Two-phase algorithms for the parametric shortest path problem
|
|
- Abraham Skinner
- 7 years ago
- Views:
Transcription
1 Two-phase algorthms for the parametrc shortest path problem Sourav Chakraborty 1, Eldar Fscher 1, Oded Lachsh 2, and Raphael Yuster 3 1 Department of Computer Scence, Technon, Hafa 32000, Israel {eldar,sourav}@cs.technon.ac.l Research supported n part by an ERC-2007-StG grant number Centre for Dscrete Mathematcs and ts Applcatons, Unversty of Warwck, Coventry, UK oded@dcs.warwck.ac.uk 3 Department of Mathematcs, Unversty of Hafa, Hafa 31905, Israel raphy@math.hafa.ac.l Abstract. A parametrc weghted graph s a graph whose edges are labeled wth contnuous real functons of a sngle common varable. For any nstantaton of the varable, one obtans a standard edgeweghted graph. Parametrc weghted graph problems are generalzatons of weghted graph problems, and arse n varous natural scenaros. Parametrc weghted graph algorthms consst of two phases. A preprocessng phase whose nput s a parametrc weghted graph, and whose output s a data structure, the advce, that s later used by the nstantaton phase, where a specfc value for the varable s gven. The nstantaton phase outputs the soluton to the (standard) weghted graph problem that arses from the nstantaton. The goal s to have the runnng tme of the nstantaton phase supersede the runnng tme of any algorthm that solves the weghted graph problem from scratch, by takng advantage of the advce. In ths paper we construct several parametrc algorthms for the shortest path problem. For the case of lnear functon weghts we present an algorthm for the sngle source shortest path problem. Its preprocessng phase runs n Õ(V 4 ) tme, whle ts nstantaton phase runs n only O(E + V log V ) tme. The fastest standard algorthm for sngle source shortest path runs n O(V E) tme. For the case of weght functons defned by degree d polynomals, we present an algorthm wth sub-exponental preprocessng tme O(V (1+log f(d)) log V ) and nstantaton tme only Õ(V ). In fact, for any par of vertces u, v, the nstantaton phase computes the dstance from u to v n only O(log 2 V ) tme. Fnally, for lnear functon weghts, we present a randomzed algorthm whose preprocessng tme s Õ(V 3.5 ) and so that for any par of vertces u, v and any nstantaton varable, the nstantaton phase computes, n O(1) tme, a length of a path from u to v that s at most (addtvely) ɛ larger than the length of a shortest path. In partcular, an all-pars shortest path soluton, up to an addtve constant error, can be computed n O(V 2 ) tme. 1 Introducton A functon-weghted graph s a graph whose edges are labeled wth contnuous real functons. When all the functons are unvarate (and all have the same varable), the graph s called a parametrc weghted graph. In other words, the graph s G = (V, E, W ) where W : E F and F s the space of all real contnuous functons wth the varable x. If G s a parametrc weghted graph, and r R s any real number, then G(r) s the standard weghted graph where the weght of an edge e s defned to be (W (e))(r). We say that G(r) s an nstantaton of G, snce the varable x n each functon s nstantated by the value r. Parametrc weghted graphs, are, therefore, a generc nstance of nfntely many nstances of weghted graphs. How can we beneft from modelng problems by a parametrc weghted graph? If we can take advantage of the generc nstance G, and use t to precompute some general generc nformaton I(G), t s plausble that for any gven nstantaton G(r), we wll be able to use the precomputed nformaton I(G) n order to speed up the tme to solve the gven problem on G(r), faster than just solvng the problem on G(r) from scratch. Let us make ths noton more precse. A parametrc weghted graph algorthm (or, for brevty, parametrc algorthm) conssts of two phases. A preprocessng phase whose nput s a parametrc weghted graph G, and whose output s a data structure (the advce) that s later used by the nstantaton phase, where a specfc value r for the varable s gven. The nstantaton phase outputs the soluton to the (standard) weghted
2 graph problem on the weghted graph G(r). Naturally, the goal s to have the runnng tme of the nstantaton phase sgnfcantly smaller than the runnng tme of any algorthm that solves the weghted graph problem from scratch, by takng advantage of the advce constructed n the preprocessng phase. Parametrc algorthms are therefore evaluated by a par of runnng tmes, the preprocessng tme and the nstantaton tme. In ths paper we show that parametrc algorthms are benefcal for one of the most natural combnatoral optmzaton problems: the shortest path problem n drected graphs. Recall that gven a drected real-weghted graph G, and two vertces u, v of G, the dstance from u to v, denoted by δ(u, v), s the length of a shortest path from u to v. The sngle par shortest path problem seeks to compute δ(u, v) and construct a shortest path from u to v. Lkewse, the sngle source shortest path problem seeks to compute the dstances and shortest paths from a gven vertex to all other vertces, and the all pars verson seeks to compute dstances and shortest paths between all ordered pars of vertces. In some of our algorthms we forgo the calculaton of the path tself to acheve a shorter nstantaton tme. In all those cases the algorthms can be easly modfed to also output a shortest path, n whch case ther nstantaton tme s the sum of the tme t takes to calculate the dstance and a tme lnear n the sze of the path to be output. Our frst algorthm s a parametrc algorthm for sngle source shortest path, n the case where the weghts are lnear functons. That s, each edge e s labeled wth a functon a e x + b e where a e and b e are reals. Such lnear parametrzaton has practcal mportance. Indeed, n many problems the cost of an edge s composed from some constant term plus a term whch s a factor of some commodty, whose cost vares (e.g. bank commssons, tax fares, vehcle mantenance costs, and so on). Our parametrc algorthm has preprocessng tme Õ(n4 ) and nstantaton tme O(m + n log n) (throughout ths paper n and m denote the number of vertces and edges of a graph, respectvely). We note that the fastest algorthm for the sngle source shortest path n real weghted drected graphs requres O(nm) tme; the Bellman-Ford algorthm [2]. The dea of our preprocessng stage s to precompute some other lnear functons, on the vertces, so that for every nstantaton r, one can quckly determne whether G(r) has a negatve cycle and otherwse use these functons to quckly produce a reweghng of the graph so as to obtan only nonnegatve weghts smlar to the weghts obtaned by Johnson s algorthm [11]. In other words, we avod the need to run the Bellman-Ford algorthm n the nstantaton phase. The Õ(n4 ) tme n the preprocessng phase comes from the use of an nteror pont algorthm that we need n order to compute the lnear vertex functons. Theorem 1. There exsts a parametrc algorthm for sngle source shortest path n graphs weghted by lnear functons, whose preprocessng tme s Õ(n4 ) and whose nstantaton tme s O(m + n log n). Our next algorthm apples to a more general settng where the weghts are polynomals of degree at most d. Furthermore, n ths case our goal s to have the nstantaton phase answer dstance queres between any two vertces n sublnear tme. Notce frst that f we allow exponental preprocessng tme, ths goal can be easly acheved. Ths s not hard to see by the fact that the overall possble number of shortest paths (when x vares over the reals) s O(n!), or from Fredman s decson tree for shortest paths whose heght s O(n 2.5 ) [8]. But can we settle for sub-exponental preprocessng tme and stll be able to have sublnear nstantaton tme? Our next result acheves ths goal. Theorem 2. There exsts a parametrc algorthm for the sngle par shortest path problem n graphs weghted by degree d polynomals, whose preprocessng tme s O(n (O(1)+log f(d)) log n ) and nstantaton tme O(log 2 n), where f(d) s the tme requred to compute the ntersecton ponts of two degree d polynomals. The sze of the advce that the preprocessng algorthm produces s O(n (O(1)+log d) log n ).
