Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We pres... more Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We present a randomized Las Vegas dynamic connectivity data structure with O(log n(log log n) 2) amortized expected update time and O(log n/ log log log n) worst case query time, which comes very close to the cell probe lower bounds of Pǎtraşcu and Demaine (2006) and Pǎtraşcu and Thorup (2011).
Since the mid-1980s it has been known that Byzantine Agreement can be solved with probability 1 a... more Since the mid-1980s it has been known that Byzantine Agreement can be solved with probability 1 asynchronously, even against an omniscient, computationally unbounded adversary that can adaptively corrupt up to f < n/3 parties. Moreover, the problem is insoluble with f ≥ n/3 corruptions. However, Bracha's [Bra87] 1984 protocol (see also Ben-Or [Ben83]) achieved f < n/3 resilience at the cost of exponential expected latency 2 Θ(n) , a bound that has never been improved in this model with f = ⌊(n − 1)/3⌋ corruptions. In this paper we prove that Byzantine Agreement in the asynchronous, full information model can be solved with probability 1 against an adaptive adversary that can corrupt f < n/3 parties, while incurring only polynomial latency with high probability. Our protocol follows earlier polynomial latency protocols of King and Saia [KS16, KS18] and Huang, Pettie, and Zhu [HPZ22], which had suboptimal resilience, namely f ≈ n/10 9 [KS16, KS18] and f < n/4 [HPZ22], respectively. Resilience f = (n−1)/3 is uniquely difficult as this is the point at which the influence of the Byzantine and honest players are of roughly equal strength. The core technical problem we solve is to design a collective coin-flipping protocol that eventually lets us flip a coin with an unambiguous outcome. In the beginning the influence of the Byzantine players is too powerful to overcome and they can essentially fix the coin's behavior at will. We guarantee that after just a polynomial number of executions of the coin-flipping protocol, either (a) the Byzantine players fail to fix the behavior of the coin (thereby ending the game) or (b) we can "blacklist" players such that the blacklisting rate for Byzantine players is at least as large as the blacklisting rate for good players. The blacklisting criterion is based on a simple statistical test of fraud detection. if any corrupt player initiates a broadcast, then either all good players accept the same value v, and only v, or all good players accept nothing. See [Bra87] for details of this primitive. Validation. The Reliable-Broadcast primitive allows us to assume that all relevant communication is public, via broadcasts. Fix any protocol P based on broadcasts. Informally, a player p validates a message m originating from q if p has already accepted and validated a set of broadcasts that, were they to be received by q, would have caused q to make a suitable state transition according to P and broadcast m. See [Bra87] for details of validation. The reliable broadcast primitive prevents the adversary from sending conflicting messages to different players, or convincing one player to accept a broadcast and another not to. The validation mechanism prevents it from making state transitions logically inconsistent with the protocol P. Note, however, that in general P is probabilistic and validation permits a series of transitions that are logically possible but statistically unlikely. In summary, the adversary is characterized by the following powers. Full Information & Scheduling. The adversary knows the internal state of all players and controls the order in which messages are delivered. It may delay messages arbitrarily. Corruption & Coin Flipping. The adversary may adaptively corrupt up to f players as the execution of the protocol progresses. Once corrupted, a player continues to follow protocol , except the adversary now chooses the outcomes of all of its coin flips. Algorithm 1 Bracha-Agreement() from the perspective of player p Require: v p ∈ {−1, 1}. 1: loop 2: Reliable-Broadcast v p and wait until n − f messages are validated from some set of players S p. set v p ← sgn(q∈Sp v q). ⊲ sgn(x) = 1 if x ≥ 0 and −1 otherwise. 3: Reliable-Broadcast v p and wait until n − f messages are validated. if more than n/2 messages have some value v * then set v p ← v * , otherwise set v p ← ⊥.
