Whom to marry, how to cook and where to buy gas: solving dilemmas of daily life, one algorithm at a...
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Transcript of Whom to marry, how to cook and where to buy gas: solving dilemmas of daily life, one algorithm at a...
Whom to marry, how to cook and where to buy gas: solving dilemmas of daily life, one
algorithm at a time
SAMIR KHULLERDept. of Computer Science
University of Maryland
A typical conversation
Person: What do you do?Me: I am a computer science professor.Person: I have a problem with my PC, can you fix it?Me: No, I don’t think I can do that.Person: You will not fix my PC?Me: I cannot fix my PC, let alone yours.Person: Then what exactly do you do?Me: I study algorithms.Person: Oh, I know that.Me: Really?Person: Yes! I learnt logarithms in high school.
Algorithms not Logarithms!
AL GO R I T H M
Al-Khowarizmi
Algorithms Introduction
Recipe for baking a cake….• 2 sticks butter• 2 cups flour• 1 cup sugar• 4 eggs• 1 cup milk• 1 tsp baking powder• Cocoa powder (1/2 pound)Mix the sugar, baking powder and flour, mix in beaten
eggs, melted butter and bake at 325F for 40 mins.
ALGORITHMS
• Set of instructions for solving a problem, to find a solution.
• What is a problem?
• What is an instruction?
• What is a solution?
Computer Science
• What is the computer actually doing?
• Its running a program (a set of instructions), but what is the program doing?
• Typically, an algorithm is what the program implements.
Outline of talk
• Algorithms and their Applications
• Whom to Marry?
• How to Cook?
• Where to buy gas?
• A few favorite projects of mine..
• Acknowledgements
Why are algorithms central to computing?
• An airport shuttle company needs to schedule pickups delivering everyone to the airport on time. Who goes in which shuttle, and in what order do the pickups occur?
• A delivery company has several customers and trucks that can carry objects. How should they schedule deliveries to the customers to minimize their cost?
This leads to interesting algorithmic problems…
There are lots of feasible solutions!
• How should we pick amongst these solutions?
• Some solutions are cheap, and others may be expensive or undesirable.
• The number of potential solutions is so large that even a fast computer cannot evaluate all these solutions.
• Algorithms tell us how to find good solutions!
A disclaimer
• I have chosen a set of problems whose algorithms are quite simple.
• Towards the end of the talk I will also mention some recent projects that are a little more involved, and its hard to really describe the algorithms and methods used since they are quite complex.
The Marriage Problem
• N men, N women
• Each person provides a ranking of the members of the opposite sex
• Can we find a “good marriage”?
• First studied by Gale and Shapley (1962)
An application: resident matching program
• Each resident rank orders the hospitals, and each hospital rank orders the residents.
• How do we choose an assignment of residents to hospitals?
• We do not want a situation that a resident prefers another hospital, and that hospital preferred this resident to the person assigned to them.
Men’s Preference Lists
Brad (B)
George(G)
Vince(V)
Jennifer(J) Laura(L) Angelina(A)1 2 3
Women’s Preference List
Jennifer(J)
Angelina(A)
Laura(L)
1 2 3
Brad(B) George(G) Vince(V)
Stable Marriage Problem
• A marriage is unstable if there is a pair of people, not married to each other, such that both prefer each other to their current partners. In other words, they have an incentive to elope….
• Can we find a “stable” marriage?
Stable marriage?(Brad, Jen)
(Vince, Angelina)
(George, Laura)
Unstable since Jen and Vince both prefer each other to their current partners.
Running the Algorithm
FIRST ROUND:
Brad proposes to Jen
Vince proposes to Laura
George proposes to Jen
Brad proposes!
Running the Algorithm
FIRST ROUND:
Brad proposes to Jen
Vince proposes to Laura
George proposes to Jen
Jen rejects George, engaged to Brad
Laura engaged to Vince
Angelina gets no proposals….
(Brad,Jen) and (Vince, Laura)
Running the Algorithm
SECOND ROUND:
George proposes to Laura
Laura breaks engagement with Vince, and gets engaged to George
(Brad,Jen) and (George,Laura)
Running the Algorithm
THIRD ROUND:
Vince proposes to Jen
Jen dumps Brad!