3 The practcal and theoretcal mportance of shortest path problems lead several researchers to consder fast algorthms that settle for an approxmate shortest path. For the general case (of real weghted dgraphs) most of the algorthms guarantee an α-stretch factor. Namely, they compute a path whose length s at most αδ(u, v). We menton here the (1 + ɛ)-stretch algorthm of Zwck for the all-pars shortest path problem, that runs n Õ(nω ) tme when the weghts are non-negatve reals [16]. Here ω < s the matrx multplcaton exponent [5]. Here we consder probablstc addtve-approxmaton algorthms, or surplus algorthms, that work for lnear weghts whch may have postve and negatve values (as long as there s no negatve weght cycle). We say that a shortest path algorthm has an ɛ-surplus f t computes paths whose lengths are at most δ(u, v)+ɛ. We are unaware of any truly subcubc algorthm that guarantees an ɛ- surplus approxmaton, and whch outperforms the fastest general all-pars shortest path algorthm [4]. In the lnear-parametrc settng, t s easy to obtan ɛ-surplus parametrc algorthms whose preprocessng tme s O(n 4 ) tme, and whose nstantaton tme, for any ordered par of quered vertces u, v s constant. It s assumed nstantatons are taken from some nterval I whose length s ndependent of n. Indeed, we can partton I nto O(n) subntervals I 1, I 2,... of sze O(1/n) each, and solve, n cubc tme (say, usng [7]), the exact all-pars soluton for any nstantaton r that s an endpont of two consecutve ntervals. Then, gven any r I j = (a j, b j ), we smply look at the soluton for b j and notce that we are (addtvely) off from the rght answer only by O(1). Standard scalng arguments can make the surplus smaller than ɛ. But do we really need to spend O(n 4 ) tme for preprocessng? In other words, can we nvest (sgnfcantly) less than O(n 4 ) tme and stll be able to answer nstantated dstance queres n O(1) tme? The followng result gves a postve answer to ths queston. Theorem 3. Let ɛ > 0, let [α, β] be any fxed nterval and let γ be a fxed constant. Suppose G s a lnear-parametrc graph that has no negatve weght cycles n the nterval [α, β], and for whch every edge weght a e + xb e satsfes a e γ. There s a parametrc randomzed algorthm for the ɛ-surplus shortest path problem, whose preprocessng tme s Õ(n3.5 ) and whose nstantaton tme s O(1) for a sngle par, and hence O(n 2 ) for all pars. We note that ths algorthm works n the restrcted addton-comparson model. We also note that gven an ordered par u, v and r [α, β], the algorthm outputs, n O(1) tme, a weght of an actual path from u to v n G(r), and ponts to a lnked lst representng that path. Naturally, f one wants to output the vertces of ths path then the tme for ths s lnear n the length of the path. The rest of ths paper s organzed as follows. The next subsecton shortly surveys related research on parametrc shortest path problems. In the three sectons followng t we prove Theorems 1,2 and 3. Secton 5 contans some concludng remarks and open problems. 1.1 Related research Karp and Orln [13], and, later, Young, Tarjan, and Orln [15] consdered a specal case of the lnear-parametrc shortest path problem. In ther case, each edge weght e s ether some fxed constant b e or s of the form b e x. It s not too dffcult to prove that for any gven vertex v, when x vares from to the largest x 0 for whch G(x 0 ) has no negatve weght cycle (possbly x 0 = ), then there are at most O(n 2 ) dstnct shortest path trees from v to all other vertces. Namely, for each r [, x 0 ] one of the trees n ths famly s a soluton for sngle-source shortest path n G(r). The results n [13, 15] cleverly and compactly compute all these trees, and the latter does t n O(nm + n 2 log n) tme. However, for general lnear functons, Carstensen [3] showed that there are constructons for whch the number of shortest path changes whle x vares over the reals s n Ω(log n). In fact, n her example each lnear functon s of the form a e + xb e and both a e and b e are postve, and x vares
4 n [0, ]. Carstensen also proved that ths s tght. In other words, for any lnear-parametrc graph the number of changes n the shortest paths s n O(log n). A smpler proof was obtaned by Nkolova et al. [14], that also supply an n O(log n) tme algorthm to compute the path breakponts. Ther method, however, does not apply to the case where the functons are not lnear, such as n the case of degree d polynomals. Gusfeld [10] studed algorthms for functon-weghted graphs but n the context of program module dstrbuton. 2 Proof of Theorem 1 The proof of Theorem 1 follows from the followng two lemmas. Lemma 1. Gven a lnear-weghted graph G = (V, E, W ), there exst α, β R { } {+ } such that G(r) has no negatve cycles f and only f α r β. Moreover α and β can be found n Õ(n 4 ) tme. Lemma 2. Let G = (V, E, W ) be a lnear-weghted graph. Also let α, β R { } {+ } be such that at least one of them s fnte and for all α r β the graph G(r) has no negatve cycle. Then for every vertex v V there exsts a lnear functon g v [α,β] such that f the new weght functon W s gven by W ((u, v)) = W ((u, v)) + g [α,β] u g [α,β] v then the new lnear-weghted graph G = (V, E, W ) has the property that for any real α r β all the edges n G (r) are non-negatve. Moreover the functons g v [α,β] for all v V can be found n O(mn) tme. So gven a lnear-weghted graph G, we frst use Lemma 1 to compute α and β. If at least one of α and β s fnte then usng Lemma 2 we compute the n lnear functons g v [α,β], one for each v V. If α = and β = +, then usng Lemma 2 we compute the 2n lnear functons g v [α,0] and g v [0,β]. These lnear functons wll be the advce that the preprocessng algorthm produces. The above lemmas guarantee us that the advce can be computed n tme Õ(n4 ), that s the preprocessng tme s Õ(n4 ). Now when computng the sngle source shortest path problem from vertex v for the graph G(r) our algorthm proceeds as follows: 1. If r < α or r > β output as there exsts a negatve cycle (such nstances are consdered nvald). 2. If α r β and at least one of α or β s fnte then compute g u (r) for all u V. Use these to re-weght the edges n the graph as n Johnson s algorthm [11]. If α = and β = + then f r 0 compute g [α,0] u (r) for all u V and f r 0 compute g u [0,β] (r) for all u V. Notce that after the reweghng we have an nstance of G (r). 3. Use Djkstra s algorthm [6] to solve the sngle source shortest path problem n G (r). Djkstra s algorthm apples snce G (r) has no negatve weght edges. The shortest paths tree returned by Djkstra s algorthms appled to G (r) s also the shortest paths tree n G(r). As n Johnson s algorthm, we use the results d (v, u) of G (r) to deduce d(v, u) n G(r) snce, by Lemma 2 d(v, u) = d (v, u) g v (r) + g u (r). The runnng tme of the nstantaton phase s domnated by the runnng tme of Djkstra s algorthm whch s O(m + n log n) [9].
