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Better Algorithms and Hardness for Broadcast Scheduling via a Discrepancy Approach

Better Algorithms and Hardness for Broadcast Scheduling via a Discrepancy Approach. N. Bansal 1 , M. Charikar 2 , R. Krishnaswamy 2 , S. Li 3 1 TU Eindhoven 2 Princeton University 3 TTIC. Broadcast Scheduling Problem. r esponse time = 2. a server holding n pages requests come over time

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Better Algorithms and Hardness for Broadcast Scheduling via a Discrepancy Approach

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  1. Better Algorithms and Hardness for Broadcast Scheduling via a Discrepancy Approach N. Bansal1, M. Charikar2, R. Krishnaswamy2, S. Li3 1TU Eindhoven 2Princeton University 3TTIC

  2. Broadcast Scheduling Problem response time = 2 • a server holding n pages • requests come over time • broadcast 1 page per time slot • all outstanding requests for the page satisfied • minimize average response time • offline version 1 3 5 2 4 5 3 4 5 1 2 4 response time = 3 requests Time 1 3 5 4 3 2 broadcasts 5 1 2 3 4

  3. Our Results *[Bansal-Coppersmith-Sviridenko08] #[Bansal-Charikar-Khanna-Naor05] ~[Erlebach-Hall 02]

  4. Discrepancy Approach • negative results (integrality gap and hardness) • connection to permutation discrepancy • positive result • Lovett-Meka algorithmic framework for discrepancy minimization

  5. Discrepancy Approach • negative results (integrality gap and hardness) • connection to permutation discrepancy • positive result • Lovett-Meka algorithmic framework for discrepancy minimization

  6. 3-Permutation Discrepancy • give 3 permutations of [n] • find a coloring χ: [n]{±1} • minimize the maximum discrepancy over all prefixes of the permutations 5 6 3 1 4 2 6 3 1 4 2 5 1 5 2 3 4 6 5 63 1 42 63 1 42 5 15 23 46 χ: 1 2 34 5 6 discrepancy = 2

  7. Why 3 Permutations? • 1 permutation : discrepancy=1, trivial • 2 permutations : discrepancy=1, easy exercise • 3 permutations? • upper bound : O(log n) • lower bound [Newman-Nikolov 11]: Ω(log n) • l ≥ 3 permutations • upper bound : O(l1/2 log n) • lower bound : max{Ω(l1/2), Ω(log n)}

  8. Negative Results Main Lemma l-permutation instance Π broadcast scheduling instance I “discrepancy” optimal response time = LP(I) = O(1) • Main + Ω(log n)-disc. for 3-perm.  Ω(log n)-int. gap • Main + Ω(l1/2)-hard. for l-perm.(new)  Ω(log1/2n)-hard.

  9. Fractional Schedule from LP response time 0.4×1+0.6×2=1.6 requests 1 3 5 2 4 5 3 4 5 1 2 4 Time integral schedule fractional schedule

  10. Main Lemma l-permutation instance Π broadcast scheduling instance I “discrepancy” optimal response time = LP(I) = O(1) • proof steps: • construction of BS instance from l permutations • Θ(1) LP value • small discrepancy  small response time • small response time  small discrepancy

  11. Construction of BS Instance • given 3 permutations π1π2π3 of size m • π1 = (5, 8, 4, 6, 3, 2, 1, 7) • π2 = (6, 7, 3, 8, 5, 1, 2, 4) • π3 = (7, 1, 3, 2, 8, 5, 6, 4) permutation interval forbidden interval P1 P2 P3 P4 P5 P6 P7 m/2 π2 π3 π1 5431 8627 5431 8627 6352 7814 6352 7814 7386 1254 7386 1254 Req:

  12. Good and Bad Requests • average response time ≈ # bad requests • new goal: minimize #bad requests • a request in Pi is good if it is satisfied at Pi or Pi+1 • otherwise, the request is bad P1 P2 P3 P4 P5 P6 P7 3 7 6 6 3 7 Brd: 3458 1276 8534 6721 4835 7216 3485 5431 8627 5431 8627 6352 7814 6352 7814 7386 1254 7386 1254 Req:

  13. Θ(1) LP Value P1 P2 P3 P4 P5 P6 P7 • LP solution • each time slot, broadcast ½ fraction of each page requested • P7: broadcast ½ fraction of the m pages arbitrarily • all requests are good: • ½ of request in Pi is satisfied immediately • remaining ½ satisfied at Pi+1 request ½ satisfied ½ satisfied 5431 8627 5431 8627 6352 7814 6352 7814 7386 1254 7386 1254 Req:

  14. How to Make All Requests Good in an Integral Schedule? P1 P2 P3 P4 P5 P6 P7 • all m pages requested in all intervals(except P7) • each P-interval has m/2 slots • solution: • m/2 pages are broadcast in P1, P3, P5, P7 • m/2 pages are broadcast in P2, P4, P6 • giving a balanced ±1 coloring of the m pages Brd: 3421 5867 4312 6785 1324 7856 3421 5431 8627 5431 8627 6352 7814 6352 7814 7386 1254 7386 1254 Req:

  15. How to Make All Requests Good in an Integral Schedule? P1 P2 P3 P4 P5 P6 P7 • enough to make all requests good? • No! Broadcast may be before the request • no bad requests only if two requests at the same time have different colors • discrepancy of 3-permutation system is 1 3 2 14 4312 Brd: 3421 5867 4312 6785 1324 7856 3421 5431 8627 5431 8627 6352 7814 6352 7814 7386 1254 7386 1254 Req:

  16. Small DiscrepancyFew Bad Requests • suppose discχ(πi) = d • πi =(1, 10, 2, 6, 8, 7, 3, 11, 5, 12, 4, 9) • χ=(1, 10, 2, 6, 8, 7, 3, 11, 5, 12, 4, 9) • order of red elements (1,6,3,5,4,9) • right rotate by d-1=1 positions: (9,1,6,3,5,4) • broadcast according to this ordering in P2i-1 • #bad quests = d-1 d = 2 requests = 128354 106711129 broadcasts = 916354 broadcast before request : bad broadcast after request : good

  17. Remarks • “discrepancy” = average discrepancy of lpermutations • size of BS instance is exponential in l • lengths of forbidden intervals grow exponentially P1 P2 P3 P4 P5 P6 P7 request bad good

  18. Discrepancy Approach • negative results (integrality gap and hardness) • connection to permutation discrepancy • positive result • Lovett-Meka algorithmic framework for discrepancy minimization

  19. Lovett-Meka Framework • A  Rm×n, x  [0,1]n,b=Ax, • λ1, λ2, …,λms.t. • output: y [0,1]n, s.t. ½ fraction of coordinates in y are integral “error” n n ±λ1||A1|| ±λ2||A2|| ±λ3||A3|| ... ±λm||Am|| A A y x b b × × = = m m

  20. Tentative Scheduling • we may broadcast more than 1 page at a time slot • tentative schedule of backlog b valid schedule, with additive b loss in the average response time 6 time slots, 11 broadcast, backlog = 5

  21. Goal • assumptions: • fractional schedule is ½-intergal • every page is broadcast ≤ Δ = O(log n) times •  # timeslots ≤ 2Δ × n • locally consistent distributions with probability 1/2 with probability 1/2

  22. Interesting Intervals λ= 0 • # time slots ≤ 2Δ × n • “error” • repeat log n times : backlog = O(log3/2n) λ= 1 64Δ λ= 2 … ……

  23. Summary • Open problems • hardness for 3-permutation(implying the same hardness for broadcast scheduling) • discrepancy of l-permutation?

  24. Thank you! Questions?

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