Problem ComplexityReview

Problem Complexity

Online Survey

The Spring term course/instructor opinion survey will be available during the period Monday, April 17th through Friday, April 28th from 6am to 11:59pm each day:

http://www.coursesurvey.gatech.edu

LB

Final Exam Schedule

• CS1311 Sections L/M/N Tuesday/Thursday 10:00 A.M.

• Exam Scheduled for 8:00 Friday May 5, 2000• Physics L1

LB

Final Exam Schedule

• CS1311 Sections E/F Tuesday/Thursday 2:00 P.M.

• Exam Scheduled for 2:50 Wednesday May 3, 2000• Physics L1

LB

Relations between Problems, Algorithms, and Programs

Problem

Algorithm Algorithm

Program ProgramProgram Program

. . . .

. . . . . . . .

Cost and Complexity

• Algorithm complexity can be expressed in Order notation, e.g. “at what rate does work grow with N?”:– O(1) Constant– O(logN) Sub-linear– O(N) Linear– O(NlogN) Nearly linear– O(N2) Quadratic– O(XN) Exponential

• But, for a given problem, how do we know if a better algorithm is possible?

The Problem of Sorting

For example, in discussing the problem of sorting:

• Two algorithms to solve:– Bubblesort – O(N2)– Mergesort – O(N Log N)

• Can we do better than O(N Log N)?

Algorithm vs. Problem Complexity

• Algorithmic complexity is defined by analysis of an algorithm

• Problem complexity is defined by– An upper bound – defined by an

algorithm– A lower bound – defined by a proof

The Upper Bound

• Defined by an algorithm• Defines that we know we can do

at least this good• Perhaps we can do better• Lowered by a better algorithm

– “For problem X, the best algorithm was O(N3), but my new algorithm is O(N2).”

The Lower Bound

• Defined by a proof• Defines that we know we can do no

better than this• It may be worse• Raised by a better proof

– “For problem X, the strongest proof showed that it required O(N), but my new, stronger proof shows that it requires at least O(N2).”

Upper and Lower Bounds

• The Upper bound is the best algorithmic solution that has been found for a problem.– “What’s the best that we know we can do?”

• The Lower bound is the best solution that is theoretically possible.– “What cost can we prove is necessary?”

Changing the Bounds

Upper bound Lowered by betteralgorithm

Lower bound Raised by betterproof

Open Problems

The upper and lower bounds differ.

Upper bound Lowered by betteralgorithm

Lower bound Raised by betterproof

Unknown

Closed Problems

The upper and lower bounds are identical.

Upper bound

Lower bound

Closed Problems

• Better algorithms are still possible

• Better algorithms will not provide an improvement detectable by “Big O”

• Better algorithms can improve the constant costs hidden in “Big O” characterizations

Tractable vs. Intractable

Problems are tractable if the upper and lower bounds have only polynomial factors.– O (log N)– O (N)

– O (NK) where K is a constant

Problems are intractable if the upper and lower bounds have an exponential factor.– O (N!)

– O (NN)

– O (2N)

Problems that “Cross the Line”

• The upper bound implies intractable • The lower bound implies a tractable• Could go either way…

Next time!!!

Terminology

• Polynomial algorithms are reasonable• Polynomial problems are tractable

• Exponential algorithms are unreasonable• Exponential problems are intractable

Terminology

Tractable

Intractable

Reasonable

Unreasonable

Problems Algorithms

Polynomial

Exponential

Questions?

Review

• The Algorithmic Model– Algorithm defined– Properties of good algorithms– How to describe algorithms– Relating problems to algorithms to programs

(hierarchy needed)

Review

• View of programming languages/algorithms

– Data vs. instructions

– Built-in vs. user-defined

– Complex vs. atomic

• Data

– Type vs. variable (Declaration and Initialization of variables)

– The 4 atomic built-in types (Num, Characters, Booleans, Pointers)

– The complex built in type String

Review

• Operations – Assignment– Arithmetic

• +, -, x, /, div, mod• Precedence rules• Using parenthesis to modify precedence

– Input and output• Print• Read

Review

• Conditionals– Purpose & defined– Relational operators (<, >, =, <>, >=, <=)– Boolean operators (AND, OR, NOT)– Boolean expressions (Simple & Complex)– Control flow of the if-then-else statement– Elseif as shorthand

• Writing an algorithm (how to begin)

Review

• Program maintenance• Software Engineering facts about program

cost• Documentation• Benefits of Constants

Review

• Procedural Abstraction• Why modularity (Benefits)

• Need for interface• Scope of variables• Contract (Pre, post, and purpose statements

for every module)• Information flow – in, out, in/out• Parameters intro (In, out, in/out)• Types of modules

Review

• Procedure• Rules when to use a procedure• Declaration• Function• Rules when to use a function• Declaration• Returning values via the “returns”

Review

• Module invocation

• Parameters

• Formal vs. actual

• Parameter matching

• How parameters work (In, Out, & In/out)

• Tracing

• Activation stack

• Frames

• Rules of parameter matching (In, Out, & In/out)