5 2.1 Proof of Lemma 1 Snce the weght on the edges of the graph G are lnear functons, we have that the weght of any drected cycle n the graph s also a lnear functon. Let C 1, C 2,..., C T be the set of all drected cycles n the graph. The lnear weght functon of a cycle C wll be denoted by wt(c ). If wt(c ) s not the constant functon, then let γ be the real number for whch the lnear equaton wt(c ) evaluates to 0. Let α and β be defned as follows: Note that f wt(c ) has a postve slope then α = max {γ wt(c ) has a postve slope}. β = mn {γ wt(c ) has a negatve slope}. γ = mn x {wt(c )(x) 0}. Thus for all x γ the value of wt(c ) evaluated at x s non-negatve. So by defnton for all x α the value of the wt(c ) s non-negatve f the slope of wt(c ) s postve, and for any x < α there exsts a cycle C such that wt(c ) has postve slope and wt(c )(x) s negatve. Smlarly, for all x β the value of the wt(c ) s non-negatve f the slope of wt(c ) s negatve and for any x > β there exsts a cycle C such that wt(c ) has negatve slope and wt(c )(x) s negatve. Ths proves the exstence of α and β. There are, however, two bad cases that we wsh to exclude. Notce that f α > β ths means that for any evaluaton at x, the resultng graph has a negatve weght cycle. The same holds f there s some cycle for whch wt(c ) s constant and negatve. Let us now show how α and β can be effcently computed whenever these bad cases do not hold. Indeed, α s the soluton to the followng Lnear Program (LP), whch has a feasble soluton f and only f the bad cases do not hold. Mnmze x under the constrants, wt(c )(x) 0. Ths s an LP on one varable, but the number of constrants can be exponental. Luckly, however, the set of constrants can be presented n a compact form: our functon-weghted graph. In other words, for any gven x, one can use the Bellman-Ford algorthm to produce a constrant that s volated (f there s one), that s produce a cycle that has negatve weght. Thus, our LP has an effcent separaton oracle. As ths LP has only one varable, and ts sze s only O(n 2 ) (n ths context the sze s defned by the length of the parameters of the separaton oracle, n our case the lnearly weghted graph), t can be solved n Õ(n4 ) tme [12]. Notce that β can be computed usng the analogous maxmzaton lnear program. In case one (and hence both) of the lnear programs has no feasble soluton we just set α = β =. 2.2 Proof of Lemma 2 Let α and β be the two numbers such that for all α r β the graph G(r) has no negatve cycles and at least one of α and β s fnte. Frst let us consder the case when both α and β are fnte. Recall that, gven any number r, Johnson s algorthm assocates a weght functon h r : V R such that, for any edge (u, v) E, W (u,v) (r) + h r (u) h r (v) 0.
6 (Johnson s algorthm computes ths weght functon by runnng the Bellman-Ford algorthm over G(r)). Defne the weght functon g v [α,β] as ( h g v [α,β] β (v) h α ) ( (v) h (x) = x + h α β (v) h α ) (v) (v) α. β α β α Ths s actually the equaton of the lne jonng (α, h α (v)) and (β, h β (v)) n R 2. Now we need to prove that for every α r β and for every (u, v) V, W (u,v) (r) + g u [α,β] (r) g v [α,β] (r) 0. Snce α r β, one can wrte r = (1 δ)α + δβ where 1 δ 0. Then for all v V, Snce W (u,v) (r) s a lnear functon we can wrte g [α,β] v (r) = (1 δ)h α (v) + δh β (v). W (u,v) (r) = (1 δ)w (u,v) (α) + δw (u,v) (β). So after re-weghtng the weght of the edge (u, v) s (1 δ)w (u,v) (α) + δw (u,v) (β) + (1 δ)h α (u) + δh β (u) (1 δ)h α (v) δh β (v). Now ths s non-negatve as by the defnton of h β and h α we know that both W (u,v) (β) + h β (u) h β (v) and W (u,v) (α) + h α (u) h α (v) are non-negatve. We now consder the case when one of α or β s not fnte. We wll prove t for the case where β = +. The case α = follows smlarly. Consder the smple weghted graph G = (V, E, W ) where the weght functon W s defned as: f the weght of the edge e s W (e) = a e x + b e then W (e) = a e. We run the Johnson s algorthm on the graph G. Let h (v) denote the weght that Johnson s algorthm assocates wth the vertex v. Then defne the weght functon g v [α, ] as g [α, ] v (x) = h α (v) + (x α)h (v). We need to prove that for every α r and for every (u, v) V, W (u,v) (r) + g u [α, ] (r) g v [α, ] (r) = W (u,v) (r) + h α (u) + (r α)h (u) h α (v) (r α)h (v) 0. Let r = α + δ where δ 0. By the lnearty of W we can wrte W (u,v) (r) = W (u,v) (α) + δa (u,v), where W (u,v) (r) = a (u,v) r + b (u,v). So the above nequalty can be restated as W (u,v) (α) + δa (u,v) + h α (u) + δh (u) h α (v) δh (v) 0. Ths now follows from the fact that both W (u,v) (α) + h α (u) h α (v) and a (u,v) + h (u) h (v) are non-negatve. Snce the runnng tme of the reweghng part of Johnson s algorthm takes O(mn) tme, the overall runnng tme of computng the functons g v [α,β] s O(mn), as clamed. 3 Proof of Theorem 2 In ths secton we construct a parametrc algorthm that computes the dstance δ(u, v) between a gven par of vertces. If one s nterested n the actual path realzng ths dstance, then t can be found wth some extra book-keepng that we omt n the proof.