Running the Algorithm
THIRD ROUND:
Vince proposes to Jen
Jen breaks engagement with Brad, and gets engaged to Vince
(Vince,Jen) and (George,Laura)
Running the Algorithm
FOURTH ROUND:
Brad proposes to Angelina
Angelina accepts and gets engaged to Brad
(Vince,Jen), (George,Laura) and (Brad,Angelina)
The couples
Stable marriage?(Brad, Angelina)
(Vince, Jen)
(George, Laura)
This solution is stable!
(Vince, Jen) (George, Laura) (Brad, Angelina)
• Vince prefers Laura to his partner Jen, but Laura would rather be with George.
• Brad prefers Jen to Angelina, but Jen would rather be with Vince.
• George prefers Jen to Laura, but Jen would rather be with Vince.
Optimal from the men’s point of view
• Each man gets the “best” possible partner in ANY stable solution.
• Unintuitive: look’s like the marriage is a good one for the women as well, or is it…?
Consider a different instanceBrad proposes to Angelina
Vince proposes to Jen
George proposes to Laura
All women accept since they onlyget one offer.
NOTE: Each woman is paired with the worst possible partner.Now run the algorithm with the womenproposing…..
Online stable marriages
• Assume that women’s preferences are known in advance. The men arrive one at a time and pick their most preferred available partner.
• This does not give a stable solution, and in fact may have MANY unstable pairs.
Paper by Khuller, Mitchell and Vazirani (1991).
What went wrong? People’s preferences change….(?)
Scheduling Problems
• Arise in many industrial applications….
• Computers schedule multiple tasks, people multi-task, complex projects have several interacting sub-parts.
• With large companies manufacturing many products, many interesting scheduling problems arise.
Cooking example
Salad:
25m prep, 0m cooking
Chicken noodle:
10m prep, 40 min cooking
Rice pudding:
15 mins prep, 20m cooking
In what order should Martha make the dishes?
• Martha can work on preparing one dish at a time, however once something is cooking, she can prepare another dish.
• How quickly can she get all the dishes ready?
• She starts at 5pm, and her guests will arrive at 6pm….
First try
5:00pm 5:25pm5:35pm
5:50pm
6:15pm 6:10pm
(25,0) (10,40) (15,20)
Prep time Cook time
Second try
5:00pm 5:10pm5:25pm 5:50pm
5:50pm 5:45pm
(10,40) (15,20)(25,0)
First work on dishes with shortest preparation time?
This rule may not work all the time
Suppose the required times are:
Bulgur (5,10) Lentils (10, 60) Lamb (15, 75)
Shortest prep time order: start at 5pm, and finish lamb at 6:45pm
Longest cooking time first: food ready at 6:30pm.
What if she had to make several dishes?
• For 3 dishes, there are only 6 possible orders. SCR,SRC,RSC,RCS,CSR,CRS.
• The number of possible orderings of 10 dishes is 3,628,800.
• For 15 dishes the number of possible orderings is 1,307,674,368,000!
• This leads to a combinatorial explosion.
Key Idea
• Order dishes in longest cooking time order.• Chicken noodle soup goes first (40 mins of cook
time), next is the Rice pudding (20 mins of cook time), followed by the Salad (0 mins of cook time).
• This is the best ordering. In other words, no other order can take less time.
• This does not work if there are very few stovetops (now the problem becomes really difficult).
What if we had a small number of burners?
• Problem becomes very difficult if we have 2, 3, 4 burners..
• Problem can be solved optimally if we only have one burner (Johnson, 1954)
Where to fill gas?
• Suppose you want to go on a road trip across the US. You start from New York City and would like to drive to San Francisco.
• You have :– roadmap– gas station locations and their gas prices
• Want to:– minimize travel cost
The Gas Station Problem (Khuller, Malekian,Mestre), Eur. Symp. of Algorithms
Finding gas prices online
• Two vertices s & t • A fixed path
• Optimal solution involves stops at every station!
• Thus we permit at most stops.
