An FPTAS for the Lead-Based Multiple Video Transmission LMVT Problem

An FPT AS for the Lead-Based Multiple Video T ransmission (LMVT) Problem Swapnoneel Roy and Atri Rudra ⋆ Department of Computer Science and Engineering, Universit y at Buffalo, The State Universit y of New Y ork, Buffalo, New Y ork 14260. { atri, sroy7 } @buffalo.edu Abstract. The Lead-Based Multiple Video T ransmissi on ( LMVT ) prob- lem is motiv ated by applications in managing the quality of exp erience (QoE) of video s treaming for mobile clients. In an earlier work, the LMVT problem has b ee n shown to be NP-hard for a sp eci fic bit- to-lea d conv ersion fun ctio n φ . In this work, we show the problem to be NP-hard even if the fun ct i on φ is linear. W e then design a fully p olynomial time approximatio n sc heme (FPT AS) for the problem. This problem is exactly equiv alent to the S ant a Clause Pro blem on whic h there h a s b ee n a lot of work don e off-late. 1 In tro duction The LMVT problem deals with multiplexing videos sim ultaneous ly slot by slot ov er a wireless channel. Ea c h slo t could be allo cated to at most o ne video. The nu mber o f bits that can b e tra nsmitted to a pa rticular video in a given slot is known. Hence the videos receive v ariable num ber of bits ov er the v arious slo ts . The time of transmission is divided ov er a num b er of ep o chs . Each such epo ch has a fix e d nu mber of slots (sa y B ) kno wn b eforehand. The go al is to allo c a te all the slots in an ep och to the n videos in a manner which maximizes the minim um nu mber of bits r eceiv ed by any video in that ep o c h. The le ad of a video is defined as the amount of time it can b e played without int err uption. An int err uption in the playing of a video o ccurs when ther e are no more frames left in the buffer to b e play ed. The lead of an y video is calcula ted at the end of a n ep och using a function φ on the n umber of bits b received by that video in that ep och. The motiv ation b ehind studying this pr oblem is to develop a slo t allo c a tion algorithm to ensure the uninterrupted play of each video in the net work. This problem has been well studied, and a num ber of pap ers hav e been published o n it recently [2], [3], [4], [5]. 2 Preliminaries In this section we present a theoretica l formulation of the problem. F or video v i and slot j we de fine the bit ra te r ij to b e the maximum num ber of bits that ca n ⋆ Researc h supp orted in part by NSF grant CCF-0844796 . 2 Roy S and R udra A be tra nsmitt ed to v i in j . The de cision version of the LMVT pro blem ca n b e presented a s follows. LMVT Problem Input: n video s in the channel, B slots in the ep och, the bit rates r ij ∈ Z + for the n videos a nd B slots, a function φ ( b ) for calculating the lead for b bits, and k ∈ Z + . Question: Is there an allo cation o f the B slots to the n videos so each video has a lead of at leas t k ? In [1 ], the LMVT pro blem has b een shown to b e NP-Hard for a specific function φ to calculate the lead based on the num b er of bits received. A natural greedy algorithm has b een designed and ha s been shown to perform w ell in practice with exp erimen tal results. In this work, we s ho w the pr o blem to remain NP- Hard even for the case in which φ is linear. Next we design an FPT AS for the problem. 