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72a7171550d17eecf3555a9268b343d3edd65f63 | subsection | 61 | 104 | Definition and Basic Properties | It follows that \mathcal {C} is in {G}_{}_{\mathrm {qf}}(\sigma ) due to closure under \le _{\mathrm {BF}}.Theorem 6.11 {G}_{}_{\mathrm {qf}}(\operatorname{<})= {G}_{}_{\mathrm {qf}}(\operatorname{<},\operatorname{+}) = {G}_{}_{\mathrm {qf}}(\operatorname{<},\operatorname{\times }) .To prove that {G}_{}_{\mathrm {qf}}(... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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83c3f3352e81f99f49dd19a191b324f70d81e63b | subsection | 62 | 104 | Definition and Basic Properties | The idea is to rearrange the (in)equation such that the variables x_1,\dots ,x_k are on one side of the (in)equation and y_1,\dots ,y_k are on the other side. This allows us to precompute the required values in the labeling of the new labeling scheme which does not use \alpha . Let us show how this works in detail when... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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6be258eabe8b4f4e7055c4814b96c54ef49f7164 | subsection | 63 | 104 | Definition and Basic Properties | We construct a labeling \ell ^{\prime } \colon V(G) \rightarrow {[n^{c^{\prime }}]}_0^2 which shows that G is in \text{gr}_{\infty }(S^{\prime }). For u \in V(G) let \ell ^{\prime }(u) = (i,j) with e_i = l_n(\ell (u)) and e_j = r_n(\ell (v)). For all u, v \in V(G) it holds that(u,v) \in E(G) & \Leftrightarrow \mathcal ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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7e55a5674192c69e9a352813fc55a4eb3e753b8c | subsection | 64 | 104 | Definition and Basic Properties | Instead, we show that (\star ) for every formula \varphi in _k(\operatorname{<}) there exists a quantifier-free formula \psi in _k(\operatorname{<},\operatorname{+}) such that \varphi and \psi are equivalent w.r.t. \mathcal {N}_n for all n \in \mathbb {N}. It immediately follows that {G}_{}(<)\subseteq {G}_{}_{\mathrm ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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e6efce19cff5f41f9606fd84c93f122ede432d98 | subsection | 65 | 104 | Definition and Basic Properties | For all in-neighbors X of Z, out-neighbors Y of Z and variables x \in X, y \in Y append `\wedge \: x + c_1 < y \wedge x \ne c_m' to \psi . Then remove the quantifier and every atom containing z from \psi . The atom x + c_1 < y ensures that the difference between x and y is at least two, which was previously expressed b... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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368ea59a5b18292450b6445615b72da19ed66c3f | subsection | 66 | 104 | Complete Graph Classes | Corollary 6.14 Let \sigma = \emptyset , or \sigma \subseteq \lbrace \operatorname{<},\operatorname{+},\operatorname{\times }\rbrace and `\operatorname{<}' is in \sigma .
A graph class \mathcal {D} is \le _{\mathrm {BF}}-complete for {G}_{}_{\mathrm {qf}}(\sigma ) iff \mathcal {D} is in {G}_{}_{\mathrm {qf}}(\sigma ) an... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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13f894eccb088fe1a4ee64ed06ce54a1367588fc | subsection | 67 | 104 | Complete Graph Classes | Therefore \mathcal {C} \subseteq f(g_1(\mathcal {D},\dots ,\mathcal {D}),\dots ,g_a(\mathcal {D},\dots ,\mathcal {D})) and thus \mathcal {C} \le _{\mathrm {BF}}\mathcal {D}.Definition 6.15 A directed graph G is dichotomic if for all u, v \in V(G) and \alpha \in \lbrace \mathrm {in},\mathrm {out}\rbrace it holds that N_... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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7b8a8b1816335b34da97fb755c1e6b788dbefd32 | subsection | 68 | 104 | Complete Graph Classes | If u_1 = v_1 then they have identical out-neighborhoods. If u_1 \ne v_1 then they have disjoint out-neighborhoods. The same applies to the in-neighborhoods and u_2,v_2. Therefore G is dichotomic.Theorem 6.17 Dichotomic graphs are \le _{\mathrm {BF}}-complete for {G}_{}(=).From the previous lemma it follows that dichoto... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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41cc907b24a8c89264bd21c35c51722a9a059e2c | subsection | 69 | 104 | Complete Graph Classes | Correctness follows from the fact that v_i^a = v_j^b iff \ell ^{\prime }(v_i)_1 = \ell ^{\prime }(v_j)_2 for all i,j \in [n] and that only numbers between 0 and n are used.Corollary 6.18 Dichotomic graphs are \le _{\mathrm {sg}}-complete for {G}_{}(=).Lemma REF states that for every self-universal and inflatable graph ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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377467584463cde2667908801a63ba7d28200af0 | subsection | 70 | 104 | Complete Graph Classes | Therefore G has a (P_{n^3},f)-representation via \ell .Definition 6.20 A directed graph G is a linear neighborhood graph if for all u, v \in V(G) and \alpha \in \lbrace \mathrm {in},\mathrm {out}\rbrace it holds that N_{\alpha }(u) \subseteq N_{\alpha }(v) or N_{\alpha }(v) \subseteq N_{\alpha }(u).Lemma 6.21 There exi... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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df41efa99997f28d9171b835b8a6f2e2588eb6e6 | subsection | 71 | 104 | Complete Graph Classes | Given two vertices u, v \in V(G) and \ell (u) = (u_1,u_2), \ell (v) = (v_1,v_2). If u_1 \le v_1 then N_{\mathrm {out}}(v) \subseteq N_{\mathrm {out}}(u). If u_1 \ge v_1 then N_{\mathrm {out}}(u) \subseteq N_{\mathrm {out}}(v). The same holds for u_2,v_2 and the in-neighborhoods of u and v. Therefore G is a linear neigh... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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3bc4f390c4f4af6ab3b90000f6f65f7c48853f53 | subsection | 72 | 104 | Complete Graph Classes | Similarly, the labels v_1^1,\dots ,v_n^1 can be mapped to new labels \bar{v}_1^1,\dots ,\bar{v}_n^1 \subseteq \lbrace 0,1,\dots ,n\rbrace such that v_i^1 < v_j^2 iff \bar{v}_i^1 < \bar{v}_j^2 for all i,j \in [n].Corollary 6.23 Linear neighborhood graphs are \le _{\mathrm {sg}}-complete for {G}_{}(<).Same argument as in... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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069c5104d2c856914ee92027e386e9994fa70864 | subsection | 73 | 104 | Complete Graph Classes | For two vertices u \ne v \in V(D_n) it holds that(u,v) \in E(D_n) \Leftrightarrow u < v \Leftrightarrow [u,u] \cap [0,v] \ne \emptyset \wedge [0,u] \cap [v,v] = \emptyset \Leftrightarrow f(A_{uv}^\ell ) = 1and therefore D_n has an (H,f)-representation via \ell . | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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762a8a79a855d492a482dff6e6d44e79be6802ce | subsection | 74 | 104 | Polynomial-Boolean Systems | In the beginning, we defined a labeling scheme independently of a model of computation. The label decoder was just a binary relation over words. In the case of logical labeling schemes we neglected this separation by identifying label decoders with logical formulas. It would have been more hygienic to say that a logica... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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ef86648ed7d487a97d6dd031010a9a1ec275a323 | subsection | 75 | 104 | Polynomial-Boolean Systems | A graph class \mathcal {C} is in {PBS}(\mathbb {X}) if there exists a PBS R such that \mathcal {C} \subseteq \text{gr}_{}(F_R^{\mathbb {X}}).Fact 6.28 kd-line segment graphs, k-ball graphs and k-dot product graphs are in {PBS}(\mathbb {Q}) for all k \in \mathbb {N}.It is intuitively clear from the definitions of these ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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23a5f0cbd066f05139c054c82ab2de5f088cf1e2 | subsection | 76 | 104 | Polynomial-Boolean Systems | Let \ell _H \colon V(H) \rightarrow \mathbb {X}^m be a labeling which shows that H is in \text{gr}_{}(F_R^{\mathbb {X}}). By combining \ell _G and \ell _H we get a labeling \ell \colon V(G) \rightarrow \mathbb {X}^{km}. Intuitively, the labeling \ell provides us with all the information required to determine adjacency ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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a539196470693828550c3ce02c0c8281976c93f8 | subsection | 77 | 104 | Polynomial-Boolean Systems | If we only consider inputs with this sign pattern then it holds that p(x,y,z) < q(x,y,z) iff \underbrace{|y| + |z|}_{p^{\prime }} < \underbrace{|x|^2|y|^3|z|}_{q^{\prime }}. For each variable in R we have two variables in R^{\prime }. (\star ) The first one is used to store the absolute value of the original variable a... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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e79e9f6fe784062502141242f4542308b2e1831a | subsection | 78 | 104 | Polynomial-Boolean Systems | Given two polynomial functions p,q \colon \mathbb {Q}_+^k \rightarrow \mathbb {Q}_+ there exist two polynomial functions p^{\prime },q^{\prime } \colon \mathbb {N}_0^{2k} \rightarrow \mathbb {N}_0 such that for all \vec{a} = (\frac{a_1}{b_1},\dots ,\frac{a_k}{b_k}) \in \mathbb {Q}_+^k it holds that p(\vec{a}) < q(\vec{... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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f1ad1a0f98c4d7c916b0e8a05cda0c4e340e6bdc | subsection | 79 | 104 | Polynomial-Boolean Systems | Given \vec{a} \in \mathbb {R}^n let \mathcal {E}(\vec{a}) = (e_1,\dots ,e_m) with e_i = 1 iff the inequation E_i(\vec{a}) holds. An element of the image of \mathcal {E} is called a sign pattern.