Structure of the Optimal Solution
sv1 v2 v3 vn t
2.99$ 2.97$2.98$ 1.00$
The Problem we want to solve
• Input: – Road map G=(V,E), source s, destination t– U: tank capacity– d: ER+
– c: VR+
: No. of stops allowed : The initial amount of gas at s
• Goal:– Minimize the cost to go from s to t.
• Output:– The chosen path– The stops– The amount of gas filled at each stop
• Gas cost is “per mile” and U is the range of the car in miles.
• We can assume we start with 0 gas at s.
s’
U -
s
c(s’)= 0
t
Dynamic Programming
• Assuming all values are integral, we can find an optimal solution in O(n2 ∆ U2) time.
• Not strongly polynomial time.The problem could be weakly NP-hard!
OPT[x,q,g] =Minimum cost to go from x to t in q stops, starting with g units of gas.
Key Property
uiui+1
c(ui) c(ui+1)
Suppose the optimal sequence of stops is u1,u2,…,u.
If c(ui) < c(ui+1) Fill up the whole tank
If c(ui) > c(ui+1) Just fill enough to reach ui+1.
Tour Gas Station problem
• Would like to visit a set of cities T.
• We have access to set of gas stations S.
• Assume gas prices are uniform.– The problem is extremely hard even with this
restriction.– We may have a deal with a particular gas
company.
The research process
Problems, Graphs and Algorithms
Is there a way to walk on every bridge exactly once and return to the starting point?
L. Euler (1707—1783)
New England Kidney Exchange• A donor’s kidney
may not match the person they wish to donate to.
• In this case, perhaps another pair has the same problem and the kidneys can be swapped.
A C
DB
Use an algorithm for Maximum matchings in graphs (Edmonds 1965).
Each node hereis a COUPLE
Re-assigning Starbucks employees to reduce commute times
Article from the Washington Post
A
BC
D
D
B
C
A
Energy Minimization
• Consider fire monitoring. Sensors have:– Fixed locations– Limited battery power
• If sensors are always on:– Full coverage over the regions– Short system life time
• Better solution:– Activating sensors in multiple
time slots• Benefits:
– Make use of overlap– Turning sensors on and off
increase their life time
Regions(Targets)
Sensors A
B
C
D
S1 S2
S3
S4
Work with A. Deshpande, A.Malekian and M. Toossi
Data Placement & Migration
• Data Layout: load balancing– Disks have constraints on space, load, etc.– User data access pattern which changes with time
Primarily joint work with L. Golubchik, S. Kashyap, Y-A. Kim, S. Shargorodskaya, J. Wan, and A. Zhu
Approximation Algorithms
• For many problems, no simple (or complex!) rules seem to work.
• Such problems arise very frequently – the famous Traveling Salesperson Problem is an example.
• How should we cope with this? • Our attempt is to develop a set of tools that would give
rise to methods for approaching such problems. Even if we cannot find the optimal solutions quickly, perhaps we can find almost optimal solutions quickly?
• Greedy Methods, LP rounding methods, Primal-Dual methods.
• In general, these methods also provide lower bounding methods
Acknowledgements Lecture dedicated to the memory of my grandfather, Prof. Ish Kumar (1902—1999). Academic Influence Prof. R. Karp, Prof. V. Vazirani, Prof. J. Mitchell, Prof. E. Arkin, Prof. U. Vishkin My wonderful co-authors, especially, B. Raghavachari, N. Young, A. Srinivasan, L.
Golubchik, B.Bhattacharjee, D. Mount, S. Mitchell, B. Schieber, A. Rosenfeld, J. Naor, R. Hassin, S. Guha, M. Charikar, R. Thurimella, R. Pless, M Shayman, G. Kortsarz.
My Ph.D. students – R. Bhatia, Y.Sussmann, R. Gandhi, Y-A. Kim, J. Wan, J. Mestre, S. Kashyap, A. Malekian.
Undergrads – K. Matherly, D. Hakim, J. Pierce, B. Wulfe, A. Zhu, S. Shargorodskaya, C. Dixon, J. Chang, M. McCutchen.
Above all, a BIG thanks to all members of my family, friends and relatives. I cannot
express my thanks deeply enough.