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is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.05530">arXiv:2403.05530</a> <span> [<a href="https://arxiv.org/pdf/2403.05530">pdf</a>, <a href="https://arxiv.org/format/2403.05530">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computation and Language">cs.CL</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> </div> </div> <p class="title is-5 mathjax"> Gemini 1.5: Unlocking multimodal understanding across millions of tokens of context </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Gemini+Team"> Gemini Team</a>, <a href="/search/cs?searchtype=author&query=Georgiev%2C+P">Petko Georgiev</a>, <a href="/search/cs?searchtype=author&query=Lei%2C+V+I">Ving Ian Lei</a>, <a href="/search/cs?searchtype=author&query=Burnell%2C+R">Ryan Burnell</a>, <a href="/search/cs?searchtype=author&query=Bai%2C+L">Libin Bai</a>, <a href="/search/cs?searchtype=author&query=Gulati%2C+A">Anmol Gulati</a>, <a href="/search/cs?searchtype=author&query=Tanzer%2C+G">Garrett Tanzer</a>, <a href="/search/cs?searchtype=author&query=Vincent%2C+D">Damien Vincent</a>, <a href="/search/cs?searchtype=author&query=Pan%2C+Z">Zhufeng Pan</a>, <a href="/search/cs?searchtype=author&query=Wang%2C+S">Shibo Wang</a>, <a href="/search/cs?searchtype=author&query=Mariooryad%2C+S">Soroosh Mariooryad</a>, <a href="/search/cs?searchtype=author&query=Ding%2C+Y">Yifan Ding</a>, <a href="/search/cs?searchtype=author&query=Geng%2C+X">Xinyang Geng</a>, <a href="/search/cs?searchtype=author&query=Alcober%2C+F">Fred Alcober</a>, <a href="/search/cs?searchtype=author&query=Frostig%2C+R">Roy Frostig</a>, <a href="/search/cs?searchtype=author&query=Omernick%2C+M">Mark Omernick</a>, <a href="/search/cs?searchtype=author&query=Walker%2C+L">Lexi Walker</a>, <a href="/search/cs?searchtype=author&query=Paduraru%2C+C">Cosmin Paduraru</a>, <a href="/search/cs?searchtype=author&query=Sorokin%2C+C">Christina Sorokin</a>, <a href="/search/cs?searchtype=author&query=Tacchetti%2C+A">Andrea Tacchetti</a>, <a href="/search/cs?searchtype=author&query=Gaffney%2C+C">Colin Gaffney</a>, <a href="/search/cs?searchtype=author&query=Daruki%2C+S">Samira Daruki</a>, <a href="/search/cs?searchtype=author&query=Sercinoglu%2C+O">Olcan Sercinoglu</a>, <a href="/search/cs?searchtype=author&query=Gleicher%2C+Z">Zach Gleicher</a>, <a href="/search/cs?searchtype=author&query=Love%2C+J">Juliette Love</a> , et al. (1112 additional authors not shown) </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2403.05530v5-abstract-short" style="display: inline;"> In this report, we introduce the Gemini 1.5 family of models, representing the next generation of highly compute-efficient multimodal models capable of recalling and reasoning over fine-grained information from millions of tokens of context, including multiple long documents and hours of video and audio. The family includes two new models: (1) an updated Gemini 1.5 Pro, which exceeds the February… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.05530v5-abstract-full').style.display = 'inline'; document.getElementById('2403.05530v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.05530v5-abstract-full" style="display: none;"> In this report, we introduce the Gemini 1.5 family of models, representing the next generation of highly compute-efficient multimodal models capable of recalling and reasoning over fine-grained information from millions of tokens of context, including multiple long documents and hours of video and audio. The family includes two new models: (1) an updated Gemini 1.5 Pro, which exceeds the February version on the great majority of capabilities and benchmarks; (2) Gemini 1.5 Flash, a more lightweight variant designed for efficiency with minimal regression in quality. Gemini 1.5 models achieve near-perfect recall on long-context retrieval tasks across modalities, improve the state-of-the-art in long-document QA, long-video QA and long-context ASR, and match or surpass Gemini 1.0 Ultra's state-of-the-art performance across a broad set of benchmarks. Studying the limits of Gemini 1.5's long-context ability, we find continued improvement in next-token prediction and near-perfect retrieval (>99%) up to at least 10M tokens, a generational leap over existing models such as Claude 3.0 (200k) and GPT-4 Turbo (128k). Finally, we highlight real-world use cases, such as Gemini 1.5 collaborating with professionals on completing their tasks achieving 26 to 75% time savings across 10 different job categories, as well as surprising new capabilities of large language models at the frontier; when given a grammar manual for Kalamang, a language with fewer than 200 speakers worldwide, the model learns to translate English to Kalamang at a similar level to a person who learned from the same content. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.05530v5-abstract-full').style.display = 'none'; document.getElementById('2403.05530v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2312.11805">arXiv:2312.11805</a> <span> [<a href="https://arxiv.org/pdf/2312.11805">pdf</a>, <a href="https://arxiv.org/format/2312.11805">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computation and Language">cs.CL</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Artificial Intelligence">cs.AI</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Computer Vision and Pattern Recognition">cs.CV</span> </div> </div> <p class="title is-5 mathjax"> Gemini: A Family of Highly Capable Multimodal Models </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Gemini+Team"> Gemini Team</a>, <a href="/search/cs?searchtype=author&query=Anil%2C+R">Rohan Anil</a>, <a href="/search/cs?searchtype=author&query=Borgeaud%2C+S">Sebastian Borgeaud</a>, <a href="/search/cs?searchtype=author&query=Alayrac%2C+J">Jean-Baptiste Alayrac</a>, <a href="/search/cs?searchtype=author&query=Yu%2C+J">Jiahui Yu</a>, <a href="/search/cs?searchtype=author&query=Soricut%2C+R">Radu Soricut</a>, <a href="/search/cs?searchtype=author&query=Schalkwyk%2C+J">Johan Schalkwyk</a>, <a href="/search/cs?searchtype=author&query=Dai%2C+A+M">Andrew M. Dai</a>, <a href="/search/cs?searchtype=author&query=Hauth%2C+A">Anja Hauth</a>, <a href="/search/cs?searchtype=author&query=Millican%2C+K">Katie Millican</a>, <a href="/search/cs?searchtype=author&query=Silver%2C+D">David Silver</a>, <a href="/search/cs?searchtype=author&query=Johnson%2C+M">Melvin Johnson</a>, <a href="/search/cs?searchtype=author&query=Antonoglou%2C+I">Ioannis Antonoglou</a>, <a href="/search/cs?searchtype=author&query=Schrittwieser%2C+J">Julian Schrittwieser</a>, <a href="/search/cs?searchtype=author&query=Glaese%2C+A">Amelia Glaese</a>, <a href="/search/cs?searchtype=author&query=Chen%2C+J">Jilin Chen</a>, <a href="/search/cs?searchtype=author&query=Pitler%2C+E">Emily Pitler</a>, <a href="/search/cs?searchtype=author&query=Lillicrap%2C+T">Timothy Lillicrap</a>, <a href="/search/cs?searchtype=author&query=Lazaridou%2C+A">Angeliki Lazaridou</a>, <a href="/search/cs?searchtype=author&query=Firat%2C+O">Orhan Firat</a>, <a href="/search/cs?searchtype=author&query=Molloy%2C+J">James Molloy</a>, <a href="/search/cs?searchtype=author&query=Isard%2C+M">Michael Isard</a>, <a href="/search/cs?searchtype=author&query=Barham%2C+P+R">Paul R. Barham</a>, <a href="/search/cs?searchtype=author&query=Hennigan%2C+T">Tom Hennigan</a>, <a href="/search/cs?searchtype=author&query=Lee%2C+B">Benjamin Lee</a> , et al. (1325 additional authors not shown) </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2312.11805v4-abstract-short" style="display: inline;"> This report introduces a new family of multimodal models, Gemini, that exhibit remarkable capabilities across image, audio, video, and text understanding. The Gemini family consists of Ultra, Pro, and Nano sizes, suitable for applications ranging from complex reasoning tasks to on-device memory-constrained use-cases. Evaluation on a broad range of benchmarks shows that our most-capable Gemini Ultr… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.11805v4-abstract-full').style.display = 'inline'; document.getElementById('2312.11805v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2312.11805v4-abstract-full" style="display: none;"> This report introduces a new family of multimodal models, Gemini, that exhibit remarkable capabilities across image, audio, video, and text understanding. The Gemini family consists of Ultra, Pro, and Nano sizes, suitable for applications ranging from complex reasoning tasks to on-device memory-constrained use-cases. Evaluation on a broad range of benchmarks shows that our most-capable Gemini Ultra model advances the state of the art in 30 of 32 of these benchmarks - notably being the first model to achieve human-expert performance on the well-studied exam benchmark MMLU, and improving the state of the art in every one of the 20 multimodal benchmarks we examined. We believe that the new capabilities of the Gemini family in cross-modal reasoning and language understanding will enable a wide variety of use cases. We discuss our approach toward post-training and deploying Gemini models responsibly to users through services including Gemini, Gemini Advanced, Google AI Studio, and Cloud Vertex AI. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2312.11805v4-abstract-full').style.display = 'none'; document.getElementById('2312.11805v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 18 December, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1802.06992">arXiv:1802.06992</a> <span> [<a href="https://arxiv.org/pdf/1802.06992">pdf</a>, <a href="https://arxiv.org/ps/1802.06992">ps</a>, <a href="https://arxiv.org/format/1802.06992">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Sublinear Algorithms for MAXCUT and Correlation Clustering </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Bhaskara%2C+A">Aditya Bhaskara</a>, <a href="/search/cs?searchtype=author&query=Daruki%2C+S">Samira Daruki</a>, <a href="/search/cs?searchtype=author&query=Venkatasubramanian%2C+S">Suresh Venkatasubramanian</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1802.06992v1-abstract-short" style="display: inline;"> We study sublinear algorithms for two fundamental graph problems, MAXCUT and correlation clustering. Our focus is