Motivation and Definitions Results Conclusion An Analysis of Call Admission Problems on Grids LSD & LAW Hans-Joachim Böckenhauer, Dennis Komm, Raphael Wegner Department of Computer Science ETH Zürich February 9, 2018 CAPG
Motivation and Definitions Results Conclusion Outline Motivation and Definitions 1 Results 2 Lower Bounds Upper Bounds Conclusion 3 CAPG
Motivation and Definitions Results Conclusion Online Problems Definition (Online Maximization Problem Π ) Sequence of requests Satisfy one request before the next one arrives Maximize the gain CAPG
Motivation and Definitions Results Conclusion The Disjoint Path Allocation Problem (DPA) 3 5 1 2 4 CAPG
Motivation and Definitions Results Conclusion The Disjoint Path Allocation Problem (DPA) 3 5 1 2 4 CAPG
Motivation and Definitions Results Conclusion The Disjoint Path Allocation Problem (DPA) 3 5 1 2 4 CAPG
Motivation and Definitions Results Conclusion The Disjoint Path Allocation Problem (DPA) 3 5 1 2 4 CAPG
Motivation and Definitions Results Conclusion The Disjoint Path Allocation Problem (DPA) 3 5 1 2 4 CAPG
Motivation and Definitions Results Conclusion Online Setting with Advice How much information are we missing . . . to be optimal? . . . to achieve some competitive ratio? = ⇒ New measure for complexity of online problems CAPG
Motivation and Definitions Results Conclusion Online Setting with Advice Definition (Online Algorithm A LG with Advice for Π ) Adversary chooses online input instance Oracle with unlimited power knows instance and chooses infinite advice string A LG can read an arbitrary long, but finite prefix q ( · ) is the advice complexity of A LG ⇐ ⇒ A LG reads at most first q ( · ) bits of advice from start Advice complexity s ( n ) of Π : maximum over a all inputs of length n , for best pair of oracle and algorithm CAPG
Motivation and Definitions Results Conclusion Online Setting with Advice Definition (Competitive ratio with advice) Π is online maximization problem A LG is online algorithm with advice for Π O PT ( I ) is an optimal (offline) solution for instance I of Π A LG is c-competitive for Π if there exists a constant α ≥ 0 such that gain ( O PT ( I )) ≤ c · gain ( A LG ( I )) + α for all instances I of Π . CAPG
Motivation and Definitions Results Conclusion Extended to Grids m . . . m − 1 v m − 1 , 3 m − 2 . . . . . . . . . . . . p ver Height: m − 1 3 Length: n − 1 . . . 2 v 2 , n − 2 1 . . . n 1 2 3 n − 2 n − 1 p hor CAPG
Motivation and Definitions Results Conclusion The Call Admission Problem on Grids (CAPG) Definition (CAPG) Online maximization problem Π CAPG Request is a pair of servers asking for a connection Every connection is fixed (no termination or modification) Only one connection per wire Goal: maximize the number of granted connections CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion Outline Motivation and Definitions 1 Results 2 Lower Bounds Upper Bounds Conclusion 3 CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion | E | − 1 Advice Bits for DPA Theorem ([BBF + 14]) To solve DPA optimally, | E | − 1 advice bits are necessary and sufficient. CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion | E | − 1 Advice Bits for DPA 0 0 1 1 0 1 P 1 Optimal solution O PT ( I ) P 2 indicated by bit string: 1 ⇐ ⇒ end one P 3 request, start another one P 4 P i contains all requests of length | E | − i + 1 P 5 which do not contradict requests in O PT ( I ) of P 6 earlier phase P 7 CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion | E | − 1 Advice Bits for DPA Optimal solution is unique 2 | E |− 1 different instances Optimal solutions S 1 , S 2 have to differ before instances I 1 , I 2 are distinct on their asked prefixes of requests ⇒ ≥ log 2 ( 2 | E |− 1 ) = | E | − 1 advice bits required for = optimality CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion Almost | E | Advice Bits for CAPG Can we just ask these instances on each column and row for CAPG? Consider long request in solution indicated by bit string If not satisfied we have much space for detours = ⇒ bit string solution is not optimal anymore Mitigation: Ask only sufficiently small requests, i.e., only last four phases = ⇒ bit string solution is optimal again Still no unique optimal solution in general CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion Almost | E | Advice Bits for CAPG Lemma There are t n + 2 bit strings of length n ∈ N which contain at most three consecutive 0 s, where t n denotes the nth tetranacci number a . a t n = t n − 1 + t n − 2 + t n − 3 + t n − 4 , t 0 = t 1 = t 2 = 0 , t 3 = 1 CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion Almost | E | Advice Bits for CAPG Proof sketch: Optimal solutions differ only in requests satisfied with paths of length 4 and have specific forms Every optimal solution has to grant a detour before rejecting a request intended by bit string = ⇒ optimal solutions are distinct before the prefixes of respective instances are different t m n · t n m instances with different optimal solutions = ⇒ ≥ m · log 2 ( t n ) + n · log 2 ( t m ) advice bits required for optimality CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion Almost | E | Advice Bits for CAPG Theorem Every optimal online algorithm with advice for CAPG on an ( m × n ) -grid G has to read at least m · log 2 ( t n ) + n · log 2 ( t m ) advice bits. CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion Almost | E | Advice Bits for CAPG Corollary Every optimal online algorithm with advice for CAPG on an ( m × n ) -grid G has to read at least 0 . 94677 · | E ( G ) | − m − n advice bits. CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion Bit Guessing for CAPG Theorem Every online algorithm with advice for CAPG which achieves a competitive ratio of c ≤ 12 11 on a grid G has to read at least � � � � � � 6 � � 6 �� | E ( G ) | 6 − 6 6 − 6 1 + log 2 + c − 5 log 2 c − 5 c c 2 bits of advice. CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion Outline Motivation and Definitions 1 Results 2 Lower Bounds Upper Bounds Conclusion 3 CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion Trivial Bound Theorem There is an optimal online algorithm with advice for CAPG which reads at most 2 | E | · ⌈ log 2 ( | V | ) ⌉ ≤ 2 | E | · log 2 ( | E | + m + n ) bits of advice for every ( m × n ) -grid G = ( V , E ) . Oracle chooses some optimal solution ≤ | E | requests Encode both endpoints of every granted request = ⇒ 2 · ⌈ log 2 ( | V | ) ⌉ bits per satisfied request Overall: ≤ 2 | E | · ⌈ log 2 ( | V | ) ⌉ CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion How can we improve? Knowing which edges are used in an optimal solution sometimes not helpful (e.g., in case all edges are used, but some requests are contradicting) Need to transmit the “membership” to a request “Neighbouring” paths need to be distinguishable = ⇒ Coloring problem in some auxiliary graph CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion The Auxiliary Graph � G Definition ( � G ( S ) = ( � V , � E ) ) ⇒ v p ∈ � Path p in S satisfying a request = V { v g , v h } ∈ � E ⇐ ⇒ g and h share some vertex in G Edges of G unused by S are split up into connected components, s.t. component q corresponds to v q ∈ � V and the chromatic number χ ( � G ( S )) is minimized CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion The Auxiliary Graph � G CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion � G Bound Theorem Let I denote all possible instances of CAPG on a grid G = ( V , E ) , and let S opt ( I ) be the set of optimal solutions for an instance I ∈ I . Then, there is an optimal online algorithm with advice for CAPG using at most S ∈S opt ( I ) ⌈| E | · log 2 ( χ ( � G )) ⌉ + 2 ⌈ log 2 ( χ ( � max min G ( S ))) ⌉ I ∈I advice bits. CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion � G Bound Oracle: Can compute all optimal solutions Selects optimal solution S , s.t. ⌈| E | · log 2 ( χ ( � G )) ⌉ + 2 ⌈ log 2 ( χ ( � G ( S ))) ⌉ is minimal Uses 2 ⌈ log 2 ( χ ( � G ( S ))) ⌉ bits to transmit χ ( � G ( S )) in a self-delimiting encoding Colors corresponding connected components of G according to χ ( � ⇒ χ ( � G ) | E | possibilities G ) = ⌈| E | · log 2 ( χ ( � G )) ⌉ bits for transmitting the coloring CAPG
Motivation and Definitions Lower Bounds Results Upper Bounds Conclusion � G Bound Algorithm (receiver): Recomputes the length of the encoding Reads off the coloring Decides accordingly CAPG
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