Warren's theorem states that the cardinality of the image of \mathcal {E} (or equivalently, the number of sign patterns of \m... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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63cb5f2d144938f690ef25f5a5e964a2631bb272 | subsection | 80 | 104 | Polynomial-Boolean Systems | We say a graph G is in \text{gr}_{}(F,s) if there exists a labeling \ell \colon V(G) \rightarrow [s(n)]_0^k such that (u,v) \in E(G) \Leftrightarrow (\ell (u),\ell (v)) \in F for all u, v \in V(G).
Let S be a set of total functions from \mathbb {N} to \mathbb {N}. We say a graph class \mathcal {C} is in {PBS}(\mathbb {... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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8329417520f7ba2a4740274bd5ea84c98380937d | subsection | 81 | 104 | Polynomial-Boolean Systems | More precisely, if a graph G on n vertices is in \text{gr}_{}( F_R^{\mathbb {Q}}, s ) via a labeling \ell \colon V(G) \rightarrow \mathbb {Q}_{s(n)} then the labeling \ell ^{\prime } derived from \ell to show that G is in \text{gr}_{}( F_{R^{\prime }}^{\mathbb {Q}_+}, s ) is only allowed to contain values from {(\mathb... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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39f811d620ff0aff151522119249ad44cd6bc95b | subsection | 82 | 104 | Polynomial-Boolean Systems | \mathcal {C} \subseteq \text{gr}_{}(F_R^{\mathbb {N}_0}). We argue that there exists a total function s \colon \mathbb {N}\rightarrow \mathbb {N} which depends on R such that \mathcal {C} \subseteq \text{gr}_{}(F_R^{\mathbb {N}_0},s). From that the claim follows.
For a graph G in \text{gr}_{}(F_R^{\mathbb {N}_0}) let \... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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46a1d7d799a4b9d018787994774e6805d803cebb | subsection | 83 | 104 | Polynomial-Boolean Systems | It holds that \text{gr}_{}(F_R^{\mathbb {N}_0},n^c) is a subset of \text{gr}_{\infty }(\varphi ,c) and thus \mathcal {C} \in {G}_{}_{\mathrm {qf}}.Corollary 6.37 {G}_{}_{\mathrm {qf}}= {PBS}(\mathbb {N}_0,n^{\mathcal {O}(1)}) = {PBS}(\mathbb {Q},n^{\mathcal {O}(1)}) \subseteq {PBS}(\mathbb {N}_0,\mathrm {Tot}) = {PBS}(... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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647ebf710bc66209f7753f537eeaeb9bd48b7622 | subsection | 84 | 104 | Polynomial-Boolean Systems | It holds that \mathcal {D} := \text{gr}_{}(F_R^{\mathbb {N}_0},n) is in {PBS}(\mathbb {N}_0,n^{\mathcal {O}(1)}).
We claim that every graph in \mathcal {C} occurs as induced subgraph of some graph in \mathcal {D} and therefore \mathcal {C} \subseteq [\mathcal {D}]_{\subseteq }.
Since {PBS}(\mathbb {N}_0,n^{\mathcal {O}... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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f041ae5bbd1da40c7c9c86fd35c976b1d4998146 | subsection | 85 | 104 | Polynomial-Boolean Systems | Let G be a graph in \mathcal {D} and let G^{\prime } be an induced subgraph of G on vertex set V^{\prime } \subseteq V(G).
There exist graphs H_1,\dots ,H_k \in \mathcal {C} on vertex set V(G) such that G = f(H_1,\dots ,H_k). It follows that G^{\prime } = f(H_1^{\prime },\dots ,H_k^{\prime }) where H_i^{\prime } is the... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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b38bc3a50b307eb2a6e4877c4d47e5c72523653c | subsection | 86 | 104 | Polynomial-Boolean Systems | If {G}_{}_{\mathrm {qf}} has an \le _{\mathrm {sg}}-complete graph class that is hereditary and inflatable then {G}_{}_{\mathrm {qf}}= {PBS}(\mathbb {N}_0).Since {G}_{}(<) has a hereditary graph class which is \le _{\mathrm {BF}}-complete, namely linear neighborhood graphs, it follows that {G}_{}(<) must be a strict su... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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4c74b669ea52220784c87a1b5d32254970379000 | subsection | 87 | 104 | Algorithmic Properties | Consider the following question: is deciding the existence of a Hamiltonian cycle W[1]-hard when parameterized by {G}_{}¶? This seems to be an ill-defined question because unlike, for example, tree-width the class {G}_{}¶ does not resemble a parameter at all. We argue that the incapability of recognizing this as a well... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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c4eaf165670d0baf42452ae55c516635832dcc86 | subsection | 88 | 104 | Algorithmic Properties | However, while it is self-evident that adjacency matrices represent graphs it is not so obvious what is represented by parameters. A different notion of boundedness helps to answer this question. Let us say a language L is bounded by a parameter \kappa if there exists a c \in \mathbb {N} such that L \subseteq \kappa _c... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
366b1d19bc70e17116e9889a09fd70f3ae2cc8b0 | subsection | 89 | 104 | Algorithmic Properties | However, not every set of languages can be interpreted as a parameter. We are only interested in sets of languages that can be represented by a parameter. To distinguish between such sets of languages and parameters let us call the former ones parameterizations.Definition 7.3 A set of languages \mathbb {K} over an alph... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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46db01c02fab04f84f0173035c9a6f05bed9a4f6 | subsection | 90 | 104 | Algorithmic Properties | Then \kappa (w) being defined as the least k such that w \in L^{\prime }_k yields the required parameter.Therefore it is more accurate to understand a parameterized problem as a tuple (L,\mathbb {K}) where L is a language and \mathbb {K} is a parameterization, both over the same alphabet. This alternative view on param... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
a55f98fc636ee2536ad3ea37c2d5af816903699c | subsection | 91 | 104 | Algorithmic Properties | A parameterized problem (L,\mathbb {K}) is in XP if for all K \in \mathbb {K} it holds that L is in ¶ if one only considers inputs from K. In contrast to FPT the degree of the polynomial that bounds the runtime is not fixed but can depend on K.
The classes FPT and XP are regarded as the analogon of ¶ and in the paramet... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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0... | |
8fbc4d3914d904baba01e2a1f6c9fcac8db27a6e | subsection | 92 | 104 | Regular Labeling Schemes | One of our main objectives is to identify suitable classes of labeling schemes against which lower bounds for hereditary graph classes can be proved.