3 LMVT problem remains NP-Hard eve n for a linear φ W e assume a linear function φ to c a lculate the lead ba sed on the num ber of received b . Let us consider a monotonic linear φ , such that φ ( b ) = b . Then the problem b ecomes o f finding a slot allo cation to maximize the minimum num b er of bits received b y any video. W e show the problem to remain N P -Hard even then. W e r educe the P ar tition P r oblem , a known NP-Hard problem to LMVT . The P ar tition Problem can b e presented as: P ar tition Problem Input: A set S ⊆ Z + with P x ∈ S x = U . Question: Ca n S be par tit ioned to S ′ and S \ S ′ such that P x ∈ S ′ x = P x ∈ S \ S ′ x = U / 2? F or the r e duct ion, consider any instance of the P ar tition Problem with S = { x 1 , x 2 , · · · , x B } , P x ∈ S x = U , and | S | = B . Now consider an insta nce of LMV T where we hav e 2 v ideos v 1 and v 2 , B slots, and the bit rates r 1 j = r 2 j = x j ∈ S , for each slo t j . Note that | S | = B = # of slots for the L MVT instanc e . Set k = U / 2. Lemma 1. The ab ove instanc e of the P ar tition Problem has a solution iff the instanc e of L MVT has a solution. Pr o of. W e show that the instance of the P ar tition Problem has a so lutio n iff we can find a slot a llo cation for the 2 video s of the L MVT instance , such that the num b er of bits r eceiv ed by each video is exactly U / 2. Suppo se the P ar tition Problem has a so lut ion. That is we hav e S ′ and S \ S ′ such that P x ∈ S ′ x = P x ∈ S \ S ′ x = U / 2. W e find a s lo t allo cation of the B An FPT A S for LMVT Pro blem 3 slots which a llocates s lo t i to v ideo v 1 if x i ∈ S ′ . Else i is allo cated to v 2 . W e note that all the B slo ts get allo cated this wa y , since | S | = B . Now since P x ∈ S ′ x = P x ∈ S \ S ′ x = U / 2 , it is easy to see tha t the num be r of bits re c eiv ed by v 1 and v 2 is exactly eq ua l to U / 2. F or the other way , supp ose we have a slot allo cation such that num ber o f bits received v 1 and v 2 = U / 2. W e partition S in the following wa y: 1. If slot i is allo cated to v 1 , then x i ∈ S ′ . 2. Els e x i ∈ S ′ \ S . Since P i r 1 i = P i r 2 i = U / 2, we have P x ∈ S ′ x = P x ∈ S \ S ′ x = U / 2, and hence a so lut ion to the insta nce of P ar tition Problem . ⊓ ⊔ Corollary 1. [1] The L MVT Problem is e asy for a c onstant bit r ate for al l the n vide os and B s lots. Pr o of. The P ar tition Problem has bee n reduced to LMVT. It is easy to see that an instance of P ar tition Problem where all the integers in S are co nstan t (equal) is ea s y to solv e. Analogously , the instance of LMVT with constant (equal) bit r ates is also easy to solve. ⊓ ⊔ 4 An FPT AS for LMVT In [1], an exact dynamic pro gramming algorithm has b e e n designed fo r LMVT . The runtime of the exact algor ithm is pseudo-p olynomial in terms of the inputs. Here w e des cribe the exa ct algorithm and then discretize the algor ithm to design an FPT AS. 4.1 The Exact Dynamic Programming Algorithm Define b i max to b e the ma xim um num ber of bits that video v i . In other words, b i max = B X j =1 r ij , is the num ber of bits v i would receive, if al l the B slots are allo cated to it. Given m slots n videos, a T x (transmiss ion) vector is an n-t u pl e < b 1 , · · · , b n > whic h tells us whether a slot a llocation is p ossible suc h that v ideo v i receives at lea st b i bits in the allo cation. The length of the T x vector is the nu mber o f videos n . W e define the predicate F ( m, T ) for m s lots and T x vector T . F ( m, T ) is tr ue iff an allo cation is p ossible to achiev e T x . F or tw o T x vectors T 1 , and T 2 , we define T 1  T 2 iff T 1 [ i ] ≤ T 2 [ i ], ∀ i . It is easy to see, if F ( m, T 2 ), then F ( m, T 1 ). In the dynamic pro gramming, we genera te n Y i =1 ( b i max + 1) p o ssible T x vectors starting from < 0 , · · · , 0 > till < b 1 max , · · · , b n max > . F or ea c h video v i , w e hav e the v alues taken fr o m the set { 0 , 1 , · · · , b i max − 1 , b i max } . W e maintain an n Y i =1 ( b i max + 1) b y n ma trix of the v ector s during the exe c ut ion of the dyna mic 4 Roy S and R udra A progra mming a lg orithm. Also, w e ha ve a truth value vector of le ngth n Y i =1 ( b i max + 1). Ea c h cell in the true v a lue vector co rrespo nds to the v alue of F ( m, T ) fo r m slots, a nd T x vector T . W e initialize the truth v alue o f < 0 , · · · , 0 > to tr ue and the r est to f al se . This signifies that we can a lw ays achiev e vector < 0 , · · · , 0 > , even without any slot allo cation. W e then s tart from m = 1 to the to tal n um b er of slots B , and ev aluate the truth v alues. The truth v alues ( F ( B , T )) at the end tell us whether that vector T was achiev able by a slo t allo cation with the B slots. W e then c ho ose the vector with the maximum minimum b i v a lue as our solution, and hav e the cor responding slot a llo cation as the o ptim al a nsw er. The wa y to ev a luate F ( m, T ) is as follows: 1. If F ( m − 1 , T ), then F ( m, T ). 2. Els e let W i be the vector wher e all the p ositions o f W i except W i [ i ] is eq ua l to T . W i [ i ] = max (0 , T [ i ] − r im ). F or i = 1 to n , if F ( m − 1 , W i ), then F ( m, T ). W e pres en t the whole alg orithm in Algorithm 1 . An FPT A S for LMVT Pro blem 5 Input : n , the n umber of v ideos, B , the num ber of slots, r ij , the ra te of video v i for slo t j Output : An allo cation of the B slots over n videos where the minimum nu mber of bits r eceiv ed by a v ideo is maximize d Generate the n Y i =1 ( b i max + 1 ) vectors whe r e b i max = B X j =1 r ij ; Construct the n Y i =1 ( b i max + 1 ) by n matr ix of the vectors; Hav e the tr ut h v alue vector of length n Y i =1 ( b i max + 1 ); Initialize the truth v a lue of < 0 , · · · , 0 > to tru e and the res t to f al se ; for m = 1 to B do foreac h T x ve ctor T do if F ( m − 1 , T ) then Set F ( m, T ) to tr ue ; end else if then for i = 1 to n do if F ( m − 1 , W i ) then Set F ( m, T ) to true ; end end end end end Return the T x vector T with the maximum min ( b i ) v alue a nd with F ( B , T ) true; Algorithm 1: The ex a ct dynamic pro gramming algo rithm Algorithm 1 has a runtime of O ( B n n Y i =1 ( b i max + 1)). Since n Y i =1 ( b i max + 1) can b e exp onen tially la rge, Algor ithm 1 has an exp onential r untime. 4.2 The FPT AS In the FPT AS, instead of consider ing all the v alues in the set { 0 , 1 , · · · , b i max − 1 , b i max } for e a c h video v i , we discr etize the set to the powers of 1 + ε , wher e ε > 0. W e define the function ψ ( i ) = ⌊ (1 + ε ) ⌊ log 1+ ε i ⌋ ⌋ . No w w e hav e the set for each video v i , as { 0 , ψ (1) , · · · , ψ ( b i max − 1) , ψ ( b i max ) } . Clearly , we have a t most l og 1+ ε ( b i max + 1) v a lues in the set. Hence we would ha ve only n Y i =1 l og 1+ ε ( b i max + 1) T x vectors to ev alua te truth v alue for , in the FPT AS. In the ev aluation of F ( m, T ), instead of considering W i as in Algorithm 1, we consider W ′ i where W ′ i 6 Roy S and R udra A is the vector where all the p ositions of W ′ i except W ′ i [ i ] is e q ual to T . W ′ i [ i ] = max (0 , ψ ( T [ i ] − r im )). W e present the FP T AS in Algor ithm 2. Input : n , the n umber of v ideos, B , the num ber of slots, r ij , the ra te of video v i for slo t j Output : An allo cation of the B slots over n videos where the minimum nu mber of bits r eceiv ed by a v ideo is maximize d Generate the n Y i =1 