7 The processng algorthm wll output the followng advce: for any par (u, v) V V the advce conssts of a set of ncreasng real numbers = b 0 < b 1 < < b t < b t+1 = and an ordered set of degree-d polynomals p 0, p 1,..., p t, such that for all b r b +1 the weght of a shortest path n G(r) from u to v s p (r). Note that each p corresponds to the weght of a path from u to v. Thus f we are nterested n computng the exact path then we need to keep track of the path correspondng to each p. Gven r, the nstantaton algorthm has to fnd the such that b r b +1 and then output p (r). So the output algorthm runs n tme O(log t). To prove our result we need to show that for any (u, v) V V we can fnd the advce n tme O(f(d)n) log n. In partcular ths wll prove that t = O(dn) log n and hence the result wll follow. Defnton 1. A mnbase s a sequence of ncreasng real numbers = b 0 < b 1 < < b t < b t+1 = and an ordered set of degree-d polynomals p 0, p 1,..., p t, such that for all b r b +1 and all j, p (r) p j (r). We call the sequence of real numbers the breaks. We call each nterval [b, b +1 ] the -th nterval of the mnbase and the polynomal p the -th polynomal. The sze of the mnbase s t. The fnal advce that the preprocessng algorthm produces s a mnbase for every par (u, v) V V where the -th polynomal has the property that p (r) s the dstance from u to v n G(r) for each b r b +1. Defnton 2. A mnbase l (u, v) s a mnbase correspondng to the ordered par u, v, where the -th polynomal p has the property that for r [b, b +1 ], p (r) s the length of a shortest path from u to v n G(r), that s taken among all paths that use at most 2 l edges. A mnbase l (u, w, v) s a mnbase correspondng to the ordered trple (u, w, v) where the -th polynomal p has the property that for each r [b, b +1 ], p (r) s the sum of the lengths of a shortest path from u to w n G(r), among all paths that use at most 2 l edges, and a shortest path from w to v n G(r), among all paths that use at most 2 l edges. Note that n both of the above defntons some of the polynomals can be + or. Defnton 3. If B 1 and B 2 are two mnbases (not necessarly of the same sze), wth polynomals p 1 and p 2 j, we say that another mnbase wth breaks b k and polynomals p k s mn(b 1 + B 2 ) f the followng holds. 1. For all k there exst, j such that p k = p1 + p2 j, and 2. For b k r b k+1 and for all, j we have p k (r) p1 (r) + p2 j (r). Defnton 4. If B 1, B 2,..., B s are s mnbases (not necessarly of the same sze), wth polynomals p 1 1, p 2 2,..., p s s, another mnbase wth breaks b k and polynomals p k s mn{b 1, B 2,..., B s } f the followng holds. 1. For all k there exst q such that p k = pq q, and 2. For b k r b k+1 and for all 1 q s and all q, we have p k (r) pq q (r). Note that usng the above defnton we can wrte the followng two equatons: { } mnbase l+1 (u, v) = mn mnbase l (u, w, v). (1) w V ( ) mnbase l (u, w, v) = mn mnbase l (u, w) + mnbase l (w, v). (2) The followng clam wll prove the result. The proof of the clam s n the Appendx, Secton 6.
8 Clam 1. If B 1 and B 2 are two mnbases of szes t 1 and t 2 respectvely, then (a) mn(b 1 + B 2 ) can be computed from B 1 and B 2 n tme O(t 1 + t 2 ). (b) mn{b 1, B 2 } can be computed from B 1 and B 2 n tme O(f(d)(t 1 + t 2 )), where f(d) s the tme requred to compute the ntersecton ponts of two degree-d polynomals. The sze of mn{b 1, B 2 } s O(d(t 1 + t 2 )). In order to compute mn{b 1,..., B s } one recursvely computes X = mn{b 1,..., B s/2 } and Y = mn{b s/2+1,..., B s } and then takes takes mn{x, Y }. If there are no negatve cycles, then the advce that the nstantaton algorthm needs from the preprocessng algorthm conssts of mnbase log n (u, v). To deal wth negatve cycles, both mnbase log n (u, v) and mnbase log n +1 (u, v) are produced, and the nstantaton algorthm compares them. f they are not equal, then the correct output s. Also note that mnbase 0 (u, v) s the trval mnbase where the breaks are and + and the polynomal s weght W ((u, v)) assocated to the edge (u, v) f (u, v) E and + otherwse. If the sze of mnbase l (u, v) s s l, then by (1), (2), and by Clam 1 the tme to compute mnbase l+1 (u, v) s O(f(d)) log n s l and the sze of mnbase l+1 (u, v) s O(d) log n s l. Thus one can compute the advce for u and v n tme (O(f(d)) log n ) log n = O(n (O(1)+log f(d)) log n ), and the length of the advce strng s O(n (O(1)+log d) log n ). 4 Proof of Theorem 3 Gven the lnear-weghted graph G = (V, E, W ), our preprocessng phase begns by verfyng that for all r [α, β], G(r) has no negatve weght cycles. From the proof of Lemma 2 we know that ths holds f and only f both G(α) and G(β) have no negatve weght cycles. Ths, n turn, can be verfed n O(mn) tme usng the Bellman-Ford algorthm. We may now assume that G(r) has no negatve cycles for any r [α, β]. Moreover, snce our preprocessng algorthm wll solve a large set of shortest path problems, each of them on a specfc nstantaton of G, we wll frst compute the reweghng functons g v [α,β] of Lemma 2 whch wll enable us to apply, n some cases, algorthms that assume nonnegatve edge weghts. Recall that by Lemma 2, the functons g v [α,β] for all v V are computed n O(mn) tme. The advce constructed by the preprocessng phase s composed of two dstnct parts, whch we respectvely call the crude-short advce and the refned-long advce. We now descrbe each of them. For each edge e E, the weght s a lnear functon w e = a e + xb e. Set K = 8(β α) max e a e. Let N 0 = K n ln n/ɛ and let N 1 = Kn/ɛ. For = 0, 1 we wll use N to defne N + 1 ponts n [α, β] and solve certan varants of shortest path problems nstantated n these ponts. Consder frst the case of splttng [α, β] nto N 0 ntervals. Let ρ 0 = (β α)/n 0 and consder the ponts α + ρ 0 for = 0,..., N 0. The crude-short part of the preprocessng algorthm solves N lmted all-pars shortest path problems n G(α + ρ 0 ) for = 0,..., N 0. Set t = 4 n ln n, and let d (u, v) denote the length of a shortest path from u to v n G(α + ρ 0 ) that s chosen among all paths contanng at most t vertces (possbly d (u, v) = f no such path exsts). Notce that d (u, v) s not necessarly the dstance from u to v n G(α + ρ 0 ), snce the latter may requre more than t vertces. It s straghtforward to compute shortest paths lmted to at most k vertces (for any 1 k n) n a real-weghted drected graph wth n vertces n tme O(n 3 log k) tme, by the repeated squarng technque. In fact, they can be computed n O(n 3 ) tme (savng the log k factor) usng the method from [1], pp Ths algorthm also constructs the predecessor data structure that represents the actual paths. It follows that for each ordered par of vertces u, v and