Suitable means that such a class should contain most of the graph classes that are known to have a labeling scheme while still possessing enough structure to be amenable ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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991862ba0cb740a7903c0cbcb1a8bc9fb65702ef | subsection | 93 | 104 | Regular Labeling Schemes | The same idea can be applied to logical labeling schemes. The logical labeling scheme for dichotomic graphs is an example of this. This trick cannot be applied to regular labeling schemes and therefore we have to externally add this ability in order to get the most out of such labeling schemes.Definition 8.2 Let S=(F,c... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
8db24d6a5051533502240933a3bf2d96db1e802b | subsection | 94 | 104 | Regular Labeling Schemes | For m \in \mathbb {N}, i \in [k] and x_i,y_i \in \lbrace 0,1\rbrace ^{cdm} we define(x_1x_2\dots x_{k^2},y_1y_2 \dots y_{k^2}) \in F^{\prime } \Leftrightarrow f \left(
\begin{matrix}
z_1 & z_2 & \dots & z_k \\
z_{k+1} & \ddots & & \\
\vdots & & & \\
z_{k^2-k+1} & & & z_{k^2}
\end{matrix}
\right) = 1with z_i := \llbrack... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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9a6ae7900d9957f656fc42258e136bfaf21f80f0 | subsection | 95 | 104 | Regular Labeling Schemes | If we plug in the definitions of \ell _{\mathrm {out}}(u) and \ell _{\mathrm {in}}(v) the right-hand side becomes\left(
\underbrace{u_1^{\mathrm {out}} \dots u_1^{\mathrm {out}}}_{k \text{ times}} \: \dots \: \underbrace{u_k^{\mathrm {out}} \dots u_k^{\mathrm {out}}}_{k \text{ times}} \text{\Large , }
v_1^{\mathrm {in}... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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0... | |
d51b3b008506a20718281a072dbaa6d05ed4c30f | subsection | 96 | 104 | Regular Labeling Schemes | Also, the truth value of each proposition can be decided by a DFA because F is regular. However, the difficulty is that a DFA does not know when x_i and y_i end and x_{i+1} and y_{i+1} begin. To resolve this one can introduce a special delimiter sign `\#' and define a new label decoder F^{\prime \prime } over the alpha... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
b3e525e610b8c1cc8b5f3402c162d496344d1054 | subsection | 97 | 104 | Regular Labeling Schemes | Then we append 0s_i\# to the label of v. The first bit tells us that v is placed in the left child of the root node. The string s_i encodes in which of the k parts of S v lies. Assume v is in the right child of r. Let s \in \lbrace 0,1\rbrace ^k be the string which has a 1 at the i-th position iff i \in R for all i \in... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
493a4d8edff7af4e262e825e09b96211dca66666 | subsection | 98 | 104 | Regular Labeling Schemes | Also, every singleton graph class is contained in {G}_{}{REG} because {G}_{}(<)\subseteq {G}_{}{REG} and {G}_{}(<) already contains every singleton graph class. The countable subset of {G}_{}{REG} such that its closure under subsets equals {G}_{}{REG} is given by the set of graph classes \text{gr}_{\mathrm {io}}(S) whe... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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... | |
dbae00ecd6eacf69b9b3a6feece903b9cb983965 | subsection | 99 | 104 | Summary and Open Questions | In Figure REF an overview of all the sets of graph classes that we have seen is given. First, we summarize the train of thought that motivated us to introduce the various concepts and what we perceive to be their importance in the context of studying the limitations of labeling schemes; this summary does not reflect th... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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1ab41071dc663fbdd05be61403b33b5f8ea81f41 | subsection | 100 | 104 | Summary and Open Questions | Going below {G}_{}_{\mathrm {qf}} we find {G}_{}(<) and {G}_{}(=) which have various complete graph classes under both types of reduction and contain a wealth of graph classes that have been intensely studied from a graph-theoretical and an algorithmic perspective. It is notable that {G}_{}(=) and {G}_{}(<) are closed ... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.0... | |
2ec7ac9d76b4365a7446108dd84207f3bf8b178c | subsection | 101 | 104 | Summary and Open Questions | The most expressive classes of labeling schemes against which proving lower bounds does not seem inconceivable (given the current state of knowledge) are {G}_{}^0, {G}_{}_{\mathrm {qf}} and {G}_{}{REG}. While we believe that there exists a small and hereditary graph class that does not reside in {G}_{}^0, we also presu... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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0.... | |
4bbf7784a449152be2ff11a1a81fe140cc4998a4 | subsection | 102 | 104 | Summary and Open Questions | The following two questions are aimed at developing such an understanding.
Let us say a graph class is f-hereditary if it is characterized by a finite set of forbidden induced subgraphs. Can the set of (undirected) f-hereditary graph classes in {G}_{}(<) be characterized in terms of their forbidden induced subgraphs? S... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
"cs.DS"
] | 2,018 | en | Computer Science | [
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7344200a785ef8825fadec8aaaf0dde58504d7c7 | subsection | 103 | 104 | Summary and Open Questions | This is equivalent to asking whether [\mathrm {Small} \cap \mathrm {Hereditary}]_{\subseteq } has a \le _{\mathrm {BF}}-complete graph class. If this were to be true then the adjacency structure of every small and hereditary graph class would just be a boolean combination of one such particular graph class.
We do not b... | {
"cite_spans": []
} | 1802.02819 | A Complexity Theory for Labeling Schemes | [
"Maurice Chandoo"
] | [
"cs.CC",
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] | 2,018 | en | Computer Science | [
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ec414e2934b4cdd6fa1306b99e3ac8c4245ff4e6 | abstract | 0 | 57 | Abstract | The consensus number of an object is the maximum number of processes among
which binary consensus can be solved using any number of instances of the
object and read-write registers. Herlihy [6] showed in his seminal work that if
an object has a consensus number of n, then there is a universal construction
for a wait-fr... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
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935a7f857da28b5ce679b7afc5e3ae63cae0b2c3 | subsection | 1 | 57 | Introduction | Any multiprocessor chip needs to support some synchronization instructions, such as compare-and-swap
or fetch-and-add, to coordinate among several concurrent processes that can take steps
asynchronously at different rates. As it is not possible to support every other synchronization
instruction on a multiprocessor, the... | {
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{
"arxiv_id": "",
"doi": "10.1145/114005.102808",
"end": 447,
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"raw": "Maurice Herlihy. Wait-free Synchronization. ACM Transactions on Programming Languages and Systems (TOPLAS), 1991.",
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"Pankaj Khanchandani",
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] | [
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4fab2ea6d58bf7dee6619fbf9227f8cc2add9b9b | subsection | 2 | 57 | Introduction | We show that
using read-write registers and registers that support half-max and max-write, we can construct a
linearizable and wait-free implementation of a compare-and-swap register so that every
compare-and-swap operation takes O(1) time.
