arXiv subjects

Yi-Jun Chang

Publications and source records attributed to Yi-Jun Chang.

66 records · Page 4Linked to original sources

arXiv2019

Quantum Fast Hitting on Glued Trees Mapped on a Photonic chip

Hitting the exit node from the entrance node faster on a graph is one of the properties that quantum walk algorithms can take advantage of to outperform classical random walk algorithms. Especially, continuous-time quantum walks on central-random glued binary trees have been investigated in theories extensively for their exponentially faster hitting speed over classical random walks. Here, using heralded single photons to represent quantum walkers and waveguide arrays written by femtosecond laser to simulate the theoretical graph, we are able to demonstrate the hitting efficiency of quantum walks with tree depth as high as 16 layers for the first time. Furthermore, we expand the graph's branching rate from 2 to 5, revealing that quantum walks exhibit more superiority over classical random walks as branching rate increases. Our results may shed light on the physical implementation of quantum walk algorithms as well as quantum computation and quantum simulation.

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arXiv2019

Improved Distributed Expander Decomposition and Nearly Optimal Triangle Enumeration

An $(ε,ϕ)$-expander decomposition of a graph $G=(V,E)$ is a clustering of the vertices $V=V_{1}\cup\cdots\cup V_{x}$ such that (1) each cluster $V_{i}$ induces subgraph with conductance at least $ϕ$, and (2) the number of inter-cluster edges is at most $ε|E|$. In this paper, we give an improved distributed expander decomposition. Specifically, we construct an $(ε,ϕ)$-expander decomposition with $ϕ=(ε/\log n)^{2^{O(k)}}$ in $O(n^{2/k}\cdot\text{poly}(1/ϕ,\log n))$ rounds for any $ε\in(0,1)$ and positive integer $k$. For example, a $(0.01,1/\text{poly}\log n)$-expander decomposition can be computed in $O(n^γ)$ rounds, for any arbitrarily small constant $γ>0$. Previously, the algorithm by Chang, Pettie, and Zhang can construct a $(1/6,1/\text{poly}\log n)$-expander decomposition using $\tilde{O}(n^{1-δ})$ rounds for any $δ>0$, with a caveat that the algorithm is allowed to throw away a set of edges into an extra part which forms a subgraph with arboricity at most $n^δ$. Our algorithm does not have this caveat. By slightly modifying the distributed algorithm for routing on expanders by Ghaffari, Kuhn and Su [PODC'17], we obtain a triangle enumeration algorithm using $\tilde{O}(n^{1/3})$ rounds. This matches the lower bound by Izumi and Le Gall [PODC'17] and Pandurangan, Robinson and Scquizzato [SPAA'18] of $\tildeΩ(n^{1/3})$ which holds even in the CONGESTED CLIQUE model. This provides the first non-trivial example for a distributed problem that has essentially the same complexity (up to a polylogarithmic factor) in both CONGEST and CONGESTED CLIQUE. The key technique in our proof is the first distributed approximation algorithm for finding a low conductance cut that is as balanced as possible. Previous distributed sparse cut algorithms do not have this nearly most balanced guarantee.

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arXiv2019

An Optimal Distributed $(Δ+1)$-Coloring Algorithm?

Vertex coloring is one of the classic symmetry breaking problems studied in distributed computing. In this paper we present a new algorithm for $(Δ+1)$-list coloring in the randomized ${\sf LOCAL}$ model running in $O(\mathsf{Det}_{\scriptscriptstyle d}(\text{poly} \log n))$ time, where $\mathsf{Det}_{\scriptscriptstyle d}(n')$ is the deterministic complexity of $(\text{deg}+1)$-list coloring on $n'$-vertex graphs. (In this problem, each $v$ has a palette of size $\text{deg}(v)+1$.) This improves upon a previous randomized algorithm of Harris, Schneider, and Su [STOC'16, JACM'18] with complexity $O(\sqrt{\log Δ} + \log\log n + \mathsf{Det}_{\scriptscriptstyle d}(\text{poly} \log n))$, and, for some range of $Δ$, is much faster than the best known deterministic algorithm of Fraigniaud, Heinrich, and Kosowski [FOCS'16] and Barenboim, Elkin, and Goldenberg [PODC'18], with complexity $O(\sqrt{Δ\log Δ}\log^\ast Δ+ \log^* n)$. Our algorithm "appears to be" optimal, in view of the $Ω(\mathsf{Det}(\text{poly} \log n))$ randomized lower bound due to Chang, Kopelowitz, and Pettie [FOCS'16], where $\mathsf{Det}$ is the deterministic complexity of $(Δ+1)$-list coloring. At present, the best upper bounds on $\mathsf{Det}_{\scriptscriptstyle d}(n')$ and $\mathsf{Det}(n')$ are both $2^{O(\sqrt{\log n'})}$ and use a black box application of network decompositions (Panconesi and Srinivasan [Journal of Algorithms'96]). It is quite possible that the true complexities of both problems are the same, asymptotically, which would imply the randomized optimality of our $(Δ+1)$-list coloring algorithm.

