Jeff Edmonds

Priority encoding transmission

We introduce a new method, called priority encoding transmission, for sending messages over lossy packet-based networks. When a message is to be transmitted, the user specifies a priority value for each part of the message. Based on the priorities, the system encodes the message into packets for transmission and sends them to (possibly multiple) receivers. The priority value of each part of the message determines the fraction of encoding packets sufficient to recover that part. Thus even if some of the encoding packets are lost en-route, each receiver is still able to recover the parts of the message for which a sufficient fraction of the encoding packets are received. For any set of priorities for a message, we define a natural quantity called the girth of the priorities. We develop systems for implementing any given set of priorities such that the total length of the encoding packets is equal to the girth. On the other hand, we give an information-theoretic lower bound that shows that for any set of priorities the total length of the encoding packets must be at least the girth. Thus the system we introduce is optimal in terms of the total encoding length. This work has immediate applications to multimedia and high-speed networks applications, especially in those with bursty sources and multiple receivers with heterogeneous capabilities. Implementations of the system show promise of being practical.

Linear time erasure codes with nearly optimal recovery

An (n,c,l,r) erasure code consists of an encoding algorithm and a decoding algorithm with the following properties. The encoding algorithm produces a set of l-bit packets of total length cn from an n-bit message. The decoding algorithm is able to recover the message from any set of packets whose total length is r, i.e., from any set of r/l packets. We describe erasure codes where both the encoding and decoding algorithms run in linear time and where r is only slightly larger than n.

Scheduling in the dark

We considered non-clairvoyant multiprocessor scheduling of jobs with arbitrary arrival times and changing execution characteristics. The problem has been studied extensively when either the jobs all arrive at time zero, or when all the jobs are fully parallelizable, or when the scheduler has considerable knowledge about the jobs. This paper considers for the first time this problem without any of these three restrictions yet when our algorithm is given more resources than the adversary. We provide new upper and lower bound techniques applicable in this more difficult scenario. The results axe of both theoretical and practical interest. In our model, a job can arrive at any arbitrary time and its execution characteristics can change through the life of the job from being anywhere from fully parallelizable to completely sequential. We assume that the scheduler has no knowledge about the jobs except for knowing when a job arrives and knowing when it completes. (This is why we say that the scheduler is completely in the dark.) Given all this, we prove that the scheduler algorithm Equi-partition, though simple, performs within a constant factor as well as the optimal scheduler as long as it is given at least twice a8 many proceesors. More over, we prove that if none of the jobs are “strictly” fully parallelizahle, then Equi-partition performs competitively with no extra processors. We also consider other variations: faster procesors; fewer preemp tions; and a wider range of execution characteristics.

The Relative Complexity of NP Search Problems

Papadimitriou introduced several classes of NP search problems based on combinatorial principles which guarantee the existence of solutions to the problems. Many interesting search problems not known to be solvable in polynomial time are contained in these classes, and a number of them are complete problems. We consider the question of the relative complexity of these search problem classes. We prove several separations which show that in a generic relativized world the search classes are distinct and there is a standard search problem in each of them that is not computationally equivalent to any decision problem. (Naturally, absolute separations would imply thatP?NP.) Our separation proofs have interesting combinatorial content and go to the heart of the combinatorial principles on which the classes are based. We derive one result via new lower bounds on the degrees of poly- nomials asserted to exist by Hilberts nullstellensatz over finite fields.

Cake cutting really is not a piece of cake

We consider the well-known cake cutting problem in which a protocol wants to divide a cake among n ≥ 2 players in such a way that each player believes that they got a fair share. The standard Robertson-Webb model allows the protocol to make two types of queries, Evaluation and Cut, to the players. A deterministic divide-and-conquer protocol with complexity O(n log n) is known. We provide the first an Ω(n log n) lower bound on the complexity of any deterministic protocol in the standard model. This improves previous lower bounds, in that the protocol is allowed to assign to a player a piece that is a union of intervals and only guarantee approximate fairness. We accomplish this by lower bounding the complexity to find, for a single player, a piece of cake that is both rich in value, and thin in width. We then introduce a version of cake cutting in which the players are able to cut with only finite precision. In this case, we can extend the Ω(n log n) lower bound to include randomized protocols.

Non-clair voy ant multiprocessor scheduling of jobs with changing execution characteristics

AbstractThis work theoretically proves that Equi-partition efficiently schedules multiprocessor batch jobs with different execution characteristics. Motwani, Phillips, and Torng (Proc. 4th Annu. ACM/SIAM Symp. on Discrete Algorithms, pp. 422–431, Austin, 1993) show that the mean response time of jobs is within two of optimal for fully parallelizable jobs. We extend this result by considering jobs with multiple phases of arbitrary nondecreasing and sublinear speedup functions. Having no knowledge of the jobs being scheduled (non-clairvoyant) one would not expect it to perform well. However, our main result shows that the mean response time obtained with Equi-partition is no more than

Speed scaling of processes with arbitrary speedup curves on a multiprocessor

2 + \sqrt 3 \approx 3.73

Multicast Pull Scheduling: When Fairness Is Fine

times the optimal. The paper also considers schedulers with different numbers of preemptions and jobs with more general classes of speedup functions. Matching lower bounds are also proved.