The size of the registers required is
logarithmic in the length of the executio... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
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5b9e02c27a378e55b57c221e90afa76a6005f063 | subsection | 3 | 57 | Related Work | One of the most central questions in concurrent computing has been to quantify the power of
synchronization instructions. Herlihy originally defined the
consensus number of an object as the maximum number of processes n that can solve consensus
using a single instance of the object and any number of read-write register... | {
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"Pankaj Khanchandani",
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] | [
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606f9d1c58eb405e72bbc057be4b97c3ab0eb7fc | subsection | 4 | 57 | Related Work | Their focus is to use read-write registers and hence wait-freedom is
impossible to achieve. Overall, there is no prior work that shows that a set of low consensus number
instructions can be as powerful and efficient as compare-and-swap registers for an arbitrary
synchronization task. | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
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3ec3308c92f4839147b8f861c8151c799a2b287e | subsection | 5 | 57 | An Overview of the Method | Our method is based on the observation that if several compare-and-swap
operations attempt to simultaneously change the value in the register, only one
of them succeeds. So, instead of updating the final value of the register for
each operation, we first determine the single operation that succeeds and update
the final... | {
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{
"arxiv_id": "",
"doi": "10.1007/978-3-642-41527-2_20",
"end": 1357,
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"raw": "Faith Ellen and Philipp Woelfel. An Optimal Implementation of Fetch-and-Increment. In 27th International Symposium on Distributed Computin... | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
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4096d7b187156e84f2dd57d057d4de149e6aff54 | subsection | 6 | 57 | An Overview of the Method | Finally, we conclude and discuss some extensions in sec:conc:alg[sec:conc]Algorithm sec:conclem[sec:conc]Lemma sec:concthm[sec:conc]Theorem sec:concln[sec:conc]Line sec:conctab[sec:conc]Table sec:conccor[sec:conc]Corollary sec:concfig[sec:conc]Figure sec:concas[sec:conc]Assumption sec:concsec[sec:conc]Section sec:concd... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
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9640932cddb4cbaf6a56bd608d3d9efd085cefb9 | subsection | 7 | 57 | Model | A sequential object is defined by the tuple (S, O, R, T). Here,
S is the set of all possible states of the object, O is the set
of operations that can be performed on the object, R is the set of
possible return values of all the operations and
T: S \times O \rightarrow S \times R is the transition function that
specifi... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
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] | 2,018 | en | Computer Science | [
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5774a4ecab29a059c79c547d568c2b52d4e78c09 | subsection | 8 | 57 | Model | Otherwise, it returns false and does not change the value. | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
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] | 2,018 | en | Computer Science | [
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bfac9fcbffdc34023caea1a4db0af68e08095166 | subsection | 9 | 57 | Algorithm | fig:sim:alg[fig:sim]Algorithm fig:simlem[fig:sim]Lemma fig:simthm[fig:sim]Theorem fig:simln[fig:sim]Line fig:simtab[fig:sim]Table fig:simcor[fig:sim]Corollary fig:simfig[fig:sim]Figure fig:simas[fig:sim]Assumption fig:simsec[fig:sim]Section fig:simdef[fig:sim]Definition fig:simlp[fig:sim]Case fig:sim[ss] shows the (sha... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
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a2b389b0d77ac3715543d2d40aca7ed5f185e90f | subsection | 10 | 57 | Algorithm | To execute the compare-and-swap function, a process starts by
reading the current value of the object (ln:rdval:alg[ln:rdval]Algorithm ln:rdvallem[ln:rdval]Lemma ln:rdvalthm[ln:rdval]Theorem ln:rdvalln[ln:rdval]Line ln:rdvaltab[ln:rdval]Table ln:rdvalcor[ln:rdval]Corollary ln:rdvalfig[ln:rdval]Figure ln:rdvalas[ln:rdva... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
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3091e2bd1c9d93502e1bac4e722d0cf83b9f7f61 | subsection | 11 | 57 | Algorithm | The process starts competing with the other concurrent processes
by trying to announce its identifier in P using the max-write operation (ln:comp:alg[ln:comp]Algorithm ln:complem[ln:comp]Lemma ln:compthm[ln:comp]Theorem ln:compln[ln:comp]Line ln:comptab[ln:comp]Table ln:compcor[ln:comp]Corollary ln:compfig[ln:comp]Figu... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
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f25a96feec5e121041baaa15cac3dec67b69c03d | subsection | 12 | 57 | Algorithm | At initialization,
we have c = 0 and V = (0 \,|\,x), where x is the initial value of the
compare-and-swap object.Once the winner of the competing processes is determined, the winner and the
value announced by it is read (Lines and ), the winner
is informed that it won after appropriate checks (Lines , )
and the curren... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
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361253a806fcd4378ecdbd7d0fcf08e2150f2afe | subsection | 13 | 57 | Analysis | Let us first define some notation. We refer to a field f of a register
X by X.f. The term X.f_k^i is the value of the field X.f just
after process i executes Line k during a call. We omit the call
identifier from the notation as it will be always clear from the context. Similarly, v_k^i is the
value of a variable v, th... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
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93e370a800d9ece3b1c503d5705d992bae67d26d | subsection | 14 | 57 | Analysis | If the process returns from ln:neqret:alg[ln:neqret]Algorithm ln:neqretlem[ln:neqret]Lemma ln:neqretthm[ln:neqret]Theorem ln:neqretln[ln:neqret]Line ln:neqrettab[ln:neqret]Table ln:neqretcor[ln:neqret]Corollary ln:neqretfig[ln:neqret]Figure ln:neqretas[ln:neqret]Assumption ln:neqretsec[ln:neqret]Section ln:neqretdef[ln... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
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6cd7cae4486b7838965210ab5b1edb929df2d502 | subsection | 15 | 57 | Analysis | This is the linearization point of the process i if its
compare-and-swap operation was successful as determined by the value of
P.pid (lp:3a:alg[lp:3a]Algorithm lp:3alem[lp:3a]Lemma lp:3athm[lp:3a]Theorem lp:3aln[lp:3a]Line lp:3atab[lp:3a]Table lp:3acor[lp:3a]Corollary lp:3afig[lp:3a]Figure lp:3aas[lp:3a]Assumption lp:... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
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34f01af96137c9fd88d90ed8915b19a07b351928 | subsection | 16 | 57 | Analysis | If V.\mathit {val}_{\ref *{ln:rdval}}^{i} = a_{\ref *{ln:arg}}^i = b_{\ref *{ln:arg}}^i, then
the linearization point is the point when i executes ln:rdval:alg[ln:rdval]Algorithm ln:rdvallem[ln:rdval]Lemma ln:rdvalthm[ln:rdval]Theorem ln:rdvalln[ln:rdval]Line ln:rdvaltab[ln:rdval]Table ln:rdvalcor[ln:rdval]Corollary ln... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
