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Efficient Local Verification of Global Properties via Proof Labeling Schemes

This paper explores the efficiency of local verification mechanisms in distributed systems through proof labeling schemes. We focus on applications such as the 3-coloring problem and self-stabilization, examining how local verification's cost compares to computation itself. The paper discusses the complexity of labeling and verification processes, presents various construction methods, and establishes lower bounds. Key examples include spanning trees in id-based graphs and orientations in anonymous trees. The work highlights the implications for designing algorithms that ensure efficient verification with minimal label sizes.

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Efficient Local Verification of Global Properties via Proof Labeling Schemes

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  1. Proof labeling schemes Amos Korman, Weizmann Shay Kutten, Technion David Peleg, Weizmann

  2. Goals • Local verification of global properties • How expensive is local verification, compared to the computation? Motivation: 3 coloring (this example uses time complexity, while we use a form of communication complexity)

  3. Informal description of model Lw Sw Lv Sv Lu Su f(Gs)=1 ? We add labels to prove that. Cost of verification = label size: capture communication (may be smaller than the node’s state!)

  4. Motivation detection Self stabilization computation Self stabilization by local detection appears in: [Afek, Kutten, Yung, 97], [Awerbuch, Patt-Shamir, Varghese, 91], [Awerbuch, Patt-Shamir, Varghese, Dolev, 94], [Awerbuch, Varghese, 91], [Arora, Gouda, 90], [Aggarwal, Kutten, 93], [Awerbuch, Kutten, Mansour, Patt-Shamir, Varghese, 93], [Dolev, Israeli, Moran, 93 and 97].

  5. This paper: Detection • Basic examples • Impossibility results • Construction methods • Lower bounds

  6. Difference between models Self stabilization: Design, compute and verify : (The computation is designed to be easily verified: enables short labels) This paper: Verify anygiven configuration (modularity) Example: Task: verify disjoint states on a path Design, compute and verify: label size = O(log n) Verify any given configuration: label size = (n)

  7. The model Graph Gswith a state at each node Graph family:  = { Gs} Function: f :  {0,1} Example: f = spanning tree states induce subgraph H eH  endpoint points at e f(Gs) = 1 the subgraph is a spanning tree.

  8. Proof labeling scheme for  and f - Marker M :Input : Gs Output: Label M(v)  vin Gs : Input : NL(v), composed of: - Decoder D Output: 0 or 1 v L(x)  x L(v)  y s L(y) v  z L(z)

  9. Requirements  Gs: f(Gs) = 1   v , D(NM(v)) = 1 f(Gs) = 0   L ,  v  Gs s.t. D(NL(v)) = 0 Complexity measure: Size of proof = max { |M(v)| : Gs  , v  Gs}

  10. Basic Examples: Example 1: Spanning Tree in Id-based Graphs Lemma: proof size is (log n). Upper bound : Given Gs s.t. f(Gs)=1 set M(v) = < id(root), distH(v,root) > root This idea was used previously, e.g., in [Arora,Gouda,90], [Aggarwal, Kutten,93], [Awerbuch, Kutten,Mansour, 93], [Afek, Dolev, 02], [Afek, Kutten, Yung, 97], [Dolev, Israeli, Moran, 97]. 0 2 1 3 2

  11. We show a matching lower bound: v v i k path f(p) = 1 Assume M(vi) < ½ log n for every i   Two pairs < M(vi),M(vi+1) > = < M(vk),M(vk+1) > M(vk) M(vi+1) M(vk-1)

  12. Example 2: Orientation in Anonymous Trees Lemma: Proof size = O(1) 0 M(v) = dist(v, root) mod 3 1 D(NL(v)) =1  (a)  u neighbor of v : 2 |L(v)-L(u)| = 1 mod 3 0 (b) L(u)+1 =L(v) mod 3 1  v u Note: the label size (O(1)) is smaller than the state size (O(log n))

  13. What can be proven locally Id based families: every computable property is provable. Anonymous families: question reduces to proving a unique leader. L1 L5 L2 L1 L3 L2 L4 L5 L4 L3 L3 L4 L5 L2 L1

  14. Cost of identities Transition from anonymous to id-based is sometimes possible but may be costly. Path: f = unique states (1 to n) Lemma : proof size = (n) Recall: if the design of the states and the verification are designed together: O(log n).

  15. Construction Methods The distributed method: For given f and , if a satisfying Gs can be computed by a distributed algorithm A of message complexity m, then construct proof labels of length O(m) “documenting” an execution of A. ~

  16. The distributed method (cont.) Formally: Consider f,  and a nondeterministic distributed algorithm A s.t  Gs  s.t f(Gs) = 1  run A’ of A s.t. : 1) A’(G) = Gs 2) # of messages v sends in A’(G) < m 3) Each message < O(log n) bits 4)  run A’, f(A’(G))=1 Lemma: Proof size of f and  is O(m(log m+log n))

  17. The distributed method (cont.) If A is syncronous and : 1) # rounds < p 2) # message of v per round < mp Lemma: Proof size of f and  is min { O(m(log p+log n)), O(pmplog n)} Example : MST Lemma: proof size of MST is O(log2n+log nlog W) Lemma: proof size of MST is (log n+log W)

  18. Construction method 2 : Composition of proofs is • k-independent set and k-clique (log n) • k- s-t vertex connectivity (log n) • k-flow O(klog n) • diameter on trees (log n+log W) • maximum matching on a path (log n+log W) Proof size of

  19. Open problems • A potentially reach field, full with open problems • - Find Lower & upper bounds for known graph problems • - Find a structure, classes? reductions?

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