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arXiv2019

Topologically Protected Quantum Entanglement

Quantum entanglement, as the strictly non-classical phenomena, is the kernel of quantum computing and quantum simulation, and has been widely applied ranging from fundamental tests of quantum physics to quantum information processing. The decoherence of quantum states restricts the capability of building quantum simulators and quantum computers in a scalable fashion. Meanwhile, the topological phase is found inherently capable of protecting physical fields from unavoidable fabrication-induced disorder, which inspires the potential application of topological protection on quantum states. Here, we present the first experimental demonstration of topologically protected quantum polarization entangled states on a photonic chip. The process tomography shows that quantum entanglement can be well preserved by the boundary states even when the chip material substantially introduces relative polarization rotation in phase space. Our work links topology, material and quantum physics, opening the door to wide applications of topological enhancement in genuine quantum regime.

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arXiv2019

The distributed complexity of locally checkable problems on paths is decidable

Consider a computer network that consists of a path with $n$ nodes. The nodes are labeled with inputs from a constant-sized set, and the task is to find output labels from a constant-sized set subject to some local constraints---more formally, we have an LCL (locally checkable labeling) problem. How many communication rounds are needed (in the standard LOCAL model of computing) to solve this problem? It is well known that the answer is always either $O(1)$ rounds, or $Θ(\log^* n)$ rounds, or $Θ(n)$ rounds. In this work we show that this question is decidable (albeit PSPACE-hard): we present an algorithm that, given any LCL problem defined on a path, outputs the distributed computational complexity of this problem and the corresponding asymptotically optimal algorithm.

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arXiv2018

The Complexity of $(Δ+ 1)$Coloring inCongested Clique, Massively Parallel Computation,and Centralized Local Computation

We present new randomized algorithms that improve the complexity of the classic $(Δ+1)$-coloring problem, and its generalization $(Δ+1)$-list-coloring, in three well-studied models of distributed, parallel, and centralized computation: Distributed Congested Clique: We present an $O(1)$-round randomized algorithm for $(Δ+1)$-list coloring in the congested clique model of distributed computing. This settles the asymptotic complexity of this problem. It moreover improves upon the $O(\log^\ast Δ)$-round randomized algorithms of Parter and Su [DISC'18] and $O((\log\log Δ)\cdot \log^\ast Δ)$-round randomized algorithm of Parter [ICALP'18]. Massively Parallel Computation: We present a $(Δ+1)$-list coloring algorithm with round complexity $O(\sqrt{\log\log n})$ in the Massively Parallel Computation (MPC) model with strongly sublinear memory per machine. This algorithm uses a memory of $O(n^α)$ per machine, for any desirable constant $α>0$, and a total memory of $\widetilde{O}(m)$, where $m$ is the size of the graph. Notably, this is the first coloring algorithm with sublogarithmic round complexity, in the sublinear memory regime of MPC. For the quasilinear memory regime of MPC, an $O(1)$-round algorithm was given very recently by Assadi et al. [SODA'19]. Centralized Local Computation: We show that $(Δ+1)$-list coloring can be solved with $Δ^{O(1)} \cdot O(\log n)$ query complexity, in the centralized local computation model. The previous state-of-the-art for $(Δ+1)$-list coloring in the centralized local computation model are based on simulation of known LOCAL algorithms.