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61a34a9cba970741b3ad12bcba63482d24e29411 | subsection | 17 | 57 | Analysis | If V.\mathit {val}_{\ref *{ln:rdval}}^{i} = a_{\ref *{ln:arg}}^i \ne b_{\ref *{ln:arg}}^i and V.\mathit {seq}_e< V.\mathit {seq}_{\ref *{ln:rdval}}^{i} + 2 , then the linearization point is at the
end, after all the other linearization points in some order.Note that we assume in lp:3:alg[lp:3]Algorithm lp:3lem[lp:3]Lem... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
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406c3ccce46326d66820968fe216cb10212d6f96 | subsection | 18 | 57 | Analysis | Say that the value of the field was changed to V.\mathit {seq}_{\ref *{ln:update}}^{i} when a process
i executed ln:update:alg[ln:update]Algorithm ln:updatelem[ln:update]Lemma ln:updatethm[ln:update]Theorem ln:updateln[ln:update]Line ln:updatetab[ln:update]Table ln:updatecor[ln:update]Corollary ln:updatefig[ln:update]F... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
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-0.041372012346982956,
... | |
d8a3776bc2c7d1ca753a7a92873738f4f6aea09f | subsection | 19 | 57 | Analysis | The linearization point as given by def:linp:alg[def:linp]Algorithm def:linplem[def:linp]Lemma def:linpthm[def:linp]Theorem def:linpln[def:linp]Line def:linptab[def:linp]Table def:linpcor[def:linp]Corollary def:linpfig[def:linp]Figure def:linpas[def:linp]Assumption def:linpsec[def:linp]Section def:linpdef[def:linp]Defi... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.01691097393631935,
-0.005769267212599516,
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0.024542279541492462,
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-0.015934167429804802,
-0.012782437726855... | |
5c4e474436821d66ee396eb726a55690e1ab02b8 | subsection | 20 | 57 | Analysis | As
V.\mathit {seq}_{\ref *{ln:rdval}}^{i} is even by lem:veven:alg[lem:veven]Algorithm lem:vevenlem[lem:veven]Lemma lem:veventhm[lem:veven]Theorem lem:vevenln[lem:veven]Line lem:veventab[lem:veven]Table lem:vevencor[lem:veven]Corollary lem:vevenfig[lem:veven]Figure lem:vevenas[lem:veven]Assumption lem:vevensec[lem:veve... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.0464436337351799,
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0.012350894510746002,
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-0.0031830824445933104,
0.00968847144395113,
0.04... | |
cb456fd4902b689b8270c7e233848ee24a8bb662 | subsection | 21 | 57 | Analysis | Consider
the process k \in S that is the first one to execute ln:chk:alg[ln:chk]Algorithm ln:chklem[ln:chk]Lemma ln:chkthm[ln:chk]Theorem ln:chkln[ln:chk]Line ln:chktab[ln:chk]Table ln:chkcor[ln:chk]Corollary ln:chkfig[ln:chk]Figure ln:chkas[ln:chk]Assumption ln:chksec[ln:chk]Section ln:chkdef[ln:chk]Definition ln:chkl... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.035498663783073425,
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0.03516290709376335,
-0.005162262823432684,
0.02302987314760685,
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0.01515484694391489,
-0.007127203978598118,
-0.04718911275267601,
-0.0003970421093981713,
... | |
45e2d30f0b3bcf4a632586a366ee0820e339ccdd | subsection | 22 | 57 | Analysis | Moreover, the process \mathit {pid}_{\ref *{ln:prd}}^{k} \in S as some process(es) (including
k) executed ln:comp:alg[ln:comp]Algorithm ln:complem[ln:comp]Lemma ln:compthm[ln:comp]Theorem ln:compln[ln:comp]Line ln:comptab[ln:comp]Table ln:compcor[ln:comp]Corollary ln:compfig[ln:comp]Figure ln:compas[ln:comp]Assumption ... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.030066844075918198,
0.012531665153801441,
-0.028617655858397484,
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0.009572270326316357,
-0.03301098570227623,
-0.01806146278977394,
... | |
3f2402d51c6e37fdc1513162dd4510a232360dd7 | subsection | 23 | 57 | Analysis | So, it holds that \mathit {ca}_{\ref *{ln:ard}}^{k} = \mathit {cp}_{\ref *{ln:prd}}^{k}
and the process k executes ln:update:alg[ln:update]Algorithm ln:updatelem[ln:update]Lemma ln:updatethm[ln:update]Theorem ln:updateln[ln:update]Line ln:updatetab[ln:update]Table ln:updatecor[ln:update]Corollary ln:updatefig[ln:update... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.034674953669309616,
-0.010240710340440273,
0.0024285439867526293,
0.007478312589228153,
-0.03333190828561783,
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0.023228555917739868,
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-0.014361413195729256,
-0.00800484698265791,
-0.02141239307820797,
-0.02716611512005329,... | |
b3d4dbdba2cc80fafbf547578261b4b1516ccc62 | subsection | 24 | 57 | Analysis | The statement is true for Cases REF and REF as the instruction
corresponding to the linearization point is executed by the process i
itself.For lp:3:alg[lp:3]Algorithm lp:3lem[lp:3]Lemma lp:3thm[lp:3]Theorem lp:3ln[lp:3]Line lp:3tab[lp:3]Table lp:3cor[lp:3]Corollary lp:3fig[lp:3]Figure lp:3as[lp:3]Assumption lp:3sec[lp... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.023092815652489662,
0.0009434399544261396,
-0.013263487257063389,
-0.006189118605107069,
-0.01530872005969286,
0.029228512197732925,
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0.03595946729183197,
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0.003527645021677017,
-0.019292345270514488,
-0.0077688321471214294,
0.026038561016321182,... | |
d2b023abd7f421b0e83dfc6722b71c74f1db4e33 | subsection | 25 | 57 | Analysis | So, the point when
ln:update:alg[ln:update]Algorithm ln:updatelem[ln:update]Lemma ln:updatethm[ln:update]Theorem ln:updateln[ln:update]Line ln:updatetab[ln:update]Table ln:updatecor[ln:update]Corollary ln:updatefig[ln:update]Figure ln:updateas[ln:update]Assumption ln:updatesec[ln:update]Section ln:updatedef[ln:update]D... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.04491741955280304,
0.02172636054456234,
-0.008376244455575943,
0.006755158770829439,
-0.02049052156507969,
0.021802647039294243,
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0.011603157967329025,
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0.005935080349445343,
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-0.02346568927168846,
-0.005431590136140585,
0... | |
b4488900bb14347005562f4de09c00323165b3a5 | subsection | 26 | 57 | Analysis | Without loss of generality assume that the process i executes ln:prd:alg[ln:prd]Algorithm ln:prdlem[ln:prd]Lemma ln:prdthm[ln:prd]Theorem ln:prdln[ln:prd]Line ln:prdtab[ln:prd]Table ln:prdcor[ln:prd]Corollary ln:prdfig[ln:prd]Figure ln:prdas[ln:prd]Assumption ln:prdsec[ln:prd]Section ln:prddef[ln:prd]Definition ln:prdl... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.05336594209074974,
0.01028263196349144,
-0.032861705869436264,
0.018368380144238472,
-0.017285196110606194,
0.009443544782698154,
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0.02904767170548439,
0.011457353830337524,
-0.013638980686664581,
-0.011281908489763737,
-0.03670625016093254,
-0.0026888928841799498,
... | |
ee5e5be0ad1633175e859b8e5d5b04644db75c5a | subsection | 27 | 57 | Analysis | To have a different value of V.\mathit {val} with x
as the value of V.\mathit {seq}, another process j must execute ln:update:alg[ln:update]Algorithm ln:updatelem[ln:update]Lemma ln:updatethm[ln:update]Theorem ln:updateln[ln:update]Line ln:updatetab[ln:update]Table ln:updatecor[ln:update]Corollary ln:updatefig[ln:updat... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.05171878635883331,
0.011831244453787804,
-0.019985726103186607,
0.013883214443922043,