l og 1+ ε ( b i max + 1 ) vectors where b i max = B X j =1 r ij ; Construct the n Y i =1 l og 1+ ε ( b i max + 1 ) by n matrix of the vectors; Hav e the tr ut h v alue vector of length n Y i =1 l og 1+ ε ( b i max + 1 ); Initialize the truth v a lue of < 0 , · · · , 0 > to tru e and the res t to f al se ; for m = 1 to B do foreac h T x ve ctor T do if F ( m − 1 , T ) then Set F ( m, T ) to tr ue ; end else if then for i = 1 to n do if F ( m − 1 , W ′ i ) then Set F ( m, T ) to true ; end end end end end Return the T x vector T with the maximum min ( b i ) v alue a nd with F ( B , T ) true; Algorithm 2: The FP T AS for LMVT Problem Algorithm 2 consider s o nly n Y i =1 l og 1+ ε ( b i max +1) to ev aluate o ut of the n Y i =1 ( b i max + 1) T x vectors ev aluated by Algorithm 1. Lemma 2. The err or gener ate d by the ro unding of W i in Algorithm 1 to W ′ i in Algo rithm 2 is at most 1 1+ γ wher e γ > 0 . Pr o of. Supp ose we hav e < b 1 , · · · , b n > ⇐ ⇒ < c 1 , · · · , c n > from the F ( m, T ) ev a luation step of Algorithm 1. In other words F ( m, T 1 ) for T 1 = < c 1 , · · · , c n > has b een ev aluated to tr ue be c ause F ( m − 1 , T 2 ) ha d b e en ev aluated to b e tru e for T 2 = < b 1 , · · · , b n > . Hence, ∃ i ∈ [ n ], such that, T 2 [ i ] = b i = max (0 , c i − r im ), and for all other pos itions, j we hav e T 2 [ j ] = T 1 [ j ]. An FPT A S for LMVT Pro blem 7 Now supp ose we have T ′ 2 = < b ′ 1 , · · · , b ′ n > and T ′ 1 = < c ′ 1 , · · · , c ′ n > in the table for Algor ithm 2, where b ′ i = ψ ( b i ), and c ′ i = ψ ( c i ), ∀ i ∈ [ n ]. W e wan t to s how that if Algorithm 2 e v aluates F ( m, T ′ 1 ) to tr ue if F ( m − 1 , T ′ 2 ) was ev aluated to tr ue at an earlier step, with an err or of a t most 1 1+ γ . In other words, < b ′ 1 , · · · , b ′ n > ⇐ ⇒ < c ′ 1 , · · · , c ′ n > in Algo rithm 2 with an err or of at most 1 1+ γ . W e obser v e that T ′ 2 [ j ] = T ′ 1 [ j ] for a ll j ∈ [ n ] \ { i } . F or j = i we hav e T ′ 2 [ i ] = b ′ i = ψ ( b i ) = m ax (0 , ψ ( c i − r im )) ≥ ψ ( c i − r im ). Algorithm 2 would calc ula te the v a lue of p osition i for vector W ′ i as W ′ i [ i ] = max (0 , ψ ( c ′ i − r im )) ≥ ψ ( c ′ i − r im ). W e hav e ψ ( c ′ i − r im ) = ψ ( ψ ( c i ) − r im ) ≥ ψ ( c i − r im 1+ ε ) ≥ 1 1+ γ ψ ( c i − r im ), where γ = 2 ε . ⊓ ⊔ Lemma 3. Algori thm 2 has a runtime of O ( B n n Y i =1 l og 1+ ε ( b i max + 1 )) . Lemma 4. The value of the solution ( S f ptas ) r eturne d by Algorithm 2 differs fr om that ( S opt ) r eturne d by Algorithm 1 at most by a factor of 1 1+ εB . Pr o of. At any s tep i , the v alue o f any p osition of an y vector of Algorithm 2 differs from the cor responding p osition o f the c o rrespo nding v ector for Algorithm 1 by a factor o f 1 (1+ ε ) i ≈ 1 1+ εi due to the rounding. W e p erform this rounding B times. Hence the v alues of the vectors after the full ex ecution would differ b y a fac t or of at most 1 1+ εB . ⊓ ⊔ Lemma 3 and 4 lea d to the fo llowing theorem. Theorem 1. Algori thm 2 is an FPT AS for L MVT Problem . References 1. Dut ta P ., S ee tharam A., A ry a V., Chetlur M., and Kaly anaraman S.: Managing Q oE of M ultiple Vide o Str e ams in Wi r eless Networks . Submitted. Manuscri pt av ailable at http://dom ino.research.ib m.com/library/CyberDig.nsf/papers/A99FBFAE927167958525787E003DA7FE/$Fil e / r e p o r t V i d e o . p d f . 2. Bansal N., Sv i ridenko M.: The Santa Claus pr oblem . STOC ’06 In Proceedings of the thirty-eigh th annual ACM symp osium on Theory of computing. 3. Bateni M. 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