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arXiv2018

Exponential Separations in the Energy Complexity of Leader Election

Energy is often the most constrained resource for battery-powered wireless devices and the lion's share of energy is often spent on transceiver usage (sending/receiving packets), not on computation. In this paper we study the energy complexity of LeaderElection and ApproximateCounting in several models of wireless radio networks. It turns out that energy complexity is very sensitive to whether the devices can generate random bits and their ability to detect collisions. We consider four collision-detection models: Strong-CD (in which transmitters and listeners detect collisions), Sender-CD and Receiver-CD (in which only transmitters or only listeners detect collisions), and No-CD (in which no one detects collisions.) The take-away message of our results is quite surprising. For randomized LeaderElection algorithms, there is an exponential gap between the energy complexity of Sender-CD and Receiver-CD, and for deterministic LeaderElection algorithms there is another exponential gap, but in the reverse direction. In particular, the randomized energy complexity of LeaderElection is $Θ(\log^* n)$ in Sender-CD but $Θ(\log(\log^* n))$ in Receiver-CD, where $n$ is the (unknown) number of devices. Its deterministic complexity is $Θ(\log N)$ in Receiver-CD but $Θ(\log\log N)$ in Sender-CD, where $N$ is the (known) size of the devices' ID space. There is a tradeoff between time and energy. We give a new upper bound on the time-energy tradeoff curve for randomized LeaderElection and ApproximateCounting. A critical component of this algorithm is a new deterministic LeaderElection algorithm for dense instances, when $n=Θ(N)$, with inverse-Ackermann-type ($O(α(N))$) energy complexity.

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arXiv2018

Distributed Triangle Detection via Expander Decomposition

We present improved distributed algorithms for triangle detection and its variants in the CONGEST model. We show that Triangle Detection, Counting, and Enumeration can be solved in $\tilde{O}(n^{1/2})$ rounds. In contrast, the previous state-of-the-art bounds for Triangle Detection and Enumeration were $\tilde{O}(n^{2/3})$ and $\tilde{O}(n^{3/4})$, respectively, due to Izumi and LeGall (PODC 2017). The main technical novelty in this work is a distributed graph partitioning algorithm. We show that in $\tilde{O}(n^{1-δ})$ rounds we can partition the edge set of the network $G=(V,E)$ into three parts $E=E_m\cup E_s\cup E_r$ such that (a) Each connected component induced by $E_m$ has minimum degree $Ω(n^δ)$ and conductance $Ω(1/\text{poly} \log(n))$. As a consequence the mixing time of a random walk within the component is $O(\text{poly} \log(n))$. (b) The subgraph induced by $E_s$ has arboricity at most $n^δ$. (c) $|E_r| \leq |E|/6$. All of our algorithms are based on the following generic framework, which we believe is of interest beyond this work. Roughly, we deal with the set $E_s$ by an algorithm that is efficient for low-arboricity graphs, and deal with the set $E_r$ using recursive calls. For each connected component induced by $E_m$, we are able to simulate congested clique algorithms with small overhead by applying a routing algorithm due to Ghaffari, Kuhn, and Su (PODC 2017) for high conductance graphs.

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arXiv2018

Hardness of RNA Folding Problem with Four Symbols

An RNA sequence is a string composed of four types of nucleotides, $A, C, G$, and $U$. The goal of the RNA folding problem is to find a maximum cardinality set of crossing-free pairs of the form $\{A,U\}$ or $\{C,G\}$ in a given RNA sequence. The problem is central in bioinformatics and has received much attention over the years. Abboud, Backurs, and Williams (FOCS 2015) demonstrated a conditional lower bound for a generalized version of the RNA folding problem based on a conjectured hardness of the $k$-clique problem. Their lower bound requires the RNA sequence to have at least 36 types of symbols, making the result not applicable to the RNA folding problem in real life (i.e., alphabet size 4). In this paper, we present an improved lower bound that works for the alphabet size 4 case. We also investigate the Dyck edit distance problem, which is a string problem closely related to RNA folding. We demonstrate a reduction from RNA folding to Dyck edit distance with alphabet size 10. This leads to a much simpler proof of the conditional lower bound for Dyck edit distance problem given by Abboud, Backurs, and Williams (FOCS 2015), and lowers the alphabet size requirement.