on constructing core-sets as well as developing streaming algorithms for these problems. Constant space algorithms are known for dense graphs for these problems, while $惟(n)$ lower bounds exist (in the streaming setting) for sparse graphs. Our goal in this paper is to bridge the gap… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.06992v1-abstract-full').style.display = 'inline'; document.getElementById('1802.06992v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1802.06992v1-abstract-full" style="display: none;"> We study sublinear algorithms for two fundamental graph problems, MAXCUT and correlation clustering. Our focus is on constructing core-sets as well as developing streaming algorithms for these problems. Constant space algorithms are known for dense graphs for these problems, while $惟(n)$ lower bounds exist (in the streaming setting) for sparse graphs. Our goal in this paper is to bridge the gap between these extremes. Our first result is to construct core-sets of size $\tilde{O}(n^{1-未})$ for both the problems, on graphs with average degree $n^未$ (for any $未>0$). This turns out to be optimal, under the exponential time hypothesis (ETH). Our core-set analysis is based on studying random-induced sub-problems of optimization problems. To the best of our knowledge, all the known results in our parameter range rely crucially on near-regularity assumptions. We avoid these by using a biased sampling approach, which we analyze using recent results on concentration of quadratic functions. We then show that our construction yields a 2-pass streaming $(1+蔚)$-approximation for both problems; the algorithm uses $\tilde{O}(n^{1-未})$ space, for graphs of average degree $n^未$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1802.06992v1-abstract-full').style.display = 'none'; document.getElementById('1802.06992v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 20 February, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages, conference</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1602.08162">arXiv:1602.08162</a> <span> [<a href="https://arxiv.org/pdf/1602.08162">pdf</a>, <a href="https://arxiv.org/format/1602.08162">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Streaming Verification of Graph Properties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Abdullah%2C+A">Amirali Abdullah</a>, <a href="/search/cs?searchtype=author&query=Daruki%2C+S">Samira Daruki</a>, <a href="/search/cs?searchtype=author&query=Roy%2C+C+D">Chitradeep Dutta Roy</a>, <a href="/search/cs?searchtype=author&query=Venkatasubramanian%2C+S">Suresh Venkatasubramanian</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1602.08162v2-abstract-short" style="display: inline;"> Streaming interactive proofs (SIPs) are a framework for outsourced computation. A computationally limited streaming client (the verifier) hands over a large data set to an untrusted server (the prover) in the cloud and the two parties run a protocol to confirm the correctness of result with high probability. SIPs are particularly interesting for problems that are hard to solve (or even approximate… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1602.08162v2-abstract-full').style.display = 'inline'; document.getElementById('1602.08162v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1602.08162v2-abstract-full" style="display: none;"> Streaming interactive proofs (SIPs) are a framework for outsourced computation. A computationally limited streaming client (the verifier) hands over a large data set to an untrusted server (the prover) in the cloud and the two parties run a protocol to confirm the correctness of result with high probability. SIPs are particularly interesting for problems that are hard to solve (or even approximate) well in a streaming setting. The most notable of these problems is finding maximum matchings, which has received intense interest in recent years but has strong lower bounds even for constant factor approximations. In this paper, we present efficient streaming interactive proofs that can verify maximum matchings exactly. Our results cover all flavors of matchings (bipartite/non-bipartite and weighted). In addition, we also present streaming verifiers for approximate metric TSP. In particular, these are the first efficient results for weighted matchings and for metric TSP in any streaming verification model. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1602.08162v2-abstract-full').style.display = 'none'; document.getElementById('1602.08162v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 October, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 25 February, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages, 2 figure, 1 table</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1509.05514">arXiv:1509.05514</a> <span> [<a href="https://arxiv.org/pdf/1509.05514">pdf</a>, <a href="https://arxiv.org/format/1509.05514">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Streaming Verification in Data Analysis </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Daruki%2C+S">Samira Daruki</a>, <a href="/search/cs?searchtype=author&query=Thaler%2C+J">Justin