• Examples

Review

• Recursion (Intro)

• Purpose – repetition

• Characteristics – calls itself, terminating condition, move closer

• 2 forms – final action or not

• Examples

• Tracing recursive modules

• Recursion (Advanced)

• Mutual recursion

• Design by Contract

Review

• Data Abstraction– Records

• Declaring records• Creating variables of record type• Accessing fields of a record variable (the ‘.’

operator)• Avoid anonymous types• Combining records (records inside records)

Review

• Static vs. dynamic memory/data

• Pointers

• Simple example of ptr toa num

• Following the pointer via the ‘^’ operator

• Dynamic data structures

• Linked Lists

• Defined/properties

• Proper Record declaration

• Accessing information in a linked list via pointers

Review

• Linked Lists (continued)– Adding nodes– Simple – no recursion (To front)– Insertion recursive method– To end– In middle (when sorted)– Deleting nodes

• Simple - no recursion (From front)• Deletion recursive method

Review

• Doubly-linked lists– Defined– Example methods (simple insertion)

• Stack– Defined/properties– Push– Pop– Tracing changes (example of algorithm using

push & pop)

Review

• Queue

• Defined/properties

• Enqueue

• Dequeue

• Tracing changes (example of algorithm using enqueue and dequeue)

• Trees

• Defined/properties

• Binary trees (Record declaration)

• Binary search trees

Review

• Trees (Continued)• Recursive insertion in BST• Deleting from BST (conceptually)

• Graphs• Defined/properties• Record definition

Review

• Static data structures• Arrays

• Defined• Need of constants• Accessing elements via the index• Multi-dimensional arrays (Declaring and

accessing)

Review

• Iteration• Defined, use for repetition• How looping works• Loop• Endloop• Exitif

• Combined types• Examples – array of lists, list of array, tree of

lists, etc.

Review

• Methods of Accessing Data Structures• Traversal

• Lists• Recursive on lists• Iterative on lists

• Trees• In-order• Pre-order• Post-order• Miscellaneous Traversals

• (Rt before Left)

Review

• Breadth-first vs. depth- first• Arrays

• Iterative on arrays• Recursive on arrays

• Search• Linear

• Recursive on lists• Traversal-search on binary tree (not BST)• Linear, iterative search on array

Review

• Binary, recursive search on BST• Binary, iterative search on sorted array

• Sort• Bubble sort• Merge sort

Review

• Conversion• Defined• Helper module advantages• List to Tree• Array to List• Tree to List

Review

• Optimization• Defined & example problems• Greedy algorithms• Dynamic programming

• Minimum spanning trees• Defined• Prim’s algorithm• Kruskal’s algorithm

Review

• Object-Oriented• Defined• Behavioral abstraction (Defined & Advantages)• Encapsulation• Abstract data types• Queue & stack examples• Need for formal programmatic construct

Review

• Classes• Defined• Advantages• Structure - public & protected• Role for each• Benefits• Examples• Scope of attributes in protected

Review

• Initialize (not built-in, but useful)• Using classes

• Declaring objects• Objects vs. classes• Accessing methods via the ‘.’ Operator

• Example Classes• Airplane example• Queue example• Pile example

Review• Generic classes

• Defined & benefits• Search and replace type then use a “magic”

keyword• Syntax• Defining objects in algorithm

• Use cases• Clone vs. copy• Inheritance

• Defined & benefits• Extension • Redefinition

Review

• Resolving method reference ambiguity• Class hierarchies• Deferred class & methods

• Polymorphism• Defined & benefits• Simple assignment example• Complex collection example

• Pure OO

Review

• Algorithm Cost and Complexity• Measuring performance (space & time)

• Measuring work (counting instructions)• Best, worst, average

• Measuring work growth• O() notation• Complex O() rules (drop constants, keep

dominant term, etc.)

Review

• Analysis– Linear vs. binary search– Traversals– Data structures

• Compare on traversals, search, & insert– Bubblesort vs. mergesort

• Exponential growth– Hanoi, Monkey tiling, wise peasant

• Reasonable vs. unreasonable• Using O() analysis in data structure design of

solution

Review

• Systems– Concurrency vs. sequential processing (what

we’ve done so far)• Defined & advantages• Multiprogramming• Multiprocessing• Multitasking• Distributed systems

Review

• Issues– Protection, mutual exclusion– Starvation, fairness– Deadlock– Time– Synchronization– Overhead costs (context switch)

Review

• Parallelism– Defined & advantages– Pipeline processing– Product complexity

• Dependencies• Precedence

– Dependence Graphs– Precedence Graphs

Review

• Problem Complexity– Defined– Problems vs. algorithms– Upper bound– Lower bound– Open vs. closed– Tractable vs. intractable

Review

• NP-Complete– Defined– Examples– Certificates– Oracles– Deterministic vs. nondeterministic

• Decidable vs. undecidable• Highly vs. partially

Questions?

Review slides available at

http://prism.gatech.edu/~wl48

Double U Ell