Graphs are important and useful mathematical representations that can be used to model different ... more Graphs are important and useful mathematical representations that can be used to model different objects in various research disciplines. For example, graphs are used for representing structures of chemical compounds in mathematical chemistry, representing phylogenetic trees in evolutionary biology, the small world concepts in social networks, and topology of a computer network. There are several properties on a graph for us to study, such as connectivity, reachability and distances. In this thesis, we study three fun algorithmic challenges and apply graph algorithms and data structures to improve them. The first challenge is to design a data structure that solves the fully dynamic graph connectivity problem. For any sequence of edge insertions and deletions to a graph, our data structure performs updates in amortized randomized O(log n(log log n) 2) time, improving Thorup's result by a O(log log n) factor for the first time since 2000. Our result leaves a small O((log log n) 2) gap to the Ω(log n) lower bound given by Pǎtraşcu and Demaine in 2006. The second topic we study includes various structures on graphs concerning distances and reachability, such as spanners, emulators, hopsets and shortcutting sets. A (β,)-hopset is a set of weighted edges added to an undirected weighted graph so that distances between any pair of vertices can then be (1 +)-approximated using a path with at most β edges (hops). We applied Thorup and Zwick's sublinear additive emulator construction to build a hopset. The construction, parameterized by k > 0, has an optimal tradeoff between the hopset size O(n 1+ 1 2 k+1 −1), the hop bound β = O(k/) k , and the approximation ratio (1 +). This optimal tradeoff is established by the work of Abboud, Bodwin, and Pettie. A shortcutting set is a reachability-preserving set of directed edges whose addition to a directed unweighted graph reduces the diameter. We show that there is a Ω(n 1/6) diameter lower bound on any shortcutting set of O(n) size. The lower bound techniques can also be applied to undirected weighted graphs. In particular, we provide a +Ω(n 1/11) additive distortion lower bound for O(n)-size spanners and a +Ω(n 1/18) additive distortion lower bound for O(n)-size emulators. The third problem we solve is the asynchronous Byzantine agreement problem. In distributed computing, reaching consensus among all n processes is a challengx ing task, especially when a subset of processes are secretly corrupted and malicious behavior may occur. We provide a protocol that solves Byzantine agreement in an asynchronous distributed network model under perhaps the most challenging assumptions. Although we assume a fully-connected authenticated network ensuring that once a message is sent it will be delivered, the protocol has to work against a full-information (no cryptography for hiding information) and adaptive (corrupting processes over time) adversary. Most Byzantine agreement protocols are developed under weaker adversary models. We propose the first polynomial time protocol that allows up to (n − 1)/4 faulty processes, improving King and Saia's 2013 result from (1.14 × 10 −9)n faulty processes. Unlike the first two topics in this thesis, the asynchronous Byzantine agreement problem is not obviously related to graphs. But interestingly, one building block in our protocol is a fractional matching algorithm with a continuous Lipschitz guarantee. xi
Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing
It has been known since the early 1980s that Byzantine Agreement in the full information, asynchr... more It has been known since the early 1980s that Byzantine Agreement in the full information, asynchronous model is impossible to solve deterministically against even one crash fault [FLP 1985], but that it can be solved with probability 1 [Ben-Or 1983], even against an adversary that controls the scheduling of all messages and corrupts up to < /3 players [Bracha 1987]. The main downside of [Ben-Or 1983, Bracha 1987] is that they terminate with 2 Θ() latency in expectation whenever = Θ(). King and Saia [KS 2016, KS 2018] developed a polynomial protocol (polynomial latency, polynomial local computation) that is resilient to < (1.14 × 10 −9) Byzantine faults. The new idea in their protocol is to detect-and blacklist-coalitions of likely-bad players by analyzing the deviations of random variables generated by those players over many rounds. In this work we design a simple collective coin-flipping protocol such that if any coalition of faulty players repeatedly does not follow protocol, then they will eventually be detected by one of two simple statistical tests. Using this coin-flipping protocol, we solve Byzantine Agreement in polynomial latency, even in the presence of up to < /4 Byzantine faults. This comes close to the < /3 upper bound on the maximum number of faults [LSP 1982, BT 1985, FLM 1986].
We prove better lower bounds on additive spanners and emulators, which are lossy compression sche... more We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We prove that any O(n)-size shortcut set cannot bring the diameter below Ω(n 1/6), and that any O(m)-size shortcut set cannot bring it below Ω(n 1/11). These improve Hesse's [16] lower bound of Ω(n 1/17). By combining these constructions with Abboud and Bodwin's [1] edge-splitting technique, we get additive stretch lower bounds of +Ω(n 1/11) for O(n)-size spanners and +Ω(n 1/18) for O(n)-size emulators. These improve Abboud and Bodwin's +Ω(n 1/22) lower bounds for both spanners and emulators. CCS Concepts: • Theory of computation → Sparsification and spanners.
