratic O(Log N) O(N) O(N Log N) O(N2) Exponential O(N!) O(NN) O(2N) Reasonable vs. Unreasonable

Reasonable algorithms have polynomial factors O (Log N) O (N) O (NK) where K is a constant Unreasonable algorithms have exponential factors O (2N) O (N!) O (NN) Reasonable vs. Unreasonable Reasonable algorithms May be usable depending upon the input size Unreasonable algorithms

Are impractical and useful to theorists Demonstrate need for approximate solutions Remember were dealing with large N (input size) Two Categories of Algorithms 10 1030 1025 1020 1015 trillion billion

million 1000 100 10 Unreasonable Runtime 35 NN

2N N5 Reasonable N Dont Care! 2 4 8 16 32 64 128 256 512 1024 Size of Input (N) Summary Reasonable algorithms feature polynomial factors in their O() and may be usable depending upon input size.

Unreasonable algorithms feature exponential factors in their O() and have no practical utility. Questions? Using O() Analysis in Design Air Traffic Control Conflict Alert Coast, add, delete

Problem Statement What data structure should be used to store the aircraft records for this system? Normal operations conducted are: Data Entry: adding new aircraft entering the area Radar Update: input from the antenna Coast: global traversal to verify that all aircraft have been updated [coast for 5 cycles, then drop] Query: controller requesting data about a specific aircraft by location Conflict Analysis: make sure no two aircraft

are too close together Air Traffic Control System #1 #2 #3 #4 #5 Program Algorithm

Freq 1. Data Entry / Exit 2. Radar Data Update 3. Coast / Drop 4. Query 5. Conflict Analysis Insert N*Search Traverse Search

Traverse*Search 15 12 60 1 12 Freq LLU 15 1 12 N^2

60 N 1 N 12 N^2 LLS N N^2 N N N^2

AU 1 N^2 N N N^2 AS N NlogN N LogN

NlogN BT 1 N^2 N N N^2 F/B BST LogN NlogN N

LogN NlogN Questions? Concurrent Systems Sequential Processing All of the algorithms weve seen so far are sequential: They have one thread of execution One step follows another in sequence One processor is all that is needed to

run the algorithm A Non-sequential Example Consider a house with a burglar alarm system. The system continually monitors: The front door The back door The sliding glass door The door to the deck The kitchen windows The living room windows The bedroom windows The burglar alarm is watching all of

these at once (at the same time). Another Non-sequential Example Your car has an onboard digital dashboard that simultaneously: Calculates how fast youre going and displays it on the speedometer Checks your oil level Checks your fuel level and calculates consumption Monitors the heat of the engine and turns on a light if it is too hot Monitors your alternator to make sure it is charging your battery

Concurrent Systems A system in which: Multiple tasks can be executed at the same time The tasks may be duplicates of each other, or distinct tasks The overall time to perform the series of tasks is reduced Advantages of Concurrency Concurrent processes can reduce duplication in code.

The overall runtime of the algorithm can be significantly reduced. More real-world problems can be solved than with sequential algorithms alone. Redundancy can make systems more reliable. Disadvantages of Concurrency Runtime is not always reduced, so careful planning is required Concurrent algorithms can be more complex than sequential algorithms Shared data can be corrupted

Communications between tasks is needed Achieving Concurrency Many computers today have more than one processor (multiprocessor machines) CPU 1 CPU 2 bus

Memory Achieving Concurrency Concurrency can also be achieved on a computer with only one processor: The computer juggles jobs, swapping its attention to each in turn Time slicing allows many users to get CPU resources

Tasks may be suspended while they wait for something, such as device I/O task 1 task 2 ZZZZ CPU task 3 ZZZZ Concurrency vs. Parallelism

Concurrency is the execution of multiple tasks at the same time, regardless of the number of processors. Parallelism is the execution of multiple processors on the same task. Types of Concurrent Systems

Multiprogramming Multiprocessing Multitasking Distributed Systems Multiprogramming Share a single CPU among many users or tasks. May have a time-shared algorithm or a priority algorithm for determining which task to run next Give the illusion of simultaneous

processing through rapid swapping of tasks (interleaving). Multiprogramming Memory User 1 User 2 CPU User1 User2

Tasks/Users Multiprogramming 4 3 2 1 1 2 CPUs

3 4 Multiprocessing Executes multiple tasks at the same time Uses multiple processors to accomplish the tasks Each processor may also timeshare among several tasks Has a shared memory that is used

CPU User2 Tasks/Users Multiprocessing 4 Shared Memory

3 2 1 1 2 CPUs 3 4

Multitasking A single user can have multiple tasks running at the same time. Can be done with one or more processors. Used to be rare and for only expensive multiprocessing systems, but now most modern operating systems can do it. Multitasking Memory User 1: Task1

3 2 1 1 2 CPUs 3 4

Distributed Systems Multiple computers working together with no central program in charge. ATM Buford ATM Student Ctr ATM Perimeter ATM North Ave