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arXiv2017

The Energy Complexity of Broadcast

Energy is often the most constrained resource in networks of battery-powered devices, and as devices become smaller, they spend a larger fraction of their energy on communication (transceiver usage) not computation. As an imperfect proxy for true energy usage, we define energy complexity to be the number of time slots a device transmits/listens; idle time and computation are free. In this paper we investigate the energy complexity of fundamental communication primitives such as broadcast in multi-hop radio networks. We consider models with collision detection (CD) and without (No-CD), as well as both randomized and deterministic algorithms. Some take-away messages from this work include: 1. The energy complexity of broadcast in a multi-hop network is intimately connected to the time complexity of leader election in a single-hop (clique) network. Many existing lower bounds on time complexity immediately transfer to energy complexity. For example, in the CD and No-CD models, we need $Ω(\log n)$ and $Ω(\log^2 n)$ energy, respectively. 2. The energy lower bounds above can almost be achieved, given sufficient ($Ω(n)$) time. In the CD and No-CD models we can solve broadcast using $O(\frac{\log n\log\log n}{\log\log\log n})$ energy and $O(\log^3 n)$ energy, respectively. 3. The complexity measures of Energy and Time are in conflict, and it is an open problem whether both can be minimized simultaneously. We give a tradeoff showing it is possible to be nearly optimal in both measures simultaneously. For any constant $ε>0$, broadcast can be solved in $O(D^{1+ε}\log^{O(1/ε)} n)$ time with $O(\log^{O(1/ε)} n)$ energy, where $D$ is the diameter of the network.

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arXiv2017

A Time Hierarchy Theorem for the LOCAL Model

The celebrated Time Hierarchy Theorem for Turing machines states, informally, that more problems can be solved given more time. The extent to which a time hierarchy-type theorem holds in the distributed LOCAL model has been open for many years. It is consistent with previous results that all natural problems in the LOCAL model can be classified according to a small constant number of complexities, such as $O(1),O(\log^* n), O(\log n), 2^{O(\sqrt{\log n})}$, etc. In this paper we establish the first time hierarchy theorem for the LOCAL model and prove that several gaps exist in the LOCAL time hierarchy. 1. We define an infinite set of simple coloring problems called Hierarchical $2\frac{1}{2}$-Coloring}. A correctly colored graph can be confirmed by simply checking the neighborhood of each vertex, so this problem fits into the class of locally checkable labeling (LCL) problems. However, the complexity of the $k$-level Hierarchical $2\frac{1}{2}$-Coloring problem is $Θ(n^{1/k})$, for $k\in\mathbb{Z}^+$. The upper and lower bounds hold for both general graphs and trees, and for both randomized and deterministic algorithms. 2. Consider any LCL problem on bounded degree trees. We prove an automatic-speedup theorem that states that any randomized $n^{o(1)}$-time algorithm solving the LCL can be transformed into a deterministic $O(\log n)$-time algorithm. Together with a previous result, this establishes that on trees, there are no natural deterministic complexities in the ranges $ω(\log^* n)$---$o(\log n)$ or $ω(\log n)$---$n^{o(1)}$. 3. We expose a gap in the randomized time hierarchy on general graphs. Any randomized algorithm that solves an LCL problem in sublogarithmic time can be sped up to run in $O(T_{LLL})$ time, which is the complexity of the distributed Lovasz local lemma problem, currently known to be $Ω(\log\log n)$ and $O(\log n)$.

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arXiv2016

An Exponential Separation Between Randomized and Deterministic Complexity in the LOCAL Model

Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish connections between the deterministic and randomized complexities in the LOCAL model. Each result has a very compelling take-away message: 1. Fast $Δ$-coloring of trees requires random bits: Building on the recent lower bounds of Brandt et al., we prove that the randomized complexity of $Δ$-coloring a tree with maximum degree $Δ\ge 55$ is $Θ(\log_Δ\log n)$, whereas its deterministic complexity is $Θ(\log_Δn)$ for any $Δ\ge 3$. This also establishes a large separation between the deterministic complexity of $Δ$-coloring and $(Δ+1)$-coloring trees. 2. Randomized lower bounds imply deterministic lower bounds: We prove that any deterministic algorithm for a natural class of problems that runs in $O(1)+o(\log_Δn)$ rounds can be transformed to run in $O(\log^*n-\log^*Δ+1)$ rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires $Ω(\log_Δn)$ time deterministically. 3. Deterministic lower bounds imply randomized lower bounds: We prove that the randomized complexity of any natural problem on instances of size $n$ is at least its deterministic complexity on instances of size $\sqrt{\log n}$. This shows that a deterministic $Ω(\log_Δn)$ lower bound for any problem implies a randomized $Ω(\log_Δ\log n)$ lower bound. It also illustrates that the graph shattering technique is absolutely essential to the LOCAL model.

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