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0.02132827788591385,
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-0.04531114920973778,
-0.040703754872083664,
... | |
308a51862044a6b2dae1ac86615be0feea06e3e2 | subsection | 28 | 57 | Analysis | As the condition in ln:chk:alg[ln:chk]Algorithm ln:chklem[ln:chk]Lemma ln:chkthm[ln:chk]Theorem ln:chkln[ln:chk]Line ln:chktab[ln:chk]Table ln:chkcor[ln:chk]Corollary ln:chkfig[ln:chk]Figure ln:chkas[ln:chk]Assumption ln:chksec[ln:chk]Section ln:chkdef[ln:chk]Definition ln:chklp[ln:chk]Case ln:chk[ss] is true for both ... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.038785696029663086,
-0.015036704950034618,
-0.030882081016898155,
-0.018721498548984528,
-0.027922039851546288,
0.0019348982023075223,
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0.021193286404013634,
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0.009650650434195995,
-0.022490210831165314,
-0.0436682403087616,
-0.02552654221653938... | |
85e4f326f4bc5ab31cc0b7e05a3644f9c5e20087 | subsection | 29 | 57 | Analysis | As \mathit {seq}_{\ref *{ln:prd}}^{i} = x, some process h modified P by executing
ln:comp:alg[ln:comp]Algorithm ln:complem[ln:comp]Lemma ln:compthm[ln:comp]Theorem ln:compln[ln:comp]Line ln:comptab[ln:comp]Table ln:compcor[ln:comp]Corollary ln:compfig[ln:comp]Figure ln:compas[ln:comp]Assumption ln:compsec[ln:comp]Secti... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.06034502759575844,
-0.0013058349722996354,
-0.0023236041888594627,
-0.011568419635295868,
-0.0360177718102932,
-0.002962690545246005,
-0.012133104726672173,
0.052683621644973755,
0.012117843143641949,
0.001465129666030407,
-0.0007168261217884719,
-0.024891942739486694,
-0.0121254744008183... | |
13b89d98051083d446f18a4b5c1b8a7d0a25441a | subsection | 30 | 57 | Analysis | Also, process h executed ln:comp:alg[ln:comp]Algorithm ln:complem[ln:comp]Lemma ln:compthm[ln:comp]Theorem ln:compln[ln:comp]Line ln:comptab[ln:comp]Table ln:compcor[ln:comp]Corollary ln:compfig[ln:comp]Figure ln:compas[ln:comp]Assumption ln:compsec[ln:comp]Section ln:compdef[ln:comp]Definition ln:complp[ln:comp]Case l... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.04641066491603851,
0.011236506514251232,
-0.013341921381652355,
0.004134546499699354,
-0.016355104744434357,
0.0033774361945688725,
-0.009092950262129307,
0.04656323045492172,
0.0014121916610747576,
-0.004527404438704252,
0.0035128386225551367,
-0.04827197268605232,
-0.006663332227617502,... | |
aecd777aa4ad786bb872d0656160050f21a85c18 | subsection | 31 | 57 | Analysis | Then, we know from
lem:vinc2:alg[lem:vinc2]Algorithm lem:vinc2lem[lem:vinc2]Lemma lem:vinc2thm[lem:vinc2]Theorem lem:vinc2ln[lem:vinc2]Line lem:vinc2tab[lem:vinc2]Table lem:vinc2cor[lem:vinc2]Corollary lem:vinc2fig[lem:vinc2]Figure lem:vinc2as[lem:vinc2]Assumption lem:vinc2sec[lem:vinc2]Section lem:vinc2def[lem:vinc2]D... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.07883765548467636,
0.007688502315431833,
0.008245307952165604,
0.02494186721742153,
-0.021326441317796707,
0.004240880254656076,
0.00007001790072536096,
0.01902294158935547,
0.008222426287829876,
-0.024224884808063507,
-0.0020479790400713682,
-0.017619485035538673,
-0.03023534081876278,
... | |
566391be219676464be89db9d81fd97e8044922d | subsection | 32 | 57 | Analysis | Thus, p is the linearization point of the process
h by def:linp:alg[def:linp]Algorithm def:linplem[def:linp]Lemma def:linpthm[def:linp]Theorem def:linpln[def:linp]Line def:linptab[def:linp]Table def:linpcor[def:linp]Corollary def:linpfig[def:linp]Figure def:linpas[def:linp]Assumption def:linpsec[def:linp]Section def:li... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.0007671607309021056,
-0.006675109267234802,
-0.01150407362729311,
0.012144884094595909,
-0.032955966889858246,
0.029172133654356003,
0.018675047904253006,
0.05773397162556648,
0.008193220011889935,
-0.011267583817243576,
-0.013518049381673336,
-0.037136491388082504,
0.019376888871192932,
... | |
1f782862655fd9817de66a50042dc00c025f2b81 | subsection | 33 | 57 | Analysis | Using lem:vseqeqthenvvaleq:alg[lem:vseqeqthenvvaleq]Algorithm lem:vseqeqthenvvaleqlem[lem:vseqeqthenvvaleq]Lemma lem:vseqeqthenvvaleqthm[lem:vseqeqthenvvaleq]Theorem lem:vseqeqthenvvaleqln[lem:vseqeqthenvvaleq]Line lem:vseqeqthenvvaleqtab[lem:vseqeqthenvvaleq]Table lem:vseqeqthenvvaleqcor[lem:vseqeqthenvvaleq]Corollary... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
0.00015248444105964154,
0.02131873182952404,
-0.024035077542066574,
0.0020696872379630804,
0.004967248998582363,
0.0014077688101679087,
0.006165187805891037,
0.01800723187625408,
0.002422583056613803,
0.004665857180953026,
0.005634890403598547,
0.00949194747954607,
0.03001714125275612,
0.0... | |
77ae37627efb05dff7471045bd4376c71fd45591 | subsection | 34 | 57 | Analysis | For k \ge 1, the
L.\mathit {ret}_{k} values are false for lp:1:alg[lp:1]Algorithm lp:1lem[lp:1]Lemma lp:1thm[lp:1]Theorem lp:1ln[lp:1]Line lp:1tab[lp:1]Table lp:1cor[lp:1]Corollary lp:1fig[lp:1]Figure lp:1as[lp:1]Assumption lp:1sec[lp:1]Section lp:1def[lp:1]Definition lp:1lp[lp:1]Case lp:1[ss],
true for lp:2:alg[lp:2]A... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.015957362949848175,
0.02935178391635418,
-0.028344914317131042,
-0.009847492910921574,
-0.006670513190329075,
0.00330855930224061,
-0.007902403362095356,
0.026254896074533463,
-0.03316568583250046,
-0.0002693567657843232,
-0.010236511006951332,
-0.03154858946800232,
0.0156827624887228,
... | |
c77f0dce27b971c48bb8f056c8d1852760d0d923 | subsection | 35 | 57 | Analysis | By induction hypothesis, it holds that L.\mathit {val}_{k^{\prime }} = V.\mathit {val}_{k^{\prime }}. By
lem:vvalmodatlp:alg[lem:vvalmodatlp]Algorithm lem:vvalmodatlplem[lem:vvalmodatlp]Lemma lem:vvalmodatlpthm[lem:vvalmodatlp]Theorem lem:vvalmodatlpln[lem:vvalmodatlp]Line lem:vvalmodatlptab[lem:vvalmodatlp]Table lem:v... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.016452521085739136,
-0.00007648924656677991,
-0.02022225596010685,
0.0007406877703033388,
-0.02921161986887455,
0.003035245928913355,
0.011026852764189243,
-0.0005785282119177282,
0.03446177393198013,
-0.007688272278755903,
0.0027471743524074554,
-0.011042114347219467,
-0.0132856396958231... | |
3b115e78b81ff4cb8965b340bb95363fc3667d53 | subsection | 36 | 57 | Analysis | Moreover, we have L.\mathit {ret}_{k} = \mathit {false} as L.\mathit {val}_{k^{\prime }}
= V.\mathit {val}_{k} \ne \mathit {a}_{\ref *{ln:arg}}^{i}.lp:2:alg[lp:2]Algorithm lp:2lem[lp:2]Lemma lp:2thm[lp:2]Theorem lp:2ln[lp:2]Line lp:2tab[lp:2]Table lp:2cor[lp:2]Corollary lp:2fig[lp:2]Figure lp:2as[lp:2]Assumption lp:2se... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.0031145622488111258,
0.005584469065070152,
-0.0095210624858737,
-0.009742304682731628,
-0.033018555492162704,
0.009902514517307281,
-0.026976343244314194,
0.006644907873123884,
0.016082050278782845,
0.006732642184942961,
-0.025786209851503372,
-0.027418827638030052,
-0.006583875510841608,... | |
11a9aeea256cda15ca7a5c723e1c63f87843b94e | subsection | 37 | 57 | Analysis | Further, we have L.\mathit {ret}_{k} = \mathit {true} as L.\mathit {val}_{k^{\prime }} = \mathit {a}_{\ref *{ln:arg}}^{i}.lp:3a:alg[lp:3a]Algorithm lp:3alem[lp:3a]Lemma lp:3athm[lp:3a]Theorem lp:3aln[lp:3a]Line lp:3atab[lp:3a]Table lp:3acor[lp:3a]Corollary lp:3afig[lp:3a]Figure lp:3aas[lp:3a]Assumption lp:3asec[lp:3a]S... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
0.008751099929213524,
0.016723984852433205,
-0.0229954794049263,
-0.03884969651699066,
-0.02513175643980503,
0.034912846982479095,
0.008285697549581528,
0.04019249975681305,
