Cliff Shaffer Computer Science Computational Complexity
Feb 25, 2016
Cliff ShafferComputer ScienceComputational Complexity
Computer Performance
Computer PerformanceDo we need to care about performance when computers keep getting faster?
Computer PerformanceDo we need to care about performance when computers keep getting faster?Our history is to do bigger problems, not the same ones faster.
Computer PerformanceDo we need to care about performance when computers keep getting faster?Our history is to do bigger problems, not the same ones faster.More complex problems are less tied to our everyday common sense experience
Algorithm AnalysisWe could compare two programs by running them side-by-side
Algorithm AnalysisWe could compare two programs by running them side-by-sideBut that means we have to implement them!
Algorithm AnalysisWe could compare two programs by running them side-by-sideBut that means we have to implement them!We want a way to easily evaluate programs before they are writtenLook at the algorithm, not the program
Algorithm AnalysisWe could compare two programs by running them side-by-sideBut that means we have to implement them!We want a way to easily evaluate programs before they are writtenLook at the algorithm, not the programAlgorithm Analysis estimates problem cost as a function of growth rate
Simple SearchFind the record with key value 1005.Sequential search: Look through each record in turn.If there are n records, we do work proportional to n (unless we are lucky).
Simple SearchFind the record with key value 1005.Sequential search: Look through each record in turn.If there are n records, we do work proportional to n (unless we are lucky).The growth rate of this problem (in the average or worst cases) is linear on n.
Sorting: Insertion SortFor each record Insert it into the sorted list made from the records already seen.Might have to look at each such record already in the (sorted) list n work.Since we do this for n records, this is n*n in the worst (and average) cases.So the cost is proportional to n2 (unless we are really lucky).
Sorting: Merge SortFor a list of n records: Split the list in half. Merge each half (using merge sort) Merge the records together (needs n work)Total cost: proportional to n log n
Sorting DemoURL: http://www.cs.ubc.ca/spider/harrison/Java/
Compare Insertion, Shell, Heap, Quick sorts
Does It Matter?1000 records:Insertion sort: 1,000,000Mergesort: 10,000Factor of 100 difference
1,000,000 recordsInsertion sort: 1,000,000,000,000Mergsort: 20,000,000Factor of 50,000 differenceHours vs. seconds on a real computer
Tractable vs. IntractableCost n is better than cost n log n, which is better than cost n2.
Tractable vs. IntractableCost n is better than cost n log n, which is better than cost n2.These are all polynomial: a faster computer gives you a bigger problem in an hour by some factor.
Tractable vs. IntractableCost n is better than cost n log n, which is better than cost n2.These are all polynomial: a faster computer gives you a bigger problem in an hour by some factor.Exponential growth: 2n.
Tractable vs. IntractableCost n is better than cost n log n, which is better than cost n2.These are all polynomial: a faster computer gives you a bigger problem in an hour by some factor.Exponential growth: 2n.Making input one unit bigger doubles the cost.
Tractable vs. IntractableCost n is better than cost n log n, which is better than cost n2.These are all polynomial: a faster computer gives you a bigger problem in an hour by some factor.Exponential growth: 2n.Making input one unit bigger doubles the cost.Running twice as fast only gives you one more problem unit.
Tractable vs. IntractableCost n is better than cost n log n, which is better than cost n2.These are all polynomial: a faster computer gives you a bigger problem in an hour by some factor.Exponential growth: 2n.Making input one unit bigger doubles the cost.Running twice as fast only gives you one more problem unit.Exponential-time algorithms are intractable.
ProblemsProblems have many algorithms (sorting)What does cost of a problem mean?
ProblemsProblems have many algorithms (sorting)What does cost of a problem mean?We say the problems cost is that of the best algorithm.But we cant know all the algorithms!
ProblemsProblems have many algorithms (sorting)What does cost of a problem mean?We say the problems cost is that of the best algorithm.But we cant know all the algorithms!It is possible (though difficult) to figure out lowest cost for any algorithm to solve the problemSorting: n log n lower bound.
Traveling Salesman ProblemGiven n cities, find a tour for all the cities that is of shortest length.
Traveling Salesman ProblemGiven n cities, find a tour for all the cities that is of shortest length.Nobody knows a polynomial-time algorithm, only exponential algorithms.