A (β,)-hopset is, informally, a weighted edge set that, when added to a graph, allows one to get ... more A (β,)-hopset is, informally, a weighted edge set that, when added to a graph, allows one to get from point a to point b using a path with at most β edges ("hops") and length (1 +) dist(a, b). In this paper we observe that Thorup and Zwick's sublinear additive emulators are also actually (O(k/) k ,)hopsets for every > 0, and that with a small change to the Thorup-Zwick construction, the size of the hopset can be made O(n 1+ 1 2 k+1 −1). As corollaries, we also shave "k" factors off the size of Thorup and Zwick's [20] sublinear additive emulators and the sparsest known (1 + , O(k/) k−1)-spanners, due to Abboud, Bodwin, and Pettie [1]. * Supported by NSF Grants CCF-1514383 and CCF-1637546. 1 Note that setting κ = 2 k+1 − 1 in the Elkin-Neiman construction gives β = O(k/) k , where log κ = log κ = k. Thus, saving any δ in the exponent of the hopset increases β significantly. In general, the statement of [9] obscures the nature of the tradeoff: there are not distinct tradeoffs for each κ ∈ {1, 2, 3,. . .}, but only for κ ∈ {1, 3, 7,. .. , 2 k+1 − 1,. . .}.
Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, 2017
Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We pres... more Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We present a new randomized dynamic connectivity structure with O(log n(log log n) 2) amortized expected update time and O(log n/ log log log n) query time, which comes within an O((log log n) 2) factor of a lower bound due to Pǎtraşcu and Demaine. The new structure is based on a dynamic connectivity algorithm proposed by Thorup in an extended abstract at STOC 2000, which left out some important details.
Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We pres... more Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We present a randomized Las Vegas dynamic connectivity data structure with O(log n(log log n) 2) amortized expected update time and O(log n/ log log log n) worst case query time, which comes very close to the cell probe lower bounds of Pǎtraşcu and Demaine (2006) and Pǎtraşcu and Thorup (2011).
Since the mid-1980s it has been known that Byzantine Agreement can be solved with probability 1 a... more Since the mid-1980s it has been known that Byzantine Agreement can be solved with probability 1 asynchronously, even against an omniscient, computationally unbounded adversary that can adaptively corrupt up to f < n/3 parties. Moreover, the problem is insoluble with f ≥ n/3 corruptions. However, Bracha's [Bra87] 1984 protocol (see also Ben-Or [Ben83]) achieved f < n/3 resilience at the cost of exponential expected latency 2 Θ(n) , a bound that has never been improved in this model with f = ⌊(n − 1)/3⌋ corruptions. In this paper we prove that Byzantine Agreement in the asynchronous, full information model can be solved with probability 1 against an adaptive adversary that can corrupt f < n/3 parties, while incurring only polynomial latency with high probability. Our protocol follows earlier polynomial latency protocols of King and Saia [KS16, KS18] and Huang, Pettie, and Zhu [HPZ22], which had suboptimal resilience, namely f ≈ n/10 9 [KS16, KS18] and f < n/4 [HPZ22], respectively. Resilience f = (n−1)/3 is uniquely difficult as this is the point at which the influence of the Byzantine and honest players are of roughly equal strength. The core technical problem we solve is to design a collective coin-flipping protocol that eventually lets us flip a coin with an unambiguous outcome. In the beginning the influence of the Byzantine players is too powerful to overcome and they can essentially fix the coin's behavior at will. We guarantee that after just a polynomial number of executions of the coin-flipping protocol, either (a) the Byzantine players fail to fix the behavior of the coin (thereby ending the game) or (b) we can "blacklist" players such that the blacklisting rate for Byzantine players is at least as large as the blacklisting rate for good players. The blacklisting criterion is based on a simple statistical test of fraud detection. if any corrupt player initiates a broadcast, then either all good players accept the same value v, and only v, or all good players accept nothing. See [Bra87] for details of this primitive. Validation. The Reliable-Broadcast primitive allows us to assume that all relevant communication is public, via broadcasts. Fix any protocol P based on broadcasts. Informally, a player p validates a message m originating from q if p has already accepted and validated a set of broadcasts