-0.004604436922818422,
0.009994717314839363,
-0.011055225506424904,
-0.007316743955016136,
0.016846058890223503,
0.0... | |
35ead3fa7030458e90d4923b3cbd0df42f34f064 | subsection | 38 | 57 | Analysis | Using
definition of lp:3a:alg[lp:3a]Algorithm lp:3alem[lp:3a]Lemma lp:3athm[lp:3a]Theorem lp:3aln[lp:3a]Line lp:3atab[lp:3a]Table lp:3acor[lp:3a]Corollary lp:3afig[lp:3a]Figure lp:3aas[lp:3a]Assumption lp:3asec[lp:3a]Section lp:3adef[lp:3a]Definition lp:3alp[lp:3a]Case lp:3a[ss], \mathit {LP}_k is the first point when ... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
0.0024834161158651114,
0.033325839787721634,
-0.02989254705607891,
-0.016693439334630966,
-0.031235346570611,
0.03463812172412872,
0.01976051554083824,
0.013344069942831993,
0.00007587818254251033,
-0.005985375959426165,
-0.018280383199453354,
0.018478751182556152,
0.02256818674504757,
0.0... | |
a61bf30155382ab497feecddf21406c4cc4ba192 | subsection | 39 | 57 | Analysis | Using definition of
lp:3a:alg[lp:3a]Algorithm lp:3alem[lp:3a]Lemma lp:3athm[lp:3a]Theorem lp:3aln[lp:3a]Line lp:3atab[lp:3a]Table lp:3acor[lp:3a]Corollary lp:3afig[lp:3a]Figure lp:3aas[lp:3a]Assumption lp:3asec[lp:3a]Section lp:3adef[lp:3a]Definition lp:3alp[lp:3a]Case lp:3a[ss], it also holds that \mathit {a}_{\ref *{... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.028208086267113686,
0.029581114649772644,
-0.037926070392131805,
-0.04122133553028107,
-0.037804022431373596,
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0.013806554488837719,
-0.0012795852962881327,
0.02082424983382225,
... | |
b6290f626e2bfebe7c3f84ccf78b6e182c209bb5 | subsection | 40 | 57 | Analysis | As \mathit {seq}_{\ref *{ln:prd}}^{j} = V.\mathit {seq}_{k} = V.\mathit {seq}_{\ref *{ln:rdval}}^{i} + 2, it is true
that some process i^{\prime } has V.\mathit {seq}_{\ref *{ln:rdval}}^{i^{\prime }} = V.\mathit {seq}_{\ref *{ln:rdval}}^{i} and that the process
executed ln:comp:alg[ln:comp]Algorithm ln:complem[ln:comp]... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.0764242634177208,
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0.021166225895285606,
-0.027712952345609665,
-0.03613671287894249,
-0.0016280978452414274,
... | |
d0e8310145d0d6d750ae9d910a413027ab933626 | subsection | 41 | 57 | Analysis | So, we have
\mathit {val}_{\ref *{ln:ard}}^{j} = \mathit {b}_{\ref *{ln:arg}}^{i} and that V.\mathit {val}_{k} = \mathit {b}_{\ref *{ln:arg}}^{i} as well. Because
\mathit {a}_{\ref *{ln:arg}}^{i} = L.\mathit {val}_{k^{\prime }} as shown before, we also have L.\mathit {ret}_{k} = \mathit {true}.lp:3b:alg[lp:3b]Algorithm... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
0.016569579020142555,
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0.002240936504676938,
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-0.0011004428379237652,
-0.009482498280704021,
... | |
a0e0c52f102733a980e1c717bbebadba7915c451 | subsection | 42 | 57 | Analysis | As process j wrote
V.\mathit {seq}_{k^{\prime }} to V.\mathit {seq}, we have V.\mathit {seq}_{\ref *{ln:rdval}}^{j} = V.\mathit {seq}_{k^{\prime }} - 2 as well. So,
we have V.\mathit {val}_{\ref *{ln:rdval}}^{i} = V.\mathit {val}_{\ref *{ln:rdval}}^{j} using lem:vseqeqthenvvaleq:alg[lem:vseqeqthenvvaleq]Algorithm lem:v... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.028864020481705666,
0.028208019211888313,
-0.02054959535598755,
0.004500468261539936,
-0.011357961222529411,
0.023249877616763115,
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0.006113771814852953,
-0.01955796778202057,
-0.01971052587032318,
-0.010045960545539856,
... | |
b5192782a7a41f8e1ed6d92afc71ab8abe2708bc | subsection | 43 | 57 | Analysis | Using
definition of lp:3a:alg[lp:3a]Algorithm lp:3alem[lp:3a]Lemma lp:3athm[lp:3a]Theorem lp:3aln[lp:3a]Line lp:3atab[lp:3a]Table lp:3acor[lp:3a]Corollary lp:3afig[lp:3a]Figure lp:3aas[lp:3a]Assumption lp:3asec[lp:3a]Section lp:3adef[lp:3a]Definition lp:3alp[lp:3a]Case lp:3a[ss] and lp:3b:alg[lp:3b]Algorithm lp:3blem[l... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
0.0177486389875412,
0.036351900547742844,
-0.057412195950746536,
-0.020678767934441566,
-0.042608942836523056,
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0.01175866462290287,
-0.00814941804856062,
-0.016405664384365082,
0.012300432659685612,
0.009606852196156979,
0.028935013338923454,
0.02... | |
deb24a0f2b9cfc6bdc7d2f1280b0aaf208deaf1b | subsection | 44 | 57 | Analysis | So, we have
V.\mathit {val}_{k^{\prime }} = V.\mathit {val}_{k} and thus L.\mathit {val}_{k} = V.\mathit {val}_{k}. Also, we have
L.\mathit {ret}_{k} = \mathit {false} as \mathit {a}_{\ref *{ln:arg}}^{i} \ne \mathit {b}_{\ref *{ln:arg}}^{j} = L.\mathit {val}_{k^{\prime }}.If the k^{th} linearization point for k\ge 1 co... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.006908657029271126,
0.01591547019779682,
-0.022431200370192528,
-0.028229743242263794,
-0.008789368905127048,
0.0032159017864614725,
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0.028931671753525734,
-0.010635746642947197,
0.007858550176024437,
-0.02992352657020092,
-0.014351390302181244,
0.01818910986185074,... | |
0913d345e783b36fa4f327cad3c5617816511743 | subsection | 45 | 57 | Analysis | Next, assume that the k^{th}
linearization point is a lp:2:alg[lp:2]Algorithm lp:2lem[lp:2]Lemma lp:2thm[lp:2]Theorem lp:2ln[lp:2]Line lp:2tab[lp:2]Table lp:2cor[lp:2]Corollary lp:2fig[lp:2]Figure lp:2as[lp:2]Assumption lp:2sec[lp:2]Section lp:2def[lp:2]Definition lp:2lp[lp:2]Case lp:2[ss] point. Then, the value return... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.007813269272446632,
0.01033121719956398,
-0.013497727923095226,
-0.02533208392560482,
-0.030352720990777016,
0.006550480145961046,
-0.008225297555327415,
0.015214511193335056,
-0.00399438152089715,
0.011361286975443363,
-0.032290779054164886,
0.021196546033024788,
0.022981999441981316,
... | |
0fbeaf934467071dc32cb8ba8bdf941b26156abc | subsection | 46 | 57 | Analysis | Say that the process j executes the operation at the linearization
point. As \mathit {pid}_{\ref *{ln:prd}}^{j} = i by definition of lp:3a:alg[lp:3a]Algorithm lp:3alem[lp:3a]Lemma lp:3athm[lp:3a]Theorem lp:3aln[lp:3a]Line lp:3atab[lp:3a]Table lp:3acor[lp:3a]Corollary lp:3afig[lp:3a]Figure lp:3aas[lp:3a]Assumption lp:3a... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.0495426207780838,
0.008493674919009209,
-0.032265279442071915,
-0.005418247077614069,
-0.02202402986586094,
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0.0476500503718853,
-0.0019746075849980116,
-0.007883167825639248,
-0.028144359588623047,
-0.027320176362991333,
0.012706170789897442,
... | |
6533e46ee8435fa2a575564e0242fd8251567845 | subsection | 47 | 57 | Analysis | So, the call returns \mathit {true} which is same as the value of
L.\mathit {ret}_{k} given by lem:sim:alg[lem:sim]Algorithm lem:simlem[lem:sim]Lemma lem:simthm[lem:sim]Theorem lem:simln[lem:sim]Line lem:simtab[lem:sim]Table lem:simcor[lem:sim]Corollary lem:simfig[lem:sim]Figure lem:simas[lem:sim]Assumption lem:simsec[... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.03122803010046482,
0.014385946094989777,
-0.023356186226010323,
-0.02364603988826275,
-0.024393560364842415,
0.010953456163406372,
-0.017940480262041092,
0.02636152133345604,
-0.013989302329719067,
0.019328732043504715,
-0.025995388627052307,
-0.00033371426980011165,
0.018077779561281204,... | |
d33c4379a507c2bea457d7ea6184ebbf5c7f4ae2 | subsection | 48 | 57 | Analysis | Now, we consider three
cases depending on the relation between \mathit {seq}_{\ref *{ln:prd}}^{j} and V.\mathit {seq}_{k}.