Traveling Salesman ProblemGiven n cities, find a tour for all the cities that is of shortest length.Nobody knows a polynomial-time algorithm, only exponential algorithms.We dont KNOW that this problem needs exponential time.
Traveling Salesman ExampleURL: http://itp.nat.uni-magdeburg.de/~mertens/TSP/TSP.html
Nearest Neighbor Heuristic
NP-CompletenessMany, many problems are like traveling salesman we know no polynomial algorithm, and have no proof they need exponential time.
NP-CompletenessMany, many problems are like traveling salesman we know no polynomial algorithm, and have no proof they need exponential time.It is possible to cheaply convert any problem from this collection into any other.
NP-CompletenessMany, many problems are like traveling salesman we know no polynomial algorithm, and have no proof they need exponential time.It is possible to cheaply convert any problem from this collection into any other.So if we had a polynomial time algorithm for any of them, wed have one for all.
NP-CompletenessMany, many problems are like traveling salesman we know no polynomial algorithm, and have no proof they need exponential time.It is possible to cheaply convert any problem from this collection into any other.So if we had a polynomial time algorithm for any of them, wed have one for all.These are called NP-complete problems.
NP-CompletenessMany, many problems are like traveling salesman we know no polynomial algorithm, and have no proof they need exponential time.It is possible to cheaply convert any problem from this collection into any other.So if we had a polynomial time algorithm for any of them, wed have one for all.These are called NP-complete problems.NP problems are those problems for which we can quickly verify that a proposed solution is correct.
Examples of NP-complete problemsFind the cheapest way to wire up telephones in a cityFind the largest clique in a graphFind a way to assign values to a boolean expression to make it trueFind the largest matching between workers and compatible jobsFind the least number of boxes needed to pack some goods
What do you do? when you must solve an NP-complete problem?
What do you do? when you must solve an NP-complete problem?ApproximationOptimization
What do you do? when you must solve an NP-complete problem?ApproximationOptimizationMany engineering problems are optimization problems
What do you do? when you must solve an NP-complete problem?ApproximationOptimizationMany engineering problems are optimization problemsExamples: Aircraft design, best decision
Why is optimization hard?Imagine a 2d problem find the highest hill.
Why is optimization hard?Imagine a 2d problem find the highest hill.Imagine a 10-parameter problemJust checking the high and low values would give 1024 combinations.
Why is optimization hard?Imagine a 2d problem find the highest hill.Imagine a 10-parameter problemJust checking the high and low values would give 1024 combinations.Imagine a 10d cube 1024 corners.The goal is to find the best point in the cube, for a complex function.
Why is optimization hard?Imagine a 2d problem find the highest hill.Imagine a 10-parameter problemJust checking the high and low values would give 1024 combinations.Imagine a 10d cube 1024 corners.The goal is to find the best point in the cube, for a complex function.Many problems have higher dimension
Why is optimization hard?Imagine a 2d problem find the highest hill.Imagine a 10-parameter problemJust checking the high and low values would give 1024 combinations.Imagine a 10d cube 1024 corners.The goal is to find the best point in the cube, for a complex function.Many problems have higher dimensionWhole branches of CS/Math/Engineering devoted to optimization
Uncomputable ProblemsNot all problems that we can think of can be solved.Abstractly, not all functions can be computed.
Uncomputable ProblemsNot all problems that we can think of can be solved.Abstractly, not all functions can be computed.The number of computable programs is countably infinite.The number of integer functions is uncountably infinite.
The Halting ProblemProblem: Given a particular program P on particular input I, does P halt when run on I?
The Halting ProblemProblem: Given a particular program P on particular input I, does P halt when run on I?Can be proved that this is impossible to determine in all cases.
The Halting ProblemProblem: Given a particular program P on particular input I, does P halt when run on I?Can be proved that this is impossible to determine in all cases.Lots of problems like this that try to determine program behavior.
The Halting ProblemProblem: Given a particular program P on particular input I, does P halt when run on I?Can be proved that this is impossible to determine in all cases.Lots of problems like this that try to determine program behavior.Does this program contain a virus?
Does this terminate for all n?(n is an integer)While n > 1 do if Odd(n) then n = 3n + 1 else n = n/2
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