Cliff Shaffer Computer Science Computational Complexity

Computer Performance

Do we need to care about performance when computers keep getting faster?

Computer Performance Do we need to care about performance when computers keep getting faster? Our history is to do bigger problems, not the same ones faster.

Computer Performance Do 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 Analysis We could compare two programs by running them side-by-side

Algorithm Analysis We could compare two programs by running them side-by-side But that means we have to implement them!

Algorithm Analysis We could compare two programs by running them side-by-side But that means we have to implement them! We want a way to easily evaluate programs before they are written Look at the algorithm, not the program

Algorithm Analysis We could compare two programs by running them side-by-side But that means we have to implement them! We want a way to easily evaluate programs before they are written Look at the algorithm, not the program Algorithm Analysis estimates problem cost as a function of growth rate

Simple Search Find the record with key value Sequential search: Look through each record in turn. If there are n records, we do work proportional to n (unless we are lucky).

Simple Search Find the record with key value 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 Sort For 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 n 2 (unless we are really lucky).

Sorting: Merge Sort For 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 Demo URL: Compare Insertion, Shell, Heap, Quick sorts

Does It Matter? 1000 records: Insertion sort: 1,000,000 Mergesort: 10,000 Factor of 100 difference 1,000,000 records Insertion sort: 1,000,000,000,000 Mergsort: 20,000,000 Factor of 50,000 difference Hours vs. seconds on a real computer

Tractable vs. Intractable Cost n is better than cost n log n, which is better than cost n 2.

Tractable vs. Intractable Cost n is better than cost n log n, which is better than cost n 2. These are all polynomial: a faster computer gives you a bigger problem in an hour by some factor.

Tractable vs. Intractable Cost n is better than cost n log n, which is better than cost n 2. These are all polynomial: a faster computer gives you a bigger problem in an hour by some factor. Exponential growth: 2 n.

Tractable vs. Intractable Cost n is better than cost n log n, which is better than cost n 2. These are all polynomial: a faster computer gives you a bigger problem in an hour by some factor. Exponential growth: 2 n. Making input one unit bigger doubles the cost.

Tractable vs. Intractable Cost n is better than cost n log n, which is better than cost n 2. These are all polynomial: a faster computer gives you a bigger problem in an hour by some factor. Exponential growth: 2 n. Making input one unit bigger doubles the cost. Running twice as fast only gives you one more problem unit.

Tractable vs. Intractable Cost n is better than cost n log n, which is better than cost n 2. These are all polynomial: a faster computer gives you a bigger problem in an hour by some factor. Exponential growth: 2 n. Making input one unit bigger doubles the cost. Running twice as fast only gives you one more problem unit. Exponential-time algorithms are “intractable”.

Problems Problems have many algorithms (sorting) ‏ What does “cost of a problem” mean?

Problems Problems have many algorithms (sorting) ‏ What does “cost of a problem” mean? We say the problem’s cost is that of the best algorithm. But we can’t know all the algorithms!

Problems Problems have many algorithms (sorting) ‏ What does “cost of a problem” mean? We say the problem’s cost is that of the best algorithm. But we can’t know all the algorithms! It is possible (though difficult) to figure out lowest cost for any algorithm to solve the problem Sorting: n log n lower bound.

Traveling Salesman Problem Given n cities, find a tour for all the cities that is of shortest length.

Traveling Salesman Problem Given 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 Problem Given n cities, find a tour for all the cities that is of shortest length. Nobody knows a polynomial-time algorithm, only exponential algorithms. We don’t KNOW that this problem needs exponential time.

Traveling Salesman Example URL: Nearest Neighbor Heuristic

NP-Completeness Many, many problems are like traveling salesman – we know no polynomial algorithm, and have no proof they need exponential time.

NP-Completeness Many, 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-Completeness Many, 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, we’d have one for all.

NP-Completeness Many, 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, we’d have one for all. These are called NP-complete problems.

NP-Completeness Many, 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, we’d 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 problems Find the cheapest way to wire up telephones in a city Find the largest clique in a graph Find a way to assign values to a boolean expression to make it true Find the largest matching between workers and compatible jobs Find 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? Approximation Optimization

What do you do? … when you must solve an NP-complete problem? Approximation Optimization Many engineering problems are optimization problems

What do you do? … when you must solve an NP-complete problem? Approximation Optimization Many engineering problems are optimization problems Examples: 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 problem Just 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 problem Just 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 problem Just 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 problem Just 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 Whole branches of CS/Math/Engineering devoted to optimization

Uncomputable Problems Not all problems that we can think of can be solved. Abstractly, not all functions can be computed.

Uncomputable Problems Not 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 Problem Problem: Given a particular program P on particular input I, does P halt when run on I?

The Halting Problem Problem: 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 Problem Problem: 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 Problem Problem: 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