First, consider that \mathit {seq}_{\ref *{ln:prd}}^{j} > V.\mathit {seq}_{k}. As \mathit {pid}_{\ref *{ln:prd}}^{j} = i and
\mathit {seq}_{\ref *{ln:prd}}^{j} is even, we have V.\mathit {seq}_{\re... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.06444862484931946,
0.03298719599843025,
-0.02879132330417633,
-0.020216483622789383,
-0.02873029187321663,
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0.006701955106109381,
-0.03371956944465637,
0.005397419445216656,
-0.022611945867538452,
0... | |
859e18dfdbcb2272ef6b75ad01189b2fb75fdd4f | subsection | 49 | 57 | Analysis | Using definition of
lp:3b:alg[lp:3b]Algorithm lp:3blem[lp:3b]Lemma lp:3bthm[lp:3b]Theorem lp:3bln[lp:3b]Line lp:3btab[lp:3b]Table lp:3bcor[lp:3b]Corollary lp:3bfig[lp:3b]Figure lp:3bas[lp:3b]Assumption lp:3bsec[lp:3b]Section lp:3bdef[lp:3b]Definition lp:3blp[lp:3b]Case lp:3b[ss], there is a process h so that \mathit {p... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.013895489275455475,
0.005711572710424662,
-0.0597941018640995,
0.006810392253100872,
-0.020175548270344734,
0.03320876881480217,
0.010888997465372086,
0.0027050802018493414,
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-0.003990851808339357,
-0.04001534730195999,
-0.0006495626294054091,
0.006421227008104324,
... | |
bc23e384670dae52937540c8121c2c2d78806ed0 | subsection | 50 | 57 | Analysis | As \mathit {pid}_{\ref *{ln:prd}}^{j} = i and
\mathit {seq}_{\ref *{ln:prd}}^{j} is even, we have V.\mathit {seq}_{\ref *{ln:rdval}}^{i} = \mathit {seq}_{\ref *{ln:prd}}^{j} - 2 using
lem:rlookup:alg[lem:rlookup]Algorithm lem:rlookuplem[lem:rlookup]Lemma lem:rlookupthm[lem:rlookup]Theorem lem:rlookupln[lem:rlookup]Line... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.08125120401382446,
0.018283046782016754,
-0.03256763145327568,
0.003159090643748641,
-0.031774044036865234,
0.0143151069059968,
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0.027043037116527557,
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0.003948863595724106,
-0.03513152897357941,
-0.018084648996591568,
-0.0366576611995697,
0.02... | |
c9bc57529500691d6a0987322ba085e40391a153 | subsection | 51 | 57 | Analysis | Thus, the process i returns \mathit {false} for lp:3b:alg[lp:3b]Algorithm lp:3blem[lp:3b]Lemma lp:3bthm[lp:3b]Theorem lp:3bln[lp:3b]Line lp:3btab[lp:3b]Table lp:3bcor[lp:3b]Corollary lp:3bfig[lp:3b]Figure lp:3bas[lp:3b]Assumption lp:3bsec[lp:3b]Section lp:3bdef[lp:3b]Definition lp:3blp[lp:3b]Case lp:3b[ss] which matche... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.005796188488602638,
0.002024277811869979,
-0.033792655915021896,
-0.014522905461490154,
-0.003542963182553649,
-0.004907108843326569,
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0.016240011900663376,
-0.026802126318216324,
0.018041066825389862,
-0.011218429543077946,
0.012195272371172905,
0.025184229016304016,... | |
da04b68375bf9e1cc8a2f01dca0f12334aab8396 | subsection | 52 | 57 | Analysis | We conclude that the compare-and-swap function as given by
alg:sim:alg[alg:sim]Algorithm alg:simlem[alg:sim]Lemma alg:simthm[alg:sim]Theorem alg:simln[alg:sim]Line alg:simtab[alg:sim]Table alg:simcor[alg:sim]Corollary alg:simfig[alg:sim]Figure alg:simas[alg:sim]Assumption alg:simsec[alg:sim]Section alg:simdef[alg:sim]D... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
0.005106783006340265,
0.0032437036279588938,
-0.013729962520301342,
-0.03151789307594299,
-0.019877934828400612,
0.01783369481563568,
0.005354685243219137,
0.024179989472031593,
0.002564833266660571,
0.020396621897816658,
0.010221194475889206,
-0.021510275080800056,
-0.009366885758936405,
... | |
35697b32bed424d22cb398d2cd64779cdaf5427f | subsection | 53 | 57 | Analysis | Then, we have
V.\mathit {val}_{k} = V.\mathit {val}_{k^{\prime }} using lem:vvalmodatlp:alg[lem:vvalmodatlp]Algorithm lem:vvalmodatlplem[lem:vvalmodatlp]Lemma lem:vvalmodatlpthm[lem:vvalmodatlp]Theorem lem:vvalmodatlpln[lem:vvalmodatlp]Line lem:vvalmodatlptab[lem:vvalmodatlp]Table lem:vvalmodatlpcor[lem:vvalmodatlp]Cor... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.004540338646620512,
-0.011942235752940178,
-0.005883363075554371,
-0.01127835363149643,
-0.010820504277944565,
0.010210039094090462,
0.023151911795139313,
0.014620653353631496,
0.022877203300595284,
0.020832141861319542,
-0.010652626864612103,
-0.015574506483972073,
0.009607204236090183,
... | |
41eda5381b2ccba08df0fdaa1d700a5ba6a82624 | subsection | 54 | 57 | Consensus Numbers | In this section, we prove that each of the max-write and the half-max primitives has consensus
number one. Note that these are two separate claims. One, that it is impossible to solve consensus
for two processes using read-write registers and registers that support the max-write and read
operation. Second, that it is i... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.030357401818037033,
0.03331686556339264,
-0.05580269917845726,
0.029716692864894867,
-0.002400751691311598,
-0.020304372534155846,
0.054277203977108,
0.018260205164551735,
0.050249889492988586,
0.009038573130965233,
0.0050875344313681126,
-0.02655891329050064,
0.04216475412249565,
-0.00... | |
348452fd0a5c0a9f70786d2e631e864075721201 | subsection | 55 | 57 | Consensus Numbers | W.l.o.g., assume that b \ge a. Then, there are following two
cases.
Operation s_b does not modify the register R. Thus, operation s_a will also
leave it unchanged as b \ge a. Also, the contents of R in C_1 s_a is same as in
C_0 because s_b did not modify R by assumption. So, the configuration C_1 s_a is
indistinguisha... | {
"cite_spans": []
} | 1802.03844 | Reducing Compare-and-Swap to Consensus Number One Primitives | [
"Pankaj Khanchandani",
"Roger Wattenhofer"
] | [
"cs.DS",
"cs.DC"
] | 2,018 | en | Computer Science | [
-0.023240311071276665,
0.028337005525827408,
-0.061038244515657425,
0.015023037791252136,
-0.0012484228936955333,
-0.026627933606505394,
0.041872236877679825,
0.04989876598119736,
0.05401884764432907,
0.009689821861684322,
-0.0029565400909632444,
-0.037538520991802216,
0.04171964153647423,
... | |
4557b54763774086dd75aba101c325a6dc94ce7a | subsection | 56 | 57 | Conclusion | The algorithm that we presented simulates a single compare-and-swap register using O(n) registers
that support the half-max, max-write, read and write primitives. If m compare-and-swap registers are to be
simulated, then a straightforward approach requires O(mn) registers. However, we can improve
this if we observe tha... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1145/277697.277735",
"end": 2576,
"openalex_id": "https://openalex.org/W1998458999",
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