9 for each = 0,..., N 0, we can compute d (u, v) and a path p (u, v) yeldng d (u, v) n G(α + ρ 0 ) n O(n 3 N 0 ) tme whch s O(n 3.5 ln n). We also mantan, at no addtonal cost, lnear functons f (u, v) whch sum the lnear functons of the edges of p (u, v). Note also that f d (u, v) = then p (u, v) and f (u, v) are undefned. Consder next the case of splttng [α, β] nto N 1 ntervals. Let ρ 1 = (β α)/n 1 and consder the ponts α + ρ 1 for = 0,..., N 1. However, unlke the crude-short part, the refned-long part of the preprocessng algorthm cannot afford to solve an all-pars shortest path algorthm for each G(α + ρ 1 ), as the overall runnng tme wll be too large. Instead, we randomly select a set H V of (at most) n vertces. H s constructed by performng n ndependent trals, where n each tral, one vertex of V s chosen to H unformly at random (notce that snce the same vertex can be selected to H more than once H n). For each h H and for each = 0,..., N 1, we solve the sngle source shortest path problem n G(α + ρ 1 ) from h, and also (by reversng the edges) solve the sngle-destnaton shortest path toward h. Notce that by usng the reweghng functons we can solve all of these sngle source problems usng Djkstra s algorthm. So, for all h H and = 0,..., N 1 the overall runnng tme s g [α,β] v O( N 1 H (m + n log n)) = O(n 1.5 m + n 2.5 log n) = O(n 3.5 ). We therefore obtan, for each h H and for each = 0,..., N 1, a shortest path tree T (h), together wth dstances d (h, v) from h to each other vertex v V, whch s the dstance from h to v n G(α + ρ 1 ). We also mantan the functons f (h, v) that sum the lnear equatons on the path n T (h) from h to v. Lkewse, we obtan a reversed shortest path tree S (h), together wth dstances d (v, h) from each v V to h, whch s the dstance from v to h n G(α + ρ 1). Smlarly, we mantan the functons f (v, h) that sum the lnear equatons on the path n S (h) from v to h. Fnally, for each ordered par of vertces u, v and for each = 0,..., N 1 we compute a vertex h u,v, H whch attans mn h H d (u, h) + d (h, u). Notce that the tme to construct the h u,v, for all ordered pars u, v and for all = 0,..., N 1 s O(n 3.5 ). Ths concludes the descrpton of the preprocessng algorthm. Its overall runtme s thus O(n 3.5 ln n). We now descrbe the nstantaton phase. Gven u, v V and r [α, β] we proceed as follows. Let be the ndex for whch the number of the form α + ρ 0 s closest to r. As we have the advce f (u, v), we let w 0 = f (u, v)(r) (recall that f (u, v) s a functon). Lkewse, let j be the ndex for whch the number of the form α + jρ 1 s closest to r. As we have the advce h = h u,v,j, we let w 1 = fj (u, h)(r) + f j (h, u)(r). Fnally, our answer s z = mn{w 0, w 1 }. Clearly, the nstantaton tme s O(1). Notce that f we also wsh to output a path of weght z n G(r) we can easly do so by usng ether p (u, v), n the case where z = w 0 or usng Sj (h) and T j (h) (we take the path from u to h n Sj (h) and concatenate t wth the path from h to v n T j (h)) n the case where z = w 1. It remans to show that, wth very hgh probablty, the result z that we obtan from the nstantaton phase s at most ɛ larger than the dstance from u to v n G(r). For ths purpose, we frst need to prove that the random set H possesses some httng set propertes, wth very hgh probablty. For every par of vertces u and v and parameter r, let p u,v,r be a shortest path n G(r) among all smple paths from u to v contanng at least t = 4 n ln n vertces (f G s strongly connected then such a path always exst, and otherwse we can just put + for all u, v pars for whch no such path exsts). The followng smple lemma s used n an argument smlar to one used n [16]. Lemma 3. For fxed u, v and r, wth probablty at least 1 o(1/n 3 ) the path p u,v,r contans a vertex from H.
10 Proof. Indeed, the path from p u,v,r by ts defnton has at least 4 n ln n vertces. The probablty that all of the n ndependent selectons to H faled to choose a vertex from ths path s therefore at most ( 1 4 ) n n ln n < e 4 ln n < 1 n n 4 = o(1/n3 ). Let us return to the proof of Theorem 3. Suppose that the dstance from u to v n G(r) s δ. We wll prove that wth probablty 1 o(1), H s such that for every u, v and r we have z δ + ɛ (clearly z δ as t s the precse length of some path n G(r) from u to v). Assume frst that there s a path p of length δ n G(r) that uses less than 4 n ln n edges. Consder the length of p n G(α + ρ 0 ). When gong from r to α + ρ 0, each edge e wth weght a e x + b e changed ts length by at most a e ρ 0. By the defnton of K, ths s at most ρ 0 K/(8(β α)). Thus, p changed ts weght by at most (4 K n ln n) ρ 0 8(β α) = (4 n ln n) K < ɛ 8N 0 2. It follows that the length of p n G(α + ρ 0 ) s less than δ + ɛ/2. But p (u, v) s a shortest path from u to v n G(α +ρ 0 ) of all the paths that contan at most t vertces. In partcular, d (u, v) δ +ɛ/2. Consder the length of p (u, v) n G(r). The same argument shows that the length of p (u, v) n G(r) changed by at most ɛ/2. But w 0 = f (u, v)(r) s that weght, and hence w 0 δ + ɛ. In partcular, z δ + ɛ. Assume next that every path of length δ n G(r) uses at least 4 n ln n edges. Let p be one such path. When gong from r to r = α + jρ 1, each edge e wth weght a e x + b e changed ts length by at most a e ρ 1. By the defnton of K, ths s at most ρ 1 K/(8(β α)). Thus, p changed ts weght by at most K n ρ 1 8(β α) = n K < ɛ 8N 1 8. In partcular, the length of p u,v,r s not more than the length of p n G(r ), whch, n turn, s at most δ + ɛ/8. By Lemma 3, wth probablty 1 o(1/n 3 ), some vertex of h appears on p u,v,r. Moreover, by the unon bound, wth probablty 1 o(1) all paths of the type p u,v,r (remember that r can hold one of O(n) possble values) are thus covered by the set H. Let h be a vertex of H appearng n p u,v,r. We therefore have d j (u, h ) + d j (h, v) δ + ɛ/8. Snce h = h u,v,j s taken as the vertex whch mnmzes these sums, we have, n partcular, d j (u, h) + d j (h, v) δ + ɛ/8. Consder the path q n G(α + jρ 1 ) realzng d j (u, h) + d j (h, v). The same argument shows that the length of q n G(r) changed by at most ɛ/8. But w 1 = fj (u, h)(r) + f j (h, v)(r) s that weght, and hence w 1 δ + ɛ/4. In partcular, z δ + ɛ/4. 5 Concludng remarks We have constructed several parametrc shortest path algorthms, whose common feature s that they preprocess the generc nstance and produce an advce that enables partcular nstantatons to be solved faster than runnng the standard weghted dstance algorthm from scratch. It would be of nterest to mprove upon any of these algorthms, ether n ther preprocessng tme or n ther nstantaton tme, or both. Perhaps the most challengng open problem s to mprove the preprocessng tme of Theorem 2 to a polynomal one, or, alternatvely, prove an hardness result for ths task. Perhaps less ambtous s the preprocessng tme n Theorem 1. The only bottleneck that prevents reducng the Õ(n4 ) tme to O(nm) s the need to solve the LP usng nteror pont methods n order to compute the range of non-negatve cycles. Perhaps ths could be crcumvented. Fnally, parametrc algorthms are of practcal mportance for other combnatoral optmzaton problems as well. It would be nterestng to fnd applcatons where, ndeed, a parametrc algorthm can be truly benefcal, as t s n the case of shortest path problems.
