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Algorithms in the Real World: From Application to Abstraction to Algorithm

Algorithms in the Real World: From Application to Abstraction to Algorithm. (and Back). Robert E. Tarjan. Princeton University and InterTrust Technologies. Research with Haim Kaplan and Kostas Tsioutsiouliklis Help from Andrew Goldberg. Outline.

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Algorithms in the Real World: From Application to Abstraction to Algorithm

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  1. Algorithms in the Real World: From Application to Abstraction to Algorithm (and Back) Robert E. Tarjan Princeton University and InterTrust Technologies Research with Haim Kaplan and Kostas Tsioutsiouliklis Help from Andrew Goldberg

  2. Outline Broadcast Scheduling Scheduling Using a Kinetic Heap [Initial Solutions] Connections to Computational Geometry [Additional Solutions] A Simple Solution (Practical?) The Discovery/Development Process Thoughts

  3. Broadcast Scheduling (Lecture by Mike Franklin research papers) requests Server: many data items Users Broadcast channel (single-item) One server, many possible items to send. One broadcast channel. Users submit requests for items. Goal: Satisfy users as well as possible, making decisions on-line.

  4. Abstractions: All items have the same broadcast time. Minimize the sum of waiting times? Scheduling Policies (heuristics) Greedy = Longest Wait first (LWF): Send item with largest sum of waiting times. (vs. number of requests or longest single waiting time) R x W: Max # requests x longest waiting time Approximations to R x W

  5. Results of Franklin and others: LWF schedules well “in practice” (in simulations) but too expensive (linear-time) This claim used to justify approximations to R x W, still linear-time but with a smaller (parameterized) constant. 5

  6. Questions (for an algorithm guy or gal) LWF does well compared to what? Try a competitive analysis _ Can we improve the cost of LWF? What data structure? _

  7. Parametic Heap A collection of items, each with an associated key. key (i) = aix + biai,, bi reals, x a real-valued parameter ai = slope, bi = constant Operations: make an empty heap h. insert item i with key ai x + bi into heap h. find an item i in heap h of minimum key for x = x0. delete item i from heap h.

  8. Kinetic Heap A parametric heap such that successive x-values of find mins are non-decreasing. (Think of x as time.) xc= largest x so far (current time) Additional operation: decrease the key of an item i, replacing it by a key that is no larger for all x >(next) xc

  9. Broadcast Scheduling via a Kinetic Heap Max-heap (replace find min by find max, decrease key by increase key = change sign of all keys) Can implement LWF orR x W or any similar policy: Broadcast decision is find min plus delete Request is insert (if first) or decrease key (if not) Only (at most) find min need be real-time, other ops can proceed concurrentlywith broadcasting Slopes are integers that count requests

  10. What is known about parametric and kinetic heaps? A key is a line _computational geometry Equivalent problems: maintain the lower envelope of a collection of lines in 2 D projective duality maintain the convex hull of a set of points in 2D under insertion and deletion “kinetic” restriction = “sweep line” constraint of queries

  11. (Seminal) Results Overmars and van Leeuwen (1981) Dynamic convex hulls and lower envelopes in O(log n) time per query, O(log2n) time per update, worst-case Basch, Guibas, and Hershberger (1997) “Kinetic” data structure paradigm (Much other work: improvements, restrictions, etc.)

  12. Our Results Four different data structures, each one best in a different setting. Simple Kinetic Heap A balanced binary tree, with items in nodes in symmetric order by key slope. The tree is a tournament on items by current key. The tree also contains swap times (times when winning keys change) and is a tournament on swap times. O(1) worst-case find min, O(log2n) amortized insert/delete Combines seminal ideas with our own Is it practical?

  13. min key min swap time swap time 3.5 2 d b d 3 2 a: x + 10 b: 2x + 7 c: 3x +2 d:4x xc = 0 A Simple Kinetic Heap

  14. The Discovery/Development Process Application model experiment apply Abstraction Algorithm develop Old/New Theory/Practice

  15. Thoughts Effective algorithm development and application in the real world requires active, repeated interaction between the application developer(s) and the algorithm guy(s)/gal(s) – teaming. Our training of students should include, or at least model, this practice. Our training should also include cross-discipline studies – rich sources of problems for algorithm guys/gals.

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