Central Bank Distributed Systems Advantages: No bottlenecks from sharing processors No central point of failure Processing can be localized for efficiency Disadvantages: Complexity Communication overhead Distributed control

Questions? Parallelism Parallelism Using multiple processors to solve a single task. Involves: Breaking the task into meaningful pieces Doing the work on many processors Coordinating and putting the

pieces back together. Parallelism Network Interface Memory CPU Parallelism Tasks

4 3 2 1 1 2 CPUs 3 4

Pipeline Processing Repeating a sequence of operations or pieces of a task. Allocating each piece to a separate processor and chaining them together produces a pipeline, completing tasks faster. input A B

C D output Example Suppose you have a choice between a washer and a dryer each having a 30 minutes cycle or A washer/dryer with a one hour cycle The correct answer depends on how much work you have to do.

wash dry wash dry wash combo combo dry

combo Examples of Pipelined Tasks Automobile manufacturing Instruction processing within a computer A 2 1 B

3 4 2 1 C 5 3

2 1 0 1 2 3 2

time 4 5 4 3 1 D

5 4 5 5 4 3 6

7 Task Queues A supervisor processor maintains a queue of tasks to be performed in shared memory. Each processor queries the queue, dequeues the next task and performs it. Task execution may involve adding more tasks to the task queue. P1 P2

Super P3 Pn Task Queue Parallelizing Algorithms How much gain can we get from parallelizing an algorithm? Parallel

Bubblesort We can use N/2 processors to do all the comparisons at once, flopping the pair-wise comparisons. 93 87 74 65 57

45 33 27 87 93 65

74 45 57 27 33 87 65

93 45 74 27 57 33

Runtime of Parallel Bubblesort 3 65 87 4 65 45

5 45 6 45 93 27 74

33 57 87 27 93 33

74 57 65 27 87 33 93

57 74 45 27 65 33

57 93 74 7 27 45 33

65 57 87 74 93 8

27 33 45 57 65 74 87

93 87 Completion Time of Bubblesort Sequential bubblesort finishes in N2 time. Parallel bubblesort finishes in N time. O(N) parallel Bubble Sort

O(N2) Product Complexity Got done in O(N) time, better than O(N2) Each time chunk does O(N) work There are N time chunks. Thus, the amount of work is still O(N2)

Product complexity is the amount of work per time chunk multiplied by the number of time chunks the total work done. Ceiling of Improvement Parallelization can reduce time, but it cannot reduce work. The product complexity cannot change or improve. How much improvement can parallelization provide? Given an O(NLogN) algorithm and Log N processors, the algorithm will take at least O(?) time.

O(N) time. Given an O(N3) algorithm and N processors, O(N2)time. time. the algorithm will take at least O(?) Number of Processors Processors are limited by hardware. Typically, the number of processors is a power of 2 Usually: The number of processors is a constant factor, 2K

Conceivably: Networked computers joined as needed (ala Borg?). Adding Processors A program on one processor Runs in X time Adding another processor Runs in no more than X/2 time Realistically, it will run in X/2 + time because of overhead At some point, adding processors will not help and could degrade performance.

Overhead of Parallelization Parallelization is not free. Processors must be controlled and coordinated. We need a way to govern which processor does what work; this involves extra work. Often the program must be written in a special programming language for parallel systems. Often, a parallelized program for one machine (with, say, 2K processors) doesnt work on other machines (with, say, 2L processors). What We Know about Tasks Relatively isolated units of computation

Should be roughly equal in duration Duration of the unit of work must be much greater than overhead time Policy decisions and coordination required for shared data Simpler algorithm are the easiest to parallelize Questions? More? Matrix Multiplication

c a b n cij aik bkj k 1 . . . .

c34 . . . . . . . . . . . . . . . c34 a31 a32 a33 a34

. . . . . . . a31b14 a32b24 a33b34 a34b44 . . . . b14

b24 b34 b44 Inner Product Procedure Procedure inner_prod(a, b, c isoftype in/out Matrix, i, j isoftype in Num) // Compute inner product of a[i][*] and b[*][j] Sum isoftype Num k isoftype Num Sum <- 0 k <- 1 loop exitif(k > n) sum <- sum + a[i][k] * b[k][j] k < k + 1 endloop endprocedure // inner_prod Matrix definesa Array[1..N][1..N] of Num N is // Declare constant defining size // of arrays Algorithm P_Demo a, b, c isoftype Matrix Shared

server isoftype Num Initialize(NUM_SERVERS) // Input a and b here // (code not shown) i, j isoftype Num i <- 1 loop exitif(i > N) server <- (i * NUM_SERVERS) DIV N j <- 1 loop exitif(j > N)

RThread(server, inner_prod(a, b, c, i, j )) j <- j + 1 endloop i <- i + 1 endloop Parallel_Wait(NUM_SERVERS) // Output c here endalgorithm // P_Demo Questions?

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