11 Acknowledgment We thank Oren Wemann and Shay Mozes for useful comments. References 1. A. V. Aho, J. E. Hopcroft, and J. Ullman, The Desgn and Analyss of Computer Algorthms, Addson-Wesley Longman Publshng Co., Boston, MA, R. Bellman, On a routng problem, Quarterly of Appled Mathematcs 16 (1958), P. Carstensen, The complexty of some problems n parametrc lnear and combnatoral programmng, Ph.D. Thess, Mathematcs Dept., U. of Mchgan, Ann Arbor, Mch., T. M. Chan, More Algorthms for All-Pars Shortest Paths n Weghted Graphs, Proceedngs of the 39 th ACM Symposum on Theory of Computng (STOC), ACM Press (2007), D. Coppersmth and S. Wnograd, Matrx multplcaton va arthmetc progressons, Journal of Symbolc Computaton 9 (1990), E. W. Djkstra, A note on two problems n connecton wth graphs, Numersche Mathematk 1 (1959), R. W. Floyd, Algorthm 97: shortest path Communcatons of the ACM 5 (1962), M. L. Fredman, New bounds on the complexty of the shortest path problem, SIAM Journal on Computng 5 (1976), M. L. Fredman and R. E. Tarjan, Fbonacc heaps and ther uses n mproved network optmzaton algorthms, Journal of the ACM 34 (1987), D. Gusfeld Parametrc combnatoral computng and a problem of program module dstrbuton, Journal of the ACM 30(3) (1983), D. B. Johnson, Effcent algorthms for shortest paths n sparse graphs, Journal of the ACM 24 (1977), N. Karmarkar, A New Polynomal Tme Algorthm for Lnear Programmng, Combnatorca 4 (1984), R. M. Karp and J. B. Orln, Parametrc shortest path algorthms for wth an applcaton to cycle staffng, Dscrete Appled Mathematcs 3 (1981), E. Nkolova, J. A. Kelner, M. Brand and M. Mtzenmacher, Stochastc Shortest Paths Va Quas-convex Maxmzaton, Proceedngs of the 14 th Annual European Symposum on Algorthms (ESA), LNCS (2006), N. E. Young, R. E. Tarjan and J. B. Orln, Faster parametrc shortest path and mnmum-balance algorthms, Networks 21 (1991), U. Zwck, All-pars shortest paths usng brdgng sets and rectangular matrx multplcaton, Journal of the ACM 49 (2002), Appendx 6 Proof of Clam 1 The Clam 1 s the followng: Clam 2. If B 1 and B 2 are two mnbases of szes t 1 and t 2 respectvely, then (a) mn(b 1 + B 2 ) can be computed from B 1 and B 2 n tme O(t 1 + t 2 ). (b) mn{b 1, B 2 } can be computed from B 1 and B 2 n tme O(f(d)(t 1 + t 2 )), where f(d) s the tme requred to compute the ntersecton ponts of two degree-d polynomals. The sze of mn{b 1, B 2 } s O(d(t 1 + t 2 )). Proof (Proof of Clam 1). Let break 1 = {b 1 0, b1 1,..., b1 t 1, b 1 t 1 +1 } and break 2 = {b 2 0, b2 1,..., b2 t 2, b 2 t 2 +1 } be the set of breaks of B 1 and B 2 respectvely. Also let {p 1 0, p1 1,..., p1 t 1 } and {p 2 0, p2 1,..., p2 t 2 } be the ordered set of polynomals for B 1 and B 2 respectvely. Proof of Part (a): Consder the sequence of breaks whose elements are break 1+2 = break 1 break 2 after sortng. Let break 1+2 = {b ,... t 1 +t 2 t 1 +t 2 +1 there exsts a p 1 1 such that p 1 1 p 1 j }. Now for an nterval [b ] for all j. Smlarly for the nterval [b ] there exsts a p2 2
12 such that p 2 2 p 2 j for all j. Thus n the nterval [b ] the polynomal p1 1 + p 2 2 s less than or equal to p 1 j + p2 k for all j, k. Thus break1+2 can be a set of breaks correspondng to mn(b 1 + B 2 ) and p p 2 2 s the polynomal correspondng to the nterval [b ]. Thus mn(b 1 + B 2 ) can be computed n tme O(t 1 + t 2 ). Proof of Part (b): Just lke n the proof of Part (a) consder the sequence of breaks whose elements are break 1+2 = break 1 break 2. Let break 1+2 = {b ,... t 1 +t 2 t 1 +t 2 +1 }. Now for an nterval [b ] there exsts a p1 1 such that p 1 1 p 1 j for all j. Smlarly for the nterval [b ] there exsts a p2 2 such that p 2 2 p 2 j for all j. Now n the nterval [b ] the two polynomals p1 1 and p 2 2 can ntersect each other at most d tmes, because they are degree d-polynomals. Let b 1, 2 1 < b 1, 2 2 < < b 1, 2 c for c d be the set of ponts where p 1 1 and p 2 2 ntersect wthn the nterval [b ]. Thus n the subntervals [b 1, 2 k, b 1, 2 k+1 ] ether p1 1 or p 2 2 s smaller and they are smaller than all other polynomals p 1 j and p2 j n ths nterval. Thus break 1+2 along wth all the ntersecton ponts of p 1 1 and p 2 2 n each nterval [b ] can be a set of breaks for mn{b 1, B 2 } and the polynomals can also be computed easly. So mn{b 1, B 2 } can be computed n tme O(f(d)(t 1 + t 2 )) where f(d) s the tme taken to compute the ntersecton ponts of two degree d polynomals. The sze of mn{b 1, B 2 } s O(d(t 1 + t 2 )).
Luby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More information1 Example 1: Axis-aligned rectangles
COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationThe Greedy Method. Introduction. 0/1 Knapsack Problem
The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationGeneralizing the degree sequence problem
Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationRing structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationBERNSTEIN POLYNOMIALS
On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationWe are now ready to answer the question: What are the possible cardinalities for finite fields?
Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationThe Development of Web Log Mining Based on Improve-K-Means Clustering Analysis
The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationProject Networks With Mixed-Time Constraints
Project Networs Wth Mxed-Tme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa
More informationTo Fill or not to Fill: The Gas Station Problem
To Fll or not to Fll: The Gas Staton Problem Samr Khuller Azarakhsh Malekan Julán Mestre Abstract In ths paper we study several routng problems that generalze shortest paths and the Travelng Salesman Problem.