that, were they to be received by q, would have caused q to make a suitable state transition according to P and broadcast m. See [Bra87] for details of validation. The reliable broadcast primitive prevents the adversary from sending conflicting messages to different players, or convincing one player to accept a broadcast and another not to. The validation mechanism prevents it from making state transitions logically inconsistent with the protocol P. Note, however, that in general P is probabilistic and validation permits a series of transitions that are logically possible but statistically unlikely. In summary, the adversary is characterized by the following powers. Full Information & Scheduling. The adversary knows the internal state of all players and controls the order in which messages are delivered. It may delay messages arbitrarily. Corruption & Coin Flipping. The adversary may adaptively corrupt up to f players as the execution of the protocol progresses. Once corrupted, a player continues to follow protocol , except the adversary now chooses the outcomes of all of its coin flips. Algorithm 1 Bracha-Agreement() from the perspective of player p Require: v p ∈ {−1, 1}. 1: loop 2: Reliable-Broadcast v p and wait until n − f messages are validated from some set of players S p. set v p ← sgn(q∈Sp v q). ⊲ sgn(x) = 1 if x ≥ 0 and −1 otherwise. 3: Reliable-Broadcast v p and wait until n − f messages are validated. if more than n/2 messages have some value v * then set v p ← v * , otherwise set v p ← ⊥.
Graphs are important and useful mathematical representations that can be used to model different ... more Graphs are important and useful mathematical representations that can be used to model different objects in various research disciplines. For example, graphs are used for representing structures of chemical compounds in mathematical chemistry, representing phylogenetic trees in evolutionary biology, the small world concepts in social networks, and topology of a computer network. There are several properties on a graph for us to study, such as connectivity, reachability and distances. In this thesis, we study three fun algorithmic challenges and apply graph algorithms and data structures to improve them. The first challenge is to design a data structure that solves the fully dynamic graph connectivity problem. For any sequence of edge insertions and deletions to a graph, our data structure performs updates in amortized randomized O(log n(log log n) 2) time, improving Thorup's result by a O(log log n) factor for the first time since 2000. Our result leaves a small O((log log n) 2) gap to the Ω(log n) lower bound given by Pǎtraşcu and Demaine in 2006. The second topic we study includes various structures on graphs concerning distances and reachability, such as spanners, emulators, hopsets and shortcutting sets. A (β,)-hopset is a set of weighted edges added to an undirected weighted graph so that distances between any pair of vertices can then be (1 +)-approximated using a path with at most β edges (hops). We applied Thorup and Zwick's sublinear additive emulator construction to build a hopset. The construction, parameterized by k > 0, has an optimal tradeoff between the hopset size O(n 1+ 1 2 k+1 −1), the hop bound β = O(k/) k , and the approximation ratio (1 +). This optimal tradeoff is established by the work of Abboud, Bodwin, and Pettie. A shortcutting set is a reachability-preserving set of directed edges whose addition to a directed unweighted graph reduces the diameter. We show that there is a Ω(n 1/6) diameter lower bound on any shortcutting set of O(n) size. The lower bound techniques can also be applied to undirected weighted graphs. In particular, we provide a +Ω(n 1/11) additive distortion lower bound for O(n)-size spanners and a +Ω(n 1/18) additive distortion lower bound for O(n)-size emulators. The third problem we solve is the asynchronous Byzantine agreement problem. In distributed computing, reaching consensus among all n processes is a challengx ing task, especially when a subset of processes are secretly corrupted and malicious behavior may occur. We provide a protocol that solves Byzantine agreement in an asynchronous distributed network model under perhaps the most challenging assumptions. Although we assume a fully-connected authenticated network ensuring that once a message is sent it will be delivered, the protocol has to work against a full-information (no cryptography for hiding information) and adaptive (corrupting processes over time) adversary. Most Byzantine agreement protocols are developed under weaker adversary models. We propose the first polynomial time protocol that allows up to (n − 1)/4 faulty processes, improving King and Saia's 2013 result from (1.14 × 10 −9)n faulty processes. Unlike the first two topics in this thesis, the asynchronous Byzantine agreement problem is not obviously related to graphs. But interestingly, one building block in our protocol is a fractional matching algorithm with a continuous Lipschitz guarantee. xi
Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing
It has been known since the early 1980s that Byzantine Agreement in the full information, asynchr... more It has been known since the early 1980s that Byzantine Agreement in the full information, asynchronous model is impossible to solve deterministically against even one crash fault [FLP 1985], but that it can be solved with probability 1 [Ben-Or 1983], even against an adversary that controls the scheduling of all messages and corrupts up to < /3 players [Bracha 1987]. The main downside of [Ben-Or 1983, Bracha 1987] is that they terminate with 2 Θ() latency in expectation whenever = Θ(). King and Saia [KS 2016, KS 2018] developed a polynomial protocol (polynomial latency, polynomial local computation) that is resilient to < (1.14 × 10 −9) Byzantine faults. The new idea in their protocol is to detect-and blacklist-coalitions of likely-bad players by analyzing the deviations of random variables generated by those players over many rounds. In this work we design a simple collective coin-flipping protocol such that if any coalition of faulty players repeatedly does not follow protocol, then they will eventually be detected by one of two simple statistical tests. Using this coin-flipping protocol, we solve Byzantine Agreement in polynomial latency, even in the presence of up to < /4 Byzantine faults. This comes close to the < /3 upper bound on the maximum number of faults [LSP 1982, BT 1985, FLM 1986].
We prove better lower bounds on additive spanners and emulators, which are lossy compression sche... more We prove better lower bounds on additive spanners and emulators, which are lossy compression schemes for undirected graphs, as well as lower bounds on shortcut sets, which reduce the diameter of directed graphs. We prove that any O(n)-size shortcut set cannot bring the diameter below Ω(n 1/6), and that any O(m)-size shortcut set cannot bring it below Ω(n 1/11). These improve Hesse's [16] lower bound of Ω(n 1/17). By combining these constructions with Abboud and Bodwin's [1] edge-splitting technique, we get additive stretch lower bounds of +Ω(n 1/11) for O(n)-size spanners and +Ω(n 1/18) for O(n)-size emulators. These improve Abboud and Bodwin's +Ω(n 1/22) lower bounds for both spanners and emulators. CCS Concepts: • Theory of computation → Sparsification and spanners.
A (β,)-hopset is, informally, a weighted edge set that, when added to a graph, allows one to get ... more A (β,)-hopset is, informally, a weighted edge set that, when added to a graph, allows one to get from point a to point b using a path with at most β edges ("hops") and length (1 +) dist(a, b). In this paper we observe that Thorup and Zwick's sublinear additive emulators are also actually (O(k/) k ,)hopsets for every > 0, and that with a small change to the Thorup-Zwick construction, the size of the hopset can be made O(n 1+ 1 2 k+1 −1). As corollaries, we also shave "k" factors off the size of Thorup and Zwick's [20] sublinear additive emulators and the sparsest known (1 + , O(k/) k−1)-spanners, due to Abboud, Bodwin, and Pettie [1]. * Supported by NSF Grants CCF-1514383 and CCF-1637546. 1 Note that setting κ = 2 k+1 − 1 in the Elkin-Neiman construction gives β = O(k/) k , where log κ = log κ = k. Thus, saving any δ in the exponent of the hopset increases β significantly. In general, the statement of [9] obscures the nature of the tradeoff: there are not distinct tradeoffs for each κ ∈ {1, 2, 3,. . .}, but only for κ ∈ {1, 3, 7,. .. , 2 k+1 − 1,. . .}.
Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, 2017
Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We pres... more Dynamic connectivity is one of the most fundamental problems in dynamic graph algorithms. We present a new randomized dynamic connectivity structure with O(log n(log log n) 2) amortized expected update time and O(log n/ log log log n) query time, which comes within an O((log log n) 2) factor of a lower bound due to Pǎtraşcu and Demaine. The new structure is based on a dynamic connectivity algorithm proposed by Thorup in an extended abstract at STOC 2000, which left out some important details.
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