problem complexity review. problem complexity online survey the spring term course/instructor...
TRANSCRIPT
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Problem ComplexityReview
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Problem Complexity
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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
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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
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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
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Relations between Problems, Algorithms, and Programs
Problem
Algorithm Algorithm
Program ProgramProgram Program
. . . .
. . . . . . . .
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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?
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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)?
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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
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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).”
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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).”
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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?”
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Changing the Bounds
Upper bound Lowered by betteralgorithm
Lower bound Raised by betterproof
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Open Problems
The upper and lower bounds differ.
Upper bound Lowered by betteralgorithm
Lower bound Raised by betterproof
Unknown
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Closed Problems
The upper and lower bounds are identical.
Upper bound
Lower bound
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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
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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)
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Problems that “Cross the Line”
• The upper bound implies intractable • The lower bound implies a tractable• Could go either way…
Next time!!!
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Terminology
• Polynomial algorithms are reasonable• Polynomial problems are tractable
• Exponential algorithms are unreasonable• Exponential problems are intractable
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Terminology
Tractable
Intractable
Reasonable
Unreasonable
Problems Algorithms
Polynomial
Exponential
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Questions?
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Review
• The Algorithmic Model– Algorithm defined– Properties of good algorithms– How to describe algorithms– Relating problems to algorithms to programs
(hierarchy needed)
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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
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Review
• Operations – Assignment– Arithmetic
• +, -, x, /, div, mod• Precedence rules• Using parenthesis to modify precedence
– Input and output• Print• Read
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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)
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Review
• Program maintenance• Software Engineering facts about program
cost• Documentation• Benefits of Constants
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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
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Review
• Procedure• Rules when to use a procedure• Declaration• Function• Rules when to use a function• Declaration• Returning values via the “returns”
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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
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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
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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)
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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
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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
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Review
• Doubly-linked lists– Defined– Example methods (simple insertion)
• Stack– Defined/properties– Push– Pop– Tracing changes (example of algorithm using
push & pop)
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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
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Review
• Trees (Continued)• Recursive insertion in BST• Deleting from BST (conceptually)
• Graphs• Defined/properties• Record definition
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Review
• Static data structures• Arrays
• Defined• Need of constants• Accessing elements via the index• Multi-dimensional arrays (Declaring and
accessing)
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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.
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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)
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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
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Review
• Binary, recursive search on BST• Binary, iterative search on sorted array
• Sort• Bubble sort• Merge sort
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Review
• Conversion• Defined• Helper module advantages• List to Tree• Array to List• Tree to List
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Review
• Optimization• Defined & example problems• Greedy algorithms• Dynamic programming
• Minimum spanning trees• Defined• Prim’s algorithm• Kruskal’s algorithm
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Review
• Object-Oriented• Defined• Behavioral abstraction (Defined & Advantages)• Encapsulation• Abstract data types• Queue & stack examples• Need for formal programmatic construct
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Review
• Classes• Defined• Advantages• Structure - public & protected• Role for each• Benefits• Examples• Scope of attributes in protected
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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
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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
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Review
• Resolving method reference ambiguity• Class hierarchies• Deferred class & methods
• Polymorphism• Defined & benefits• Simple assignment example• Complex collection example
• Pure OO
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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.)
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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
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Review
• Systems– Concurrency vs. sequential processing (what
we’ve done so far)• Defined & advantages• Multiprogramming• Multiprocessing• Multitasking• Distributed systems
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Review
• Issues– Protection, mutual exclusion– Starvation, fairness– Deadlock– Time– Synchronization– Overhead costs (context switch)
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Review
• Parallelism– Defined & advantages– Pipeline processing– Product complexity
• Dependencies• Precedence
– Dependence Graphs– Precedence Graphs
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Review
• Problem Complexity– Defined– Problems vs. algorithms– Upper bound– Lower bound– Open vs. closed– Tractable vs. intractable
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Review
• NP-Complete– Defined– Examples– Certificates– Oracles– Deterministic vs. nondeterministic
• Decidable vs. undecidable• Highly vs. partially
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Questions?
Review slides available at
http://prism.gatech.edu/~wl48
Double U Ell
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