More informationn + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)
MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total
More information1. Math 210 Finite Mathematics
1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationGraph Calculus: Scalable Shortest Path Analytics for Large Social Graphs through Core Net
Graph Calculus: Scalable Shortest Path Analytcs for Large Socal Graphs through Core Net Lxn Fu Department of Computer Scence Unversty of North Carolna at Greensboro Greensboro, NC, U.S.A. lfu@uncg.edu
More informationSngle Snk Buy at Bulk Problem and the Access Network
A Constant Factor Approxmaton for the Sngle Snk Edge Installaton Problem Sudpto Guha Adam Meyerson Kamesh Munagala Abstract We present the frst constant approxmaton to the sngle snk buy-at-bulk network
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationINSTITUT FÜR INFORMATIK
INSTITUT FÜR INFORMATIK Schedulng jobs on unform processors revsted Klaus Jansen Chrstna Robene Bercht Nr. 1109 November 2011 ISSN 2192-6247 CHRISTIAN-ALBRECHTS-UNIVERSITÄT ZU KIEL Insttut für Informat
More informationSPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:
SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and
More informationThursday, December 10, 2009 Noon - 1:50 pm Faraday 143
1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More informationA Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression
Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,
More informationFeature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College
Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure
More informationPractical Issues and Algorithms for Analyzing Terrorist Networks 1
Practcal Issues and Algorthms for Analyzng Terror Networks 1 Tam Carpenter, George Karakoas, and Davd Shallcross Telcorda Technologes 445 South Street Morrown, NJ 07960 Keywords: socal network analyss,
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More information) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance
Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell
More informationLogical Development Of Vogel s Approximation Method (LD-VAM): An Approach To Find Basic Feasible Solution Of Transportation Problem
INTERNATIONAL JOURNAL OF SCIENTIFIC & TECHNOLOGY RESEARCH VOLUME, ISSUE, FEBRUARY ISSN 77-866 Logcal Development Of Vogel s Approxmaton Method (LD- An Approach To Fnd Basc Feasble Soluton Of Transportaton
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationForecasting the Direction and Strength of Stock Market Movement
Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems
More informationCS 2750 Machine Learning. Lecture 3. Density estimation. CS 2750 Machine Learning. Announcements
Lecture 3 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 5329 Sennott Square Next lecture: Matlab tutoral Announcements Rules for attendng the class: Regstered for credt Regstered for audt (only f there
More informationJoint Scheduling of Processing and Shuffle Phases in MapReduce Systems
Jont Schedulng of Processng and Shuffle Phases n MapReduce Systems Fangfe Chen, Mural Kodalam, T. V. Lakshman Department of Computer Scence and Engneerng, The Penn State Unversty Bell Laboratores, Alcatel-Lucent
More informationNPAR TESTS. One-Sample Chi-Square Test. Cell Specification. Observed Frequencies 1O i 6. Expected Frequencies 1EXP i 6
PAR TESTS If a WEIGHT varable s specfed, t s used to replcate a case as many tmes as ndcated by the weght value rounded to the nearest nteger. If the workspace requrements are exceeded and samplng has
More informationPower-of-Two Policies for Single- Warehouse Multi-Retailer Inventory Systems with Order Frequency Discounts
Power-of-wo Polces for Sngle- Warehouse Mult-Retaler Inventory Systems wth Order Frequency Dscounts José A. Ventura Pennsylvana State Unversty (USA) Yale. Herer echnon Israel Insttute of echnology (Israel)
More informationLecture 2: Single Layer Perceptrons Kevin Swingler
Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCulloch-Ptts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses
More informationA Probabilistic Theory of Coherence
A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want
More informationAvailability-Based Path Selection and Network Vulnerability Assessment
Avalablty-Based Path Selecton and Network Vulnerablty Assessment Song Yang, Stojan Trajanovsk and Fernando A. Kupers Delft Unversty of Technology, The Netherlands {S.Yang, S.Trajanovsk, F.A.Kupers}@tudelft.nl
More informationA Lyapunov Optimization Approach to Repeated Stochastic Games
PROC. ALLERTON CONFERENCE ON COMMUNICATION, CONTROL, AND COMPUTING, OCT. 2013 1 A Lyapunov Optmzaton Approach to Repeated Stochastc Games Mchael J. Neely Unversty of Southern Calforna http://www-bcf.usc.edu/
More informationProduction. 2. Y is closed A set is closed if it contains its boundary. We need this for the solution existence in the profit maximization problem.
Producer Theory Producton ASSUMPTION 2.1 Propertes of the Producton Set The producton set Y satsfes the followng propertes 1. Y s non-empty If Y s empty, we have nothng to talk about 2. Y s closed A set
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product
More informationHow To Calculate An Approxmaton Factor Of 1 1/E
Approxmaton algorthms for allocaton problems: Improvng the factor of 1 1/e Urel Fege Mcrosoft Research Redmond, WA 98052 urfege@mcrosoft.com Jan Vondrák Prnceton Unversty Prnceton, NJ 08540 jvondrak@gmal.com
More informationFisher Markets and Convex Programs
Fsher Markets and Convex Programs Nkhl R. Devanur 1 Introducton Convex programmng dualty s usually stated n ts most general form, wth convex objectve functons and convex constrants. (The book by Boyd and
More informationImplementation of Deutsch's Algorithm Using Mathcad
Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"
More informationAn Analysis of Central Processor Scheduling in Multiprogrammed Computer Systems
STAN-CS-73-355 I SU-SE-73-013 An Analyss of Central Processor Schedulng n Multprogrammed Computer Systems (Dgest Edton) by Thomas G. Prce October 1972 Techncal Report No. 57 Reproducton n whole or n part
More informationJ. Parallel Distrib. Comput.
J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationMatrix Multiplication I
Matrx Multplcaton I Yuval Flmus February 2, 2012 These notes are based on a lecture gven at the Toronto Student Semnar on February 2, 2012. The materal s taen mostly from the boo Algebrac Complexty Theory
More informationFrom Selective to Full Security: Semi-Generic Transformations in the Standard Model
An extended abstract of ths work appears n the proceedngs of PKC 2012 From Selectve to Full Securty: Sem-Generc Transformatons n the Standard Model Mchel Abdalla 1 Daro Fore 2 Vadm Lyubashevsky 1 1 Département
More informationA hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm
Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):1884-1889 Research Artcle ISSN : 0975-7384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel
More informationEnergies of Network Nastsemble
Supplementary materal: Assessng the relevance of node features for network structure Gnestra Bancon, 1 Paolo Pn,, 3 and Matteo Marsl 1 1 The Abdus Salam Internatonal Center for Theoretcal Physcs, Strada
More informationFormulating & Solving Integer Problems Chapter 11 289
Formulatng & Solvng Integer Problems Chapter 11 289 The Optonal Stop TSP If we drop the requrement that every stop must be vsted, we then get the optonal stop TSP. Ths mght correspond to a ob sequencng
More informationGeneral Auction Mechanism for Search Advertising
General Aucton Mechansm for Search Advertsng Gagan Aggarwal S. Muthukrshnan Dávd Pál Martn Pál Keywords game theory, onlne auctons, stable matchngs ABSTRACT Internet search advertsng s often sold by an
More informationRate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Priority-based scheduling. States of a process
Dsadvantages of cyclc TDDB47 Real Tme Systems Manual scheduler constructon Cannot deal wth any runtme changes What happens f we add a task to the set? Real-Tme Systems Laboratory Department of Computer
More informationProduct-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks
Bulletn of Mathematcal Bology (21 DOI 1.17/s11538-1-9517-4 ORIGINAL ARTICLE Product-Form Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set