Thaler</a>, <a href="/search/cs?searchtype=author&query=Venkatasubramanian%2C+S">Suresh Venkatasubramanian</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1509.05514v1-abstract-short" style="display: inline;"> Streaming interactive proofs (SIPs) are a framework to reason about outsourced computation, where a data owner (the verifier) outsources a computation to the cloud (the prover), but wishes to verify the correctness of the solution provided by the cloud service. In this paper we present streaming interactive proofs for problems in data analysis. We present protocols for clustering and shape fitting… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1509.05514v1-abstract-full').style.display = 'inline'; document.getElementById('1509.05514v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1509.05514v1-abstract-full" style="display: none;"> Streaming interactive proofs (SIPs) are a framework to reason about outsourced computation, where a data owner (the verifier) outsources a computation to the cloud (the prover), but wishes to verify the correctness of the solution provided by the cloud service. In this paper we present streaming interactive proofs for problems in data analysis. We present protocols for clustering and shape fitting problems, as well as an improved protocol for rectangular matrix multiplication. The latter can in turn be used to verify $k$ eigenvectors of a (streamed) $n \times n$ matrix. In general our solutions use polylogarithmic rounds of communication and polylogarithmic total communication and verifier space. For special cases (when optimality certificates can be verified easily), we present constant round protocols with similar costs. For rectangular matrix multiplication and eigenvector verification, our protocols work in the more restricted annotated data streaming model, and use sublinear (but not polylogarithmic) communication. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1509.05514v1-abstract-full').style.display = 'none'; document.getElementById('1509.05514v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 September, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2015. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1304.4243">arXiv:1304.4243</a> <span> [<a href="https://arxiv.org/pdf/1304.4243">pdf</a>, <a href="https://arxiv.org/format/1304.4243">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Computational Geometry">cs.CG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Data Structures and Algorithms">cs.DS</span> </div> </div> <p class="title is-5 mathjax"> Range Counting Coresets for Uncertain Data </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/cs?searchtype=author&query=Abdullah%2C+A">Amirali Abdullah</a>, <a href="/search/cs?searchtype=author&query=Daruki%2C+S">Samira Daruki</a>, <a href="/search/cs?searchtype=author&query=Phillips%2C+J+M">Jeff M. Phillips</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1304.4243v1-abstract-short" style="display: inline;"> We study coresets for various types of range counting queries on uncertain data. In our model each uncertain point has a probability density describing its location, sometimes defined as k distinct locations. Our goal is to construct a subset of the uncertain points, including their locational uncertainty, so that range counting queries can be answered by just examining this subset. We study three… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1304.4243v1-abstract-full').style.display = 'inline'; document.getElementById('1304.4243v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1304.4243v1-abstract-full" style="display: none;"> We study coresets for various types of range counting queries on uncertain data. In our model each uncertain point has a probability density describing its location, sometimes defined as k distinct locations. Our goal is to construct a subset of the uncertain points, including their locational uncertainty, so that range counting queries can be answered by just examining this subset. We study three distinct types of queries. RE queries return the expected number of points in a query range. RC queries return the number of points in the range with probability at least a threshold. RQ queries returns the probability that fewer than some threshold fraction of the points are in the range. In both RC and RQ coresets the threshold is provided as part of the query. And for each type of query we provide coreset constructions with approximation-size tradeoffs. We show that random sampling can be used to construct each type of coreset, and we also provide significantly improved bounds using discrepancy-based approaches on axis-aligned range queries. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1304.4243v1-abstract-full').style.display = 'none'; document.getElementById('1304.4243v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 April, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages, 3 figures, 29th Annual ACM Symposium on Computational Geometry (SoCG)- June 2013</span> </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a 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