More informationTHE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES
The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered
More informationOn Lockett pairs and Lockett conjecture for π-soluble Fitting classes
On Lockett pars and Lockett conjecture for π-soluble Fttng classes Lujn Zhu Department of Mathematcs, Yangzhou Unversty, Yangzhou 225002, P.R. Chna E-mal: ljzhu@yzu.edu.cn Nanyng Yang School of Mathematcs
More information+ + + - - This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationLogistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification
Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson
More informationPOLYSA: A Polynomial Algorithm for Non-binary Constraint Satisfaction Problems with and
POLYSA: A Polynomal Algorthm for Non-bnary Constrant Satsfacton Problems wth and Mguel A. Saldo, Federco Barber Dpto. Sstemas Informátcos y Computacón Unversdad Poltécnca de Valenca, Camno de Vera s/n
More informationQuantization Effects in Digital Filters
Quantzaton Effects n Dgtal Flters Dstrbuton of Truncaton Errors In two's complement representaton an exact number would have nfntely many bts (n general). When we lmt the number of bts to some fnte value
More informationLoop Parallelization
- - Loop Parallelzaton C-52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I-,J]+B[I-,J-] ED FOR ED FOR analyze
More informationCalculating the high frequency transmission line parameters of power cables
< ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,
More informationPeriod and Deadline Selection for Schedulability in Real-Time Systems
Perod and Deadlne Selecton for Schedulablty n Real-Tme Systems Thdapat Chantem, Xaofeng Wang, M.D. Lemmon, and X. Sharon Hu Department of Computer Scence and Engneerng, Department of Electrcal Engneerng
More informationFault tolerance in cloud technologies presented as a service
Internatonal Scentfc Conference Computer Scence 2015 Pavel Dzhunev, PhD student Fault tolerance n cloud technologes presented as a servce INTRODUCTION Improvements n technques for vrtualzaton and performance
More informationComplete Fairness in Secure Two-Party Computation
Complete Farness n Secure Two-Party Computaton S. Dov Gordon Carmt Hazay Jonathan Katz Yehuda Lndell Abstract In the settng of secure two-party computaton, two mutually dstrustng partes wsh to compute
More informationStochastic Games on a Multiple Access Channel
Stochastc Games on a Multple Access Channel Prashant N and Vnod Sharma Department of Electrcal Communcaton Engneerng Indan Insttute of Scence, Bangalore 560012, Inda Emal: prashant2406@gmal.com, vnod@ece.sc.ernet.n
More informationMinimal Coding Network With Combinatorial Structure For Instantaneous Recovery From Edge Failures
Mnmal Codng Network Wth Combnatoral Structure For Instantaneous Recovery From Edge Falures Ashly Joseph 1, Mr.M.Sadsh Sendl 2, Dr.S.Karthk 3 1 Fnal Year ME CSE Student Department of Computer Scence Engneerng
More informationHow To Solve A Problem In A Powerline (Powerline) With A Powerbook (Powerbook)
MIT 8.996: Topc n TCS: Internet Research Problems Sprng 2002 Lecture 7 March 20, 2002 Lecturer: Bran Dean Global Load Balancng Scrbe: John Kogel, Ben Leong In today s lecture, we dscuss global load balancng
More informationEfficient Reinforcement Learning in Factored MDPs
Effcent Renforcement Learnng n Factored MDPs Mchael Kearns AT&T Labs mkearns@research.att.com Daphne Koller Stanford Unversty koller@cs.stanford.edu Abstract We present a provably effcent and near-optmal
More informationPSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12
14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationNON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia
To appear n Journal o Appled Probablty June 2007 O-COSTAT SUM RED-AD-BLACK GAMES WITH BET-DEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate
More informationA Secure Password-Authenticated Key Agreement Using Smart Cards
A Secure Password-Authentcated Key Agreement Usng Smart Cards Ka Chan 1, Wen-Chung Kuo 2 and Jn-Chou Cheng 3 1 Department of Computer and Informaton Scence, R.O.C. Mltary Academy, Kaohsung 83059, Tawan,
More informationEnabling P2P One-view Multi-party Video Conferencing
Enablng P2P One-vew Mult-party Vdeo Conferencng Yongxang Zhao, Yong Lu, Changja Chen, and JanYn Zhang Abstract Mult-Party Vdeo Conferencng (MPVC) facltates realtme group nteracton between users. Whle P2P
More information1. Measuring association using correlation and regression
How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a
More informationResearch Article Enhanced Two-Step Method via Relaxed Order of α-satisfactory Degrees for Fuzzy Multiobjective Optimization
Hndaw Publshng Corporaton Mathematcal Problems n Engneerng Artcle ID 867836 pages http://dxdoorg/055/204/867836 Research Artcle Enhanced Two-Step Method va Relaxed Order of α-satsfactory Degrees for Fuzzy
More informationAn MILP model for planning of batch plants operating in a campaign-mode
An MILP model for plannng of batch plants operatng n a campagn-mode Yanna Fumero Insttuto de Desarrollo y Dseño CONICET UTN yfumero@santafe-concet.gov.ar Gabrela Corsano Insttuto de Desarrollo y Dseño
More informationData Broadcast on a Multi-System Heterogeneous Overlayed Wireless Network *
JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 24, 819-840 (2008) Data Broadcast on a Mult-System Heterogeneous Overlayed Wreless Network * Department of Computer Scence Natonal Chao Tung Unversty Hsnchu,
More informationEnergy Efficient Routing in Ad Hoc Disaster Recovery Networks
Energy Effcent Routng n Ad Hoc Dsaster Recovery Networks Gl Zussman and Adran Segall Department of Electrcal Engneerng Technon Israel Insttute of Technology Hafa 32000, Israel {glz@tx, segall@ee}.technon.ac.l
More informationSoftware project management with GAs
Informaton Scences 177 (27) 238 241 www.elsever.com/locate/ns Software project management wth GAs Enrque Alba *, J. Francsco Chcano Unversty of Málaga, Grupo GISUM, Departamento de Lenguajes y Cencas de
More informationThe Geometry of Online Packing Linear Programs
The Geometry of Onlne Packng Lnear Programs Marco Molnaro R. Rav Abstract We consder packng lnear programs wth m rows where all constrant coeffcents are n the unt nterval. In the onlne model, we know the
More informationUsing Series to Analyze Financial Situations: Present Value
2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated
More informationOptimal resource capacity management for stochastic networks
Submtted for publcaton. Optmal resource capacty management for stochastc networks A.B. Deker H. Mlton Stewart School of ISyE, Georga Insttute of Technology, Atlanta, GA 30332, ton.deker@sye.gatech.edu
More informationAbteilung für Stadt- und Regionalentwicklung Department of Urban and Regional Development
Abtelung für Stadt- und Regonalentwcklung Department of Urban and Regonal Development Gunther Maer, Alexander Kaufmann The Development of Computer Networks Frst Results from a Mcroeconomc Model SRE-Dscusson
More informationOPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004
OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL Thomas S. Ferguson and C. Zachary Glsten UCLA and Bell Communcatons May 985, revsed 2004 Abstract. Optmal nvestment polces for maxmzng the expected
More informationSchedulability Bound of Weighted Round Robin Schedulers for Hard Real-Time Systems
Schedulablty Bound of Weghted Round Robn Schedulers for Hard Real-Tme Systems Janja Wu, Jyh-Charn Lu, and We Zhao Department of Computer Scence, Texas A&M Unversty {janjaw, lu, zhao}@cs.tamu.edu Abstract
More informationSection 5.4 Annuities, Present Value, and Amortization
Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today
More information"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *
Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789-794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC
More informationPerformance Analysis of Energy Consumption of Smartphone Running Mobile Hotspot Application
Internatonal Journal of mart Grd and lean Energy Performance Analyss of Energy onsumpton of martphone Runnng Moble Hotspot Applcaton Yun on hung a chool of Electronc Engneerng, oongsl Unversty, 511 angdo-dong,
More informationAnswer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy
4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.
More information